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SWIPE

Permanent Link: http://ufdc.ufl.edu/UFE0021589/00001

Material Information

Title: SWIPE A Sawtooth Waveform Inspired Pitch Estimator for Speech and Music
Physical Description: 1 online resource (116 p.)
Language: english
Creator: Camacho, Arturo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: estimation, frequency, fundamental, harmonics, music, numbers, pitch, prime, sawtooth, speech, tracking
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A Sawtooth Waveform Inspired Pitch Estimator (SWIPE) has been developed for processing speech and music. SWIPE is shown to outperform existing algorithms on several publicly available speech/musical-instruments databases and a disordered speech database. SWIPE estimates the pitch as the fundamental frequency of the sawtooth waveform whose spectrum best matches the spectrum of the input signal. A decaying cosine kernel provides an extension to older frequency-based, sieve-type estimation algorithms by providing smooth peaks with decaying amplitudes to correlate with the harmonics of the signal. An improvement on the algorithm is achieved by using only the first and prime harmonics, which significantly reduces subharmonic errors commonly found in other pitch estimation algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Arturo Camacho.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Harris, John G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021589:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021589/00001

Material Information

Title: SWIPE A Sawtooth Waveform Inspired Pitch Estimator for Speech and Music
Physical Description: 1 online resource (116 p.)
Language: english
Creator: Camacho, Arturo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: estimation, frequency, fundamental, harmonics, music, numbers, pitch, prime, sawtooth, speech, tracking
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A Sawtooth Waveform Inspired Pitch Estimator (SWIPE) has been developed for processing speech and music. SWIPE is shown to outperform existing algorithms on several publicly available speech/musical-instruments databases and a disordered speech database. SWIPE estimates the pitch as the fundamental frequency of the sawtooth waveform whose spectrum best matches the spectrum of the input signal. A decaying cosine kernel provides an extension to older frequency-based, sieve-type estimation algorithms by providing smooth peaks with decaying amplitudes to correlate with the harmonics of the signal. An improvement on the algorithm is achieved by using only the first and prime harmonics, which significantly reduces subharmonic errors commonly found in other pitch estimation algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Arturo Camacho.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Harris, John G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021589:00001

Full Text





SWIPE: A SAWTOOTH WAVEFORM INSPIRED PITCH ESTIMATOR
FOR SPEECH AND MUSIC




















By

ARTURO CAMACHO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007




































O 2007 Arturo Camacho




























Dedico esta disertaci6n a mis queridos abuelos
Hugo y Flory









ACKNOWLEDGMENTS

I thank my grandparents for all the support they have given to me during my life, my wife

for her support during the years in graduate school, and my daughter who was in my arms when

the most important ideas expressed here came to my mind. I also thank Dr. John Harris for his

guidance during my research and for always pushing me to do more, Dr. Rahul Shrivastay for his

support and for introducing me to the world of auditory system models, and Dr. Manuel

Bermudez for his unconditional support all these years.












TABLE OF CONTENTS



page

ACKNOWLEDGMENT S .............. ...............4.....


LI ST OF T ABLE S ................. ...............7................


LI ST OF FIGURE S .............. ...............8.....


LI ST OF AB BREVIAT IONS ................. ................. 11......... ....


AB S TRAC T ............._. .......... ..............._ 12...


CHAPTER


1 INTRODUCTION ................. ...............13.......... ......


1.1 Pitch Background ................. ...............14...............
1.1.1 Conceptual Definition .............. ...............14....
1.1.2 Operational Definition............... ...............15
1.1.3 Strength .................. ...............16...............
1.1.4 Duration Threshold............... .. ... ................. 19
1.2 Illustrative Examples and Pitch Determination Hypotheses .............. .....................2
1.1.2 Pure Tone............... .. ....... .... .. .... .. .. .......2
1.2.2 Sawtooth Waveform and the Largest Peak Hypothesis .............. ....................2
1.2.3 Missing Fundamental and the Components Spacing Hypothesis...........................21
1.2.4 Square Wave and the Maximum Common Divisor Hypothesis ............................22
1.2.5 Alternating Pulse Train............... ...............24.
1.2.6 Inharmonic Signals............... ...............25
1.3 Loudness ................. ... ......... ............. ...... .........2
1.4 Equivalent Rectangular Bandwidth ................ ...............27........... ...
1.5 Dissertation Organization ................ ...............29........... ...
1.6 Summary ................. ...............3.. 0.............

2 PITCH ESTIMATION ALGORITHMS: PROBLEMS AND SOLUTIONS ................... .....31


2.1 Harmonic Product Spectrum (HPS)............... ...............32.
2.2 Sub-harmonic Summation (SHS) .............. ...............34....
2.3 Subharmonic to Harmonic Ratio (SHR) ................. ...............36..............
2.4 Harmonic Sieve (HS)............... ...............37..
2.5 Autocorrelation (AC) .............. .. ............ .. .. ......... ......... ..... .......3
2.6 Average Magnitude and Squared Difference Functions (AMDF, ASDF) .......................42
2.7 Cepstrum (CEP) ................. ...............44................
2.8 Summary ................. ...............46......... ......












3 THE SAWTOOTH WAVEFORM INSPIRED PITCH ESTIMATOR ................ ...............47


3.1 Initial Approach: Average Peak-to-Valley Distance Measurement .............. .................47
3.2 Blurring of the Harmonics ................. ...............49........... ..
3.3 Warping of the Spectrum ................. ...............51........... ..
3.4 Weighting of the Harmonics. ................ ................. .................. ..........53
3.5 Number of Harmonics ................. ...............55..............
3.6 Warping of the Frequency Scale ................. ...............55......___ ..
3.7 Window Type and Size............... ...............57..
3.8 SW IPE .............. ...............63....
3.9 SW IPE' ......... .. .. ............... .... .................6
3.9. 1 Pitch Strength of a Sawtooth Waveform ................. ...............69 .......... .
3.10 Reducing Computational Cost .................. ......... ...............71....
3.10.1 Reducing the Number of Fourier Transforms .............. ...............71....
3.10.1.1 Reducing window overlap................ ...............72
3.12. 1.2 Using only power-of-two window sizes. ................ ......... ...............74
3.10.2 Reducing the Number of Spectral Integral Transforms .............. ...................81
3.11 Summary ................. ...............86........... ...


4 EVALUATION .............. ...............87....


4.1 Al gorithm s .............. ...............87....
4.2 Databases .............. ...............8 8....
4.3 M ethodology ................. ...............89........... ....
4.4 Results............... ...............89
4.5 Discussion............... ...............9


5 CONCLU SION................ ..............9


APPENDIX


A MATLAB IMPLEMENTATION OF SWIPE' ................ ...............99...............


B DETAILS OF THE EVALUATION ................. ...............102........... ...


B.1 D databases .............. ... .......... .............10
B. 1.1 Paul Bagshaw' sDatabase ................ ...............102.............
B. 1.2 Keele Pitch Database ................. ...............102.......... ..
B.1.3 Disordered Voice Database .............. ...............103....
B. 1.4 Musical Instruments Database ................. ...............104........... ..
B.2 Evaluation Using Speech ................ .....___ ...............105 ....
B.3 Evaluation Using Musical Instruments .................._.._.. ......... ............0


C GROUND TRUTH PITCH FOR THE DISORDERED VOICE DATABASE ................... 110


REFERENCES ................. ...............112....... ......


BIOGRAPHICAL SKETCH ........._..... ...............116....... ......











LIST OF TABLES

Table page

3-1 Common windows used in signal processing ................. ...............62........... ..

4-1 Gross error rates for speech .............. ...............90....

4-2 Proportion of overestimation errors relative to total gross errors ........._.. .....................90

4-3 Gross error rates by gender ........._. ........_. ...............91.....

4-4 Gross error rates for musical instruments .............. ...............92....

4-5 Gross error rates by instrument family .............. ...............92....

4-6 Gross error rates for musical instruments by octave ................. ................ ......... .93

4-7 Gross error rates for musical instruments by dynamic .............. ...............94....

4-8 Gross error rates for variations of SWIPE' ............. ...............95.....

C-1 Ground truth pitch values for the disordered voice database ................. ............... .....110












LIST OF FIGURES


Figure page

1-1 Sawtooth waveform .............. ...............18....


1-2 Pure tone .............. ...............20....


1-3 Missing fundamental ................. ...............22........... ....


1-4 Square wave ................. ...............23........... ....

1-5 Pulse train............... ...............24.


1-6 Alternating pulse train............... ...............25.


1-7 Inharmonic signal............... ...............26.


1-8 Equivalent rectangular bandwidth. ............. ...............28.....


1-9 Equivalent-rectangul ar-b andwi dth scale ...._.._.._ ..... .._._. ....._.._.........2


2-2 Harmonic product spectrmm................ ...............3

2-3 Sub harmonic summation .............. ...............34....


2-4 Sub harmonic summation with decay ........................._ ...............35.....

2-5 Subharmonic to harmonic ratio............... ...............37.


2-6 Harmonic sieve .............. ...............38....


2-7 Autocorrelation .............. ...............40....


2-8 Comparison between AC, BAC, ASDF, and AMDF. ............. ...............42.....


2-9 Cepstrum ................. ...............44.......... .....


2-10 Problem caused to cepstrum by cosine lobe at DC ................. ...............45.............


3-1 Average-peak-to-valley-di stance kernel ................. ...............48................


3-3 Necessity of strictly convex kernels .............. ...............50....


3-4 Kernels formed from concatenations of truncated squarings, Gaussians, and cosines......5 1


3-5 Warping of the spectrum ................. ...............52........... ...


3-6 Weighting of the harmonics ................. ...............54........... ...











3-7 Fourier transform of rectangular window ...._.._.._ ........__. ...._.._ ...........5

3-8 Cosine lobe and square-root of the spectrum of rectangular window ........._.... ..............59

3 -9 H ann wind ow ..........._.__......... ...............60....

3-10 Fourier transform of the Hann window ................ ........................ ..............61


3-11 Cosine lobe and square-root of the spectrum of Hann window ........._...... ........_........61

3-12 SWIPE kernel............... ...............64.


3-13 Most common pitch estimation errors .............. ...............66....

3-14 SWIPE' kernel ........._...... ...............69._.._. ......


3-15 Pitch strength of sawtooth waveform .............. ...............70....

3 -16 Wind ows overlapping ................. ...............73................

3-17 Idealized spectral lobes ................. ...............75................

3-18 If-normalized inner product between template and idealized spectral lobes ...................77

3-19 Individual and combined pitch strength curves .............. ...............78....

3-20 Pitch strength loss when using suboptimal window sizes .............. ....................7

3-21 Coefficients of the pitch strength interpolation polynomial ............__.. ......__.........84

3-22 Interpolated pitch strength .............. ...............85....










LIST OF OBJECTS


Obl ect

Object 1-1.

Object 1-2.

Object 1-3.

Object 1-4.

Object 1-5.

Object 1-6.

Object 1-7.

Obj ect 2-1.

Obj ect 2-2.

Obj ect 3-1.


page

Sawtooth waveform (WAV file, 32 KB). ............. ...............18.....

Pure tone (WAV file, 32 KB). ............. ...............20.....

Missing fundamental (WAV file, 32 KB)............... ...............22..

Square wave (WAV file, 32 KB). .............. ...............23....

Pulse train (WAV file, 32 KB). ............. ...............24.....

Alternating pulse train (WAV file, 32 KB). ............. ...............25.....

Inharmonic signal (WAV file, 32 KB). ............. ...............26.....

Bandpass filtered /u/ (WAV file 6 KB) .............. ...............33....

Signal with strong second harmonic (WAV file, 32 KB) ................. ............... ....42

Beating tones (WAV file, 32 KB)............... ...............50..










LIST OF ABBREVIATIONS


AC Autocorrelation

AMDF Average magnitude difference function

APVD Average peak-to-valley distance

ASDF Average squared difference function

BAC Biased autocorrelation

CEP Cepstrum

ERB Equivalent rectangular bandwidth

ERBs Equivalent-rectangular-bandwidth scale

FFT Fast Fourier transform

HPS Harmonic product spectrum

HS Harmonic sieve

IP Inner product

IT Integral transform

ISL Idealized spectral lobe

Kt-NIP Kt-normalized inner product

NIP Normalized inner product

O-WS Optimal window size

P2-WS Power-of-two window size

SHS Subharmoni c- summati on

SHR Subharmoni c-to-harmoni c rati o

STFT Short-time Fourier transform

SWIPE Sawtooth Waveform Inspired Pitch Estimator

UAC Unbiased autocorrelation

WS Window size









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SWIPE: A SAWTOOTH WAVEFORM INSPIRED PITCH ESTIMATOR
FOR SPEECH AND MUSIC


By

Arturo Camacho

December 2007

Chair: John G. Harris
Major: Computer Engineering

A Sawtooth Waveform Inspired Pitch Estimator (SWIPE) has been developed for

processing speech and music. SWIPE is shown to outperform existing algorithms on several

publicly available speech/musical-instruments databases and a disordered speech database.

SWIPE estimates the pitch as the fundamental frequency of the sawtooth waveform whose

spectrum best matches the spectrum of the input signal. A decaying cosine kernel provides an

extension to older frequency-based, sieve-type estimation algorithms by providing smooth peaks

with decaying amplitudes to correlate with the harmonics of the signal. An improvement on the

algorithm is achieved by using only the first and prime harmonics, which significantly reduces

subharmonic errors commonly found in other pitch estimation algorithms.









CHAPTER 1
INTTRODUCTION

Pitch is an important characteristic of sound, providing information about the sound's

source. In speech, pitch helps to identify the gender of the speaker (pitch tends to be higher for

females than for males) (Wang and Lin, 2004), gives additional meaning to words (e.g., a group

of words can be interpreted as a question depending on whether the pitch is rising or not), and

may help to identify the emotional state of the speaker (e.g., joy produces high pitch and a wide

pitch range, while sadness produce normal to low pitch and a narrow pitch range) (Murray and

Arnott, 1993). Pitch is also important in music because it determines the names of the notes

(Sethares, 1998).

Pitch estimation also has applications in many areas that involve processing of sound:

music, communications, linguistics, and speech pathology. In music, one of the main

applications of pitch estimation is automatic music transcription. Musicologists are often faced

with music for which no transcription exists. Therefore, automated tools that extract the pitch of

a melody, and from there the individual musical notes, are invaluable tools for musicologists

(Askenfelt, 1979). Automated transcription has also been used in query-by-humming systems

(e.g., Dannenberg et al., 2004). These systems allow people to search for music in databases by

singing or humming the melody rather than typing the title of the song, which may be unknown

for the user or the database.

In communications, pitch estimation is used for speech coding (Spanias, 1994). Many

speech coding systems are based on the source-filter model (Fant, 1960), which models speech

as a filtered source signal. In some implementations, the source is either a periodic sequence of

glottal pulses (for voiced sound) or white noise (for unvoiced sound). Therefore, the correct

estimation of the glottal pulse rate is crucial for the correct coding of voiced speech.









Pitch estimators are useful in linguistics for the recognition of intonation patterns, which

are used, for example, in the acquisition of a second language (de Bot, 1983). Pitch estimators

are also used in speech pathology to determine speech disorders, which are characterized by high

levels of noise in the voice. Since most methods used to estimate noise are based on the

fundamental frequency of the signal (e.g., Yomoto and Gould, 1982), pitch estimators are of vital

importance in this area.

The goal of our work is to develop an automatic pitch estimator that operates on both

speech and music. The algorithm should be competitive with the best known pitch estimators,

and therefore be suitable for the many applications mentioned above. Furthermore, the algorithm

should provide a measure to determine if a pitch exists or not in each region of the signal. The

remaining sections of this chapter present several psychoacoustics definitions and phenomena

that will be used to explain the operation and rationale of the algorithm.

1.1 Pitch Background

1.1.1 Conceptual Definition

Several conceptual definitions of pitch have been proposed. The American Standard

Association (ASA, 1960) definition of pitch is

"Pitch is that attribute of auditory sensation in terms of which sounds may be ordered on a
musical scale,"

and the American National Standards Institute (ANSI, 1994) definition of pitch is

"Pitch is that auditory attribute of sound according to which sounds can be ordered on a
scale from low to high. Pitch depends mainly on the frequency content of the sound
stimulus, but it also depends on the sound pressure and the waveform of the stimulus."

These definitions mention an attribute that allows ordering sounds in a scale, but they say

nothing about what that attribute is.









We will propose another definition of pitch, which is based on the fundamental frequency

of a signal. The fundamental frequency fo of a signal (sound or no sound) exists only for periodic

signals, and is defined as the inverse of the period of the signal, where the period To of the signal

(a.k.a. fundamental period) is the minimum repetition interval of the signal x(t), i.e.,

T,, = min(T > 0 |f t: x(t) = x(t +T)\I. (1-1)

It is also possible, to define the fundamental frequency in the frequency domain:


= max >0 rO c k;i;~h k Sin(2nkft +95k) (1-2)

Although both equations are mathematically equivalent (i.e., it can be shown that fo = 1/To), they

are conceptually different: Equation 1-1 looks at the signal in the time domain, while

Equation 1-2 looks at the signal as a combination of sinusoids using a Fourier series expansion.

The key element for periodicity in Equation 1-1 is the equality in x(t) = x(t + T), and the key

element for periodicity in Equation 1-2 is the existence of components only at multiples of the

fundamental frequency. Unfortunately, no signal in nature is perfectly periodic because of

natural variations in frequency and amplitude, and contamination produced by noise.

Nevertheless, when listening to many natural signals, we perceive pitch. This suggests that, to

determine pitch, the brain probably uses either a modified version of Equation 1-1, where the

equality x(t) = x(t + T) is substituted by an approximation, or a modified version of Equation 1-2,

where noise and fluctuations in the frequency of the components are allowed. Based on this

suggestion, we define pitch as the perceived "fundamental frequency" of a sound, in other words,

as the estimate our brain does of the (quasi) fundamental frequency of a sound.

1.1.2 Operational Definition

Since the previous definitions of pitch do not indicate how to measure it, they are of no

practical use, and an operational definition of pitch is required. The usual way in which pitch is










measured is the following. A listener is presented with two sounds: a target sound, for which the

pitch is to be determined, and a matching sound. The matching sound is usually a pure tone,

although sometimes harmonic complex tones are used as well. The levels of the target and the

matching sounds are usually equalized to avoid any effect of differences in level in the

perception of pitch. The sounds are presented sequentially, simultaneously, or in any

combination of them, depending on the design of the experiment. The listener is asked to adjust

the fundamental frequency of the matching sound such that it matches the target sound, in the

sense of the conceptual definitions of pitch presented above. The fundamental frequency of the

matching sound is recorded and the experiment is repeated several times and with different

listeners. The data is summarized, and if the distribution of fundamental frequencies shows a

clear peak around a certain frequency, the target sound is said to have a pitch corresponding to

that frequency.

1.1.3 Strength

Some sounds elicit a strong pitch sensation, and some do not. For example, when we

speak, some sounds are highly periodic and elicit a strong pitch sensation (e.g., vowels), but

some do not (e.g., some consonants: /s/, /sh/, /p/, and /k/). In the case of musical instruments, the

attack tends to contain transient components that obscure the pitch, but they disappear quickly

letting the pitch show more clearly. The quality of the sound that allows us to determine whether

pitch exists is called pitch strength. Pitch strength is not a categorical variable but a continuum.

Also, pitch strength is independent of pitch: two sounds may have the same pitch and differ in

pitch strength. For example, a pure tone and a narrow band of noise centered at the frequency of

the tone have the same pitch, however, the pure tone elicit a stronger pitch sensation than the

noise.









Unfortunately, not much research exists on pitch strength, and the few studies that exist

have concentrated mostly on noise (Yost, 1996; Wiegrebe and Patterson, 1998), although some

have explored harmonic sounds as well (Fastl and Stoll, 1979; Shofner and Selas, 2002). In terms

of variety of sounds, the most complete study is probably Fastl and Stoll's, which included pure

tones, complex tones, and several types of noises. In that study, pure tones were reported to have

the strongest pitch among all sounds. However, other studies have found that pitch identification

improves as harmonics are added (Houtsma, 1990), which suggests that pitch strength increases

as well.

We hypothesize that our brain determines pitch by searching for a match between our

voice, produced or imagined, and the target signal for which pitch is to be determined, probably

based on their spectra. This hypothesis agrees with studies of pitch determination in which

subjects have been allowed to hum the target sound to facilitate pitch matching tasks (Houtgast,

1976). Based on this hypothesis, we believe that the higher the similarity of the target signal with

our voice, the higher its pitch strength. If the similarity is based on the spectrum, a signal will

have maximum pitch strength when its spectrum is closest to the spectrum of a voiced sound. If

we assume that voiced sounds have harmonic spectra with envelopes that decay on average as 1/f

(i.e., inversely proportional to frequency) (Fant, 1960), then a signal will have maximum pitch

strength if its spectrum has that structure.

An example of a signal with such property is a sawtooth waveform, which is exemplified

in Figure 1-1. A sawtooth waveform is formed by adding sines with frequencies that are

multiples of a common fundamental fo, and whose amplitude decays inversely proportional to

frequency:

S1
x(t) =- sin 2ikft (1-3)
k=1 k










Though sawtooth waveforms play a key role in our research, their importance resides in

their spectrum, and not in their time-domain waveform. In particular, the phase of the

components can be manipulated (destroying its sawtooth waveform shape) and the signal would

still play the same role in our work as the sawtooth waveform. In other words, it is assumed in

this work that what matters to estimate pitch and its strength is the amplitude of the spectral

components of the sound, and not their phase, which in fact is ignored here. However, phase

does play a role in pitch perception, as have been shown by some researchers (Moore, 1977;

Shackleton and Carlyon, 1994; Galembo, et al., 2001). These researchers have created pairs of

sounds that have the same spectral amplitudes but significantly different pitches, by choosing the

phases of the components appropriately. Nevertheless, it is not the aim of this research to cover

the whole range of pitch phenomena, but to concentrate only on the most common speech and











0 10 20 30 40 50
Time; (ms)





S0.

0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 1-1. Sawtooth waveform. A) Signal. B) Spectrum.

Object 1-1. Sawtooth waveform (WAV file, 32 KB).










musical instruments sounds. As we will see later, good pitch predictions are obtained for these

types of sounds based solely on the amplitude of their spectral components.

1.1.4 Duration Threshold

Doughty and Garner (1947) studied the minimum duration required to perceive a pitch for

a pure tone. They found that there are two duration thresholds with two different properties.

Tones with durations below the shorter threshold are perceived as a click, and no pitch is

perceived. Tones with durations between the two thresholds are perceived as having pitch, and

an increase in their duration causes an increase in their pitch strength. Tones with durations

above the largest threshold are also perceived as having pitch, but further increases in their

duration do not increase their pitch strength.

These thresholds are not constant, but approximately proportional to the pitch period of the

tone. In other words, the threshold corresponds to a certain number of periods of the tone.

However, there is some interaction between pitch and the minimum number of cycles required to

perceive it (lower frequencies have a tendency to require fewer cycles to elicit a pitch). The

shorter threshold is approximately two to four cycles, and the larger threshold is approximately

three to ten cycles. For frequencies above 1 kHz the thresholds become constant: 4 ms the

shorter and 10 ms the larger, regardless of their corresponding number of cycles.

Robinson and Patterson (1995a; 1995b) studied note discriminability as a function of the

number of cycles in the sound using strings, brass, flutes, and harpsichords. A large increase in

discriminability can be observed in their data as the number of cycles increases from one to

about ten, but beyond ten cycles the discriminability of the notes does not seem to increase

much. This trend agrees with the thresholds for pure tones mentioned above, which suggests that

the thresholds are also valid for musical instruments, and probably for sawtooth waveforms as

well.










1.2 Illustrative Examples and Pitch Determination Hypotheses

In previous sections, conceptual and operational definitions of pitch were given. From a practical

point of view, both types of definitions are of limited use since the conceptual definitions are too

abstract and the operational definition requires a human to determine the pitch. In this section we

propose more algorithmic ways for determining pitch, through the search for cues that may give

us hints regarding the pitch. These cues, hereafter referred as hypotheses, are illustrated with

examples of sounds in which they are valid, and examples in which they are not.

1.1.2 Pure Tone

From a frequency domain point of view, the simplest periodic sound is a pure tone. A pure

tone with a frequency of 100 Hz and its spectrum is shown in Figure 1-2. Based on our

operational definition of pitch (i.e., the one that uses a pure tone as matching tone presented at

the same intensity level as the testing tone), the pitch of a pure tone is its frequency, and











0 10 20 30 40 50
Time (ms)






0 '100 200 300 400 500
Frequency (Hz)

Figure 1-2. Pure tone. A) Signal. B) Spectrum.

Object 1-2. Pure tone (WAV fie, 32 KB).









therefore frequency determines pitch in this case. This may not be true if the tones are presented

at different levels. Intriguingly, the pitch of a pure tone may change with intensity level (Stevens,

1935): as intensity increases, the pitch of high frequency tones tends to increase, and the pitch of

low frequency tones tends to decrease. However, this change is usually less than 1% or 2%

(Verschuure and van Meeteren, 1975), occurs at very disparate intensity levels, and varies

significantly from person to person.

Since the goal in this research is to predict pitch for sounds represented in a computer as a

sequence of numbers without knowing the level at which the sound will be played, it will be

assumed that the sound will be played at a "comfortable" level, and therefore the algorithm will

be designed to predict the pitch at that level. Nevertheless, variations of pitch with level are

small, and have little effect even for complex tones (Fastl, 2007), otherwise, music would

become out of tune as we change the volume.

1.2.2 Sawtooth Waveform and the Largest Peak Hypothesis

The sawtooth waveform presented in Section 1.1.3 was shown to have a harmonic

spectrum with components whose amplitude decays inversely proportional to frequency (see

Figure 1-1). The computational determination of the pitch of a sawtooth waveform is not as easy

as it is for a pure tone because its spectrum has more than one component. Since the pitch of a

sawtooth waveform corresponds to its fundamental frequency, and the fundamental frequency in

this case is the component with the highest energy, one possible hypothesis for the derivation of

the pitch is that the pitch corresponds to the largest peak in the spectrum. However, as we will

show in the next section, this hypothesis does not always hold.

1.2.3 Missing Fundamental and the Components Spacing Hypothesis

This section shows that it is possible to create a periodic sound with a pitch corresponding

to a frequency at which there is no energy in the spectrum. A sound with such property is said to


















0 10 20 30 40 50
Time; (ms)







0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 1-3. Missing fundamental. A) Signal. B) Spectrum.

Object 1-3. Missing fundamental (WAV file, 32 KB).



have a missing fund amental.~dd~~dd~~dd It is easy to build such a signal: just take a sawtooth waveform and

remove its fundamental, as shown in Figure 1-3. Certainly, the timbre of the sound will change,

but not its pitch. This fact disproves the hypothesis that the pitch corresponds to the largest peak

in the spectrum.

After it was realized that the pitch of a complex tone was unaffected by removing the

fundamental frequency, it was hypothesized that the pitch corresponds to the spacing of the

frequency components. However, this hypothesis is not always valid, as we will show in the next

section.

1.2.4 Square Wave and the Maximum Common Divisor Hypothesis

The previous section hypothesized that the pitch corresponds to the spacing between the

frequency components. However, it is easy to find an example for which this hypothesis fails: a

















0 10 20 30 40 50
Time; (ms)





S0.

0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 1-4. Square wave. A) Signal. B) Spectrum.

Object 1-4. Square wave (WAV file, 32 KB).



square wave. A square wave is similar to a sawtooth wave, but does not have even order

harmonics:

S1
x(t) = C sin 2jt(2k 1) fat (1-4)
k=1 (2k 1)

A square wave with a fundamental frequency of 100 Hz and its spectrum is shown in

Figure 1-4. The components are located at odd multiples of 100 Hz, producing a spacing of 200

Hz between them. However, the fundamental frequency, and indeed its pitch, is 100 Hz. Thus,

the components spacing hypothesis is invalid.

A hypothesis that seems to work for this example, and all the previous ones, is that the

pitch must correspond to the maximum common divisor of the frequency components. As shown

in Equation 1-2, this is equivalent to saying that the pitch corresponds to the fundamental

frequency. However, we will show in the next section that this hypothesis is also wrong.


























































0 10 20 30 40 50 60 70 80 90 1 {
Time; (ms)


1.2.5 Alternating Pulse Train


A pulse train is a sum of pulses separated by a constant time interval To:



x(t)= [3(t -k%), (1-5)



where 3 is the delta or pulse function, a function whose value is one if its argument is zero, and


zero otherwise. A pulse train with a fundamental frequency of 100 Hz (fundamental period of 10


ms) and its spectrum are shown in Figure 1-5. The spectrum of a pulse train is another pulse train


with pulses at multiples of the fundamental frequency, which corresponds to the pitch. If the


signal is modified by decreasing the height of every other pulse in the time domain to 0.7, as


shown in Figure 1-6, the period of the signal will change to 20 ms. This will be reflected in the


spectrum as a change in the fundamental frequency from 100 Hz to 50 Hz. However, although


this change may cause an effect on the timbre (depending on the overall level of the signal), the


m
rr
sg
r~


0.5



0b 100 200 300 400 500 600 700 800 900 10(
Frequency (Hz)


Figure 1-5. Pulse train. A) Signal. B) Spectrum.

Object 1-5. Pulse train (WAV file, 32 KB).


E
2
fi
a,
8,
V)


00














1- 1


m
rr
sg
r~


0 10 20 30 40 50 60 70 80 90 100
Time; (ms)


1 .



0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 1-6. Alternating pulse train. A) Signal. B) Spectrum.

Object 1-6. Alternating pulse train (WAV file, 32 KB).



pitch will remain the same: 100 Hz, refuting the hypothesis that the pitch of a sound corresponds

to its fundamental frequency (i.e., the maximum common divisor of the frequency components).

1.2.6 Inharmonic Signals

This section shows another example of a signal whose pitch does not correspond to its

fundamental frequency (i.e., the maximum common divisor of its frequency components).

Consider a signal built from the 13th, 19th, and 25th harmonics of 50 Hz (i.e., 650, 950, and

1250 Hz), as shown in Figure 1-7. Its fundamental frequency is 50 Hz, but its pitch is 334 Hz

(Patel and Balaban, 2001). This is interesting since the ratios between the components and the

pitch are far from being integer multiples: 1.95, 2.84, and 3.74. In any case, the pitch of the

signal no longer corresponds to its fundamental frequency. Although the true period of the signal

is To = 20 ms, the signal peaks about every 3 ms, which corresponds to the pitch period of the


















0 10 20 30 40 50
Time; (ms)







650 950 1250
Frequency (Hz)

Figure 1-7. Inharmonic signal. A) Signal. To corresponds to the fundamental period of the signal
and to corresponds to the pitch period. B) Spectrum.

Object 1-7. Inharmonic signal (WAV file, 32 KB).



signal to (see Panel A). These type of signals for which the components are not integer multiples

of the pitch are called inharmonic signals.

1.3 Loudness

Loudness is another perceptual quality of sound that provides us with information about its

source. It is important for pitch because the unification of the components of a sound into a

single entity, for which we identify a pitch, may be mediated by the relative loudness of the

components of the sound.

A conceptual definition of loudness is (Moore, 1997)

"...that attribute of auditory sensation in terms of which sounds can be ordered on a scale
extending from quiet to loud."









The most common unit to measure loudness is the sone. A sone is defined as the loudness

elicited by a 1 k
usually modeled as a power function of the sound pressure P of the tone, i.e.,

L =k P", (1-6)

where k is a constant that depends on the units and a is the exponent of the power law.

In a review of loudness studies, Takeshima et. al (2003) found that the value of a is

usually reported to be within the range 0.4-0.6. They also reviewed more elaborate models with

many more parameters, but for simplicity, in this work we will use the simpler power model, and

for reasons we will explain later, we choose the value of a to be 0.5. In other words, we model

the loudness of a tone as being proportional to the square-root of its amplitude.

1.4 Equivalent Rectangular Bandwidth

The bandwidth and the distribution of the filters used to extract the spectral components of

a sound are important issues that may affect our perception of pitch. Since each point of the

cochlea responds better to certain frequencies than others, the cochlea acts as a spectrum

analyzer. The bandwidth of the frequency response of each point of the cochlea is not constant

but varies with frequency, being almost proportional to the frequency of maximum response at

each point (Glasberg and Moore, 1990).

The concept of Equivalent Rectangular Bandwidth (ERB) was introduced as a description

of the spread of the frequency response of a filter. The ERB of a filter F is defined as the

bandwidth (in Hertz) of a rectangular filter R centered at the frequency of maximum response of

F, scaled to have the same output as F at that frequency, and passing the overall same amount of

white noise energy as F. In other words, when the power responses of F and R are plotted as a











Auldltory filter
---- ERB filter













0 0.5 1 1.5
Frequency (knz)

Figure 1-8. Equivalent rectangular bandwidth.



function of frequency, as in Figure 1-8, the central frequency of R corresponds to the mode of F,

and both curves have the same height and area.

Glasberg and Moore (1990) studied the response of auditory filters at different frequencies,

and proposed the following formula to approximate the ERB of the filters:

ERB( f) = 24.7 + 0. 108 f (1-7)

Another property of the cochlea is that the relation between frequency and site of

maximum excitation in the cochlea is not linear. If the distance between the apex of the cochlea

and the site of maximum excitation of a pure tone is plotted as a function of frequency of the

tone, it will be found that a displacement of 0.9 mm in the cochlea corresponds approximately to

one ERB (Moore, 1986). Therefore, it is possible to build a scale to measure the position of

maximum response in the cochlea for a certain frequency fby integrating Equation 1-7 to obtain

the number of ERBs below f and then multiplying it by 0.9 mm to obtain the position. However,


























0 2 4 6 8 10
Frequency (knz)

Figure 1-9. Equivalent-rectangular-bandwidth scale.



it is common practice in psychoacoustics to merely compute the number of ERBs below f which

can be computed as

ERB s(f) = 21.4 log,, (1 + f /229)(18

This scale is shown in Figure 1-9, and it will be the scale used by SWIPE to compute spectral

similarity.

1.5 Dissertation Organization

The rest of this dissertation is organized as follows. Chapter 2 presents previous pitch

estimation algorithms that are related to SWIPE, their problems and possible solutions to these

problems. Chapter 3 will discuss how these problems, plus some ones and their solutions, lead to

SWIPE. Chapter 4 evaluates SWIPE using publicly available speech/music databases and a

disordered speech database. Publicly available implementations of other algorithms are also

evaluated on the same databases, and their performance is compared against SWIPE' s.









1.6 Summary

Here we have presented the motivations and applications for pitch estimation. Then, we

presented conceptual and operational definitions of pitch, together with the related concept of

pitch strength and the duration threshold to perceive pitch. Next, we presented examples of

signals and their pitch, together with hypotheses about how pitch is determined. The sawtooth

waveform was highlighted, since it plays a key role in the development of SWIPE.

Psychoacoustic concepts such as inharmonic signals, loudness, and the ERB scale were also

introduced since they are also relevant for the development of SWIPE.









CHAPTER 2
PITCH ESTIMATION ALGORITHMS: PROBLEMS AND SOLUTIONS

This chapter presents some well known pitch estimation algorithms that appear in the

literature. These algorithms were chosen because of their influence upon the creation of SWIPE.

We will present the algorithms in a very basic form with the intent to capture their essence in a

simple expression, although their actual implementations may have extra details that we do not

present here. The purpose of those details is usually to fine tune the algorithms, but the actual

power of the algorithms is based on the essence we describe here.






Signal





Windows a-


STFTI

Spectrum


~ITI "'

Score



SPitch


Figure 2-1. General block diagram of pitch estimators.









Traditionally, there have been two types of pitch estimation algorithms (PEAs): algorithms

based on the spectrmm of the signal, and algorithms based on the time-domain representation of

the signal. The time-domain based algorithms presented in this chapter can also be formulated

based on the spectr-um of the signal, which will be the approach followed here.

The basic steps that most PEAs perform to track the pitch of a signal are shown in the

block diagram of Figure 2-1. First, the signal is split into windows. Then, for each window the

following steps are performed: (i) the spectrum is estimated using a short-time Fourier transform

(STFT), (ii) a score is computed for each pitch candidate within a predefined range by computing

an integral transform (IT) over the spectrum, and (iii) the candidate with the highest score is

selected as the estimated pitch. The algorithms will be presented in an order that is convenient

for our purposes, but does not necessarily correspond to the chronological order in which they

were developed.

2.1 Harmonic Product Spectrum (HPS)

The first algorithm to be presented is Harmonic Product Spectrum (HPS) (Schroeder,

1968). This algorithm estimates the pitch as the frequency that maximizes the product of the

spectrum at harmonics of that frequency, i.e. as


p =arg max | X(klf ) | (2-1)
f k=1

where X is the estimated spectrum of the signal, n is the number of harmonics to be used

(typically between 5 and 11), and p is the estimated pitch. The purpose of limiting the number of

harmonics to n is to reduce the computational cost, but there is no logical reason behind this

limit; it is hard to believe that the n-th harmonic is useful for pitch estimation, but not the n+1-th.

SSince all the pitch estimators presented here use the magnitude of the spectrum but not its phase, the words
"magnitude of' will be omitted, and the word spectrum should be interpreted as magnitude of the spectrum unless
explicitly noted otherwise.











Spectrum
Kernel

1-











0 0.5 1 1.5
Frequency (kHz()

Figure 2-2. Harmonic product spectrum.

Obj ect 2-1. Bandpass filtered /u/ (WAV Hile 6 KB)



Since the logarithm is an increasing function, an equivalent approach is to estimate the

pitch as the frequency that maximizes the logarithm of the product of the spectrum at harmonics

of that frequency. Since the logarithm of a product is equal to the sum of the logarithms of the

terms, HPS can also be written as


p =arg max flog| X(kf) | (2-2)
I k=1

or using an integral transform, as


p = arg max~ log |X( f')| f3(f'-kf ) df '. (2-3)
f k=1

Figure 2-2 shows the kernel of this integral for a pitch candidate with frequency 190 Hz.

A pitfall of this algorithm is that if any of the harmonics is missing (i.e., its energy is zero),

the product will be zero (equivalently, the sum of the logarithms will be minus infinity) for the

candidate corresponding to the pitch, and therefore the pitch will not be recognized. Figure 2-2










also shows the spectrum of the vowel /u/ (as in good) with a pitch of 190 Hz (Object 2-1). This

sample was passed through a filter with a bandpass range of 300-3400 Hz to simulate telephone-

quality speech. Therefore, the fundamental is missing and HPS is not able to recognize the pitch

of this signal. Another salient characteristic of this sample is its intense second harmonic at 380

Hz, caused probably by the first formant of the vowel, which is on average around 380 Hz as

well (Huang, Acero, and Hon, 2001).

2.2 Sub-harmonic Summation (SHS)

An algorithm that has no problem with missing harmonics is Sub-Harmonic Summation

(SHS) (Hermes, 1988), which solves the problem by using addition instead of multiplication.

Therefore, if any harmonic is missing, it will not contribute to the total, but will not bring the

sum to zero either. In mathematical terms, SHS estimates the pitch as


p =argmaxf| X(kf) |, (2-4)
f k=1




Spectrulm
Kernel













0 0.5 1 1.5
Frequency (kHz)

Figure 2-3. Subharmonic summation.








































Spectrulm
- Kernel


or using an integral transform as


p = arg max |X( f')| G( f'-kf) df '. (2-5)
f k=1

An example of the kernel of this integral is shown in Figure 2-3.

A pitfall of this algorithm is that since it gives the same weight to all the harmonics,

subharmonics of the pitch may have the same score as the pitch, and therefore they are valid

candidates for being recognized as the pitch. For example, suppose that a signal has a spectrum

consisting of only one component at fHz. By definition, the pitch of the signal is fHz as well.

However, since the algorithm adds the spectrum at n multiples of the candidate, each of the

subharmonics f/2, f/3,..., f/n will have the same score asJ; and therefore they are equally valid to

be recognized as the pitch.


0 0.5 1 1.5 2
Frequency (kHz)

Figure 2-4. Subharmonic summation with decay.









This problem can be solved by introducing a monotonically decaying weighting factor for

the harmonics. SHS implements this idea by weighting the harmonics with a geometric

progression as


p = arg max | IX( f') | r k-1 '-kf )df (2-6)
f k=1

where the value of r was empirically set to 0.84 based on experiments using speech. The kernel

of this integral is shown in Figure 2-4. SHS is the only algorithm in this chapter that solves the

subharmonic problem by applying this decay factor. Later, another algorithm will be presented

(Biased Autocorrelation) which solves this problem in a different way.

2.3 Subharmonic to Harmonic Ratio (SHR)

A drawback of the algorithms presented so far is that they examine the spectrum only at

the harmonics of the fundamental, ignoring the contents of the spectrum everywhere else. An

example will illustrate why this is a problem. Suppose that the input signal is white noise (i.e., a

signal with a flat spectrum). This signal is perceived as having no pitch. However, the previous

algorithms will produce the same score for each pitch candidate, making each of them a valid

estimate for the pitch.

This problem is solved by the Subharmonic to Harmonic Ratio algorithm (SHR) (Sun,

2000), which not only adds the spectrum at harmonics of the pitch candidate, but also subtracts

the spectrum at the middle points between harmonics. However, this algorithm uses the

logarithm of the spectrum, and therefore has the problem previously discussed for HPS. Also,

this algorithm gives the same weight to all the harmonics and therefore it suffers from the

subharmonics problem. SHR can be written as


p = arg maxI log |X(f f')| 3(f'-kf) (f'-(k 1 2) f) df' .(2-7)
f k=1











Spectrum
-- Kernel


-1
0 0.5 1 1.5 2
Frequency (kHz()

Figure 2-5. Subharmonic to harmonic ratio.



The kernel of the integral is shown in Figure 2-5. Notice that SHR will produce a positive score

for a signal with a harmonic spectrum and a score of zero for white noise. However, this

algorithm has a problem that is shared by all the algorithms presented so far: since they examine

the spectrum only at harmonic locations, they cannot recognize the pitch of inharmonic signals.

Before we move on to the next algorithm, we wish to add some insight to SHR. If we

further divide the sum in Equation 2-7 by n, the algorithm would compute the average peak-to-

valley ratio, where the peaks are expected to be at the harmonics of the candidate, and the valleys

are expected to be at the middle point between harmonics. This idea will be exploited later by

SWIPE, albeit with some refinements: the average will be weighted, the ratio will be replaced

with the distance, and the peaks and valleys will be examined over wider and blurred regions.

2.4 Harmonic Sieve (HS)

One algorithm that is able to recognize the pitch of some inharmonic signals is the

Harmonic Sieve (HS) (Duifhuis and Willems, 1982). This algorithm is similar to SHS, but has










two key differences: instead of using pulses it uses rectangles, and instead of computing the inner

product between the spectrum and the rectangles, it counts the number of rectangles that contain

at least one component (a rectangle is said to contain a component if the component fits within

the rectangle and its amplitude exceeds a certain threshold T). The rectangles are centered at the

harmonics of the pitch candidates, and their width is 8% of the frequency of the harmonics. This

algorithm can be expressed mathematically as


p =argmax T < max |X(f'| (28
=-p,-k=1 7 f't(0.96k~f,1.04k~f) xf ] 2

where [-] is the Iverson bracket (i.e., produces a value of one if the bracketed proposition is true,

and zero otherwise). Notice that the expression in the sum is a non-linear function of the

spectrum, and therefore this algorithm cannot be written using an integral transform. Figure 2-6

shows the kernel used by this algorithm.





Spectrulm
Kernel



1









0 0.5 1 1.5
Frequency (kHz)

Figure 2-6. Harmonic sieve.










A pitfall of HS is that, when a component is close to an edge of a rectangle, a slight change

in its frequency could put it in or out of the rectangle, possibly changing the estimated pitch

drastically. Such radical changes do not typically occur in pitch perception, where small changes

in the frequency of the components lead to small changes in the perceived pitch, as mentioned in

Section 1.2.6. This problem can be solved by using smoother boundaries to decide whether a

component should be considered as a harmonic or not, as done by the next algorithm.

2.5 Autocorrelation (AC)

One of the most popular methods for pitch estimation is autocorrelation. The

autocorrelation function r(t) of a signal x(t) measures the correlation of the signal with itself after

a lag of size t, i.e.,


-T 2


The Wiener-Khinchin theorem shows that autocorrelation can also be computed as the inverse

Fourier cosine transform of the squared spectrum of the signal, i.e., as


r(t)=j |X( f) |cos(2xft) df .(2-10)


The autocorrelation-based pitch estimation algorithm (AC) estimates the pitch as the frequency

whose inverse maximizes the autocorrelation function of the signal, i.e., as


p = arg max |X( f') | cos(2nf' /f) df', (2-11)
f~fn0

where the parameter fma is introduced to avoid the maximum that the integral has at infinity. The

kernel for this integral is shown in Figure 2-7. It is easy to see that as f increases, the kernel

stretches without limit, and since the cosine starts with a value of one and decays smoothly,

eventually it will give a weight of one to the whole spectrum, producing a maximum at infinity.











Spectrum
Kernel













0 1 2 3 0 1 2
Frequency (kHz)

Figure 2-7. Autocorrelation.



Notice that this problem can be easily solved by removing the first quarter of the first cycle of

the cosine (i.e., setting it to zero). Since the DC of a signal (i.e., it zero-frequency component)

only adds a constant to the signal, ignoring the DC should not affect the pitch estimation of a

periodic signal.

Because of the frequency domain representation of autocorrelation, we can see that there is

a large resemblance between AC and SHR (compare the kernel of Figure 2.7 with the kernel of

Figure 2.5), although with three main differences. First, instead of using an alternating sequence

of pulses, AC uses a cosine, which adds a smooth interpolation between the pulses. Second, AC

adds an extra lobe at DC, which was already shown to have a negative effect. Third, AC uses the

power of the spectrum (i.e., the squared spectrum) instead of the logarithm of the spectrum.

Therefore, both algorithms measure the average peak-to-valley distance, one in the power

domain and the other in the logarithmic domain, although AC does it in a much smoother way.









There is also a similarity between AC and HS (compare the kernel of Figure 2.7 with the

kernel of Figure 2.6). HS allows for inharmonicity of the components of the signal by

considering as harmonic any component within a certain distance from a harmonic of the

candidate pitch. AC does the same in a smoother way by assigning to a component a weight that

is a function of its distance to the closer harmonic of the candidate pitch; the smaller the distance,

the larger the weight, and the further the distance, the smaller the weight. In fact, if the

component is too far from any harmonic, its weight can be negative.

Like all the algorithms presented so far, except SHS, AC exhibits the subharmonics problem

caused by the equal weight given to all the harmonics (see Section 2.2). To solve this problem, it

is common to take the local maximum of highest frequency rather than the global maximum.

However, this technique sometimes fails. For example, consider a signal with fundamental

frequency 200 Hz (i.e., period of 5 ms) and first four harmonics with amplitudes 1,6,1,1, as

shown in Figure 2-8A (Object 2-2). Except at very low intensity levels, the four components are

audible, and the pitch of the signal corresponds to its fundamental frequency. However, as shown

in Figure 2-8C, AC has its first non-zero local maximum at 2.5 ms, which corresponds to a pitch

of 400 Hz.

Another common solution is to use the biased autocorrelation (BAC) (Sondhi, 1968;

Rabiner, 1977), which introduces a factor that penalizes the selection of low pitch values. This

factor gives a weight of one to a pitch period of zero and decays linearly to zero for a pitch

period corresponding to the window size T. This can be written as


p = arg maxfc/f,, 1 (f)|'cs2f /f f (2-12)

































0


- -

0 200 400 600 800 101
Frequency (Hz)


/i Y
Y
Y


15


E
2 6
04q
c~O0


A

00


m
r
rnY


VI


,i


O i
Q


I~


20


5


10
Time (ms)


Figure 2-8. Comparison between AC, BAC, ASDF, and AMDF. A) Spectrum of a signal with
pitch and fundamental frequency of 200 Hz. B) Waveform of the signal with a
fundamental period of 5 msec. C) AC has a maximum at every multiple of 5 ms,
making it hard to choose the best candidate. The first (non-zero) local maxima is at
2.5 ms, making the "first peak" criteria to fail. D) BAC has its first peak and its non-
zero largest local maximum at 2.5 ms. E) ASDF is an inverted, shifted, ands scaled
AC. F) AMDF is similar to ASDF.

Obj ect 2-2. Signal with strong second harmonic (WAV file, 32 KB)



However, the combination of this bias and the squaring of the spectrum may introduce new

problems. For example, if T= 20 ms as in the BAC function of Figure 2-8D, the bias will make

the height of the peak at 2.5 ms larger than the height of the peak at 5 ms, consequently causing

an incorrect pitch estimate.


B










2.6 Average Magnitude and Squared Difference Functions (AMDF, ASDF)

Two functions similar to autocorrelation (in the sense that they compare the signal with

itself after a lag of size t) are the magnitude difference function (AMDF) and the average

squared difference function (ASDF). The AMDF is defined as

T/2
d t) = x(t') -x(t'+t)| dt', (2-13)
T/2

and the ASDF as

T/2
st) = [x(t') -x(t'+t)]' dt'. (2-14)
T/2

It is easy to show that ASDF and autocorrelation are related through the equation (Ross, 1974)

s~t)= 2((0)- r~)),(2-15)

and therefore, s(t) is just an inverted, shifted, and scaled version of autocorrelation. Therefore, as

illustrated in the panels C (or D) and E of Figure 2-8, where (biased) autocorrelation has peaks,

s(t) has dips. Thus, an ASDF-based algorithm must look for minima instead of maxima to

estimate pitch.

It has also been shown (Ross, 1974) that d(t) can be approximated as


d(t) P(t) [s(t)] /. (2-16)

Although the relation between d(t) and s(t) depends on t through P(t), it is found in practice that

this factor does not play a significant role, and a large similarity between d(t) and s(t) exists, as

observed in panels E and F of Figure 2-8. Therefore, since the functions r(t), s(t), and d(t) are so

strongly related, none of them is expected to offer much more than the others for pitch

estimation. However, modifications to these functions, which cannot be expressed in terms of the

other functions, have been used successfully to improve their performance on pitch estimation.










An example is given by YIN (de Cheveigne, 2002), which uses a variation of s(t) to avoid the dip

at lag zero, improving its performance. Another variation is the one we proposed in the previous

section (i.e., the removal of the first quarter of the cosine) to avoid the maximum at zero lag for

autocorrelation.

2.7 Cepstrum (CEP)

An algorithm similar to AC is the cepstrum-based pitch estimation algorithm (CEP) (Noll,

1967). The cepstrum c(t) of a signal x(t) is very similar to its autocorrelation. The only difference

is that it uses the logarithm of the spectrum instead of its square, i.e.,



c(t) = log | X( f) |cos(2xft) df .(2-17)


CEP estimates the pitch as the frequency whose inverse maximizes the cepstrum of the signal,

i.e., as





Spectrulm
Kernel













0 0.5 1 1.5
Frequency (kHz)

Figure 2-9. Cepstrum.











p = arg maxj log |X(f') |cos(2f'/ f) df'. (2-18)
f
The kernel of this integral is shown in Figure 2-9. Like AC, CEP exhibits the subharmonics

problem and the problem of having a maximum at a large value of f The maximum is not

necessarily at infinity because, depending on the scaling of the signal, the logarithm of the

spectrum may be negative at large frequencies, and therefore assigning a positive weight to that

region may in fact decrease the score. Figure 2-10 shows the spectrum of the speech signal that

has been used in previous figures and the kernel that produces the highest score for that

spectrum, which corresponds to a candidate pitch of about 10 k
the spectrum was arbitrarily set to zero for frequencies below 300 Hz because its original value

(minus infinity) would make unfeasible the evaluation of the integral in Equation 2-18. This

problem of the use of the logarithm when there are missing harmonics was already discussed in

Section 2.1.





Spectrulm
Kernel













0 0.5 1 1.5
Frequency (kHz)

Figure 2-10. Problem caused to cepstrum by cosine lobe at DC.









2.8 Summary

In this chapter we presented pitch estimation algorithms that have influenced the creation

of SWIPE. The most common problems found in these algorithms were the inability to deal with

missing harmonics (HPS, SHR, and CEP) and inharmonic signals (HPS, SHS, and SHR), and the

tendency to produce high scores for subharmonics of the pitch (all the algorithms, although to a

lesser extent SHS and BAC). Solutions to these problems were either found in other algorithms

or were proposed by us.









CHAPTER 3
THE SAWTOOTH WAVEFORM INSPIRED PITCH ESTIMATOR

Aiming to improve upon the algorithms presented in Chapter 2, we propose the Sawtooth

Waveform Inspired Pitch Estimator (SWIPE)2. The seed of SWIPE is the implicit idea of the

algorithms presented in Chapter 2: to find the frequency that maximizes the average peak-to-

valley distance at harmonics of that frequency. However, this idea will be implemented trying to

avoid the problem-causing features found in those algorithms. This will be achieved by avoiding

the use of the logarithm of the spectrum, applying a monotonically decaying weight to the

harmonics, observing the spectrum in the neighborhood of the harmonics and middle points

between harmonics, and using smooth weighting functions.

3.1 Initial Approach: Average Peak-to-Valley Distance Measurement

If a signal is periodic with fundamental frequency J; its spectrum must contain peaks at

multiples of fand valleys in between. Since each peak is surrounded by two valleys, the average

peak-to-valley distance (APVD) for the k-th peak is defined as

dkf) |X~kf)|-|X((k-1/2)f)| ]+1 | Xkf)|-|X((k+1/2)f)|
2 2


= |X(kf )l | 1 X((k -1/2)1 f |+ |X((k +1/2) f ) l | (3-1)


Averaging over the first n peaks, the global APVD is

1
D,, ( f )= d
YZk=1

11 1
Z |Xf2) |X((n+1/2)f)|+i |X(kf.)|-|X((k-1/2)f)| (3-2)





2 The name of the algorithm will become clear in a posterior section.












Spectrum
Kernel




0.5-





-0.5-


-1,
0 0.5 1 1.5
Frequency (kHz()


Figure 3-1. Average-peak-to-valley-di stance kernel.




Our first approach to estimate pitch is to find the frequency that maximizes the APVD. Staying

with the integral transform notation used in Chapter two, and dropping the unnecessary 1/n term,

the algorithm can be expressed as



p = argmax | X(f')| K,(f,f') df.', (3-3)
f
where


K,(, f)=1(f/ ) 1 s((n+1/2)f'/ )~(~'f)+ 6(kf' )l(-/2)f'/ f). (3-4)
2 2 k=1

The kernel K,,(~ff ') for f= 190 Hz is shown in Figure 3-1 together with the spectrum of the

sample vowel /u/ used in Chapter 2, which will be used extensively in this chapter as well. The

kernel is a function not only of the frequencies but also of n, the number of harmonics to be used.

Each positive pulse in the kernel has a weight of 1, each negative pulse between positive pulses

has a weight of -1, and the first and last negative pulses have a weight of -1/2. This kernel is










similar to the kernel used by SHR (see Chapter 2), with the only difference that in K,, the first

negative pulse has a weight of -1/2 and K,, has an extra negative pulse at the end, also with a

weight of -1/2.

3.2 Blurring of the Harmonics

The previous method of measuring the APVD works if the signal is harmonic, but not if it

is inharmonic. To allow for inharmonicity, our first approach was to blur the location of the

harmonics by replacing each pulse with a triangle function with base f/2,

f /4-|I f '| ,if | f '|< f /4
A, ( ') =(3-5)
0 otherwise.

The base of the triangle was set to f/2 to produce a triangular wave as shown in Figure 3-2. To be

consistent with the APVD measure, the first and last negative triangles were given a height of

1/2. One reason for using a base that is proportional to the candidate pitch is that it allows for a

pitch-independent handling of inharmonicity, as seems to be done in the auditory system (see

section 1.2.6).



Spectrum
Kernel






-1 I;





0 0.5 1 1.5
Frequency (kHz()

Figure 3-2. Triangular wave kernel.

















0.5-




-1 i



0 100 200 300 400 500 600
Frequency (Hz)

Figure 3-3. Necessity of strictly convex kernels.

Object 3-1. Beating tones (WAV file, 32 KB)



The triangular kernel approach was abandoned because it was found that the components

of the kernel must be strictly concave (i.e., must have a continuous second derivative) at their

maxima. The following example will illustrate why this is necessary. Suppose we have a signal

with components at 200 and 220 Hz, as shown in Figure 3-3 (Object 3-1). This signal is

perceived as a loudness-varying tone with a pitch of 210 Hz, phenomena known as beating.

However, the triangular kernel produces the same score for each candidate between 200 and 220

Hz. This is easy to see by slightly stretching or compressing the kernel such that its first positive

peak is within that range. Such stretching or compression would cause an increment on the

weight of one of the components and a decrement of the same amount on the other, keeping the

score constant.

Therefore, the triangle was discarded and concatenations of truncated squarings,

Gaussians, and cosines were explored. The squaring function was truncated at its fixed point, and


























0 f 2f 3f
Frequency (Hz)

Figure 3-4. Kernels formed from concatenations of truncated squarings, Gaussians, and cosines.



the Gaussian and the cosine functions were truncated at their inflection points. The Gaussian was

truncated at the inflection points to ensure that the concatenation of positive and negative

Gaussians have a continuous second derivative. The same can be said about the cosine, but

furthermore, the concatenation of positive and negative cosine lobes produces a cosine, which

has all order derivatives.

Concatenations of these three functions, stretched or compressed to form the desired

pattern of maxima at multiples of the candidate pitch, are illustrated in Figure 3-4. Although

informal tests showed no significant differences in pitch estimation performance among the

three, the cosine was preferred because of its simplicity. Notice also that this kernel is the one

used by the AC and CEP pitch estimators (see Chapter 2).

3.3 Warping of the Spectrum

As mentioned in Chapter 2, the use of the logarithm of the spectrum in an integral

transform is inconvenient because there may be regions of the spectrum with no energy, which



















r ~


1






1 =







-1


0.5 1






0. 1


0.5 1


0


0 0.5 1
Frequency (kHz)


Figure 3-5. Warping of the spectrum.


would prevent the evaluation of the integral, since the logarithm of zero is minus infinity. But

even if there is some small energy in those regions, the large absolute value of the logarithm

could make the effect of these low energy regions on the integral larger than the effect of the

regions with the most energy, which is certainly inconvenient.

To avoid this situation, the use of the logarithm of the spectrum was discarded and other

commonly used functions were explored: square, identity, and square-root. Figure 3-5 shows

how these functions warp the spectrum of the vowel /u/ used in Chapter 2. As mentioned earlier,

this spectrum has two particularities: it has a missing fundamental, and it has a salient second









harmonic. The missing fundamental is evident in panel B, which shows that the logarithm of the

spectrum in the region of 190 Hz is minus infinity. The salient second harmonic at 380 Hz shows

up clearly in the other three panels, but especially in panel C, where the spectrum has been

squared. Panel D shows the square-root of the spectrum, which neither overemphasizes the

missing fundamental (as the logarithm does) nor the salient second harmonic (as the square

does).

We believe the square-root warping of the spectrum is more convenient for three reasons.

First, it matches better the response of the auditory system to amplitude, which is close to a

power function with an exponent in the range 0.4-0.6 (see Chapter 2); second, it allows for a

weighting of the harmonics proportional to their amplitude, as we will show in the next section;

and third, it produces better pitch estimates, as found tests presented later.

3.4 Weighting of the Harmonics

To avoid the subharmonics problem presented in Chapter 2, a decaying weighting factor was

applied to the harmonics. The types of decays explored were exponential and harmonic. For

exponential decays, a weight of r k was applied to the k-th harmonic (k= 1, 2, ..., n, and

r = 0.9, 0.7, 0.5) through the multiplication of the kernel by the envelope r f-, as shown in

Figure 3-6. For harmonic decays, a weight of 1/k P was applied to the k-th harmonic

(k= 1, 2, ..., n, and p= 1/2, 1, 2) through the multiplication of the kernel by the envelope

(f/ f' '), as shown in Figure 3-6. In informal tests, the best results were obtained using harmonic

decays with p = 1/2, which matches the decay of the square-root of the average spectrum of

vowels (see Chapter 2). In other words, better pitch estimates were obtained when computing the

inner product (IP) of the square-root of the input spectrum and the square-root of the expected

spectrum, than when computing the IP's over the raw spectra.












Exponent al- r =0 9
1.8 Exponent al- r; = 0
Exponent al. r =0 5
1.6 ,) Harmonic. p; =O5
--- Harmonic. p= 1
1 A -_ Harmonlc- p 2



to 0.8C Harmonic: p=05E


0.4



0 100 200 300 400 500 600 700 800 900 1000
Frequency (kHz)

Figure 3-6. Weighting of the harmonics.




One explanation for this is that when the input spectrum matches its corresponding

template (i.e., the expected spectrum for that pitch), the use of the square-root of the spectra in

the IP gives to each harmonic a weight proportional to its amplitude. For example, if the input

spectrum has the expected shape for a vowel, i.e., the amplitude of the harmonics decay as 1, 1/2,

1/3, etc., then their square root decays as 1, 1/-\2, 1/-\3, etc. Since the terms in the sum of the IP

are the squares of these values (i.e., 1, 1/2, 1/3, etc.), then the relative contribution of each

harmonic is proportional to its amplitude. Conversely, if we compute the IP over the raw spectra,

the terms of the sum will be 1, 1/4, 1/9, etc., which are not proportional to the amplitude of the

components, but to their square. This would make the contribution of the strongest harmonics too

large and the contribution of the weakest too small. The situation would be even worse if we

would compute the IP over the energy of the spectrum (i.e., its square). The expected energy of

the harmonics for a vowel follows the pattern 1, 1/4, 1/16, etc., and computing the IP of the










energy of the harmonics with itself produces the terms 1, 1/16, 1/256, etc, which gives too much

weight to the first harmonic and almost no weight to the other harmonics.

In the ideal case in which there is a perfect match between the input and the template, any

of the previous types of warping would produce the same result: a normalized inner product

(NIP) equal to 1. However, the likelihood of a perfect match is low, and the warping may play a

big role in the determination of the best match, as we found in informal tests, which show that

the use of the square-root of the spectrum produces better pitch estimates.

3.5 Number of Harmonics

An important issue is the number of harmonics to be used to analyze the pitch. HPS, SHS,

SHR, and HS use a Eixed Einite number of harmonics, and CEP and AC use all the available

harmonics (i.e., as many as the sampling frequency allows). In informal tests the best results

were obtained when using as many harmonics as available, although it was found that going

beyond 3.3 k
significantly. Thus, to reduce computational cost it is reasonable to set these limits.

3.6 Warping of the Frequency Scale

As mentioned in Section 3.4, if the input matches perfectly any of the templates, their NIP

will be equal to 1, regardless of the type of warping used on the spectrum. The same applies to

the frequency scale. However, since a perfect match will rarely occur, a warping of the frequency

scale may play a role in determining the best match.

For the purposes of computing the integral of a function, we can think of a warping of the

scale as the process of sampling the function more Einely in some regions than others, effectively

giving more emphasis to the more Einely sampled regions. In our case, since we are computing

an inner product to estimate pitch, it makes sense to sample the spectrum more Einely in the

region that contributes the most to the determination of pitch. It seems reasonable to assume that









this region is the one with the most harmonic energy. In the case of speech, and assuming that

the amplitude of the harmonics decays inversely proportional to frequency, it seems reasonable

to sample the spectrum more finely in the neighborhood of the fundamental and decrease the

granularity as we move up in frequency, following the expected 1/fpattern for the amplitude of

the harmonics. A decrease in granularity should also be performed below the fundamental

because no harmonic energy is expected below it. However, the determination of the frequency

at which this decrease should begin is non-trivial, since we do not know a-priori the fundamental

frequency of the incoming sound (that is precisely what we wish to determine).

As we did for the selection of the warping of the amplitude of the spectrum, we appeal to

the auditory system and borrow the frequency scale it seems to use: the ERB scale (see

Chapter 1). Therefore, to compute the similarity between the input spectrum and the template,

we sample both of them uniformly in the ERB scale, whose formula is given in Equation 1-8.

This scale has several of the characteristics we desire (see Figure 1-9): it has a logarithmic

behavior as increases, tends toward a constant as decreases, and the frequency at which the

transition occurs (229 Hz) is close to the mean fundamental frequency of speech, at least for

females (Bagshaw, 1994; Wang and Lin, 2004; Schwartz and Purves, 2004). It does not produce

a decrease of granularity as approaches zero, but at least does not increase without bound either,

as a pure logarithmic scale does.

The convenience of the use of the ERB scale for pitch estimation over the Hertz and

logarithmic scales was confirmed in informal tests, since better results were obtained when using

the ERB scale. Two other common psychoacoustic scales, the Mel and Bark scales, were also

explored, but they produced worse results than the ERB scale.









3.7 Window Type and Size

Along this chapter we have been mentioning our wish to obtain a perfect match (i.e., NIP

equal to 1) between the input spectrum and the template corresponding to the pitch of the input.

This section deals with the feasibility of achieving such goal.

First of all, since the input is non-negative but the template has negative regions, a perfect

match is impossible. One solution would be to set the negative part of the template to zero, but

this would leave us without the useful property that the negative weights have: the production of

low scores for noisy signals (see Section 2.3). Instead, the solution we adopt is to preserve the

negative weights, but ignore them when computing the norm of the template. In other words, we

normalize the kernel using only the norm of its positive part

K' (f )= max(0, K(f )) (3-6)

Hereafter, we will refer to this normalization as F-normalization.

To obtain a Kt-normalized inner product (Kt-NIP) close to 1, we must direct our efforts to

make the shape of the spectral peaks match the shape of the positive cosine lobe used as base

element of the template, and also to force the template have a value of zero in the negative part

of the cosine. Since the shape of the spectral peaks is the same for all peaks, it is enough to

concentrate our efforts on one of them, and for simplicity we will do it for the peak at zero

frequency.

The shape of the spectral peaks is determined by the type of window used to examine the

signal. The most straightforward window is the rectangular window, which literally acts like a

window: it allows seeing the signal inside the window but not outside it. More formally, the

rectangular window multiplies the signal by a rectangular function of the form














0.75



0.25






-5/T -4KT -3KT -2/T -1/T 0 17T 27T 3/T 4/T 5rT
Frequency (Hz)


Figure 3-7. Fourier transform of rectangular window.



1/T ,if It|I 0, (t) =(3-7)
iti~0 otherwise,

where Tis the window size.

If a rectangular window is used to extract a segment of a sinusoid of frequency f Hz to

compute its Fourier transform, the support of this transform will not be concentrated at a single

point but will be smeared in the neighborhood of f This effect is shown in Figure 3-7 for f= 0, in

other words, the figure shows the Fourier transform of Hr (t). This transform can be written as

sinc(Tf~ ), where the since function is defined as

sin( )~
sinc(q$)= (3-8)


This function consists of a main lobe centered at zero and small side lobes that extend towards

both sides of zero. For any other value of f its Fourier transform is just a shifted version of this

function, centered at/:
















0.5








-fi -l40 f4 fi2
Frequency (Hz)

Figure 3-8. Cosine lobe and square-root of the spectrum of rectangular window.



Since the height of the side lobes is small compared to the height of the main lobe, the

most obvious approach to try to maximize the match between the input and the template is to

match the width of the main lobe, 2/T, to the width of the cosine lobe, f/2, and solve for the free

variable T. This produces an "optimal" window size, hereafter denoted T*, equal to T= 4/f:

Figure 3-8 shows the square-root of the spectrum of a rectangular window of size T = T* = 4/f and

a cosine with period f (i.e., the template used to recognize a pitch of f Hz). The Kt-NIP of the

main lobe of the spectrum and the cosine positive lobe (i.e., from -f/4 to f/4) sampled at 128

equidistant points is 0.9925, which seems satisfactorily high. However, the Kt-NIP computed

over the whole period of the cosine (i.e., from -f/2 to f/2) sampled at 128 equidistant points is

only 0.5236, which is not very high. This low Kt-NIP is caused by the relatively large side lobes,

which reach a height of almost 0.5.






























































3 This time-frequency relation may not be obvious at first sight, but it can be shown using Fourier analysis.


A window with much smaller side lobes is the Hann window. The shorter side lobes are

achieved by attenuating the time-domain window down towards zero at the edgeS3. The formula

for this window is


h7(t)=11+ cos ,I] (3-9)


where T is the window size (i.e., the size of its support). This window is simply one period of a

raised cosine centered at zero, as illustrated in Figure 3-9.

The Fourier transform of a Hann window of size T is


1 1
H, ( f) = sinc(Tf) + -sinc(Tf 1) + -sinc(Tf + 1),
2 2


(3-10)


a sum of three since functions, as illustrated in Figure 3-10. The width of the main lobe of this

transform is 4/T, twice as large as the main lobe of the spectrum of the rectangular window.


O
Time (s)

Figure 3-9. Hann window.













H(T,f)

sinc(Trf)


sinc(TT+1l) sinc(Tf-1)


-0-3rT -2T -1T 0 1T
Frequency : Hz)


2r 3r


Figure 3-10. Fourier transform of the Hann window. The FT of the Hann window consists of a
sum of three since functions.




Equalizing this width to the width of the cosine lobe, f/2, and solving for T, we obtain an optimal


window size of T*


---Squaret-root spectrum of Hann-win
-- Cosine


0
Fnrquency Hz)


fl4 f/2


Figure 3-11i. Cosine lobe and square-root of the spectrum of Hann window.










Figure 3-11 shows the square-root of the spectr-um of a Hann window of size T= T* = 8/f

and a cosine with period f: The similarity between the main lobe and the positive lobe of the

cosine is remarkable. Using Equations 3-8 and 3-10 it can be shown that they match at 5 points:

0, +/- f/8, and +/- f/4, with values cos(0) = 1, cos(n/4) = 1/-\2, and cos(n/2) = 0, respectively. The

f-NIP of the main lobe of the spectrum and the positive part of the cosine sampled at 128

equidistant points is 0.9996, and the Kt-NIP computed over the whole period of the cosine

sampled at 256 equidistant points is 0.8896, much larger than the one obtained with the

rectangular window.

The same approach can be used to obtain the optimal window size for other window types.

For the most common window types used in signal processing, it can be shown that the width of

the main lobe is 2k/T, where the parameter k depends on the window type (see Oppenheim,

Schafer, and Buck, 1999) and is tabulated in Table 3-1. For these windows, the optimal window



Table 3-1. Common windows used in signal processing*
Kt-NIP
Window type k Positive lobe Whole period
Bartlett 2 0.9984 0.7959
Bartlett-Hann 2 0.9995 0.8820
Blackman 3 0.9899 0.9570
Blackman-Harris 4 0.9738 0.9689
Bohman 3 0.9926 0.9474
Flat top 5 0.9896 0.9726
Gauss 3.14 0.9633 0.8744
Hamming 2 0.9993 0.9265
Hann 2 0.9996 0.8896
Nuttall 4 0.9718 0.9682
Parzen 4 0.9627 0.9257
Rectangular 1 0.9925 0.5236
Triangular 2 0.9980 0.8820
* The IC-NIP values were computed using 128 equidistant samples for the positive lobe and 256 equidistant
samples for the whole period.









size to analyze a signal with pitch fHz can be obtained by equalizing 2k/T to the width of the

cosine lobe, f/2, to produce T* = T= 4k/f:

Table 3-1 also shows the Kt-NIPs between the square-root of the spectrum and the cosine

computed over the positive lobe of the cosine (from -f/4 to f/4) and over the whole period of the

cosine (from -f/2 to f/2). The window that produces the largest K -NIP over the whole period is

the flat-top window. However, its size is so large compared to other windows that the increase in

Kt-NIP is probably not worth the increase in computational cost; similar results are obtained

with the Blackman-Harris window, which is 4/5 its size. If computational cost is a serious issue,

a good compromise is offered by the Hamming window, which requires half the size of the

Blackman-Harris window, and produces a f-NIP of about 0.93. This f-NIP is larger than the

one produced by the Hann window, with no increased computational cost (k-2 in both cases).

However, since the difference in performance between them is not large, we prefer the

analytically simpler Hann window.

3.8 SWIPE

Putting all the previous sections together, the SWIPE estimate of the pitch at time t can be

formulated as

ERBs(f, )
SK( f, q(s)) | Xt, f (s))|1/2 de
p(t)= argmax o )12 (3-11)
SERBs(f,,) ERBs(f,,)



where

cos(27tf'/f) ,if 3/4 < f'/f < n(f)+1/4,

K(f f ')= 1- cos(27rf'/ f ) ,if 1/4 < f '/f < 3/4 or n(f )+1/4 < f '/f < n(f) )+3/4, (-2

0 ,otherwise,











X(t, f,f')= jw-k/Jr xe_' d -ydfdt, (3-13)


E is frequency in ERBs, 9 (-) converts frequency from ERBs into Hertz, ERBs(-) converts

frequency from Hertz into ERBs, K (-) is the positive part of K(-) {i.e., max[0, K(-)]}, fmax is the

maximum frequency to be used (typically the Nyquist frequency, although 5 k
most applications), n(f)= L fmax/lf-3/4 and we .-(t) is one of the window functions in

Table 3-1, with size 4k/f: The kernel corresponding to a candidate with frequency 190 Hz is

shown in Figure 3-12. Panel A shows the kernel in the Hertz scale and Panel B in the ERB scale,

the scale used to compute the integral.

Although the initial approach of measuring a smooth average peak to valley distance has

been used everywhere in this chapter, we can make a more precise description of the algorithm.




0.2
Spectrum
E~- Kernel




0 0.5 1 1.5 2
Frequency (Hz)
0.2
Spectrum
: -- Kernel




0 5 10 15 20
Frequency (ERBs)

Figure 3-12. SWIPE kernel. A) The SWIPE kernel consists of a cosine that decays as 1/J; with a
truncated DC lobe and halved first and last negative lobes. B) SWIPE kernel in the
ERB scale.









It can be described as the computation of the similarity between the square-root of the spectrum

of the signal and the square-root of the spectrum of a sawtooth waveform, using a pitch-

dependant optimal window size. This description gave rise to the name Sawtooth-Waveform

Inspired Pitch Estimator (SWIPE).

3.9 SWIPE'

So far in this chapter we have concentrated our efforts on maximizing the similarity

between the input and the desired template, but we have not done anything explicitly to reduce

the similarity between the input and the other templates, which will be the goal of this section.

The first fact we want to mention is that most of the mistakes that pitch estimators make,

including SWIPE, are not random: they consist of estimations of the pitch as multiples or

submultiples of the pitch. Therefore, a good source of error to attack is the score (pitch strength)

of these candidates.

A good feature to reduce supraharmonic errors is to use negative weights between

harmonics. When analyzing a pitch candidate, if there is energy between any pair of consecutive

harmonics of the candidate, this suggests that the pitch, if any, is a lower candidate. This idea is

implemented by the negative weights, which reduce the score of the candidate if there is any

energy between its harmonics. This feature is used by algorithms like SHR, AC, CEP, and

SWIPE.

The effect of negative weights on supraharmonics of the pitch is illustrated in

Figure 3-13A. It shows the spectrum of a signal with fundamental at 100 Hz and all its

harmonics at the same amplitude (vertical lines). (Only harmonics up to 1 k
signal contains harmonics up to 5 k
visualization, but in general they will be wider, with a width that depends on the window size.


















0 100 20 300 400 500 600 70 800 900 1000

- : .: _~i _I ,I_ __


-


1

1
0.5
0


1
I 05
+0


-0.5
1
-0.5
II
~L 0
+
-0.5


-


50 100 200


Frequency (Hz)


Figure 3-13. Most common pitch estimation errors. A) Harmonic signal with 100 Hz
fundamental frequency and all the harmonics at the same amplitude, and 200 Hz
kernel with positive (continuous lines) and negative (dashed lines) cosine lobes. B)
Same signal and 50 Hz kernel. C) Scores using only positive cosine lobes (exhibits
peaks at sub and supraharmonics). D) Scores using both positive and negative cosine
lobes (exhibits peaks at subharmonics). E) Scores using both positive and negative
cosine lobes at the first and prime harmonics (exhibits a maj or peaks only at the
fundamental)



Panel A also shows the positive cosine lobes (continuous curves) used to recognize a pitch of

200 Hz and the negative cosine lobes that reside in between (dashed curves). The positive cosine

lobes at the harmonics of 200 Hz produce a positive contribution towards the score of the 200 Hz

candidate, but the negative cosine lobes at the odd multiples of 100 Hz cancel out this

contribution. Panel C shows the score for each pitch candidate using as kernel only the positive









cosine lobes, whereas Panel D shows the scores using both the positive and the negative cosine

lobes. The effect on the 200 Hz peak is definite: it has disappeared. The same effect is obtained

for higher order multiples of 100 Hz (not shown in the figure).

To reduce subharmonic errors, two techniques were presented in Chapter 2: the use of a

decaying weighting factor for the harmonics, and the use of a bias to penalize the selection of

low frequency candidates. The former is used by SHS and SWIPE, and the latter by AC.

Although these techniques have an effect in reducing the score of subharmonics, significant

peaks are nevertheless present at submultiples of the pitch, as shown in Figure 3-13D.

To further reduce the height of the peaks at subharmonics of the pitch we propose to

remove from the kernel the lobes located at non-prime harmonics, except the lobe at the first

harmonic. Figure 3-13B helps to show the intuition behind this idea. This figure shows the same

spectrum as in Figure 3-13A and the kernel corresponding to the 50 Hz candidate. This kernel

has positive lobes at each multiple of 50 Hz and therefore at each multiple of 100 Hz, producing

a high score for the 50 Hz candidate, as shown in Panel D. Notice that this candidate gets all of

its credit from its 2nd, 4th, 6th, etc., harmonics, i.e., 100 Hz, 200 Hz, 300 Hz, etc., frequencies that

suggest a fundamental frequency (and pitch) of 100 Hz. The same situation occurs with the

candidate at 33 Hz (kernel not shown), but in this case its credit comes from its 3rd, 6th, 9th, etc.,

harmonics.

If we use only the first and prime lobes of the kernel, the candidates at subharmonics of

100 Hz would get credit only from their harmonic at 100 Hz, but not from any other. In general,

it can be shown that with this approach, no candidate below 100 Hz can get credit from more

than one of the harmonics of 100 Hz. In other words, if there is a match between one of the

prime harmonics of this candidate and a harmonic of 100 Hz, no other prime harmonic of the









candidate can match another harmonic of 100 Hz, and therefore the score of all the candidates

below 100 Hz has to be low compared to the score of the 100 Hz candidate. This effect is evident

in Figure 3-13E, which shows the scores of the pitch candidates when using only their first and

prime harmonics. Certainly, there are peaks below 100 Hz, but they are relatively small

compared to the peak at 100 Hz. Contrast this with Panels C and D, where the score of 50 Hz is

relatively high, and therefore the risk of selecting this candidate is high.

An extra step needs to be done to avoid bias in the scores. Remember from the beginning

of this chapter that the central idea of SWIPE was to compute the average peak-to valley distance

at harmonic locations in the spectrum. When computing this average for a single peak, the

weight of the peak was twice as large as the weight of its valleys, as expressed in Equation 3-1.

Since the global average is the average of this equation over all the peaks, and since each valley

is associated to two peaks too, the weight of the valleys, except the first and the last ones, was

the same as the weight of the peaks, as expressed in Equation 3-2. However, if we use only the

first and prime harmonics, the weight of the valleys will not be necessarily -1, but will depend on

whether the valleys are between the first or prime harmonics. The only valleys with a weight

of-1 will be the valley between the first and second harmonics, and the valley between the

second and third harmonics; all the other valleys will have a weight of -1/2, before applying the

decaying weighting factor, of course.

This variation of SWIPE in which only the first and prime harmonics are used to estimate

the pitch will be denominated SWIPE' (read SWIPE prime). Its kernel is defined as


K(f, f')= CK,(f, f'), (3-14)
76(1) P

where P is the set of prime numbers, and











SSpectrumerl









0' 5i 10 15 20 2



Frequency (ERBs)

Figure 3-14. SWIPE' kernel. Similar to the SWIPE kernel but includes only the first and prime
harmonics.




cos(2x ~f'/ f) ,if | f'/ f i |< 1/4,

K, (f, f')= 1cos(2x ~f'/ f) ,if 1/4 < | f' f i |< 3/4, (3-1_5)

0 otherwise.

Notice that the SWIPE kernel can also be written as in Equation 3-14, by including all the

harmonics in the sum. The SWIPE' kernel corresponding to a pitch candidate of 190 Hz (5.6

ERBs) is shown in Figure 3-14. The numbers on top of the peaks show the harmonic number

they correspond to.

3.9.1 Pitch Strength of a Sawtooth Waveform

Since the template used by SWIPE' has peaks only at the first and prime harmonics, a

perfect match between the template and the spectrum of a sawtooth waveform is impossible

(unless fmax is so small relative to the pitch that the template contains no more than three













312 Hz I


1625 Hz


0.9 iA 0.9 i




r 0.8 *0.8
co 1 3 7 15 31 63 127 255 1 3 7 15 31 63 127 255
---SWIPE
~- SWIPE'

.5 H 78.1 Hz



0.9 C 0.9 D




0.8 0.8
1 3 7 15 31 63 127 255 1 3 7 15 31 63 127 255

Number of harmonics


Figure 3-15. Pitch strength of sawtooth waveform. A) 625 Hz. B) 312 Hz. C) 156 Hz.
D) 78.1 Hz.




harmonics). Therefore, it would be interesting to analyze the f-NIP between the spectrum and

the template as a function of the number of harmonics. Figure 3-15 shows the pitch strength (Kt-

NIP) obtained using SWIPE and SWIPE' for different pitches and different number of

harmonics. The pitches shown are 625, 312, 156, and 78.1 Hz. They were chosen because their

optimal window sizes are powers of two for the sampling rates used: 2.5, 5, 10, 20, and 40 k
In each case, fmax was set to the Nyquist frequency.

The pitch strength estimates produced by SWIPE are larger than the ones produced by

SWIPE', except when the number of harmonics is less than four, in which case both algorithms

use all the harmonics. The pitch strength estimates produced by SWIPE in Figure 3-15 have a










mean of 0.93 and a variance of 5.1x10 This mean is significantly larger than the Kt-NIP

reported in Table 3-1 for the Hann window. The reason of the mismatch is that the granularity

used to produce the data in Table 3-1 and the data in Figure 3-15 is different. The f-NIP values

in Table 3-1 are based on a sampling of 128 points per spectral lobe, while the data in Figure 3-

15 is based on a sampling of 10 points per ERB, which depending on the pitch and the harmonic

being sampled, may correspond to a range of about 0 to 40 points per spectral lobe.

On the other hand, the mean of the pitch strength estimates produced by SWIPE' is 0.87

and the variance is 1.0x10-3. The smaller mean is expected since the template of SWIPE'

includes only the first and prime harmonics, while a sawtooth waveform has energy at each of its

harmonics. The larger variance is also expected since the prime numbers become sparser as they

become larger, causing a reduction in the similarity of the template and the spectrum of the

sawtooth waveform as the number of harmonics increases.

It would be useful to have a lower bound for the pitch strength estimates produced by

SWIPE', but an analytical formulation for it is intractable. However, the data in Figure 3-15,

which is representative of a wide range of pitches and number of harmonics, suggests that the

pitch strength produced by SWIPE' for a sawtooth waveform does not go below 0.8.

3.10 Reducing Computational Cost

3.10.1 Reducing the Number of Fourier Transforms

The computation of Fourier transforms is one of the most computationally expensive

operations of SWIPE and SWIPE'. Therefore, to reduce computational cost it is important to

reduce the number of Fourier transforms. There are two strategies to achieve this: to reduce the

window overlap and to share Fourier transforms among several candidates.










3.10.1.1 Reducing window overlap

The most common windows used in signal processing are the ones that are attenuated

towards zero at their edges (e.g., Hann and Hamming windows). A disadvantage of this

attenuation is that it is possible to overlook short events if these events are located at the edges of

the windows. To avoid this situation, it is common to use overlapping windows, which increases

the coverage of the signal, at the cost of an increase in computation. However, after a certain

point, overlapping windows start to produce redundancy in the analysis, without adding any

significant benefit. The goal of this section is to propose a schema obtain a good balance

between signal coverage and computational cost.

As mentioned in Section 1.1.4, depending on frequency, a minimum of two to four cycles

are necessary to perceive the pitch of a pure tone. Based on the similarity of the data used to

arrive to this conclusion and data obtained using musical instruments, it is reasonable to assume

that these results are applicable to more general waveforms, in particular, to sawtooth

waveforms. To avoid the interaction between the number of cycles and pitch, for purposes of the

algorithm, we set the minimum number of cycles necessary to determine pitch to four, the

maximum among the minimum number of cycles required over all frequencies.

Since SWIPE and SWIPE' are designed to produce maximum pitch strength for a sawtooth

waveform4 and zero pitch strength for a flat spectrmm', a natural choice to decide whether a

sound has pitch is to use as threshold half the pitch strength of a sawtooth waveform. (In Section

3.9.1 it was found that the pitch strength of a sawtooth waveform is about 0.93 for SWIPE and

between 0.83 and 0.93 for SWIPE'.) To make these algorithms produce maximum pitch strength,


4 In fact, SWIPE' produces maximum pitch strength for sawtooth waveforms with the non-prime harmonics
removed (except the first one), but we believe this type of signal is unlikely to occur in nature.

5 The pitch strength of a flat spectrum is in fact negative because of the decaying kernel envelope.










a perfect match between the kernel and the spectrum of the signal is necessary, which requires

that the window contains eight cycles of the sawtooth waveform, when using a Hann window. If

the signal contains exactly eight cycles (i.e., if it is zero outside the window) and is shifted

slightly with respect to the window, the pitch strength decreases, and it reaches a limit of zero

when the signal gets completely out of the window. Although hard to show analytically, it is easy

to show numerically that that the relation between the shift and pitch strength is linear.

Therefore, if the window contains four or more cycles of the sawtooth waveform, the pitch

strength is at least half the maximum attainable pitch strength (i.e., the one achieved when the

window is full of the sawtooth waveform), and if the window contains less than four cycles of

the sawtooth waveform, the pitch strength is less than half the maximum attainable pitch

strength.






















Figure 3-16. Windows overlapping.

Object 3-2. Four cycles of a 100 Hz sawtooth waveform (WAV file, 2 KB)









Therefore, if we determine the existence of pitch based on a pitch strength threshold equal

to half the maximum attainable pitch strength, to determine as pitched a signal consisting of four

cycles of a sawtooth waveform, we need to ensure that there exists at least one window whose

coverage includes the whole signal. It is straightforward to show that to achieve this goal, we

need to distribute the windows such that their separation in no larger than four cycles of the pitch

period of the signal. In other words, the windows must overlap by at least 50%.

This situation is illustrated in Figure 3-16, which shows a signal consisting of four cycles

of a sawtooth waveform (listen to Object 3-2) and two Hann windows centered at the beginning

and the end of the signal. The windows are separated at a distance of four cycles, and the support

of each of them overlaps with the whole signal, making it possible for each window to reach the

pitch strength threshold. If the signal is slightly shifted in any direction, one of the windows will

cover less than four periods, but the other will cover the four periods.

This would not be true if the separation of the windows is larger than four cycles. If the

support of one of the windows overlaps completely with the signal but the separation of the

windows is larger than four cycles, the other window will not cover the signal completely, and

therefore a small shift of the signal towards the latter window would not necessarily put the

whole signal inside the window, making it impossible for any of the windows to produce a pitch

strength larger than the threshold.

3.12.1.2 Using only power-of-two window sizes

There is a problem with the optimal window size (O-WS) proposed in Section 3.7: each

pitch candidate has its own, which means that a different STFT must be computed for each

candidate. If we separate the candidates at a distance of 1/8 semitone over a range of 5 octaves

(appropriate for music, for example), we will need to compute 8*12*5 = 480 STFTs for each










pitch estimate. Not only that, for some WSs it may be inefficient to use an FFT (recall that the

FFT is more efficient for windows sizes that are powers of two).

To alleviate this problem, we propose to substitute the O-WS with the power-of-two (P2)

WS that produces the maximum f-NIP between the square-root of the main lobe of the

spectrum and the cosine kernel. To find such a WS, it is convenient to have a closed-form

formula for the f-NIP of these functions, but this involves integrating the product of a cosine

and the square-root of the sum of three since functions, which is analytically intractable.

As an alternative, we approximate the square-root of the spectral lobe with an idealized

spectral lobe (ISL) consisting of the function it approximates: a positive cosine lobe. Figure 3-17

shows a f-normalized cosine whose positive part has a width of f/2 (i.e., the cosine template

used by an fHz pitch candidate), and two normalized ISLs whose widths are half and twice the

width of the positive part of the cosine. Since the cosine and the ISLs are symmetric around zero,

the Kt-NIP can be computed using only the positive frequencies. Hence, the Kt-NIP





SK -normalized cos 27Eflf (Template)
---- K -normalized -os 4xflf (|f| K -normalized cos /f# (|lf|










-fl2 -3fl8 -1/4 -fl8 0 f/8 f4 3f/8 fl2
Frequency Hz)

Figure 3-17. Idealized spectral lobes.









of the central positive lobe of a cosine with period rf (the ISL) and a cosine with period f (the

template) can be computed as
f/4r
Scos(2irf'/ f) cos(2rf'/ f) df'
P(r) = 0 /
f/4r f/4
cos2(2nrf'/ f) df' cos2(2xf'/ f) df'1

f/4r
cos ,,[2,(1+,,r) f'/ fcos[2n(1- r) f'/ f] df'

[f/8rj1/2 //8]1/2


2ZF sin[2K(1+r) f'/ f ] sin[2K(1-r) f '/f] f'f4f
1+r 1-r


2- sin@(1 +r)/2r] sin @(1-r) /2r]
171+ II (3-16)

It is convenient to transform the input of this function to a base-2 logarithmic scale,

Ai = log2(r), and then redefine the function as

21+1/2 Sin(2 +1)r/22 si[(2" -1)r1/2 (-7
H(A) = + .(-7
xi 1+2 1-

Figure 3-18A shows H(il) for ii between -1 and 1 (i.e., r = 2" between 1/2 and 2). As ii departs

from zero, H(il) departs from 1, as expected. However, the distribution is not symmetric: a

decrease in ii has a larger effect on H(il) than an increase in ii. This make sense since a decrease

in ii corresponds to a widening of the ISL, which puts part of it in the region where the cosine

template is negative (see wider ISL in Figure 3-1), producing a large decrease in H(il). On the

other hand, narrowing the ISL keeps it in the positive region of the cosine template, producing a

smaller decrease in H(il).

















0.6





-0.2
01 0. 03 040.5 0 07 0 0. 0. 1


Fiue -8.fnomlie inrprdctbtwe tmlaead dalzd pcta lbs

Fiue31Acnb hlfli idn heP-Sta rdce h ags -I

bewe heILad h eplt.I teOW frtetepaeisTscnd n hesmln












shows te diffe~nrencied betwen r anod ctI(-1 safnto f2 o between 0eplt a ielz sctand 1.From



tefigure wecan ianfe that foru A'eten 0andin 0.56 P2we should use the larger t P2-S, ndfo

Between 0.56IS and 1,e we shold use the smaller P2-WS Howevaer Figur ecd 3-18 shows amlso ha

rthee isf, hnotmuhe loss in thele f-IP by chosn 0.5 asic threshold rthe thaO n 0.56 Theurefore, to




simplify sthealgorihwe ecdd to seo ht the toi' thrsodat 0.5. Ipnd other wors tlodestermine the











~_ I+ Larger WS
r I oSmaller WS


0 A



160 180 200 220 240 260 280

.t + Larger WS
3 1I oSmaller WS
CombinedB





160 180 200 220 240 260 280
Frequency (Hz)

Figure 3-19. Individual and combined pitch strength curves.



P2-WS to use for a pitch candidate, we transform the O-WS and the P2-WSs to a logarithmic

scale, and choose the P2-WS closest to the optimal.

Unfortunately, this approach produces discontinuities in the pitch strength (PS) curves, as

illustrated in Figure 3-19A. The PS values marked with a plus sign were produced using a WS

larger than the WS than the ones marked with a circle. To emphasize the effect, the pitch of the

signal (220 Hz) was chosen to match the point at which the change of WS occurs. Since the PS

values produced by the larger window in the neighborhood of the pitch are larger than the ones

produced by the smaller window, the pitch could be biased toward a lower value.

Although an effort was made to find an appropriate value for the threshold, it was based on

an idealized spectrum, which does not have the side lobes found in real spectra. This problem

can be alleviated by using a threshold larger than 0.56, determined through trial and error, but we










found a better solution: to compute the PS as a linear combination of the PS values produced by

the two closest P2-WSs, where the coefficients of the combination are proportional to the log-

distance between the P2-WSs and the O-WS.

Concretely, to determine the P2-WSs used to compute the PS of a candidate with

frequency f Hz, the O-WS is written as a power of two, N* = 2L+h, where L is an integer and

0< A i<1. Then, the PS values So~f) and S1Cf) are computed using windows of size 2L and 2L+1,

respectively. Finally, these PSs are combined into a single one to produce the final PS

S(f) = (1- A) Sn (f) + Ai S, (f) (3-18)

Figure 3-19B shows how this combination of PS curves smoothes the discontinuity found in

Figure 3-19A.

It would be interesting to know how much is lost in PS by using the formula proposed in

Equation 3-18, when the O-WS is not a power of two. This lost can be approximated by finding












0.~94




0.3 '5

U 0.2 0.4 0.6 0.8


Figure 3-20. Pitch strength loss when using suboptimal window sizes.









the minimum of the linear combination (1-ii) TI(A) + Ai T(Al-1) for 0 < Ai < 1, which is plotted in

Figure 3-20. It can be seen that it has a minimum of 0.93 at around Al= 0.4. Therefore, the

maximum loss when computing PS using the two closest P2-WSs is 7%. Since the minimum PS

of a sawtooth waveform when using an O-WS is about 0.92 for SWIPE and 0.83 for SWIPE'

(see Figure 3-15), the minimum pitch strength of a sawtooth waveform when using the two

closest P2-WSs is about 0.86 for SWIPE and 0.77 for SWIPE'.

Besides using an optimal window size for the FFT computation, the approximation of

O-WSs using P2-WSs has another advantage that is probably more important: the same FFT can

be shared by several pitch candidates, more precisely, by all the candidates within an octave of

the optimal pitch for that FFT. Going back to the example that started this section, the

replacement of the O-WS with the closest P2-WSs reduces the number of FFTs required to

estimate the pitch from 480 to just 5: a huge save in computation.

Using this approach, and translating the algorithm to a discrete-time domain (necessary to

compute an FFT), we can write the SWIPE' estimate of the pitch at the discrete-time index r as

p[r]= argmax (1-Al(f)) SL(f)(r, f) + Al(f) SL(f)+1(r, f), (3-19)


where

A( f) = L*( f) L( f), (3 -20)

L( f )= L'(f ) (3-21)

L*(f)= log,(4kf,/ f), (3 -22)












SL 1z f 1 ,(3-23)

r ERBs( fmax)1 K+f ~a> ERBs( fmx)1



X, [r f '] =I( (0,...,N -1), X [r, (0,...,N -1) ], f 'N / f, ), (3-24)


X,[r, C]= w,'[r'-r] xC[r] e"' "~, (3-25)


As is the ERB scale step size (0.1 gives good enough resolution), I(@,E,#) is an interpolating

function that uses the functional relations Ek = F(O'k) to predict the value of F(#), and XN[r,p]l

(p= 0, 1,..., N-1) is the discrete Fourier transform (computed via FFT) of the discrete signal

x[r'], multiplied by the size-N windowing function wN[r'], centered at z. The other variables,

constants, and functions are defined as before (see Section 3.8). A Matlab implementation of this

algorithm is given in Appendix A.

3.10.2 Reducing the Number of Spectral Integral Transforms

The pitch resolution of SWIPE and SWIPE' depends on the granularity of the pitch

candidates. Therefore, to achieve high pitch resolution, a large number of pitch candidates must

be used, and since the pitch strength of each candidate is determined by computing a Kt-NIP

between its kernel and the spectrum, the computational cost of the algorithm would increase

enormously. To avoid this situation, we propose to compute Kt-NIPs only for certain candidates,

and then use interpolation to estimate the pitch strength of the other candidates.

As noted by de Cheveigne (2002), the AC of a signal is the Fourier transform of its power

spectrum, and therefore the AC is a sum of cosines that can be approximated around zero by

using a Taylor series expansion with even powers. If the signal is periodic, its AC is also










periodic, and therefore the shape of the AC around the pitch period is the same as the shape

around zero, and therefore it can also be approximated by the same Taylor series, centered at the

pitch period. If the width of the spectral lobes is narrow and the energy of the high frequency

components is small, the terms of order 4 in the series vanish as the independent variable

approaches the pitch period, and therefore the series can be approximated using a parabola.

Since SWIPE perform an inner product between the spectrum and a kernel consisting of

cosine lobes, a similar argument can be applied to the pitch strength curves produce by SWIPE.

However, the quality of the fit of a parabola is not guaranteed for two reasons: first, the width of

the spectral lobes produced by SWIPE are not narrow, in fact, they are as wide as the positive

lobes of the cosine; and second, the use of the square-root of the spectrum rather than its energy

makes the contribution of the high frequency components large, violating the requirement of low

contribution of high frequency components. Nevertheless, parabolic interpolation produces a

good fit to the pitch strength curve in the neighborhood of the SWIPE peaks, as we will proceed

to show.

Let's derive an approximation to the pitch strength curve cr(t) produced by SWIPE for a

sawtooth waveform with fundamental frequency fo = 1/To Hz in the neighborhood of the pitch

period To. To simplify the equations, let's define the scaling transformations ro~= 27 and

r= 2xit/To. To make the calculations tractable, let's use idealized spectral lobes (i.e. cosine lobes)

and let's ignore the normalization factors and the change of width of the spectral lobe with

change of window size caused by a change of pitch candidate. Let's also replace the continuous

decaying envelope of the kernel with a decaying step function that gives a weight of 1/-\k to the

k-th harmonic. With all this simplifications, the pitch strength of a candidate with scaled pitch









period r in the neighborhood of 2xi (i.e., when the non-scaled pitch period t is in the

neighborhood of To) can be approximated as


ocr)= [akcr), (3-26)
k=1

where

k+1 4
k1/
k1/

21 Iicos[(t-2n)w]+cos[(t+2n)i] del~
2k /


1k sin[(t -2)il]+ sin[(t + 2r)o] C

2kt-2xi t+2xi


21 sin[(k +1/ 4)(t 2x)~- sin[(k +1/ 4)(t i1-2 x)



sin[(k +1/ 4)(t +2x)]~- sin[(k +1/ 4)(t +2x)~t2i1(-7


Since we are interested in approximating this function in the neighborhood of 2x,: we can

equivalently shift the function 2xi units to the left by defining dk F) Gk(r+ 2x),) and then

approximate dk (r) in the neighborhood of zero. Since sin(x) / x = 1 x2/3!i + x4/5!i O(x6) in the

neighborhood of zero, it is useful to express dk(T) aS

k +1/4 sin[(k +1/4)r] k-1/4 sin[(k-1/4)r]
2k (k +1/ 4)r 2k (k -1/ 4)r

1 sin [k 1/ 4)r]- sin [k + 1/ 4)r]
+ -(3-28)
2k + 4xi

which has the Taylor series expansion











k+1/ (k 1/) (k 14)
2k 3! 5!




2k 3! 5!


(3-29)


in the neighborhood of zero. Finally, the approximation of the pitch strength curve in the shifted-

time domain is


k=1


(3-30)


0.8



0.4


C
1=

a,
m
a,









Figure 3-21.


0.2I




-0.

-0.

-0.6
1 3 57
Number of harmonics


Coefficients of the pitch strength interpolation polynomial.


1
+ [(k -1/4)r -
2k(r + 47)


(k -1/ 4) ()
3!


(k +1/4)r- rk +O/4r ) u
3!


illl
ii I
~ 111
111










Figure 3-21 shows the relative value of the coefficients of the expansion as a function of the

number of harmonics in the signal. As the number of harmonics increases, the relative weight of

the order-4 coefficient increases. However, as r approaches zero, its fourth power becomes so

small that its overall contribution to the sum is small compared to the contribution of the order-2

term.

This effect is clear in Figure 3-22, which shows o'(r) for a sawtooth waveform with 15

harmonics using polynomials of order 2 and order 4 in the range +/- 0.045, which corresponds to

+/- 1/8 semitones. The curve has been scaled to have a maximum of 1. The large circles

correspond to candidates separated by 1/8 semitones, which is the interval used in our

implementation of SWIPE and SWIPE' for the distance between pitch candidates for which the

pitch strength is computed directly. The other markers correspond to candidates separated by

1/64 semitones, which is the resolution used to fine tune the pitch strength curve based on the

pitch strength of the candidates for which the pitch strength is computed directly. As observed in












t 0.99




[]ss I C sntplate (odr2
Interpolated (order 4)

-0.04 -0.02 0 0.02 0.04
Nomalized time z (2xfTO-2x)

Figure 3-22. Interpolated pitch strength.










the figure, for such small values of r, the pitch strength values obtained with an order 2

polynomial (squares) are indistinguishable from the ones obtained with an order 4 polynomial

(diamonds). Hence, a parabola is good enough to estimate the pitch strength between candidates

separated at distances as small as 1/8 semitones.

3.11 Summary

This chapter described the SWIPE algorithm and its variation SWIPE'. The initial

approach of the algorithm was the search for the frequency that maximizes the average peak-to-

valley distance at harmonic locations. Several modifications to this idea were applied to improve

its performance: the locations of the harmonics were blurred, the spectral amplitude and the

frequency scale were warped, an appropriate window type and size were chosen, and

simplifications to reduce computational cost were introduced. After these modifications, SWIPE

estimates the pitch as the fundamental frequency of the sawtooth waveform whose spectrum best

matches the spectrum of the input signal. Its variation, SWIPE', uses only the first and prime

harmonics of the signal.









CHAPTER 4
EVALUATION

To asses the relevance of SWIPE and SWIPE', they were compared against other

algorithms using two speech databases and a musical instruments database. This chapter presents

a brief description of these algorithms, databases, and the evaluation process. A more detailed

description is given in Appendix B.

4.1 Algorithms

The algorithms with which SWIPE and SWIPE' were compared were the following:

* AC-P: This algorithm (Boersma, 1993) computes the autocorrelation of the signal and
divides it by the autocorrelation of the window used to analyze the signal. It uses post-
processing to reduce discontinuities in the pitch trace. It is available with the Praat System
at (http://www.fon.hum.uva.nl/praat). The name of the function is ac.

* AC-S: This algorithm uses the autocorrelation of the cubed signal. It is available with the
Speech Filing System at (http://www.phon.ucl .ac.uk/resource/sfs). The name of the
function is fxac.

* ANAL: This algorithm (Secrest and Doddington, 1983) uses autocorrelation to estimate
the pitch, and dynamic programming to remove discontinuities in the pitch trace. It is
available with the Speech Filing System at (http://www.phon.ucl .ac.uk/resource/sfs). The
name of the function is fxanal.

* CATE: This algorithm uses a quasi autocorrelation function of the speech excitation signal
to estimate the pitch. We implemented it based on its original description (Di Martino,
1999). The dynamic programming component used to remove discontinuities in the pitch
trace was not implemented.

* CC: This algorithm uses cross-correlation to estimate the pitch and post-processing to
remove discontinuities in the pitch trace. It is available with the Praat System at
(http://www.fon.hum.uva.nl/praat). The name of the function is cc.

* CEP: This algorithm (Noll, 1967) uses the cepstrum of the signal and is available with the
Speech Filing System at (http://www.phon.ucl .ac.uk/resource/sfs). The name of the
function is fxcep.

* ESRPD: This algorithm (Bagshaw, 1993; Medan, 1991) uses a normalized
cross-correlation to estimate the pitch, and post-processing to remove discontinuities
in the pitch trace. It is available with the Festival Speech Filing System at
(http://www.cstr. ed.ac.uk/proj ects/festival). The name of the function i s pda~.










* RAPT: This algorithm (Secrest and Doddington, 1983) uses a normalized cross-
correlation to estimate the pitch, and dynamic programming to remove discontinuities
in the pitch trace. It is available with the Speech Filing System at
(http:.//www.phon.ucl_ ac.uk/resource/sfs). The name of the function is fxrapt.

* SHS: This algorithm (Hermes, 1988) uses subharmonic summation. It is available with the
Praat System at (http ://www.fon.hum.uva.nl/praat). The name of the function is shs.

* SHR: This algorithm (Sun, 2000) uses the subharmonic-to-harmonic ratio. It is available at
Matlab Central (http ://www.mathworks.com/matlabcentral) under the title "Pitch
Determination Algorithm". The name of the function is ship.

* TEMPO: This algorithm (Kawahara et al., 1999) uses the instantaneous frequency of the
outputs of a filterbank. It is available with the STRAIGHT System at its author web page
(http ://www.wakayama-u.ac.j p/~kawahara). The name of the function is exstraightsource.

* YIN: This algorithm (de Cheveigne and Kawahara, 2002) uses a modified version of the
average squared difference function. It is available from its author web page at
(http ://www.ircam.fr/pcm/cheveign/sw/yin.zip). The name of the function is yin.

4.2 Databases

The databases used to test the algorithms were the following:

* DVD: Disordered Voice Database. This database contains 657 samples of sustained
vowels produced by persons with disordered voice. It can be bought from Kay Pentax
(http ://www.kayelemetrics.com).

* KPD: Keele Pitch Database. This speech database was collected by Plante et. al (1995) at
Keele University with the purpose of evaluating pitch estimation algorithms. It contains
about 8 minutes of speech spoken by five males and five females. Laryngograph data was
recorded simultaneously with speech, and was used to produce estimates of the
fundamental frequency. It is publicly available at (ftp://ftp.cs.keele. ac.uk/pub/pitch).

* MIS: M~usical hIstruntents Samples. This database contains more than 150 minutes of
sound produced by 20 different musical instruments. It was collected at the University of
lowa Electronic Music Studios, directed by Lawrence Fritts, and is publicly available at
(http:.//theremin. music.uiowa. edu).

* PBD: Paul Bagshaw 's Database for evaheating pitch determination algorithms. This
database contains about 8 minutes of speech spoken by one male and one female.
Laryngograph data was recorded simultaneously with speech, and was used to produce
estimates of the fundamental frequency. It was collected by Paul Bagshaw at the
University of Edinburg (Bagshaw et. al 1993; Bagshaw 1994), and is publicly available at
(http://www.cstr. ed. ac.uk/research/proj ects/fda).









4.3 Methodology

The algorithms were asked to produce a pitch estimate every millisecond. The search range

was set to 40-800 Hz for speech and 30-1666 Hz for musical instruments. The algorithms were

given the freedom to decide if the sound was pitched or not. However, to compute our statistics,

we considered only the time instants at which all the algorithms agreed that the sound was

pitched.

Special care was taken to account for time misalignments. Specifically, the pitch estimates

were associated to the time corresponding to the center of their respective analysis windows, and

when the ground truth pitch varied over time (i.e., for PBD and KPD), the estimated pitch time

series were shifted within a range of 100 ms to find the best alignment with the ground truth.

The performance measure used to compare the algorithms was the gross error rate (GER).

A gross error occurs when the estimated pitch is off from the reference pitch by more than 20%.

At first glance this margin of error seems too large, but considering that most of the errors pitch

estimation algorithms produce are octave errors (i.e., halving or doubling the pitch), this is a

reasonable metric. On the other hand, this tolerance gives room for dealing with misalignments.

The GER measure has been used previously to test PEAs by other researchers (Bagshaw, 1993;

Di Martino, 1999; de Cheveigne and Kawahara, 2002).

4.4 Results

Table 4-1 shows the GERs for each of the algorithms over each of the speech databases.

Both the rows and the columns are sorted by average GER: the best algorithms are at the top, and

the more difficult databases are at the right. The best algorithm overall is SWIPE', followed by

SHS and SWIPE. Although on average SHS performs better than SWIPE, the only database in

which SHS beats SWIPE is in the disordered voice database, which indicates that SWIPE

performs better than SHS on normal speech.





Table 4-1. Gross error rates for speech*


Gross error (%)
KPD DVD


Algorithm PBD
SWIPE' 0.13
SHS 0.15
SWIPE 0.15
RAPT 0.75
TEMPO 0.32
YINT 0.33
SHR 0.69
ESRPD 1.40
CEP 6.10
AC-P 0.73
CATE 2.60
CC 0.48
ANAL 0.83
AC-S 8.80
Average 1.70
* Values computed using two significant digits.


Average
0.53
0.75
0.91
1.40
1.40
2.10
3.50
5.00
5.90
6.70
6.60
2.40
13.00
19.00
4.90


0.83
1.00
0.87
1.00
1.90
1.40
1.50
3.90
4.20
2.90
10.00
3.60
2.00
7.00
3.00


0.63
1.10
1.70
2.40
2.00
4.50
5.10
4.60
14.00
16.00
7.20
5.00
35.00
40.00
9.90


Table 4-2. Proportion of overestimation

Algorithm DVD
CC 0.0
SHS 0.0
RAPT 0.0
SHR 0.0
AC 0.0
AC 0.0
ANAL 0.0
CEP 0.4
SWIPE' 0.0
SWIPE 0.1
YINT 0.1
TEMPO 0.1
CATE 0.5
ESRPD 0.5
Average 0.1
* Values computed using one significant digit.


errors relative to total gross errors*
Proportion of overestimations
PBD KPD


Average


Table 4-2 shows the proportion of GEs caused by overestimations of the pitch with respect

to the total number of GEs. The proportion of GEs caused by underestimation of the pitch is just



































One minus the values shown in the table. Algorithms at the top have a tendency to underestimate

the pitch while algorithms at the bottom have a tendency to overestimate it. Most algorithms tend

to underestimate the pitch in the disordered voice database while the errors are more balanced in

the normal speech databases.

Table 4-3 shows the pitch estimation performance as a function of gender for the two

databases for which we had access to this information: PVD and KPD. The error rates are on

average larger for female speech than for male speech.

Table 4-4 shows the GERs for the musical instruments database. Some of the algorithms

were not evaluated on this database because they did not provide a mechanism to set the search

range, and the range they covered was smaller that the pitch range spanned by the database. The

two algorithms that performed the best were SWIPE' and SWIPE.


Table 4-3. Gross error rates by gender*

Algorithm Male
SWIPE'
SHS
SWIPE
RAPT
TEMPO
SHR

AC-P
CEP
CC
ESRPD
ANAL
AC-S
CATE
Average
* Values computed using two significant digits.


Gross error (%)
Female
2.40
2.50
2.70
2.90
3.10
3.60
3.20
3.60
4.20
4.50
3.90
5.90
10.00
4.20
4.00


Average


0.36
0.55
0.49
0.42
0.67
0.61
1.10
2.10
1.80
2.40
3.10
1.30
3.20
11.00
2.10










Table 4-4. Gross error rates for musical instruments*


Gross error (%)
Overestimates
0.10
0.02
1.00
1.70
0.83
0.00
0.00
1.50
5.30
1.20


Algorithm Underestimates
SWIPE' 1.00
SWIPE 1.30
SHS 0.88
TEMPO 0.29
YINT 1.60
AC-P 3.20
CC 3.60
ESRPD 5.30
SHR 15.00
Average 3.60
* Values computed using two significant digits.


Total


1.10
1.30
1.90
2.00
2.40
3.20
3.60
6.80
20.00
4.70


Table 4-5. Gross error rates by instrument family*
Gross error (%)
Bowed


Plucked
Strings
8.80
11.00
4.00
14.00
8.10
26.00
28.00
11.00
15.00
14.00


Algorithm
SWIPE'
SWIPE
TEMPO

SHS
AC-P
CC
ESRPD
SHR
Average


Brass
0.01
0.00
0.00
0.03
0.02
0.03
0.07
4.00
22.00
2.90


Strings
0.19
0.22
2.60
1.10
1.50
0.56
0.83
6.90
25.00
4.30


Woodwinds
0.14
0.23
1.40
1.50
0.72
0.80
1.00
7.10
38.00
5.60


Piano
2.20
0.02
7.30
0.36
12.00
0.36
0.36
6.00
26.00
6.10


Average
2.30
2.30
3.10
3.40
4.50
5.60
6.00
7.00
25.00
6.60


* Values computed using two significant digits. Brass: French horn, bass/tenor trombones, trumpet, and tuba.
Bowed strings: double bass, cello, viola, and violin. Woodwinds: flute, bass/alto flutes, bass/Bb/Eb clarinets,
alto/soprano saxes. Plucked strings: cello and violin.


Table 4-5 shows the GERs by instrument family. The two best algorithms are SWIPE' and

SWIPE. SWIPE' tends to perform better than SWIPE except for the piano, for which SWIPE

produces almost no error. On the other hand, SWIPE' performance on piano is relatively bad

compared to correlation based algorithms. The family for which fewer errors were obtained was

the brass family; many algorithms achieved almost perfect performance for this family. The










Table 4-6. Gross error rates for musical instruments by octave*
Gross error (%)


46.2 Hz
+/- 1/2
oct.
1.20
0.08
3.20
0.24
7.80
0.26
15.00
7.90
37.00
8.10


92.5 Hz
+/- 1/2
Oct.
1.00
1.20
0.95
2.00
2.60
2.60
2.80
2.60
0.60
1.80


185 Hz
+/- 1/2
Oct.
2.30
3.00
5.30
7.80
3.20
8.20
2.00
4.80
1.80
4.30


370 Hz
+/- 1/2
Oct.
0.89
1.00
1.80
2.50
1.20
2.70
1.10
4.20
27.00
4.70


740 Hz
+/- 1/2
Oct.
0.13
0.25
0.69
0.71
0.23
0.93
0.52
12.00
70.00
9.50


1480 Hz
+/- 1/2
Oct.
0.29
0.38
0.96
0.30
0.14
0.40
0.31
32.00
81.00
13.00


Algorithm
SWIPE'
SWIPE

AC-P
SHS
CC
TEMPO
ESRPD
SHR
Average


Average
0.97
0.99
2.20
2.30
2.50
2.50
3.60
11.00
36.00
6.90


* Values computed using two significant digits.


family for which more errors were produced was the strings family playing pizzicato, i.e., by

plucking the strings. Indeed, pizzicato sounds were the ones for which the performers produced

more errors and the ones that were hardest for us to label (see Appendix B).

Table 4-6 shows the GERs as a function of octave. The best performance on average was

achieved by SWIPE' and SWIPE. The results of the algorithms with an average GER less than





Pitch (Hz)
46.2 92.5 185 370 740 1480
100.0

f ~SWIPE'
10.0-SWP


[E 1.01 b -X-- AC-P
0 prt SHS
0.1-
--+ TEMPO

0.0

Figure 4-1. Gross error rates for musical instruments as a function of pitch.










Table 4-7. Gross error rates for musical instruments by dynamic*
Gross error (%)
Algorithm pp mf ff Average
SWIPE' 1.30 1.20 0.92 1.10
SWIPE 1.40 1.40 1.20 1.30
SHS 1.50 2.30 2.00 1.90
TEMPO 2.00 1.90 2.00 2.00
YINT 2.20 2.50 2.40 2.40
AC-P 3.30 3.20 3.30 3.30
CC 3.60 3.30 3.80 3.60
ESRPD 5.70 7.10 7.60 6.80
SHR 27.00 29.00 29.00 28.00
Average 5.30 5.80 5.80 5.60
* Values computed using two significant digits.


10% is reproduced in Figure 4-1. All algorithms have approximately the same tendency, except

at the lowest octave, where a larger variance in the GERs can be observed.

Table 4-7 shows the GERs as a function of dynamic (i.e., loudness). In general, there is no

significant variation of GERs with changes in loudness, although SWIPE' has a tendency to

reduce the GER as loudness increases [i.e., as the dynamic moves from pianissimo (pp) to

fortissimo (ff) ].

As a final test, we wanted to validate the choices we made in Chapter 3, i.e., shape of the

kernel, warping of the spectrum, weighting of the harmonics, warping of the frequency scale, and

selection of window type and size. For this purpose, we evaluated SWIPE' replacing every time

one of its features with a more standard feature, i.e., smooth vs. pulsed kernels, square-root vs.

raw spectrum, decaying vs. flat kernel envelope, ERB vs. Hertz frequency scale, and pitch-

optimized vs. fixed window size. We varied each of these variables independently and obtained

the results shown in Table 4-8. Although some of the variations made SWIPE' improve in some

of the databases, overall SWIPE' worked better with the features we proposed in Chapter 3.










Table 4-8. Gross error rates for variations of SWIPE'*
Gross error (%)
Variation PBD KPD DVD MIS Average
Original 0.13 0.83 0.63 1.10 0.67
Flat envelope 0.16 1.00 1.40 0.60 0.79
Hertz scale 0.23 1.70 1.40 0.37 0.93
Pulsed kernel 0.21 0.84 3.00 2.60 1.70
Raw spectrum2 0.25 2.10 1.60 4.90 2.20
Fixed WS3 0.15 0.77 1.70 9.10 2.90
* Values computed using two significant digits. FFTs were computed using optimal window sizes and the
spectrum was inter/extrapolated to frequency bins separated at 5 Hz.2 The use of the raw spectrum rather than the
square root of the spectrum implies the use of a kernel whose envelope decays as 1/frather than 1/-\/f to match the
spectral envelope of a sawtooth waveform.3 The power-of-two window size whose optimal pitch was closest to the
geometric mean pitch of the database was used in each case. A window of size 1024 samples was used for the
speech databases and a window of size of 256 samples was used for the musical instruments database.


4.5 Discussion

SWIPE' showed the best performance in all categories. SWIPE was the second best ranked

for musical instruments and normal speech but not for disordered speech, for which SHS

performed better (see Table 4-1). One possible reason is that it is common for disordered voices

to have energy at multiples of subharmonics of the pitch, and therefore algorithms that apply

negative weights to the spectral regions between harmonics (e.g., SWIPE, SWIPE', and all

autocorrelation based algorithms) are prone to produce low scores for the pitch. Although

SWIPE' is among this group, its use of only the first and prime harmonics, reduces substantially

the score subharmonics of the pitch, producing most of the time a larger score for the pitch than

for its subharmonics.

The rankings of the algorithms are relatively stable in all the tables except for SHR, which

showed a good performance for speech but not for musical instruments. We believe this is

caused by the wide pitch range spanned by the musical instruments. This is suggested by the

results in Table 4-6, which show that SHR performs well in the octaves around 92.5 Hz and 185










Hz, which corresponds to the pitch region of speech, but performs very bad as the pitch moves

from this region.

Figure 4-1 shows that the relative trend on performance with pitch for musical instruments

is about the same for all the algorithms except in the lowest region, where a large variance in

performance was observed. However, this variance may be caused by a significant reduction in

the numbers of samples in this region (about 4% of the data). The figure also shows an overall

increase in GER in the octave around 185 Hz. We believe this is caused by the presence of a set

of difficult sounds in the database with pitches in that region, since it is hard to believe that there

is an inherent difficulty of the algorithms to recognize pitch in that region.









CHAPTER 5
CONCLUSION

The SWIPE pitch estimator has been developed. SWIPE estimates the pitch as the

fundamental frequency of the sawtooth waveform whose spectrum best matches the spectrum of

the input signal. The schematic description of the algorithm is the following:

1. For each pitch candidate within a pitch range fmin-fmax, COmpute its pitch strength as follows:

a. Compute the square-root of the spectrum of the signal.

b. Normalize the square-root of the spectrum and apply an integral transform using a
normalized cosine kernel whose envelope decays as 1/-\f:

2. Estimate the pitch as the candidate with highest strength.

An implicit objective of the algorithm was to find the frequency for which the average

peak-to-valley distance at its harmonics is maximized. To achieve this, the kernel was set to zero

below the first negative lobe and above the last negative lobe, and to avoid bias, the magnitude

of these two lobes was halved.

To make the contribution of each harmonic of the sawtooth waveform proportional to its

amplitude and not to the square of its amplitude, the square-root of the spectrum was taken

before applying the integral transform.

To make the kernel match the normalized square-root spectrum of the sawtooth waveform,

a 1/-\f envelope was applied to the kernel. The kernel was normalized using only its positive

part.

To maximize the similarity between the kernel and the square-root of the input spectrum,

each pitch candidate required its own window size, which in general is not a power of two, and

therefore not ideal to compute an FFT. To reduce computational cost, the two closest power-of-

two window sizes were used, and their results are combined to produce a single pitch strength

value. This had the extra advantage of allowing an FFT to be shared by many pitch candidates.









Another technique used to reduce computational cost was to compute a coarse pitch strength

curve and then fine tune it by using parabolic interpolation. A last technique used to reduce

computational cost was to reduce the amount of window overlap while allowing the pitch of a

signal as short as four cycles to be recognized.

The ERB frequency scale was used to compute the spectral integral transform since the

density of this scale decreases almost proportionally to frequency, which avoids wasting

computation in regions where there little spectral energy is expected.

SWIPE', a variation of SWIPE, uses only the first and prime harmonics of the signal,

producing a large reduction in subharmonic errors by reducing significantly the scores of

subharmonics of the pitch.

Except for the obvious architectural decisions that must be taken when creating an

algorithm (e.g., selection of the kernel), there are no free parameters in SWIPE and SWIPE', at

least in terms of "magic numbers".

SWIPE and SWIPE' were tested using speech and musical instruments databases and their

performance was compared against twelve other algorithms which have been cited in the

literature and for which free implementations exist. SWIPE' was shown to outperform all the

algorithms on all the databases. SWIPE was ranked second in the normal speech and musical

instruments databases, and was ranked third in the disordered speech database.











APPENDIX A
MATLAB IMPLEMENTATION OF SWIPE'


This is a Matlab implementation of SWIPE'. To convert it into SWIPE just replace


[ 1 primes(n) ] in the for loop of the function pirId.r Il;lo egthneCa~ndidate with [ 1:n ].

function [p,t,s] = swipep(x,fs,plim,dt,dlog2p,dERBs,sTHR)
SSWIPEP Pitch estimation using SWIPE'.
SP = SWIPEP(X,Fs,[PMIN PMAX],DT,DLOG2F,DERBS,STHR) estimates the pitch of
Sthe vector signal X with sampling frequency Fs (in Hertz) every DT
Seconds. The pitch is estimated by sampling the spectrum in the ERB scale
Using a step of size DERBS ERBs. The pitch is searched within the range
S[PMIN PMAX] (in Hertz) sampled every DLOG2P units in a base-2 logarithmic
Scale of Hertz. The pitch is fine tuned by using parabolic interpolation
With a resolution of 1/64 of semitone (approx. 1.6 cents). Pitches with a
Strength lower than STHR are treated as undefined.

S[P,T,S] = SWIPEP(X,Fs,[PMIN PMAX],DT,DLOG2P,DERBS,S/thr) returns the times
ST at which the pitch was estimated and their corresponding pitch strength.

SP = SWIPEP(X,Fs) estimates the pitch using the default settings PMIN =
S30 Hz, PMAX = 5000 Hz, DT = 0.01 s, DLOG2P = 1/96 (96 steps per octave),
SDERBS = 0.1 ERBs, and STHR = -Inf.

SP = SWIPEP(X,Fs,...[],...) uses the default setting for the parameter
Replaced with the placeholder [].

SEXAMPLE: Estimate the pitch of the signal X every 10 ms within the
range 75-500 Hz using the default resolution (i.e., 96 steps per
Octavee, sampling the spectrum every 1/20th of ERB, and discarding
samples with pitch strength lower than 0.4. Plot the pitch trace.
S[x,Fs] = wavread(filename);
S[p,t,s] = swipep(x,Fs,[75 500],0.01,[],1/20,0.4);
plot(1000*t,p)
Sxlabel('Time (ms)')
ylabel('Pitch (Hz)')if ~ exist( 'plim' ) | isempty(plim), plim = [30
5000]; end
if ~ exist( 'dt' ) | isempty(dt), dt = 0.01; end
if ~ exist( 'dlog2f' ) | isempty(dlog2f), dlog2f = 1/96; end
if ~ exist( 'dERBs' ) | isempty(dERBs), dERBs = 0.1; end
if ~ exist( 'sTHR' ) | isempty(sTHR), sTHR = -Inf; end
t = [ 0: dt: length(x)/fs ]'; X Times
dc = 4; X Hop size (in cycles)
K = 2; X Parameter k for Hann window
g Define pitch candidates
log2pc =[ log2(plim(1)): dlog2f: log2(plim(end)) ]';
pc = 2 .^ log2pc;
S = zeros( length(pc), length(t) ); X Pitch strength matrix
3 Determine P2-WSs
logWs = round( log2( 4*K fs ./ plim ))
ws = 2.^[ logWs(1): -1: logWs(2) ]; X P2-WSs
pO = 4*K fs ./ ws; X Optimal pitches for P2-WSs
g Determine window sizes used by each pitch candidate
d = 1 + log2pc log2( 4*K*fs./ws(1) );











% Create ERBs spaced frequencies (in Hertz)
fERBs = erbs2hz([ hz2erbs Opc(1)/4): dERBs: hz2erbs(fs/2) ])
for i = 1 : length(ws)
dn = round( dc fs / pO(i) ); % Hop size (in samples)
% Zero pad signal
xzp =[ zeros( ws(i)/2, 1 ); x(:); zeros( dn + ws(i)/2, 1 ) ];
% Compute spectrum
w = hanning( ws(i) ); % Hann window
o = max( 0, round( ws(i) dn ) ); % Window overlap
[X, f, ti ]= specgram( xzp, ws(i), fs, w, O )
% Interpolate at equidistant ERBs steps
M = max( 0, interpl( f, abs (X), fERBs, 'spline', 0) ); % Magnitude
L = sqrt( M ); % Loudness
% Select candidates that use this window size
if i==1ength(ws); j=find(d-i>-1); k=find(d(j)-i<0);
elseif i==1; j=find(d-i<1); k=find(d(j)-i>0);
else j=find(abs(d-i)<1); k=1:1ength(j);
end
Si = pitchStrengthAllCandidates( fERBs, L, pc(j) );
% Interpolate at desired times
if size(Si,2) > 1
Si = interpl( ti, Si', t, 'linear', NaN )';
else
Si = repmat( NaN, length(Si), length(t) );
end
lambda = d( j (k) ) i;
mu = ones( size(j) );
mu(k) = 1 abs( lambda )
S(j,:) = S(j,:) + repmat(mu,1,size(Si,2)) .* Si;
end
% Fine-tune the pitch using parabolic interpolation
p = repmat( NaN, size(S,2), 1 );
s = repmat( NaN, size(S,2), 1 )
for j = 1 : size(S,2)
[s(j), i ]= max( S(:,j) )
if s(j) < sTHR continue, end
if i==1, p(j)=pc(1); elseif i==1ength(pc), p(j)=pc(1); else

tc = 1 ./ pc(I);
ntc = ( tc/tc(2) 1 ) 2*pi;
c = polyfit( ntc, S(I,j), 2 );
ftc = 1 ./ 2.^[ log2 Opc(I(1))): 1/12/64: log2 Opc(I(3))) ];
nftc = ( ftc/tc(2) 1 ) 2*pi;
[s(j) k] = max( polyval( c, nftc ) );
p(j) = 2 ^ ( log2 Opc(I(1))) + (k-1)/12/64 )
end
end

function S = pitchStrengthAllCandidates( f, L, pc)
% Normalize loudness
warning off MATLAB:divideByZero
L = L ./ repmat( sqrt( sum(L.*L) ), size(L,1), 1 );
warning on MATLAB:divideByZero
% Create pitch salience matrix
S =zers(length(pc), size(L,2) )
for j = 1 : length(pc)
S(j,:) = pitchStrengthOneCandidate( f, L, pc(j) );











end


function S = pitchStrengthOneCandidate( f, L, pc)
n = fix( f(end)/pc 0.75 ); % Number of harmonics
k = zeros( size(f) ); % Kernel
q = f / pc; % Normalize frequency w.r.t. candidate
for i = [1 primes (n)]
a = abs( q i )
% Peak's weigth
p= a< .25;
k(p) = cos( 2*pi q(p) );
% Valleys' weights
v = .25 < a & a < .75;
k(v) = k(v) + cos( 2*pi q(v) ) / 2;
end
% Apply envelope
k = k .* sqrt( 1./f )
% K+-normalize kernel
k = k / norm( k(k>0) );
% Compute pitch strength
S = k' L;

function erbs = hz2erbs (hz)
erbs = 21.4 logl0( 1 + hz/229 );

function hz = erbs2hz(erbs)
hz = ( 10 .^ (erbs./21.4) 1 ) 229;










APPENDIX B
DETAILS OF THE EVALUATION

B.1 Databases

All the databases used in this work are free and publicly available on the Internet, except

the disordered voice database. Besides speech recordings, the speech databases contain

simultaneous recordings of laryngograph data, which facilitates the computation of the

fundamental frequency. The authors of these databases used them to produce ground truth pitch

values, which are also included in the databases. The disordered voice database includes

fundamental frequency estimates, but as it will be explained later, a different ground truth data

set was used. The musical instruments database contains the names of the notes in the names of

the files.

B.1.1 Paul Bagshaw's Database

Paul Bagshaw's database (PBD) for evaluation of pitch determination algorithms

(Bagshaw et. al 1993; Bagshaw 1994) was collected at the University of Edinburgh, and is

available at (http://www. cstr. ed.ac.uk/research/proj ects/fda). The speech and laryngograph

signals of this database were sampled at 20 k
computed by estimating the location of the glottal pulses in the laryngograph data and taking the

inverse of the distance between each pair of consecutive pulses. Each fundamental frequency

estimate is associated to the time instant in the middle between the pair of pulses used to derive

the estimate.

B.1.2 Keele Pitch Database

The Keele Pitch Database (KPD) (Plante et. al, 1995) was created at Keele University and

is available at (ftp://ftp. cs.keele.ac.uk/pub/pitch). The speech and laryngograph signals were

sampled at 20 k









26.5 ms window shifted at intervals of 10 ms. Windows where the pitch is unclear are marked

with special codes.

Both of these speech databases PBD and KPD have been reported to contain errors (de

Cheveigne, 2002), especially at the end of sentences, where the energy of speech decays and

malformed pulses may occur. We will explain later how we deal with this problem.

B.1.3 Disordered Voice Database

The disordered voice database (DVD) was collected by Kay Pentax

(http ://www.kayelemetrics.com). It includes 657 disordered voice samples of the sustained

vowel "ah" sampled at 25 k
patients with a wide variety of organic, neurological, traumatic, psychogenic, and other voice

di orders.

The database includes fundamental frequency estimates, but by definition, they do not

necessarily match their pitch. Therefore we estimated the pitch by ourselves by listening to the

samples through earphones, and matching the pitch to the closest note, using as reference a

synthesizer playing sawtooth waveforms. Assuming that we chose one of the two closest notes

every time, this procedure should introduce an error no larger than 6%, which is smaller than the

20% necessary to produce a GE (see Chapter 4).

There were some samples for which the pitch ranged over a perfect fourth or more (i.e., the

higher pitch was more than 33% higher than the lower pitch). Since this range is large compared

to the permissible 20%, these samples were excluded. Samples for which the range did not span

more than a maj or third (i.e., the higher pitch was no more than 26% higher than the lower pitch)

were preserved, and they were assigned the note corresponding to the median of the range. If the

median was between two notes, it was assigned to any of them. This should introduce an error no










larger than two semitones (12%), which is about half the maximum permissible error of 20%.

There were 30 samples for which we could not perceive with confidence a pitch, so they were

excluded as well.

Since the ground truth data was based on the perception of only one listener (the author), it

could be argued that this data has low validity. To alleviate this, we excluded the samples for

which the minimum error produced by any algorithm was larger than 50%.

After excluding the non-pitch, variable pitch, and samples at which the algorithms

disagreed with the ground truth, we ended up with 612 samples out of the original 657.

Appendix C shows the ground truth used for each of these 612 samples.

B.1.4 Musical Instruments Database

The musical instruments samples database was collected at the University of Iowa, and is

available at (http ://theremin.music.uiowa.edu). The recordings were made using CD quality

sampling at a rate of 44,100 k
computational cost. No noticeable change of perceptual pitch was perceived by doing this, even

for the highest pitch sounds. This database contains recordings of 20 instruments, for a total of

more than 150 minutes and 4,000 notes. The notes are played in sequence using a chromatic

scale with silences in between. Each file usually spans one octave and is labeled with the name

of the initial and final notes, plus the name of the instrument, and other details (e.g.,

Violin.pizz.mf. sulG. C4B4.aiff).

In order to test the algorithms, the files were split into separate files containing each of

them a single note with no leading or trailing silence. This process was done in a semi-automatic

way by using a power-based segmentation method, and then checking visually and auditively the

quality of the segmentation.









While doing this task it was discovered that some of the note labels were wrong. The

intervals produced by the performers were sometimes larger than a semitone, and therefore the

names of the files did not correspond to the notes that were in fact played. This situation was

common with string instruments, especially when playing in pizzicato.

Therefore, after splitting the files, we listened to each of them, and manually corrected the

wrong names by using as reference an electronic keyboard. This procedure sometimes

introduced name conflicts (i.e., there were repeated notes played by the same instrument, same

dynamic, etc.), and when this occurred, we removed the repeated notes trying to keep the closest

note to the target. When the conflicting notes were equally close to the target, the "best quality"

sound was preserved. This removal of files was done to avoid the overhead of having to add

extra symbols to the file names to allow for repetitions, which would have complicated the

generation of scripts to test the algorithms.

Since this process of manually correcting the names of the notes was very tedious,

especially for the pizzicato sounds, after fixing all the pizzicato bass and violin notes, the process

was abandoned and the cello and viola pizzicato sounds were excluded from our evaluation.

Arguably, except for the bass, pizzicato sounds are not very common in music, and therefore

leaving the cello and viola pizzicato sounds out did not affect the representativeness of the

sample significantly.

B.2 Evaluation Using Speech

Whenever possible, each of the algorithms was asked to give a pitch estimate every

millisecond within the range 40-800 Hz, using the default settings of the algorithm (an exception

was made for ESRPD: instead of using the default settings in the Festival implementation, the

recommendations suggested by the author of the algorithm were followed). The range 40-800

was used to make the results comparable to the results published by de Cheveigne (2002).









However, a full comparison is not possible since some other variables were treated differently in

that study.

The commands issued for each of the algorithms were the following

* AC-P: To Pitch (ac)... 0.001 40 15 no 0.03 0.45 0.01 0.35 0.14 800
* AC-S: fxac input~file
* ANAL: fxanal input~file
* CC: To Pitch (cc)... 0.001 40 15 no 0.03 0.45 0.01 0.35 0.14 800
* CEP: fxcep input~file
* ESRPD: pda input~file -o output~file -L -d 1 -shift 0.001 -length 0.0384 -fmax 800 -fmin
40 -lpfilter 600
* RAPT: fxrapt input~file
* SHS: To Pitch (shs)... 0.001 40 15 1250 15 0.84 800 48
* SHR: [t, p ] =shrp( x, fs, [40 800], 40, 1, 0.4, 1250, 0, 0 );
* SWIPE: [ p, t ] = swipe( x, fs, [40 800], 0.001, 1/96, 0.1, -Inf );
* SWIPE': [ p, t ] = swipep( x, fs, [40 800], 0.001, 1/96, 0.1i, -Inf );
* TEMPO: f0raw = exstraightsource( x, fs );
* YIN: p.minf0 = 40; p.maxf0 = 800; p.hop = 20; p.sr = fs; r = yin( x, p );

where x is the input signal and f, is the sampling rate in Hertz.

An important issue that had to be considered was the time associated to each pitch

estimate. Since all algorithms use symmetric windows, a reasonable choice was to associate each

estimate to the time at the center of the window. For CATE, ESRPD, and SHR, the user is

allowed to determine the size of the window, so we followed the recommendation of their

authors and we set the window sizes to 51.2, 38.4, and 40 ms, respectively. YIN uses a different

window size for each pitch candidate, but the windows are always centered at the same time

instant, and the largest window size is two periods of the largest expected pitch period. For the

Praat's algorithms AC-P, CC, and SHS, through trial and error we found that they use windows

of size 3, 1, and 2 times the largest expected pitch period, respectively. For AC-S, ANAL, CEP,

RAPT, and TEMPO, the user is not allowed to set up the window size, but the algorithms output



6 The command for CATE is not reported because we used our own implementation.









the time instants associated to each pitch estimate, so we used these times hoping that they

correspond to the centers of the analysis windows used to determine the pitch.

The times associated to the pitch ground truth series are explicitly given in the PBD

database, but not in the KPD database. For KPD, each pitch value was associated to the center of

the window. Therefore, since the ground truth pitch values were computed using 26.5 msec

windows separated at a distance of 10 msec, the first pitch estimate was assigned a time of 13.25

msec, and the time associated to each successive pitch estimate added 10 msec to the time of the

previous estimate. For the DVD databases, each vowel was assumed to have a constant pitch, so

the ground truth pitch time series was assumed to be constant.

The purpose of the evaluation was to compare the pitch estimates of the algorithms, but not

their ability to distinguish the existence of pitch. Therefore, we included in the evaluation only

the regions of the signal at which all algorithms and the ground truth data agreed that pitch

existed. To achieve this, we took the time instants of the ground truth values and the time

instants produced by all the algorithms that estimated the pitch every millisecond (9 out of 13

algorithms), rounded them to the closest multiple of 1 millisecond, and took the intersection.

This intersection would form the set of times at which all the algorithms would be evaluated. The

algorithms that produced pitch estimates at a rate lower than 1,000 per second were not

considered for finding the intersection because that would reduce the time granularity of our

evaluation, which was desired to be one millisecond.

As suggested in the previous paragraph, some algorithms do not necessarily produce pitch

estimates at times that are multiples of one millisecond, i.e., they may produce the estimates at

the times t+ At ms, where t is an integer and |At| < 1. Thus, to evaluate them at multiples of one

millisecond, the pitch values at the desired times were inter/extrapolated in a logarithmic scale.









In other words, we took the logarithm of the estimated pitches, inter/extrapolated them to the

desired times, and took the exponential of the inter/extrapolated pitches. Inter/extrapolation in

the logarithmic domain was preferred because we believe this is the natural scale for pitch. This

is what allows us to recognize a song even if it is sung by a male or a female.

An important issue that must be considered when using simultaneous recordings of the

laryngograph and speech signals is that the latter are typically delayed with respect to the former.

An attempt to correct this misalignment was reported by the authors of KPD, but the success was

not warranted. No attempt of correction was reported for PBD. Since pitch in speech is time-

varying, such misalignment could increase the estimation error significantly. To alleviate this

problem, each pitch time series produced by each algorithm was delayed or advanced, in steps of

1 msec, and up to 100 msec, in order to find the best match with the ground truth data.

B.3 Evaluation Using Musical Instruments

Considering that many algorithms were designed for speech, the pitch range of the MIS

database is probably too large for them to handle. To alleviate this, we excluded the samples that

were outside the range 30-1666 Hz, which is nevertheless large, compared to the pitch range of

speech. Since the range 30-1666 Hz was found to be too large for the Speech Filing System

algorithms (AC-S, ANAL, CEP, and RAPT) these algorithms were not evaluated on the MIS

database. The commands issued for each of the algorithms were the following:

* AC-P: To Pitch (ac)... 0.001 30 15 no 0.03 0.45 0.01 0.35 0.14 1666
* CC: To Pitch (cc)... 0.001 30 15 no 0.03 0.45 0.01 0.35 0.14 1666
* ESRPD: pda input~file -o output~file -P -d 1 -shift 0.001 -length 0.0384 -fmax 1666 -fmin
30 -n 0 -m 0
* SHS: To Pitch (shs)... 0.001 30 15 5000 15 0.84 1666 48
* SHR: [ t, p ] = shrp( x, fs, [30 1666], 40, 1, 0.4, 5000, 0, O );
* SWIPE: [ p, t ] = swipe( x, fs, [30 1666], 0.001, 1/96, 0. 1, -Inf );
* SWIPE': [ p, t ] = swipep( x, fs, [30 1666], 0.001, 1/96, 0.1, -Inf );
* YIN: p.minf0 = 30; p.maxf0 = 1666; p.hop = 10; p.sr = 10000; r = yin(x,p);









Besides the widening of the pitch range, the only difference with respect to the commands

used for the speech databases were for ESRPD and SHS. For both of them, the low-pass filtering

was removed in order to use as much information from the spectrum as possible. This was

convenient because the sounds were already low-pass filtered at 5 k
pitch sounds (around 1666 Hz) had no more than three harmonics in the spectrum. The second

change was the use of the ESRPD peak-tracker (option -P) as an attempt to make the algorithm

improve upon its results with speech.

The evaluation process was very similar to the one followed for speech: the time instants

of the ground truth and the pitch estimates were rounded to the closest millisecond, the

intersection of all the times was taken, and the statistics were computed only at the times of this

intersection. However, there was an issue that was necessary consider in this database. Some

instruments played much longer notes than others. The range of durations goes from tenths of

second for strings playing in pizzicato, to several seconds for some notes of the piano. If the

overall error is computed without taking this into account, the results will be highly biased

toward the performance produced with the instruments that play the largest notes.

To account for this, the GER was computed independently for each sample, and then

averaged over all the samples. However, this introduced an undesired effect: some samples had

very few pitch estimates (only one estimate in some cases), and therefore this procedure would

give them too much weight, which potentially would introduce noise in our results. Therefore,

we discarded the samples for which the time instants at which the algorithms were evaluated

were less than half the duration of the sample (in milliseconds). This discarded 164 samples,

resulting in a total of 3459 samples, which was nevertheless a significant amount of data to

quantify the performance of the algorithms.











APPENDIX C
GROUND TRUTH PITCH FOR THE DISORDERED VOICE DATABASE


Table C-1. Ground truth pitch values for the disordered voice database
AAKO2 220.0 AAS16 123.5 ABBO9 246.9 ABGO4 116.5 ACGl3 207.7 ACG20 164.8
ACH16 185.0 ADM14 138.6 ADPO2 155.6 ADP11 116.5 AEAO3 220.0 AFR17 246.9
AHKO2 110.0 AHS20 196.0 AJFl2 110.0 AJMO5 138.6 AJM29 123.5 AJP25 233.1
AL~Bl8 123.5 AL~W27 174.6 AL~W28 220.0 AMB22 146.8 AMC14 92.5 AMC16 146.8
AMC23 196.0 AMDO7 130.8 AMJ23 123.5 AMK25 77.8 AMPl2 220.0 AMvT11 246.9
AMV23 185.0 ANAl5 155.6 ANA20 155.6 ANB28 196.0 AOS21 110.0 ASK21 116.5
ASR20 92.5 ASR23 130.8 AWEO4 155.6 AXD11 174.6 AXDI9 196.0 AXLO4 196.0
AXL22 196.0 AXSO8 155.6 AXT 11 185.0 AXT 13 196.0 BAH13 98.0 BAS 19 293.7
BAT19 185.0 BBR24 164.8 BCMO8 233.1 BEFO5 185.0 BGSO5 246.9 BJHO5 174.6
BJKl6 174.6 BJK29 103.8 BKBl3 87.3 BLBO3 110.0 BMKO5 246.9 BMMO9 233.1
BPFO3 116.5 BRT 18 311.1 BSD30 130.8 BSGl3 174.6 BXD17 138.6 CAC10 185.0
CAHO2 196.0 CAK25 196.0 CAL~12 92.5 CAL~28 261.6 CAR10 196.0 CBD17 164.8
CBDI9 174.6 CBD21 207.7 CBR29 174.6 CCMI5 110.0 CDWO3 146.8 CEN21 92.5
CER16 185.0 CER30 174.6 CFWO4 155.6 CJB27 116.5 CJP10 98.0 CLE29 116.5
CLS31 185.0 CMAO6 123.5 CMA22 103.8 CMR0 1 185.0 CMRO6 110.0 CMR26 174.6
CMS 10 196.0 CMS25 185.0 CNPO7 196.0 CNR0 1 185.0 CPKl9 155.6 CPK21 174.6
CPW28 220.0 CRM12 185.0 CSJl6 233.1 CSYO1 110.0 CTB30 146.8 CTYO3 130.8
CXLO8 174.6 CXMO7 130.8 CXM14 220.0 CXM18 146.8 CXPO2 207.7 CXR13 146.8
CXTO8 155.6 DAC26 155.6 DAG0 1 185.0 DAMO8 174.6 DAPl7 130.8 DAS 10 146.8
DAS24 146.8 DAS30 87.3 DAS40 77.8 DBAO2 220.0 DBFl18 155.6 DBGl4 103.8
DFBO9 233.1 DFS23 293.7 DFS24 293.7 DGL30 207.7 DGOO3 110.0 DHDO8 123.5
DJF23 146.8 DJM14 130.8 DJM28 185.0 DJPO4 110.0 DLB25 261.6 DLL25 174.6
DLTO9 207.7 DLWO4 130.8 DMCO3 185.0 DMF ll 293.7 DMGO7 146.8 DMG24 196.0
DMG27 155.6 DMPO4 123.5 DMR27 233.1 DMS0 1 146.8 DOA27 92.5 DRC 15 196.0
DRGl9 116.5 DSC25 277.2 DSW14 138.6 DVDI9 164.8 DWKO4 130.8 DXS20 123.5
EAB27 164.8 EALO06 207.7 EAS11 110.0 EAS15 138.6 EAW21 207.7 EBJO3 146.8
EDGl9 196.0 EEB24 164.8 EECO4 196.0 EEDO7 554.4 EFCO8 130.8 EGK30 196.0
EGTO3 138.6 EGW23 220.0 EJB0 1 92.5 EJMO4 123.5 ELLO4 116.5 EMDO8 82.4
EML18 370.0 EMP27 174.6 EOWO4 164.8 EPWO4 164.8 EPWO7 123.5 ERSO7 185.0
ESL28 207.7 ESMO5 138.6 ESPO4 138.6 ESSO5 174.6 ESS24 220.0 EWWO5 174.6
EXEO6 146.8 EXH21 185.0 EXIO4 110.0 EXIO5 116.5 EXSO7 207.7 EXW12 164.8
FAHO 1 164.8 FGR15 130.8 FJL23 116.5 FLL27 207.7 FLW13 207.7 FMCO8 196.0
FMM21 207.7 FMM29 207.7 FMQ20 155.6 FMR17 116.5 FRH18 146.8 FSPl3 155.6
FXC12 110.0 FXE24 196.0 FXI23 103.8 GCU31 123.5 GEA24 130.8 GEKO2 138.6
GJWO9 174.6 GLB0 1 77.8 GLB22 98.0 GMMO6 196.0 GMMO7 207.7 GMSO3 110.0
GMSO5 261.6 GMW18 146.8 GRS20 110.0 GSB ll 164.8 GSLO4 116.5 GTN21 130.8
GXL21 196.0 GXT 10 155.6 GXX13 164.8 HBS12 196.0 HED26 123.5 HJHO7 130.8
HLC16 110.0 HLK01 116.5 HLKl5 130.8 HLM24 138.6 HMGO3 185.0 HML26 207.7
HWRO4 164.8 HXB20 196.0 HXI29 82.4 HXL58 116.5 HXR23 116.5 IGDO8 196.0
IGDI6 174.6 JABO8 130.8 JAB30 164.8 JAFl15 146.8 JAJ10 207.7 JAJ22 155.6
JAJ31 155.6 JAL~O5 174.6 JAM0 1 207.7 9-Jan 130.8 JAPO2 138.6 JAPl7 174.6
JAP25 174.6 JBPl4 98.0 JBR26 110.0 JBS17 82.4 JBW14 130.8 JCCO8 164.8
JCC10 207.7 JCH13 110.0 JCH21 116.5 JCL12 174.6 JCL20 146.8 JCR01 233.1
JDMO4 110.0 JEG29 246.9 JES29 123.5 JFC28 82.4 JFGO8 138.6 JFG26 138.6
JFM24 174.6 JFN11 110.0 JFN21 116.5 JHW29 146.8 JIJ30 146.8 JJDO6 174.6
JJD11 185.0 JJD29 138.6 JJIO3 110.0 JJM28 220.0 JLCO8 185.0 JLD24 233.1
JLHO3 174.6 JLM18 207.7 JLM27 123.5 JLS11 130.8 JLS18 138.6 JMC18 138.6
JME23 164.8 JMH22 155.6 JMJO4 207.7 JMZl6 196.0 JOPO7 130.8 JPBO7 98.0
JPBl7 164.8 JPB30 98.0 JPM25 110.0 JPP27 207.7 JRF30 123.5 JRP20 110.0
JSGl8 207.7 JTMO5 87.3 JTSO2 103.8 JWE23 185.0 JWK27 98.0 JWMI5 116.5
JXBl16 110.0 JXB26 116.5 JXC21 220.0 JXD0 1 138.6 JXDO8 138.6 JXD30 123.5
JXF ll 246.9 JXF29 103.8 JXGO5 138.6 JXM30 146.8 JXSO9 110.0 JXS14 146.8
JXS23 98.0 JXS39 146.8 JXZll 123.5 KABO3 185.0 KACO7 246.9 KAOO9 261.6
KASO9 233.1 KAS14 220.0 KCG23 246.9 KCG25 220.0 KDB23 220.0 KEP27 87.3












Table C-1. Continued


KEW22
KJMO8
KMC22
KWD22
LAIO4
LGK25
LLM22
LPN14
LXC01
LXSO5
MAT26
MCB20
MEW15
MJLO2
MLC23
MMM12
MPC21
MPS26
MRR22
MXS10
NGAl6
NML 15
ORS18
PDOll
PMC26
PTS01
RBCO9
RFC28
RJC24
RJZl6
RML 13
RTH15
RXG29
SAM25
SCH15
SEKO6
SHCO7
SLM27
SRB31
SXH10
TAR18
TNC14
TRF21
VMSO4
WDK13
WJFl5
WTGO7
JEC18


220.0
130.8
207.7
185.0
174.6
110.0
277.2
146.8
207.7
196.0
261.6
174.6
246.9
130.8
174.6
246.9
207.7
220.0
174.6
233.1
116.5
196.0
98.0
110.0
92.5
130.8
155.6
116.5
98.0
185.0
233.1
87.3
98.0
138.6
207.7
164.8
164.8
87.3
174.6
185.0
155.6
207.7
98.0
277.2
220.0
174.6
130.8
196.0


KGM22
KJS28
KMC27
KXA21
LAPO5
LGM 1
LMB18
LRD21
LXC11
MABO6
MAT 28
MCW14
MFC20
MJMO4
MLF13
MMR 1
MPF25
MRBl 1
MSM20
MYWO4
NJSO6
NMR29
OWHO4
PEEO9
PMD25
RABO8
RBDO3
RFH18
RJFl4
RL~M21
RMM13
RTL 17
RXMI5
SAR14
SECO2
SEM27
SHD17
SMD22
SRR24
SXM27
TCD26
TPMO4
TRS28
VMSO5
WDKl7
WJP20
WXEO4
TMD12


220.0
207.7
207.7
164.8
116.5
185.0
116.5
116.5
207.7
196.0
233.1
277.2
123.5
207.7
196.0
138.6
110.0
98.0
77.8
220.0
207.7
123.5
233.1
185.0
130.8
185.0
155.6
155.6
164.8
123.5
246.9
87.3
110.0
207.7
196.0
116.5
220.0
207.7
130.8
196.0
138.6
155.6
185.0
246.9
130.8
123.5
123.5
349.2


KJBl9
KJWO7
KMS29
KXBl7
LARO5
LHLO8
LMMO4
LRMO3
LXC28
MABl 1
MBMO5
MCW21
MGM28
MJZl8
MLG10
MMS29
MPH 12
MRB25
MWD28
MYW14
NLCO8
NMVO7
OWPO2
PFMO3
PMFO3
RAB22
RCC11
RFH19
RJF22
RMBO7
RPC 14
RWC23
RXPO2
SAV18
SEF10
SFD17
SHT20
SMKO4
SWBl4
SXS16
TESO3
TPP11
VFMll
VRS01
WDK47
WPB30
WXHO2
SMAO8


164.8
103.8
155.6
246.9
116.5
207.7
185.0
293.7
207.7
146.8
155.6
196.0
220.0
196.0
233.1
130.8
220.0
98.0
110.0
207.7
185.0
207.7
246.9
103.8
233.1
196.0
233.1
130.8
174.6
98.0
174.6
98.0
138.6
277.2
98.0
116.5
138.6
370.0
123.5
220.0
220.0
220.0
220.0
164.8
146.8
123.5
103.8
220.0


KJI23
KL~CO6
KMWO5
KXH19
LBA24
LJHO6
LMM17
LSBl8
LXD22
MACO3
MBM21
MECO6
MGV 1
MKL~31
MMD 1
MNHO4
MPSO9
MRB30
MXC 10
NAC21
NMB28
NXM18
PAM01
PGBl6
PSA21
RAE12
REC 19
RGE19
RJL28
RMCO7
RPJl5
RWFO6
RXS13
SBFll
SEGl8
SFD23
SJD28
SMK23
SWSO4
SXZ01
TLPl3
TPP24
VJVO2
WBR12
WFCO7
WPKll
WXS21
SHDO4


138.6
207.7
311.1
246.9
220.0
207.7
196.0
174.6
207.7
185.0
196.0
196.0
103.8
123.5
233.1
207.7
246.9
92.5
233.1
98.0
185.0
185.0
92.5
110.0
155.6
110.0
233.1
82.4
92.5
155.6
116.5
146.8
130.8
207.7
130.8
87.3
123.5
146.8
155.6
87.3
233.1
185.0
130.8
277.2
116.5
110.0
110.0
349.2


KJI24 130.8
KLD26 164.8
KPS25 103.8
LACO2 164.8
LCW30 196.0
LJM24 196.0
LMPl2 196.0
LVD28 261.6
LXGl7 116.5
MAMO8 207.7
MBM25 185.0
MEC28 174.6
MHL19 138.6
MLBl6 196.0
MMD15 233.1
MNH14 261.6
MPS21 233.1
MRC20 174.6
MXN24 233.1
NAP26 92.5
NMC22 233.1
NXRO8 185.0
PAT10 110.0
PJM12 98.0
PT0l8 98.0
RAM30 261.6
REW16 110.0
RHGO7 220.0
RJR15 110.0
RMC18 196.0
RPQ20 103.8
RWR14 110.0
SAC10 103.8
SBF24 207.7
SEH26 174.6
SFM22 92.5
SLC23 220.0
SMW17 77.8
SXCO2 146.8
TAB21 174.6
TLSO8 185.0
TPS16 116.5
VJVO9 110.0
WCB24 174.6
WJBO6 233.1
WSBO6 110.0
LMEO7 659.3
KXH30 174.6


KJnll 116.5
KMC19 207.7
KTJ26 220.0
LADI3 130.8
LDJll 82.4
LJS31 220.0
LNC11 98.0
LWR18 220.0
LXR15 103.8
MAM21 220.0
MCAO7 164.8
MEH26 196.0
MIDO8 174.6
MLCO8 233.1
MMG27 246.9
MPB23 103.8
MPS23 311.1
MRMI6 155.6
MXSO6 246.9
NFGO8 207.7
NMFO4 164.8
OAB28 69.3
PCL24 110.0
PLW14 207.7
PTO22 98.0
RAN30 261.6
RFC19 233.1
RHPl2 196.0
RJR29 116.5
RMFl4 196.0
RSM20 130.8
RWR16 116.5
SAE01 164.8
SCC15 138.6
SEH28 246.9
SGN18 138.6
SLGO5 196.0
SPM26 92.5
SXG23 174.6
TAC22 207.7
TMKO4 261.6
TRFO6 116.5
VMBl8 174.6
WDKO4 110.0
WJBl2 110.0
WST20 87.3
EAMO5 146.8
VAWO7 174.6










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3335.









BIOGRAPHICAL SKETCH

Arturo Camacho was born in San Jose, Costa Rica, on October 21, 1972. He did his

elementary school at Centro Educativo Roberto Cantillano Vindas and his high school at Liceo

Salvador Umafia Castro. After that, he studied Music at the Universidad Nacional, and at the

same time he performed as pianist in some of the most popular Costa Rican Latin music bands in

the 1990's. He also studied Computer and Information Science at the Universidad de Costa Rica,

where he obtained his B.S. degree in 2001. He worked for a short time as a software engineer in

Banco Central de Costa Rica during that year, but soon he moved to the United States to pursue

graduate studies in Computer Engineering at the University of Florida. He received his M. S. and

Ph.D. degrees in 2003 and 2007, respectively.

Arturo's research interests span all areas of automatic music analysis, from the lowest level

tasks like pitch estimation and timbre identification, to the highest levels tasks like analysis of

harmony and gender. His dream is to have one day a computer program that allows him (and

everyone) to analyze music as well or better than a well-trained musician would do.

Currently, Arturo lives happily with his loved wife Alexandra, who is another Ph. D. gator

in Computer Engineering and who he married in 2002, and their loved daughter Melissa, who

was born in 2006.





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1 SWIPE: A SAWTOOTH WAVEFORM INSPIRED PITCH ESTIMATOR FOR SPEECH AND MUSIC By ARTURO CAMACHO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Arturo Camacho

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3 Dedico esta disertaci n a mis queridos abuelos Hugo y Flory

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4 ACKNOWLEDGMENTS I thank my grandparents for all the support they have given to me during my life, my wife for her support during the years in graduate scho ol, and my daughter who was in my arms when the most important ideas expressed here came to my mind. I also thank Dr. John Harris for his guidance during my research and for always pushi ng me to do more, Dr. Rahul Shrivastav for his support and for introducing me to the world of auditory system models, and Dr. Manuel Bermudez for his unconditional support all these years.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 LIST OF ABBREVIATIONS........................................................................................................11 ABSTRACT....................................................................................................................... ............12 CHAPTER 1 INTRODUCTION................................................................................................................. .13 1.1 Pitch Background........................................................................................................... ...14 1.1.1 Conceptual Definition............................................................................................14 1.1.2 Operational Definition............................................................................................15 1.1.3 Strength................................................................................................................. ..16 1.1.4 Duration Threshold.................................................................................................19 1.2 Illustrative Examples and P itch Determination Hypotheses............................................20 1.1.2 Pure Tone................................................................................................................ 20 1.2.2 Sawtooth Waveform and the Largest Peak Hypothesis.........................................21 1.2.3 Missing Fundamental and the Components Spacing Hypothesis...........................21 1.2.4 Square Wave and the Maximu m Common Divisor Hypothesis............................22 1.2.5 Alternating Pulse Train...........................................................................................24 1.2.6 Inharmonic Signals.................................................................................................25 1.3 Loudness................................................................................................................... ........26 1.4 Equivalent Rectangular Bandwidth..................................................................................27 1.5 Dissertation Organization.................................................................................................2 9 1.6 Summary.................................................................................................................... .......30 2 PITCH ESTIMATION ALGORITHMS : PROBLEMS AND SOLUTIONS........................31 2.1 Harmonic Product Spectrum (HPS)..................................................................................32 2.2 Sub-harmonic Summation (SHS).....................................................................................34 2.3 Subharmonic to Harmonic Ratio (SHR)...........................................................................36 2.4 Harmonic Sieve (HS)........................................................................................................ 37 2.5 Autocorrelation (AC)....................................................................................................... .39 2.6 Average Magnitude and Squared Di fference Functions (AMDF, ASDF).......................42 2.7 Cepstrum (CEP)............................................................................................................. ...44 2.8 Summary.................................................................................................................... .......46

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6 3 THE SAWTOOTH WAVEFORM IN SPIRED PITCH ESTIMATOR.................................47 3.1 Initial Approach: Average Peak -to-Valley Distance Measurement.................................47 3.2 Blurring of the Harmonics................................................................................................49 3.3 Warping of the Spectrum..................................................................................................51 3.4 Weighting of the Harmonics.............................................................................................53 3.5 Number of Harmonics......................................................................................................55 3.6 Warping of the Frequency Scale.......................................................................................55 3.7 Window Type and Size.....................................................................................................57 3.8 SWIPE...................................................................................................................... ........63 3.9 SWIPE .............................................................................................................................65 3.9.1 Pitch Strength of a Sawtooth Waveform................................................................69 3.10 Reducing Computational Cost........................................................................................71 3.10.1 Reducing the Number of Fourier Transforms......................................................71 3.10.1.1 Reducing window overlap..........................................................................72 3.12.1.2 Using only power-of-two window sizes.....................................................74 3.10.2 Reducing the Number of Spec tral Integral Transforms.......................................81 3.11 Summary................................................................................................................... ......86 4 EVALUATION................................................................................................................... ...87 4.1 Algorithms................................................................................................................. .......87 4.2 Databases.................................................................................................................. ........88 4.3 Methodology................................................................................................................ .....89 4.4 Results.................................................................................................................... ...........89 4.5 Discussion................................................................................................................. ........95 5 CONCLUSION................................................................................................................... ....97 APPENDIX A MATLAB IMPLEMENTATION OF SWIPE .......................................................................99 B DETAILS OF THE EVALUATION....................................................................................102 B.1 Databases.................................................................................................................. .....102 B.1.1 Paul Bagshaw’s Database....................................................................................102 B.1.2 Keele Pitch Database...........................................................................................102 B.1.3 Disordered Voice Database.................................................................................103 B.1.4 Musical Instruments Database.............................................................................104 B.2 Evaluation Using Speech...............................................................................................105 B.3 Evaluation Using Musical Instruments..........................................................................108 C GROUND TRUTH PITCH FOR THE DISORDERED VOICE DATABASE...................110 REFERENCES..................................................................................................................... .......112 BIOGRAPHICAL SKETCH.......................................................................................................116

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7 LIST OF TABLES Table page 3-1 Common windows used in signal processing....................................................................62 4-1 Gross error rates for speech............................................................................................... 90 4-2 Proportion of overestimation errors relative to total gross errors......................................90 4-3 Gross error rates by gender................................................................................................ 91 4-4 Gross error rates for musical instruments..........................................................................92 4-5 Gross error rates by instrument family..............................................................................92 4-6 Gross error rates for mu sical instruments by octave..........................................................93 4-7 Gross error rates for musical instruments by dynamic......................................................94 4-8 Gross error rates for variations of SWIPE ........................................................................95 C-1 Ground truth pitch values for the disordered voice database...........................................110

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8 LIST OF FIGURES Figure page 1-1 Sawtooth waveform.......................................................................................................... .18 1-2 Pure tone.................................................................................................................. ..........20 1-3 Missing fundamental........................................................................................................ ..22 1-4 Square wave................................................................................................................ .......23 1-5 Pulse train................................................................................................................ ...........24 1-6 Alternating pulse train.................................................................................................... ....25 1-7 Inharmonic signal.......................................................................................................... .....26 1-8 Equivalent rectangular bandwidth.....................................................................................28 1-9 Equivalent-rectangular-bandwidth scale............................................................................29 2-2 Harmonic product spectrum...............................................................................................33 2-3 Subharmonic summation...................................................................................................34 2-4 Subharmonic summation with decay.................................................................................35 2-5 Subharmonic to harmonic ratio..........................................................................................37 2-6 Harmonic sieve............................................................................................................. .....38 2-7 Autocorrelation............................................................................................................ ......40 2-8 Comparison between AC, BAC, ASDF, and AMDF........................................................42 2-9 Cepstrum................................................................................................................... .........44 2-10 Problem caused to cepstru m by cosine lobe at DC............................................................45 3-1 Average-peak-to-valley-distance kernel............................................................................48 3-3 Necessity of strictly convex kernels..................................................................................50 3-4 Kernels formed from concatenations of truncated squarings, Gaussians, and cosines......51 3-5 Warping of the spectrum.................................................................................................... 52 3-6 Weighting of the harmonics...............................................................................................54

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9 3-7 Fourier transform of rectangular window..........................................................................58 3-8 Cosine lobe and square-root of the spectrum of rectangular window...............................59 3-9 Hann window................................................................................................................ .....60 3-10 Fourier transform of the Hann window.............................................................................61 3-11 Cosine lobe and square-root of the spectrum of Hann window.........................................61 3-12 SWIPE kernel.............................................................................................................. .......64 3-13 Most common pitch estimation errors...............................................................................66 3-14 SWIPE kernel....................................................................................................................69 3-15 Pitch strength of sawtooth waveform................................................................................70 3-16 Windows overlapping....................................................................................................... .73 3-17 Idealized spectral lobes.................................................................................................. ....75 3-18 K+-normalized inner product between temp late and idealized spectral lobes...................77 3-19 Individual and combined pitch strength curves.................................................................78 3-20 Pitch strength loss when using suboptimal window sizes.................................................79 3-21 Coefficients of the pitch st rength interpola tion polynomial..............................................84 3-22 Interpolated pitch strength............................................................................................... ..85

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10 LIST OF OBJECTS Object page Object 1-1. Sawtooth waveform (WAV file, 32 KB)...................................................................18 Object 1-2. Pure tone (WAV file, 32 KB)....................................................................................20 Object 1-3. Missing fundam ental (WAV file, 32 KB)..................................................................22 Object 1-4. Square wave (WAV file, 32 KB)...............................................................................23 Object 1-5. Pulse trai n (WAV file, 32 KB)..................................................................................24 Object 1-6. Alternating pulse train (WAV file, 32 KB)...............................................................25 Object 1-7. Inharmonic si gnal (WAV file, 32 KB)......................................................................26 Object 2-1. Bandpass filter ed /u/ (WAV file 6 KB).....................................................................33 Object 2-2. Signal with strong se cond harmonic (WAV file, 32 KB)..........................................42 Object 3-1. Beating tones (WAV file, 32 KB)..............................................................................50

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11 LIST OF ABBREVIATIONS AC Autocorrelation AMDF Average magnitude difference function APVD Average peak-to-valley distance ASDF Average squared difference function BAC Biased autocorrelation CEP Cepstrum ERB Equivalent rectangular bandwidth ERBs Equivalent-rectangular-bandwidth scale FFT Fast Fourier transform HPS Harmonic product spectrum HS Harmonic sieve IP Inner product IT Integral transform ISL Idealized spectral lobe K+-NIP K+-normalized inner product NIP Normalized inner product O-WS Optimal window size P2-WS Power-of-two window size SHS Subharmonic-summation SHR Subharmonic-to-harmonic ratio STFT Short-time Fourier transform SWIPE Sawtooth Waveform Inspired Pitch Estimator UAC Unbiased autocorrelation WS Window size

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12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SWIPE: A SAWTOOTH WAVEFORM INSPIRED PITCH ESTIMATOR FOR SPEECH AND MUSIC By Arturo Camacho December 2007 Chair: John G. Harris Major: Computer Engineering A Sawtooth Waveform Inspired Pitch Esti mator (SWIPE) has been developed for processing speech and music. SWIPE is shown to outperform existing algorithms on several publicly available speech/musical-instruments databases and a disordered speech database. SWIPE estimates the pitch as the fundamental frequency of the sawtooth waveform whose spectrum best matches the spectrum of the input signal. A decaying cosine kernel provides an extension to older frequency-based, sieve-type estimation algorithms by providing smooth peaks with decaying amplitudes to correlate with the harmonics of the signal. An improvement on the algorithm is achieved by using only the first a nd prime harmonics, which significantly reduces subharmonic errors commonly found in other pitch estim ation algorithms.

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13 CHAPTER 1 INTRODUCTION Pitch is an important characteristic of sound, providing information about the sound’s source. In speech, pitch helps to identify the gender of the speaker (pitch tends to be higher for females than for males) (Wang and Lin, 2004), gives additional meaning to words (e.g., a group of words can be interpreted as a question depend ing on whether the pitch is rising or not), and may help to identify the emotional state of th e speaker (e.g., joy produces high pitch and a wide pitch range, while sadness produce normal to low pitch and a narrow pitch range) (Murray and Arnott, 1993). Pitch is also important in music because it determines the names of the notes (Sethares, 1998). Pitch estimation also has appl ications in many areas that involve processing of sound: music, communications, linguistics, and speech pathology. In music, one of the main applications of pitch estimation is automatic mu sic transcription. Musicologists are often faced with music for which no transcription exists. Therefore, automated tools that extract the pitch of a melody, and from there the individual musical notes, are invaluable t ools for musicologists (Askenfelt, 1979). Automated transcription has also been used in query-by-humming systems (e.g., Dannenberg et al ., 2004). These systems allow people to search for music in databases by singing or humming the melody rather than ty ping the title of the song, which may be unknown for the user or the database. In communications, pitch estimation is us ed for speech coding (Spanias, 1994). Many speech coding systems are based on the source-f ilter model (Fant, 1960), which models speech as a filtered source signal. In some implementations, the source is either a periodic sequence of glottal pulses (for voiced sound) or white no ise (for unvoiced sound). Therefore, the correct estimation of the glottal pulse rate is cr ucial for the correct coding of voiced speech.

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14 Pitch estimators are useful in linguistics for the recognition of intonation patterns, which are used, for example, in the acquisition of a second language (de Bot, 1983). Pitch estimators are also used in speech pathology to determine speech disorders, which are characterized by high levels of noise in the voice. Since most methods used to es timate noise are based on the fundamental frequency of the signal (e.g., Yomoto and Gould, 1982), pitch es timators are of vital importance in this area. The goal of our work is to develop an auto matic pitch estimator that operates on both speech and music. The algorithm should be comp etitive with the best known pitch estimators, and therefore be suitable for the many applicat ions mentioned above. Furthermore, the algorithm should provide a measure to determine if a pitch exists or not in each region of the signal. The remaining sections of this chapter present se veral psychoacoustics definitions and phenomena that will be used to explain the oper ation and rationale of the algorithm. 1.1 Pitch Background 1.1.1 Conceptual Definition Several conceptual definitions of pitch ha ve been proposed. The American Standard Association (ASA, 1960) de finition of pitch is “Pitch is that attribute of a uditory sensation in terms of which sounds may be ordered on a musical scale,” and the American National Standards Instit ute (ANSI, 1994) definition of pitch is “Pitch is that auditory attribute of sound acco rding to which sounds can be ordered on a scale from low to high. Pitch depends main ly on the frequency content of the sound stimulus, but it also depends on the sound pre ssure and the waveform of the stimulus.” These definitions mention an attr ibute that allows ordering s ounds in a scale, but they say nothing about what th at attribute is.

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15 We will propose another definition of pitch, wh ich is based on the fundamental frequency of a signal. The f undamental frequency f0 of a signal (sound or no sound ) exists only for periodic signals, and is defined as the inverse of the period of the signal, where the period T0 of the signal (a.k.a. fundamental period) is the mini mum repetition interval of the signal x ( t ), i.e., ) ( ) ( : | 0 min0T t x t x t T T (1-1) It is also possible, to define the funda mental frequency in the frequency domain: 0 0) 2 sin( ) ( : } { }, { | 0 maxk k k k kkft c t x c f f (1-2) Although both equations are ma thematically equivalent (i .e., it can be shown that f0 = 1/ T0), they are conceptually different: Equation 1-1 looks at the signal in the time domain, while Equation 1-2 looks at the signal as a combination of sinusoids using a Four ier series expansion. The key element for periodic ity in Equation 1-1 is the equality in x ( t ) = x ( t + T ), and the key element for periodicity in Equation 1-2 is the existence of components only at multiples of the fundamental frequency. Unfortunately, no signal in nature is perfectly periodic because of natural variations in frequency and amplitude, and contamination produced by noise. Nevertheless, when listening to many natural signals we perceive pitch. Th is suggests that, to determine pitch, the brain probabl y uses either a modified vers ion of Equation 1-1, where the equality x ( t ) = x ( t + T ) is substituted by an approximation, or a modified version of Equation 1-2, where noise and fluctuations in the frequenc y of the components are allowed. Based on this suggestion, we define pitch as the perceived “fundamental freque ncy” of a sound, in other words, as the estimate our brain does of the ( quasi) fundamental frequency of a sound. 1.1.2 Operational Definition Since the previous definitions of pitch do not indicate how to measure it, they are of no practical use, and an operational definition of pi tch is required. The usual way in which pitch is

PAGE 16

16 measured is the following. A liste ner is presented with two sounds : a target sound, for which the pitch is to be determined, and a matching s ound. The matching sound is usually a pure tone, although sometimes harmonic complex tones are used as well. The levels of the target and the matching sounds are usually equalized to avoid any effect of differen ces in level in the perception of pitch. The sounds are presente d sequentially, simultane ously, or in any combination of them, depending on the design of the experiment. The listener is asked to adjust the fundamental frequency of the matching sound such that it matches the target sound, in the sense of the conceptual definitions of pitch pr esented above. The fundamental frequency of the matching sound is recorded and the experiment is repeated several times and with different listeners. The data is summarize d, and if the distribution of f undamental frequencies shows a clear peak around a certain frequency, the target sound is said to have a pitch corresponding to that frequency. 1.1.3 Strength Some sounds elicit a strong pitch sensation, and some do not. For example, when we speak, some sounds are highly periodic and elicit a strong pitch sensation (e.g., vowels), but some do not (e.g., some consonants: / s /, / sh /, / p /, and / k /). In the case of musical instruments, the attack tends to contain transien t components that obscu re the pitch, but they disappear quickly letting the pitch show more clear ly. The quality of the sound that allows us to determine whether pitch exists is called pitch strength Pitch strength is not a categor ical variable but a continuum. Also, pitch strength is independe nt of pitch: two sounds may have the same pitch and differ in pitch strength. For example, a pure tone and a na rrow band of noise centered at the frequency of the tone have the same pitch, however, the pure tone elicit a stronger p itch sensation than the noise.

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17 Unfortunately, not much research exists on pi tch strength, and the fe w studies that exist have concentrated mostly on noise (Yost, 1996; Wiegrebe and Patterson, 1998), although some have explored harmonic sounds as well (Fastl and St oll, 1979; Shofner and Selas, 2002). In terms of variety of sounds, the most co mplete study is probably Fastl a nd Stoll’s, which included pure tones, complex tones, and several types of noises. In th at study, pure tones were reported to have the strongest pitch among all s ounds. However, other studies have found that pitch identification improves as harmonics are added (Houtsma, 1990) which suggests that pitch strength increases as well. We hypothesize that our brain determines p itch by searching for a match between our voice, produced or imagined, and the target signal for which pitch is to be determined, probably based on their spectra. This hypothe sis agrees with st udies of pitch determination in which subjects have been allowed to hum the target sound to facilitate pitch matching tasks (Houtgast, 1976). Based on this hypothesis, we believe that the higher the similarity of the target signal with our voice, the higher its pitch stre ngth. If the similarity is base d on the spectrum, a signal will have maximum pitch strength when its spectrum is closest to the spectrum of a voiced sound. If we assume that voiced sounds have harmonic spect ra with envelopes that decay on average as 1/ f (i.e., inversely proportional to frequency) (Fan t, 1960), then a signal will have maximum pitch strength if its spectrum has that structure. An example of a signal with such property is a sawtooth waveform, which is exemplified in Figure 1-1. A sawtooth waveform is formed by adding sines with frequencies that are multiples of a common fundamental f0, and whose amplitude decays inversely proportional to frequency: 1 02 sin 1 ) (kt kf k t x. (1-3)

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18 Though sawtooth waveforms play a key role in our research, their importance resides in their spectrum, and not in their time-domain waveform. In particul ar, the phase of the components can be manipulated (destroying its sa wtooth waveform shape) and the signal would still play the same role in our work as the sawt ooth waveform. In other words, it is assumed in this work that what matters to estimate pitch and its strength is the amplitude of the spectral components of the sound, and not their phase, whic h in fact is ignored here. However, phase does play a role in pitch perception, as have been shown by some researchers (Moore, 1977; Shackleton and Carlyon, 1994; Gale mbo, et al., 2001). These resear chers have created pairs of sounds that have the same spectral amplitudes but significantly different pitches, by choosing the phases of the components appropriately. Nevertheless, it is not the aim of this research to cover the whole range of pitch phenomena, but to con centrate only on the most common speech and Figure 1-1. Sawtooth waveform A) Signal. B) Spectrum. Object 1-1. Sawtooth wa veform (WAV file, 32 KB).

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19 musical instruments sounds. As we will see late r, good pitch predictions are obtained for these types of sounds based solely on the amp litude of their spectral components. 1.1.4 Duration Threshold Doughty and Garner (1947) studied the minimum duration required to perceive a pitch for a pure tone. They found that ther e are two duration thresholds w ith two different properties. Tones with durations below the shorter threshol d are perceived as a click, and no pitch is perceived. Tones with durations between the two thresholds are perceived as having pitch, and an increase in their duration causes an increase in their pitch strength. Tones with durations above the largest threshold are al so perceived as having pitch, but further increases in their duration do not increase their pitch strength. These thresholds are not consta nt, but approximately proportiona l to the pitch period of the tone. In other words, the threshold corresponds to a certain number of periods of the tone. However, there is some interaction between pitc h and the minimum number of cycles required to perceive it (lower frequencies have a tendency to require fewer cycles to elicit a pitch). The shorter threshold is appr oximately two to four cycles, and th e larger threshold is approximately three to ten cycles. For frequencies above 1 kH z the thresholds become constant: 4 ms the shorter and 10 ms the larger, regardless of their corresponding number of cycles. Robinson and Patterson (1995a; 1995b) studied note discriminability as a function of the number of cycles in the sound us ing strings, brass, flutes, and ha rpsichords. A large increase in discriminability can be observed in their data as the number of cycles increases from one to about ten, but beyond ten cycles the discriminab ility of the notes does not seem to increase much. This trend agrees with th e thresholds for pure tones men tioned above, which suggests that the thresholds are also valid for musical inst ruments, and probably for sawtooth waveforms as well.

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20 1.2 Illustrative Examples and Pitch Determination Hypotheses In previous sections, conceptual and operational de finitions of pitch were given. From a practical point of view, both types of definitions are of lim ited use since the conceptual definitions are too abstract and the operational definiti on requires a human to determine the pitch. In this section we propose more algorithmic ways for determining pitch, through the search for cues that may give us hints regarding the pitch. Th ese cues, hereafter referred as hypotheses, are illustrated with examples of sounds in which they are va lid, and examples in which they are not. 1.1.2 Pure Tone From a frequency domain point of view, the si mplest periodic sound is a pure tone. A pure tone with a frequency of 100 Hz and its spec trum is shown in Figure 1-2. Based on our operational definition of pitch (i.e., the one that uses a pure tone as matching tone presented at the same intensity level as the testing tone), the pitch of a pure tone is its frequency, and Figure 1-2. Pure tone. A) Signal. B) Spectrum. Object 1-2. Pure tone (WAV file, 32 KB).

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21 therefore frequency determines pitch in this case. This may not be true if the tones are presented at different levels. Intriguingly, the pitch of a pure tone may change with intensity level (Stevens, 1935): as intensity increases, the pi tch of high frequency tones tends to increase, and the pitch of low frequency tones tends to decrease. However, this change is usually less than 1% or 2% (Verschuure and van Meeteren, 1975) occurs at very disparate intensity levels, and varies significantly from person to person. Since the goal in this research is to predict pitch for sounds represented in a computer as a sequence of numbers without knowing the level at which the sound will be played, it will be assumed that the sound will be played at a “comfor table” level, and therefore the algorithm will be designed to predict the pitch at that level. Nevertheless, variations of pitch with level are small, and have little effect even for comp lex tones (Fastl, 2007), otherwise, music would become out of tune as we change the volume. 1.2.2 Sawtooth Waveform and the Largest Peak Hypothesis The sawtooth waveform presented in Sect ion 1.1.3 was shown to have a harmonic spectrum with components whose amplitude decay s inversely proportional to frequency (see Figure 1-1). The computational dete rmination of the pitch of a sawt ooth waveform is not as easy as it is for a pure tone because its spectrum has more than one component. Since the pitch of a sawtooth waveform corresponds to its fundamental frequency, and the fundamental frequency in this case is the component with the highest en ergy, one possible hypothesis for the derivation of the pitch is that the pitch corres ponds to the largest peak in the spectrum. However, as we will show in the next section, this hypothesis does not always hold. 1.2.3 Missing Fundamental and the Components Spacing Hypothesis This section shows that it is possible to crea te a periodic sound w ith a pitch corresponding to a frequency at which there is no energy in the spectrum. A sound w ith such property is said to

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22 Figure 1-3. Missing fundamenta l. A) Signal. B) Spectrum. Object 1-3. Missing fundamental (WAV file, 32 KB). have a missing fundamental It is easy to build such a signal: just take a sawtooth waveform and remove its fundamental, as shown in Figure 1-3. Certainly, the timbre of the sound will change, but not its pitch. This fact disproves the hypothesis that the pitch corresponds to the largest peak in the spectrum. After it was realized that the pitch of a complex tone was unaffected by removing the fundamental frequency, it was hypothesized that the pitch corr esponds to the spacing of the frequency components. However, this hypothesis is not always valid, as we will show in the next section. 1.2.4 Square Wave and the Maximum Common Divisor Hypothesis The previous section hypothesized that the pitch corresponds to the spacing between the frequency components. However, it is easy to find an example for which this hypothesis fails: a

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23 Figure 1-4. Square wave. A) Signal. B) Spectrum. Object 1-4. Square wave (WAV file, 32 KB). square wave. A square wave is similar to a sawtooth wave, but does not have even order harmonics: 1 0) 1 2 ( 2 sin ) 1 2 ( 1 ) (kt f k k t x. (1-4) A square wave with a fundamental frequenc y of 100 Hz and its spectrum is shown in Figure 1-4. The components are located at odd multiples of 100 Hz, producing a spacing of 200 Hz between them. However, the fundamental freque ncy, and indeed its pi tch, is 100 Hz. Thus, the components spacing hypothesis is invalid. A hypothesis that seems to work for this exampl e, and all the previous ones, is that the pitch must correspond to the maximum common divi sor of the frequency components. As shown in Equation 1-2, this is equivalent to saying that the pitch correspo nds to the fundamental frequency. However, we will show in the next section that this hypothesis is also wrong.

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24 1.2.5 Alternating Pulse Train A pulse train is a sum of pulses se parated by a constant time interval T0: 1 0) ( ) (kkT t t x, (1-5) where is the delta or pulse function, a function w hose value is one if its argument is zero, and zero otherwise. A pulse train with a fundamental frequency of 100 Hz (fundamental period of 10 ms) and its spectrum are shown in Figure 1-5. The spectrum of a pulse train is another pulse train with pulses at multiples of the fundamental freque ncy, which corresponds to the pitch. If the signal is modified by decreasing the height of ever y other pulse in the time domain to 0.7, as shown in Figure 1-6, the period of the signal will change to 20 ms. This will be reflected in the spectrum as a change in the fundamental fre quency from 100 Hz to 50 Hz. However, although this change may cause an effect on the timbre (dep ending on the overall level of the signal), the Figure 1-5. Pulse train. A) Signal. B) Spectrum. Object 1-5. Pulse tr ain (WAV file, 32 KB).

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25 Figure 1-6. Alternating pulse tr ain. A) Signal. B) Spectrum. Object 1-6. Alternating pul se train (WAV file, 32 KB). pitch will remain the same: 100 Hz, refuting the hypothesis that the pitch of a sound corresponds to its fundamental frequency (i.e., the maximu m common divisor of the frequency components). 1.2.6 Inharmonic Signals This section shows another example of a signal whose pitch does not correspond to its fundamental frequency (i.e., the maximum co mmon divisor of its frequency components). Consider a signal built from the 13th, 19th, and 25th harmonics of 50 Hz (i.e., 650, 950, and 1250 Hz), as shown in Figure 1-7. Its fundamental frequency is 50 Hz, but its pitch is 334 Hz (Patel and Balaban, 2001). This is interesting since the ratios between the components and the pitch are far from being integer multiples: 1.95, 2.84, and 3.74. In any case, the pitch of the signal no longer corresponds to it s fundamental frequency. Although the true period of the signal is T0 = 20 ms, the signal peaks about every 3 ms, whic h corresponds to the pi tch period of the

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26 Figure 1-7. Inharmonic signal. A) Signal. T0 corresponds to the fundamental period of the signal and t0 corresponds to the pitc h period. B) Spectrum. Object 1-7. Inharmonic signal (WAV file, 32 KB). signal t0 (see Panel A). These type of signals for which the components are not integer multiples of the pitch are called inharmonic signals 1.3 Loudness Loudness is another perceptual qu ality of sound that provides us with information about its source. It is important for pi tch because the unification of th e components of a sound into a single entity, for which we identify a pitch, ma y be mediated by the relative loudness of the components of the sound. A conceptual definition of loudness is (Moore, 1997) “…that attribute of auditory sensation in terms of which s ounds can be ordered on a scale extending from quiet to loud.”

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27 The most common unit to measure loudness is the sone A sone is defined as the loudness elicited by a 1 kHz tone presented at 40 dB sound pressure level. The loudness L of a pure tone is usually modeled as a power function of the sound pressure P of the tone, i.e., P k L (1-6) where k is a constant that depends on the units and is the exponent of the power law. In a review of loudness studies, Takeshima et. al (2003) found that the value of is usually reported to be within th e range 0.4-0.6. They also review ed more elaborate models with many more parameters, but for simplicity, in this work we will use the simpler power model, and for reasons we will explain later, we choose the value of to be 0.5. In other words, we model the loudness of a tone as being proportio nal to the square-root of its amplitude. 1.4 Equivalent Rectangular Bandwidth The bandwidth and the distributi on of the filters used to extr act the spectral components of a sound are important issues that may affect our perception of pitch. Since each point of the cochlea responds better to cert ain frequencies than others, th e cochlea acts as a spectrum analyzer. The bandwidth of the frequency response of each point of the cochlea is not constant but varies with frequency, being almost propor tional to the frequency of maximum response at each point (Glasberg and Moore, 1990). The concept of Equivalent R ectangular Bandwidth (ERB) was introduced as a description of the spread of the frequency response of a filter. The ERB of a filter F is defined as the bandwidth (in Hertz) of a rectangular filter R centered at the frequenc y of maximum response of F scaled to have the same output as F at that frequency, and passi ng the overall same amount of white noise energy as F In other words, when the power responses of F and R are plotted as a

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28 Figure 1-8. Equivalent rectangular bandwidth. function of frequency, as in Figu re 1-8, the central frequency of R corresponds to the mode of F and both curves have the same height and area. Glasberg and Moore (1990) studied the response of auditory filters at different frequencies, and proposed the following formula to approximate the ERB of the filters: f f 108 0 7 24 ) ( ERB (1-7) Another property of the cochlea is that th e relation between frequency and site of maximum excitation in the cochlea is not linear. If the distance between the apex of the cochlea and the site of maximum excitati on of a pure tone is plotted as a function of frequency of the tone, it will be found that a disp lacement of 0.9 mm in the coch lea corresponds approximately to one ERB (Moore, 1986). Therefore, it is possibl e to build a scale to measure the position of maximum response in the cochlea for a certain frequency f by integrating Equation 1-7 to obtain the number of ERBs below f and then multiplying it by 0.9 mm to obtain the position. However,

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29 Figure 1-9. Equivalent-rect angular-bandwidth scale. it is common practice in psychoacoustics to merely compute the number of ERBs below f which can be computed as ) 229 / 1 ( log 4 21 ) ( ERBs10f f (1-8) This scale is shown in Figure 19, and it will be th e scale used by SWIPE to compute spectral similarity. 1.5 Dissertation Organization The rest of this dissertation is organized as follows. Chapter 2 presents previous pitch estimation algorithms that are related to SWIPE, their problems and possible solutions to these problems. Chapter 3 will discuss how these proble ms, plus some ones and their solutions, lead to SWIPE. Chapter 4 evaluates SWIPE using publicly available speech/music databases and a disordered speech database. Publicly available implementations of other algorithms are also evaluated on the same databa ses, and their performance is compared against SWIPE’s.

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30 1.6 Summary Here we have presented the motivations and applications for pitch estimation. Then, we presented conceptual and operational definitions of pitch, together with the related concept of pitch strength and the duration threshold to perc eive pitch. Next, we presented examples of signals and their pitch, together with hypothese s about how pitch is determined. The sawtooth waveform was highlighted, since it plays a key role in the development of SWIPE. Psychoacoustic concepts such as inharmonic signals, loudness, and the ERB scale were also introduced since they are also releva nt for the development of SWIPE.

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31 CHAPTER 2 PITCH ESTIMATION ALGORITHMS : PROBLEMS AND SOLUTIONS This chapter presents some well known pitch estimation algorithms that appear in the literature. These algorithms were chosen because of their influence upon the creation of SWIPE. We will present the algorithms in a very basic form with the intent to ca pture their essence in a simple expression, although their actual implementations may have extra details that we do not present here. The purpose of those details is usua lly to fine tune the algorithms, but the actual power of the algorithms is based on the essence we describe here. Figure 2-1. General block di agram of pitch estimators.

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32 Traditionally, there have been two types of pitch estimation algorithms (PEAs): algorithms based on the spectrum1 of the signal, and algorithms based on the time-domain representation of the signal. The time-domain based algorithms presented in this chapter can also be formulated based on the spectrum of the signal, which will be the approach followed here. The basic steps that most PEAs perform to track the pitch of a signal are shown in the block diagram of Figure 2-1. First, the signal is split into windows. Th en, for each window the following steps are performed: (i) the spectrum is estimated using a short-time Fourier transform (STFT), (ii) a score is computed for each pitch candidate within a predefined range by computing an integral transform (IT) over the spectrum, and (iii) the candidate with the highest score is selected as the estimated pitch. The algorithms w ill be presented in an order that is convenient for our purposes, but does not necessarily corre spond to the chronological order in which they were developed. 2.1 Harmonic Product Spectrum (HPS) The first algorithm to be presented is Ha rmonic Product Spectrum (HPS) (Schroeder, 1968). This algorithm estimates the pitch as the frequency that maximizes the product of the spectrum at harmonics of that frequency, i.e. as | ) ( | max arg1n k fkf X p, (2-1) where X is the estimated sp ectrum of the signal, n is the number of harmonics to be used (typically between 5 and 11), and p is the estimated pitch. The purpose of limiting the number of harmonics to n is to reduce the computati onal cost, but there is no lo gical reason behind this limit; it is hard to believe that the n -th harmonic is useful for pitch estimation, but not the n +1-th. 1 Since all the pitch estimators presented here use the magnitude of the spectrum but not its phase, the words “magnitude of” will be omitted, and the word spectrum should be interpreted as magnitude of the spectrum unless explicitly noted otherwise.

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33 Figure 2-2. Harmonic product spectrum. Object 2-1. Bandpass filt ered /u/ (WAV file 6 KB) Since the logarithm is an increasing function, an equivalent approach is to estimate the pitch as the frequency that maximizes the logari thm of the product of the spectrum at harmonics of that frequency. Since the loga rithm of a product is equal to th e sum of the logarithms of the terms, HPS can also be written as | ) ( | log max arg1n k fkf X p, (2-2) or using an integral transform, as 0 1' ) ( )| ( | log max arg df kf f f X pn k f. (2-3) Figure 2-2 shows the kernel of this integral for a pitch candi date with frequency 190 Hz. A pitfall of this algorithm is that if any of the harmonics is missing (i .e., its energy is zero), the product will be zero (equivalently, the sum of the logarithms will be minus infinity) for the candidate corresponding to the pitch, and theref ore the pitch will not be recognized. Figure 2-2

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34 also shows the spectrum of the vowel /u/ (as in good ) with a pitch of 190 Hz (Object 2-1). This sample was passed through a filter with a bandpass range of 300 3400 Hz to simulate telephonequality speech. Therefore, the f undamental is missing and HPS is not able to recognize the pitch of this signal. Another salient characteristic of this sample is its intense second harmonic at 380 Hz, caused probably by the first formant of th e vowel, which is on average around 380 Hz as well (Huang, Acero, and Hon, 2001). 2.2 Sub-harmonic Summation (SHS) An algorithm that has no problem with mi ssing harmonics is Sub-Harmonic Summation (SHS) (Hermes, 1988), which solves the problem by using addition instead of multiplication. Therefore, if any harmonic is missing, it will no t contribute to the total, but will not bring the sum to zero either. In mathematical terms, SHS estimates the pitch as | ) ( | max arg1n k fkf X p, (2-4) Figure 2-3. Subharmonic summation.

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35 or using an integral transform as 0 1' ) ( )| ( | max arg df kf f f X pn k f. (2-5) An example of the kernel of this integral is shown in Figure 2-3. A pitfall of this algorithm is that since it gives the same weight to all the harmonics, subharmonics of the pitch may have the same scor e as the pitch, and therefore they are valid candidates for being recognized as the pitch. Fo r example, suppose that a signal has a spectrum consisting of only one component at f Hz. By definition, the pitch of the signal is f Hz as well. However, since the algorithm adds the spectrum at n multiples of the candidate, each of the subharmonics f /2, f /3,…, f / n will have the same score as f and therefore they are equally valid to be recognized as the pitch. Figure 2-4. Subharmonic summation with decay.

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36 This problem can be solved by introducing a monotonically decaying weighting factor for the harmonics. SHS implements this idea by weighting the harmonics with a geometric progression as 0 1 1' ) ( | ) ( | max arg df kf f r f X pn k k f, (2-6) where the value of r was empirically set to 0.84 based on experiments using speech. The kernel of this integral is shown in Figure 2-4. SHS is the only algorithm in this chapter that solves the subharmonic problem by applying this decay fact or. Later, another algorithm will be presented (Biased Autocorrelation) which solves this problem in a different way. 2.3 Subharmonic to Harmonic Ratio (SHR) A drawback of the algorithms presented so fa r is that they examine the spectrum only at the harmonics of the fundamental, ignoring the c ontents of the spectrum everywhere else. An example will illustrate why this is a problem. Suppos e that the input signal is white noise (i.e., a signal with a flat spectrum). This signal is perceived as having no pitch. However, the previous algorithms will produce the same score for each pitch candidate, making each of them a valid estimate for the pitch. This problem is solved by the Subharmoni c to Harmonic Ratio algorithm (SHR) (Sun, 2000), which not only adds the spectrum at harmonics of the pitch candidate, but also subtracts the spectrum at the middle points between ha rmonics. However, this algorithm uses the logarithm of the spectrum, and therefore has th e problem previously discussed for HPS. Also, this algorithm gives the same weight to all the harmonics and therefore it suffers from the subharmonics problem. SHR can be written as 0 1' ) ) 2 / 1 ( ( ) ( )| ( | log max arg df f k f kf f f X pn k f (2-7)

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37 Figure 2-5. Subharmonic to harmonic ratio. The kernel of the integral is shown in Figure 2-5. Notice that SHR will produce a positive score for a signal with a harmonic spectrum and a scor e of zero for white noise. However, this algorithm has a problem that is shared by all the algorithms presented so far: since they examine the spectrum only at harmonic locat ions, they cannot recognize the pitch of inharmonic signals. Before we move on to the next algorithm, we wish to add some insight to SHR. If we further divide the sum in Equation 2-7 by n the algorithm would compute the average peak-tovalley ratio, where the peaks are ex pected to be at the harmonics of the candidate, and the valleys are expected to be at the middle point between harmonics. This idea will be exploited later by SWIPE, albeit with some refinements: the averag e will be weighted, the ratio will be replaced with the distance, and the peaks and valleys w ill be examined over wider and blurred regions. 2.4 Harmonic Sieve (HS) One algorithm that is able to recognize the pitch of some inharmonic signals is the Harmonic Sieve (HS) (Duifhuis and Willems, 1982). This algorithm is similar to SHS, but has

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38 two key differences: instead of using pulses it us es rectangles, and instead of computing the inner product between the spectrum and the rectangles, it counts the number of r ectangles that contain at least one component (a rectangle is said to contain a component if th e component fits within the rectangle and its amplitude exceeds a certain threshold T ). The rectangles are centered at the harmonics of the pitch candidates, and their width is 8% of the frequency of the harmonics. This algorithm can be expressed mathematically as n k kf kf f ff X T p1 ) 04 1 96 0 ( '| ) ( | max max arg, (2-8) where [ ] is the Iverson bracket (i.e., produces a value of one if th e bracketed proposition is true, and zero otherwise). Notice that the expressi on in the sum is a non-linear function of the spectrum, and therefore this algorithm cannot be written using an integral transform. Figure 2-6 shows the kernel used by this algorithm. Figure 2-6. Harmonic sieve.

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39 A pitfall of HS is that, when a component is close to an edge of a rectangle, a slight change in its frequency could put it in or out of the rectangle, possibly cha nging the estimated pitch drastically. Such radical changes do not typically occur in pitch perception, where small changes in the frequency of the components lead to small changes in the perceived pitch, as mentioned in Section 1.2.6. This problem can be solved by using smoother boundaries to decide whether a component should be considered as a harmoni c or not, as done by the next algorithm. 2.5 Autocorrelation (AC) One of the most popular methods for p itch estimation is au tocorrelation. The autocorrelation function r ( t ) of a signal x ( t ) measures the correlation of the signal with itself after a lag of size t i.e., 2 / 2 /' ) ( ') ( 1 lim ) (T T Tdt t t x t x T t r. (2-9) The Wiener-Khinchin theorem shows that autocorre lation can also be computed as the inverse Fourier cosine transform of the squa red spectrum of the signal, i.e., as 0 2) 2 cos( | ) ( | ) ( df ft f X t r. (2-10) The autocorrelation-based pitch estimation algorithm (AC) estimates the pitch as the frequency whose inverse maximizes the autocorrela tion function of the signal, i.e., as 0 2' ) / 2 cos( | ) ( | max argmaxdf f f f X pf f, (2-11) where the parameter fmax is introduced to avoid the maximum that the integral has at infinity. The kernel for this integral is shown in Fi gure 2-7. It is easy to see that as f increases, the kernel stretches without limit, and sinc e the cosine starts with a va lue of one and decays smoothly, eventually it will give a weight of one to the whole spectrum, producing a maximum at infinity.

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40 Figure 2-7. Autocorrelation. Notice that this problem can be easily solved by removing the first quarter of the first cycle of the cosine (i.e., setting it to zero). Since the DC of a signal (i.e., it zero-frequency component) only adds a constant to the signa l, ignoring the DC should not a ffect the pitch estimation of a periodic signal. Because of the frequency domain representation of autocorrelation, we can see that there is a large resemblance between AC and SHR (compare the kernel of Figure 2.7 with the kernel of Figure 2.5), although with three main differences. First, instead of using an alternating sequence of pulses, AC uses a cosine, which adds a smoot h interpolation between the pulses. Second, AC adds an extra lobe at DC, which was already shown to have a negative effect. Third, AC uses the power of the spectrum (i.e., the squared spectrum ) instead of the logarithm of the spectrum. Therefore, both algorithms measure the averag e peak-to-valley distance, one in the power domain and the other in the logarithmic domai n, although AC does it in a much smoother way.

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41 There is also a similarity between AC and HS (compare the kernel of Figure 2.7 with the kernel of Figure 2.6). HS allows for inharm onicity of the compone nts of the signal by considering as harmonic any component within a certain distance from a harmonic of the candidate pitch. AC does the same in a smoothe r way by assigning to a component a weight that is a function of its distance to the closer harmonic of the candidate pitch; the smaller the distance, the larger the weight, and the further the distance, the smaller the weight. In fact, if the component is too far from any harmoni c, its weight can be negative. Like all the algorithms presented so far, excep t SHS, AC exhibits th e subharmonics problem caused by the equal weight given to all the harmoni cs (see Section 2.2). To solve this problem, it is common to take the local maximum of highest frequency rather than the global maximum. However, this technique sometimes fails. For example, consider a si gnal with fundamental frequency 200 Hz (i.e., period of 5 ms) and first four harmonics with amplitudes 1,6,1,1, as shown in Figure 2-8A (Object 2-2). Except at very low intensity levels, the four components are audible, and the pitch of the si gnal corresponds to its fundamental frequency. However, as shown in Figure 2-8C, AC has its firs t non-zero local maximum at 2.5 ms which corresponds to a pitch of 400 Hz. Another common solution is to use the biased autocorrelation (BAC) (Sondhi, 1968; Rabiner, 1977), which introduces a factor that pe nalizes the selection of low pitch values. This factor gives a weight of one to a pitch period of zero and decays linearly to zero for a pitch period corresponding to the window size T This can be written as 0 2 ) / 1 (' ) / 2 cos( | ) ( | 1 1 max argmaxdf f f f X Tf pf T f. (2-12)

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42 Figure 2-8. Comparison between AC, BAC, ASDF, and AMDF. A) Spectrum of a signal with pitch and fundamental frequency of 200 Hz B) Waveform of the signal with a fundamental period of 5 msec. C) AC ha s a maximum at every multiple of 5 ms, making it hard to choose the best candidate. The first (non-zero) local maxima is at 2.5 ms, making the “first peak ” criteria to fail. D) BAC has its first peak and its nonzero largest local maximum at 2.5 ms. E) ASDF is an inverted, shifted, ands scaled AC. F) AMDF is similar to ASDF. Object 2-2. Signal with strong second harmonic (WAV file, 32 KB) However, the combination of this bias and the squaring of the spectrum may introduce new problems. For example, if T = 20 ms as in the BAC function of Figure 2-8D, the bias will make the height of the peak at 2.5 ms larger than th e height of the peak at 5 ms, consequently causing an incorrect pitch estimate.

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43 2.6 Average Magnitude and Squared Difference Functions (AMDF, ASDF) Two functions similar to autocorrelation (in the sense that they compare the signal with itself after a lag of size t ) are the magnitude difference f unction (AMDF) and the average squared difference function (ASDF) The AMDF is defined as 2 / 2 /' )| ( ) ( | 1 ) (T Tdt t t x t x T t d, (2-13) and the ASDF as 2 / 2 / 2' )] ( ) ( [ 1 ) (T Tdt t t x t x T t s. (2-14) It is easy to show that ASDF and autocorrelation are related through the equation (Ross, 1974) ) ( ) 0 ( 2 ) ( t r r t s (2-15) and therefore, s ( t ) is just an inverted, shifte d, and scaled version of autocorrelation. Therefore, as illustrated in the panels C (or D) and E of Figure 2-8, where (biased) autocorrelation has peaks, s ( t ) has dips. Thus, an ASDF-based algorithm mu st look for minima instead of maxima to estimate pitch. It has also been shown (Ross, 1974) that d ( t ) can be approximated as )] ( [ ) ( ) (2 / 1t s t t d (2-16) Although the relation between d ( t ) and s ( t ) depends on t through ( t ), it is found in practice that this factor does not play a significant role, and a large similarity between d ( t ) and s ( t ) exists, as observed in panels E and F of Figure 2-8. Therefore, since the functions r ( t ), s ( t ), and d ( t ) are so strongly related, none of them is expected to offer much more than the others for pitch estimation. However, modifications to these functions, which cannot be expressed in terms of the other functions, have been used successfully to improve their performance on pitch estimation.

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44 An example is given by YIN (de Chevei gne, 2002), which uses a variation of s ( t ) to avoid the dip at lag zero, improving its performance. Another va riation is the one we proposed in the previous section (i.e., the removal of the first quarter of the cosine) to avoid the maximum at zero lag for autocorrelation. 2.7 Cepstrum (CEP) An algorithm similar to AC is the cepstrum -based pitch estimation algorithm (CEP) (Noll, 1967). The cepstrum c ( t ) of a signal x ( t ) is very similar to its auto correlation. The only difference is that it uses the logarithm of the spectrum instead of its square, i.e., 0) 2 cos( | ) ( | log ) ( df ft f X t c. (2-17) CEP estimates the pitch as the frequency whose inverse maximizes the cepstrum of the signal, i.e., as Figure 2-9. Cepstrum.

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45 0' ) / 2 cos( | ) ( | log max argmaxdf f f f X pf f. (2-18) The kernel of this integral is shown in Figur e 2-9. Like AC, CEP e xhibits the subharmonics problem and the problem of having a maximum at a large value of f The maximum is not necessarily at infinity because, depending on the scaling of the signal, the logarithm of the spectrum may be negative at larg e frequencies, and therefore assi gning a positive weight to that region may in fact decrease the score. Figure 210 shows the spectrum of the speech signal that has been used in previous figures and the ke rnel that produces the highest score for that spectrum, which corresponds to a candidate pitch of about 10 kHz. Notice that the logarithm of the spectrum was arbitrarily set to zero for fre quencies below 300 Hz because its original value (minus infinity) would make unfeasible the eval uation of the integral in Equation 2-18. This problem of the use of the logarithm when ther e are missing harmonics wa s already discussed in Section 2.1. Figure 2-10. Problem caused to cepstrum by cosine lobe at DC.

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46 2.8 Summary In this chapter we presented pitch estimation algorithms that have influenced the creation of SWIPE. The most common problem s found in these algorithms were the inability to deal with missing harmonics (HPS, SHR, and CEP) and inha rmonic signals (HPS, SHS, and SHR), and the tendency to produce high scores for subharmonics of the pitch (all the algorithms, although to a lesser extent SHS and BAC). Solutions to these problems were either found in other algorithms or were proposed by us.

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47 CHAPTER 3 THE SAWTOOTH WAVEFORM IN SPIRED PITCH ESTIMATOR Aiming to improve upon the algorithms presente d in Chapter 2, we propose the Sawtooth Waveform Inspired Pitch Estimator (SWIPE)2. The seed of SWIPE is the implicit idea of the algorithms presented in Chapter 2: to find the frequency that maximizes the average peak-tovalley distance at harmonics of that frequency. Ho wever, this idea will be implemented trying to avoid the problem-causing features found in those algorithms. This will be achieved by avoiding the use of the logarithm of the spectrum, a pplying a monotonically decaying weight to the harmonics, observing the spectrum in the ne ighborhood of the harmonics and middle points between harmonics, and using smooth weighting functions. 3.1 Initial Approach: Average Peak -to-Valley Distance Measurement If a signal is periodic w ith fundamental frequency f its spectrum must contain peaks at multiples of f and valleys in between. Since each peak is surrounded by two valleys, the average peak-to-valley distance (APVD) for the k -th peak is defined as | ) ) 2 / 1 (( | | ) ( | 2 1 | ) ) 2 / 1 (( | | ) ( | 2 1 ) ( f k X kf X f k X kf X f dk | ) ) 2 / 1 (( | | ) ) 2 / 1 (( | 2 1 | ) ( | f k X f k X kf X (3-1) Averaging over the first n peaks, the global APVD is n k k nf d n f D1) ( 1 ) ( | ) ) 2 / 1 (( | | ) ( | | ) ) 2 / 1 (( | 2 1 | ) 2 / ( | 2 1 11 n kf k X kf X f n X f X n (3-2) 2 The name of the algorithm will become clear in a posterior section.

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48 Figure 3-1. Average-peak-t o-valley-distance kernel. Our first approach to estimate pitch is to find the frequency that maximizes the APVD. Staying with the integral transform notation used in Chapter two, and dropping the unnecessary 1/ n term, the algorithm can be expressed as 0' ') ( | ) ( | max argmaxdf f f K f X pn f f, (3-3) where n k nf f k f kf f f n f f f f K1) / ) 2 / 1 (( ) / ( ) / ) 2 / 1 (( 2 1 ) 2 / ( 2 1 ) ( (3-4) The kernel Kn ( f f ) for f = 190 Hz is shown in Figure 3-1 t ogether with the spectrum of the sample vowel / u / used in Chapter 2, which will be used extensively in this chapter as well. The kernel is a function not only of the frequencies but also of n the number of harmonics to be used. Each positive pulse in the kernel has a weight of 1, each negative pulse between positive pulses has a weight of -1, and the first and last negativ e pulses have a weight of -1/2. This kernel is

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49 similar to the kernel used by SHR (see Chapter 2), with the only difference that in Kn the first negative pulse has a weight of -1/2 and Kn has an extra negative pulse at the end, also with a weight of -1/2. 3.2 Blurring of the Harmonics The previous method of measuring the APVD wo rks if the signal is harmonic, but not if it is inharmonic. To allow for inharmonicity, our first approach was to blur the location of the harmonics by replacing each pulse with a triangle function with base f /2, otherwise. 0 4 / | | if | | 4 / ) ( f f f f ff (3-5) The base of the triangle was set to f /2 to produce a triangular wave as shown in Figure 3-2. To be consistent with the APVD measure, the first and last negative triangles we re given a height of 1/2. One reason for using a base th at is proportional to the candidate pitch is that it allows for a pitch-independent handling of inha rmonicity, as seems to be done in the auditory system (see section 1.2.6). Figure 3-2. Triangul ar wave kernel.

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50 Figure 3-3. Necessity of strictly convex kernels. Object 3-1. Beating tones (WAV file, 32 KB) The triangular kernel approach was abandone d because it was found that the components of the kernel must be strictly concave (i.e., must have a continuous second derivative) at their maxima. The following example will illustrate why this is nece ssary. Suppose we have a signal with components at 200 and 220 Hz, as shown in Figure 3-3 (Object 3-1). This signal is perceived as a loudness-varying tone with a pitch of 210 Hz, phenomena known as beating However, the triangular kernel produces the sa me score for each candidate between 200 and 220 Hz. This is easy to see by slightly stretching or compressing the kernel such that its first positive peak is within that range. Such stretching or compression would cau se an increment on the weight of one of the components and a decremen t of the same amount on the other, keeping the score constant. Therefore, the triangle was discarded and concatenations of truncated squarings, Gaussians, and cosines were explored. The squari ng function was truncated at its fixed point, and

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51 Figure 3-4. Kernels formed from concatenations of truncated s quarings, Gaussians, and cosines. the Gaussian and the cosine func tions were truncated at their in flection points. The Gaussian was truncated at the inflection points to ensure that the concatenation of positive and negative Gaussians have a continuous second derivative. The same can be said about the cosine, but furthermore, the concatenation of positive and negative cosine lobes produces a cosine, which has all order derivatives. Concatenations of these three functions, st retched or compressed to form the desired pattern of maxima at multiples of the candidate pitch, are illustrated in Figure 3-4. Although informal tests showed no significant differenc es in pitch estimation performance among the three, the cosine was preferred because of its si mplicity. Notice also that this kernel is the one used by the AC and CEP pitch estimators (see Chapter 2). 3.3 Warping of the Spectrum As mentioned in Chapter 2, the use of the logarithm of the spectrum in an integral transform is inconvenient because there may be regions of the spectrum with no energy, which

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52 Figure 3-5. Warping of the spectrum. would prevent the evaluation of the integral, sin ce the logarithm of zero is minus infinity. But even if there is some small energy in those re gions, the large absolute value of the logarithm could make the effect of these low energy regions on the integral larger than the effect of the regions with the most energy, wh ich is certainly inconvenient. To avoid this situation, the use of the loga rithm of the spectrum was discarded and other commonly used functions were explored: square identity, and square-root. Figure 3-5 shows how these functions warp th e spectrum of the vowel / u / used in Chapter 2. As mentioned earlier, this spectrum has two particularities: it has a missing fundamental, and it has a salient second

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53 harmonic. The missing fundamental is evident in panel B, which s hows that the logarithm of the spectrum in the region of 190 Hz is minus infini ty. The salient second harmonic at 380 Hz shows up clearly in the other three panels, but especia lly in panel C, where the spectrum has been squared. Panel D shows the square-root of th e spectrum, which neit her overemphasizes the missing fundamental (as the logarithm does) nor the salient second harmonic (as the square does). We believe the square-root warping of the sp ectrum is more convenient for three reasons. First, it matches better the response of the aud itory system to amplitude, which is close to a power function with an exponent in the range 0.4-0.6 (see Chapter 2); second, it allows for a weighting of the harmonics proportional to their amp litude, as we will show in the next section; and third, it produces better pitch esti mates, as found tests presented later. 3.4 Weighting of the Harmonics To avoid the subharmonics problem presented in Chapter 2, a decaying weighting factor was applied to the harmonics. The types of decays explored were exponential and harmonic. For exponential decays, a weight of r k 1 was applied to the k -th harmonic ( k = 1, 2, …, n and r = 0.9, 0.7, 0.5) through the multiplication of the kernel by the envelope r f / f -1, as shown in Figure 3-6. For harmonic decays, a weight of 1/ k p was applied to the k -th harmonic ( k = 1, 2, …, n and p = 1/2, 1, 2) through the multiplication of the kernel by the envelope ( f / f ) p, as shown in Figure 3-6. In in formal tests, the best results were obtained using harmonic decays with p = 1/2, which matches the decay of the square-root of the average spectrum of vowels (see Chapter 2). In other wo rds, better pitch estimates were obtained when computing the inner product (IP) of the square-r oot of the input spectrum and th e square-root of the expected spectrum, than when computing the IP’s over the raw spectra.

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54 Figure 3-6. Weighting of the harmonics. One explanation for this is that when the input spectrum matches its corresponding template (i.e., the expected spectrum for that pitc h), the use of the square-root of the spectra in the IP gives to each harmonic a weight proportiona l to its amplitude. For example, if the input spectrum has the expected shape for a vowel, i.e., th e amplitude of the harmonics decay as 1, 1/2, 1/3, etc., then their square root decays as 1, 1/ 2, 1/ 3, etc. Since the terms in the sum of the IP are the squares of these values (i.e., 1, 1/2, 1/ 3, etc.), then the rela tive contribution of each harmonic is proportional to its amplitude. Conversely, if we compute the IP over the raw spectra, the terms of the sum will be 1, 1/4, 1/9, etc., which are not proportional to the amplitude of the components, but to their square. This would make the contribution of the strongest harmonics too large and the contribution of the weakest too sma ll. The situation would be even worse if we would compute the IP over the energy of the spect rum (i.e., its square). The expected energy of the harmonics for a vowel follows the pattern 1, 1/4, 1/16, etc., and computing the IP of the

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55 energy of the harmonics with itsel f produces the terms 1, 1/16, 1/ 256, etc, which gives too much weight to the first harmonic and almo st no weight to the other harmonics. In the ideal case in wh ich there is a perfect match between the input and the template, any of the previous types of warping would produ ce the same result: a no rmalized inner product (NIP) equal to 1. However, the likelihood of a perfect match is lo w, and the warping may play a big role in the determination of the best match, as we found in informal tests, which show that the use of the square-root of the spec trum produces better pitch estimates. 3.5 Number of Harmonics An important issue is the number of harmonics to be used to analyze the pitch. HPS, SHS, SHR, and HS use a fixed finite number of ha rmonics, and CEP and AC use all the available harmonics (i.e., as many as the sampling frequency allows). In informal tests the best results were obtained when using as many harmonics as available, although it was found that going beyond 3.3 kHz for speech and 5 kHz for musica l instruments did not improve the results significantly. Thus, to reduce computational co st it is reasonable to set these limits. 3.6 Warping of the Frequency Scale As mentioned in Section 3.4, if the input matc hes perfectly any of th e templates, their NIP will be equal to 1, regardless of the type of warping used on the spectrum. The same applies to the frequency scale. However, since a perfect matc h will rarely occur, a warping of the frequency scale may play a role in determining the best match. For the purposes of computing the integral of a function, we can think of a warping of the scale as the process of sampling the function more finely in some regions than others, effectively giving more emphasis to the more finely sample d regions. In our case, since we are computing an inner product to estimate pitch, it makes sens e to sample the spectrum more finely in the region that contributes the most to the determinati on of pitch. It seems reasonable to assume that

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56 this region is the one with the most harmonic energy. In the case of speech, and assuming that the amplitude of the harmonics decays inversel y proportional to frequency, it seems reasonable to sample the spectrum more finely in the neighborhood of the fundamental and decrease the granularity as we move up in fr equency, following the expected 1/ f pattern for the amplitude of the harmonics. A decrease in gr anularity should also be pe rformed below the fundamental because no harmonic energy is expected below it. However, the determination of the frequency at which this decrease should be gin is non-trivial, since we do not know a-priori the fundamental frequency of the incoming sound (that is pr ecisely what we wish to determine). As we did for the selection of the warping of the amplitude of the spectrum, we appeal to the auditory system and borrow the frequenc y scale it seems to use: the ERB scale (see Chapter 1). Therefore, to compute the similarity between the input spectrum and the template, we sample both of them uniformly in the ERB s cale, whose formula is given in Equation 1-8. This scale has several of the characteristics we desire (see Figure 1-9): it has a logarithmic behavior as f increases, tends toward a constant as f decreases, and the fr equency at which the transition occurs (229 Hz) is close to the mean fundamental frequency of speech, at least for females (Bagshaw, 1994; Wang and Lin, 2004; Sc hwartz and Purves, 2004). It does not produce a decrease of granularity as f approaches zero, but at least doe s not increase without bound either, as a pure logarithmic scale does. The convenience of the use of the ERB scal e for pitch estimation over the Hertz and logarithmic scales was confirmed in informal test s, since better results we re obtained when using the ERB scale. Two other common psychoacoustic scales, the Mel and Bark scales, were also explored, but they produced worse results than the ERB scale.

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57 3.7 Window Type and Size Along this chapter we have been mentioning our wish to obtain a perf ect match (i.e., NIP equal to 1) between the input spectrum and the template corresponding to the pitch of the input. This section deals with the feasib ility of achieving such goal. First of all, since the input is non-negative but the template has negative regions, a perfect match is impossible. One solution would be to set the negative part of the template to zero, but this would leave us without the useful property that the negative weights have: the production of low scores for noisy signals (see Section 2.3). Instead, the solution we adopt is to preserve the negative weights, but ignore them when computin g the norm of the template. In other words, we normalize the kernel using only the norm of its positive part )) ( 0 max( ) ( f K f K (3-6) Hereafter, we will refer to this normalization as K+-normalization. To obtain a K+-normalized inner product ( K+-NIP) close to 1, we must direct our efforts to make the shape of the spectral peaks match the sh ape of the positive cosine lobe used as base element of the template, and also to force the te mplate have a value of zero in the negative part of the cosine. Since the shape of the spectral p eaks is the same for all peaks, it is enough to concentrate our efforts on one of them, and for simplicity we will do it for the peak at zero frequency. The shape of the spectral peaks is determined by the type of window used to examine the signal. The most straightforward window is th e rectangular window, whic h literally acts like a window: it allows seeing the signal inside the window but not outside it. More formally, the rectangular window multiplies the signal by a rectangular function of the form

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58 Figure 3-7. Fourier transf orm of rectangular window. otherwise, 0 2 / | | if / 1 ) ( T t T tT (3-7) where T is the window size. If a rectangular window is used to extr act a segment of a sinusoid of frequency f Hz to compute its Fourier transform, the support of this transform will not be concentrated at a single point but will be smeared in the neighborhood of f This effect is s hown in Figure 3-7 for f = 0, in other words, the figure show s the Fourier transform of T ( t ). This transform can be written as sinc( Tf ), where the sinc function is defined as ) sin( ) ( sinc (3-8) This function consists of a main lobe centered at zero and small side lobes that extend towards both sides of zero. For any other value of f its Fourier transform is just a shifted version of this function, centered at f

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59 Figure 3-8. Cosine lobe and square-root of the spect rum of rectangular window. Since the height of the side l obes is small compared to the height of the main lobe, the most obvious approach to try to maximize the ma tch between the input and the template is to match the width of the main lobe, 2/ T to the width of the cosine lobe, f /2, and solve for the free variable T This produces an “optimal” window size, hereafter denoted T*, equal to T = 4/ f Figure 3-8 shows the square-r oot of the spectrum of a rectangular window of size T = T* = 4/ f and a cosine with period f (i.e., the template used to recognize a pitch of f Hz). The K+-NIP of the main lobe of the spectrum and the cosine positive lobe (i.e., from f /4 to f /4) sampled at 128 equidistant points is 0.9925, which seem s satisfactorily high. However, the K+-NIP computed over the whole period of the cosine (i.e., from – f /2 to f /2) sampled at 128 equidistant points is only 0.5236, which is not very high. This low K+-NIP is caused by the rela tively large side lobes, which reach a height of almost 0.5.

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60 A window with much smaller side lobes is the Hann window. The shorter side lobes are achieved by attenuating the time-domain window down towards zero at the edges3. The formula for this window is T t T t hT2 cos 1 1 ) (, (3-9) where T is the window size (i.e., the size of its suppor t). This window is simply one period of a raised cosine centered at zero, as illustrated in Figure 3-9. The Fourier transform of a Hann window of size T is ) 1 ( sinc 2 1 ) 1 ( sinc 2 1 ) ( sinc ) ( Tf Tf Tf f HT, (3-10) a sum of three sinc functions, as illustrated in Figure 310. The width of the main lobe of this transform is 4/ T twice as large as the main lobe of the spectrum of the rectangular window. Figure 3-9. Hann window. 3 This time-frequency relation may not be obvious at fi rst sight, but it can be shown using Fourier analysis.

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61 Figure 3-10. Fourier transform of the Hann wi ndow. The FT of the Hann window consists of a sum of three sinc functions. Equalizing this width to the width of the cosine lobe, f /2, and solving for T we obtain an optimal window size of T* = 8/ f Figure 3-11. Cosine lobe and square -root of the spectrum of Hann window.

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62 Figure 3-11 shows the square-root of th e spectrum of a Hann window of size T = T* = 8/ f and a cosine with period f The similarity between the main lobe and the positive lobe of the cosine is remarkable. Using Equations 3-8 and 310 it can be shown that th ey match at 5 points: 0, +/f /8, and +/f /4, with values cos(0) = 1, cos( /4) = 1/ 2, and cos( /2) = 0, respectively. The K+-NIP of the main lobe of the spectrum and th e positive part of the cosine sampled at 128 equidistant points is 0.9996, and the K+-NIP computed over the whole period of the cosine sampled at 256 equidistant poin ts is 0.8896, much larger th an the one obtained with the rectangular window. The same approach can be used to obtain the optimal window size for other window types. For the most common window types used in signal processing, it can be shown that the width of the main lobe is 2 k / T where the parameter k depends on the window type (see Oppenheim, Schafer, and Buck, 1999) and is tabulated in Ta ble 3-1. For these windows, the optimal window Table 3-1. Common windows used in signal processing* K+-NIP Window type k Positive lobe Whole period Bartlett 2 0.99840.7959 Bartlett-Hann 2 0.99950.8820 Blackman 3 0.98990.9570 Blackman-Harris 4 0.97380.9689 Bohman 3 0.99260.9474 Flat top 5 0.98960.9726 Gauss 3.14 0.96330.8744 Hamming 2 0.99930.9265 Hann 2 0.99960.8896 Nuttall 4 0.97180.9682 Parzen 4 0.96270.9257 Rectangular 1 0.99250.5236 Triangular 2 0.99800.8820 The K+-NIP values were computed using 128 equidistant samples for the positive lobe and 256 equidistant samples for the whole period.

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63 size to analyze a signal with pitch f Hz can be obtaine d by equalizing 2 k / T to the width of the cosine lobe, f /2, to produce T* = T = 4 k / f Table 3-1 also shows the K+-NIPs between the square-root of the spectrum and the cosine computed over the positive lobe of the cosine (from f /4 to f /4) and over the whole period of the cosine (from f /2 to f /2). The window that produces the largest K+-NIP over the whole period is the flat-top window. However, its si ze is so large compared to othe r windows that the increase in K+-NIP is probably not worth the increase in computational cos t; similar results are obtained with the Blackman-Harris window, wh ich is 4/5 its size. If computa tional cost is a serious issue, a good compromise is offered by the Hamming window, which requires half the size of the Blackman-Harris window, and produces a K+-NIP of about 0.93. This K+-NIP is larger than the one produced by the Hann window, with no increased computational cost ( k =2 in both cases). However, since the difference in performance between them is not large, we prefer the analytically simpler Hann window. 3.8 SWIPE Putting all the previous sections together, the SWIPE estimate of the pitch at time t can be formulated as 2 / 1 ) ( ERBs 0 2 / 1 ) ( ERBs 0 2 ) ( ERBs 0 2 / 1 2 / 1max max max))| ( ( | ))] ( ( [ ) ( 1 | )) ( ( | )) ( ( ) ( 1 max arg ) ( f f f fd f t X d f K d f t X f K t p (3-11) where otherwise, 0 3/4, ) ( 1/4 ) ( or 3/4 1/4 if ) / 2 cos( 2 1 1/4, ) ( 3/4 if ) / 2 cos( ) ( f n /f f f n /f f f f f n /f f f f f f K (3-12)

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64 ') ( ) ( ) (' 2 / 4dt e t x t t w f f t Xf j f k, (3-13) is frequency in ERBs, ( ) converts frequency from ERBs into Hertz, ERBs( ) converts frequency from Hertz into ERBs, K+( ) is the positive part of K ( ) {i.e., max[0, K ( )]}, fmax is the maximum frequency to be used (typically the Nyquist frequency, although 5 kHz is enough for most applications), n( f ) = fmax / f 3/4 and w4 k / f ( t ) is one of the window functions in Table 3-1, with size 4 k / f The kernel corresponding to a ca ndidate with frequency 190 Hz is shown in Figure 3-12. Panel A shows the kernel in the Hertz scale and Pane l B in the ERB scale, the scale used to compute the integral. Although the initial approach of measuring a smooth average peak to valley distance has been used everywhere in this chapter, we can ma ke a more precise descri ption of the algorithm. Figure 3-12. SWIPE kernel. A) The SWIPE kern el consists of a cosine that decays as 1/ f with a truncated DC lobe and halved first and last negative lobes. B) SWIPE kernel in the ERB scale.

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65 It can be described as the computation of the si milarity between the square-root of the spectrum of the signal and the square-r oot of the spectrum of a sa wtooth waveform, using a pitchdependant optimal window size. This descriptio n gave rise to the na me Sawtooth-Waveform Inspired Pitch Estimator (SWIPE). 3.9 SWIPE So far in this chapter we have concentrat ed our efforts on maximizing the similarity between the input and the desired template, but we have not done anything explicitly to reduce the similarity between the input and the other temp lates, which will be the goal of this section. The first fact we want to mention is that most of the mistakes that pitch estimators make, including SWIPE, are not random: they consist of estimations of the pitch as multiples or submultiples of the pitch. Therefore, a good source of error to attack is the score (pitch strength) of these candidates. A good feature to reduce supraharmonic errors is to use negative weights between harmonics. When analyzing a pitch candidate, if th ere is energy between a ny pair of consecutive harmonics of the candidate, this suggests that the p itch, if any, is a lower candidate. This idea is implemented by the negative weight s, which reduce the score of th e candidate if there is any energy between its harmonics. This feature is used by algorithms like SHR, AC, CEP, and SWIPE. The effect of negative weights on supraharmonics of the pitch is illustrated in Figure 3-13A. It shows the spectrum of a si gnal with fundamental at 100 Hz and all its harmonics at the same amplitude (vertical lines) (Only harmonics up to 1 kHz are shown, but the signal contains harmonics up to 5 kHz.) The components are shown as lines to facilitate visualization, but in general th ey will be wider, with a width that depends on the window size.

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66 Figure 3-13. Most common pitch estimati on errors. A) Harmonic signal with 100 Hz fundamental frequency and all the harmoni cs at the same amplitude, and 200 Hz kernel with positive (continuous lines) and negative (dashed lines) cosine lobes. B) Same signal and 50 Hz kernel. C) Scores us ing only positive cosine lobes (exhibits peaks at sub and supraharmonics). D) Scor es using both positive and negative cosine lobes (exhibits peaks at subharmonics). E) Scores using both positive and negative cosine lobes at the first and prime harm onics (exhibits a major peaks only at the fundamental) Panel A also shows the positive cosine lobes (continuous curves) used to recognize a pitch of 200 Hz and the negative cosine lobes that reside in between (dashed curves). The positive cosine lobes at the harmonics of 200 Hz produce a positive contribution towards the score of the 200 Hz candidate, but the negative cosine lobes at the odd multiples of 100 Hz cancel out this contribution. Panel C shows the score for each pitc h candidate using as kernel only the positive

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67 cosine lobes, whereas Panel D shows the scores using both the positive and the negative cosine lobes. The effect on the 200 Hz peak is definite : it has disappeared. The same effect is obtained for higher order multiples of 100 Hz (not shown in the figure). To reduce subharmonic errors, two techniques we re presented in Chapter 2: the use of a decaying weighting factor for the harmonics, and th e use of a bias to pe nalize the selection of low frequency candidates. The former is used by SHS and SWIPE, and the latter by AC. Although these techniques have an effect in reducing the scor e of subharmonics, significant peaks are nevertheless present at submultiples of the pitch, as shown in Figure 3-13D. To further reduce the height of the peaks at subharmonics of the pitch we propose to remove from the kernel the lobes located at no n-prime harmonics, except the lobe at the first harmonic. Figure 3-13B helps to show the intuiti on behind this idea. This figure shows the same spectrum as in Figure 3-13A and the kernel corresponding to the 50 Hz candidate This kernel has positive lobes at each multiple of 50 Hz a nd therefore at each multiple of 100 Hz, producing a high score for the 50 Hz candidate, as shown in Pa nel D. Notice that this candidate gets all of its credit from its 2nd, 4th, 6th, etc., harmonics, i.e., 100 Hz, 200 Hz 300 Hz, etc., frequencies that suggest a fundamental frequency (and pitch) of 100 Hz. The same situation occurs with the candidate at 33 Hz (kernel not shown), but in this case its credit comes from its 3rd, 6th, 9th, etc., harmonics. If we use only the first and prime lobes of the kernel, the candidate s at subharmonics of 100 Hz would get credit only from their harmonic at 100 Hz, but not from any other. In general, it can be shown that with this approach, no candidate below 100 Hz can get credit from more than one of the harmonics of 100 Hz. In other wo rds, if there is a ma tch between one of the prime harmonics of this candida te and a harmonic of 100 Hz, no other prime harmonic of the

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68 candidate can match another harmonic of 100 Hz, and therefore the score of all the candidates below 100 Hz has to be low compared to the score of the 100 Hz candidate. This effect is evident in Figure 3-13E, which shows the scores of the pitch candidates when using only their first and prime harmonics. Certainly, th ere are peaks below 100 Hz, but they are relatively small compared to the peak at 100 Hz. Contrast this w ith Panels C and D, where the score of 50 Hz is relatively high, and therefore the risk of selecting this candidate is high. An extra step needs to be done to avoid bias in the scores. Remember from the beginning of this chapter that the central idea of SWIPE wa s to compute the average peak-to valley distance at harmonic locations in the spectrum. When computing this average for a single peak, the weight of the peak was twice as large as the wei ght of its valleys, as ex pressed in Equation 3-1. Since the global average is the average of this equation over all the peaks, and since each valley is associated to two peaks too, the weight of the valleys, except the first and the last ones, was the same as the weight of the peaks, as expres sed in Equation 3-2. However, if we use only the first and prime harmonics, the weight of the vall eys will not be necessarily -1, but will depend on whether the valleys are between the first or pr ime harmonics. The only valleys with a weight of -1 will be the valley between the first a nd second harmonics, and the valley between the second and third harmonics; all the other valleys will have a weight of -1 /2, before applying the decaying weighting factor, of course. This variation of SWIPE in which only the fi rst and prime harmonics are used to estimate the pitch will be denominated SWIPE (read SWIPE prime). Its kernel is defined as P i if f K f f K} 1 {) ( ) (, (3-14) where P is the set of prime numbers, and

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69 Figure 3-14. SWIPE kernel. Similar to the SWIPE kernel but includes only the first and prime harmonics. otherwise. 0 3/4, | | 1/4 if ) / 2 cos( 2 1 1/4, | | if ) / 2 cos( ) ( i f / f f f i f / f f f f f Ki (3-15) Notice that the SWIPE kernel can also be written as in Equa tion 3-14, by including all the harmonics in the sum. The SWIPE kernel corresponding to a pitch candidate of 190 Hz (5.6 ERBs) is shown in Figure 3-14. The numbers on top of the p eaks show the harmonic number they correspond to. 3.9.1 Pitch Strength of a Sawtooth Waveform Since the template used by SWIPE has peaks only at the first and prime harmonics, a perfect match between the template and the sp ectrum of a sawtooth waveform is impossible (unless fmax is so small relative to the pitch that the template contains no more than three

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70 Figure 3-15. Pitch strength of sawtooth waveform. A) 625 Hz. B) 312 Hz. C) 156 Hz. D) 78.1 Hz. harmonics). Therefore, it would be interesting to analyze the K+-NIP between the spectrum and the template as a function of the number of harmonics. Figure 3-15 shows the pitch strength ( K+NIP) obtained using SWIPE and SWIPE for different pitches an d different number of harmonics. The pitches shown are 625, 312, 156, and 78.1 Hz. They were chosen because their optimal window sizes are powers of two for the sampling rates used: 2.5, 5, 10, 20, and 40 kHz. In each case, fmax was set to the Nyquist frequency. The pitch strength estimates produced by SW IPE are larger than the ones produced by SWIPE except when the number of harmonics is less than four, in which case both algorithms use all the harmonics. The pitch strength esti mates produced by SWIPE in Figure 3-15 have a

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71 mean of 0.93 and a variance of 5.1 10 5. This mean is significantly larger than the K+-NIP reported in Table 3-1 for the Hann window. The reas on of the mismatch is that the granularity used to produce the data in Table 3-1 and the data in Figure 3-15 is different. The K+-NIP values in Table 3-1 are based on a sampling of 128 points per spectral lobe, while the data in Figure 315 is based on a sampling of 10 points per ERB, which depending on the pitch and the harmonic being sampled, may correspond to a range of about 0 to 40 points per spectral lobe. On the other hand, the mean of the p itch strength estimates produced by SWIPE is 0.87 and the variance is 1.0 10 3. The smaller mean is expected since the template of SWIPE includes only the first and prime harmonics, while a sawtooth waveform has energy at each of its harmonics. The larger variance is also expected since the prime numbers become sparser as they become larger, causing a reduction in the similari ty of the template and the spectrum of the sawtooth waveform as the number of harmonics increases. It would be useful to have a lower bound for the pitch strength estimates produced by SWIPE but an analytical formulation for it is intr actable. However, the data in Figure 3-15, which is representative of a wi de range of pitches and number of harmonics, suggests that the pitch strength produced by SWIPE for a sawtooth waveform does not go below 0.8. 3.10 Reducing Computational Cost 3.10.1 Reducing the Number of Fourier Transforms The computation of Fourier transforms is one of the most computationally expensive operations of SWIPE and SWIPE Therefore, to reduce computati onal cost it is important to reduce the number of Fourier transforms. There ar e two strategies to achieve this: to reduce the window overlap and to share Fourier tr ansforms among several candidates.

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72 3.10.1.1 Reducing window overlap The most common windows used in signal processing are the ones that are attenuated towards zero at their edges (e.g., Hann a nd Hamming windows). A disadvantage of this attenuation is that it is possible to overlook short events if these events are located at the edges of the windows. To avoid this situation, it is co mmon to use overlapping windows, which increases the coverage of the signal, at the cost of an increase in computation. However, after a certain point, overlapping windows star t to produce redundancy in the analysis, without adding any significant benefit. The goal of this section is to propose a schema obtain a good balance between signal coverage and computational cost. As mentioned in Section 1.1.4, depending on fr equency, a minimum of two to four cycles are necessary to perceive the pitch of a pure tone. Based on the similarity of the data used to arrive to this conclusion and data obtained using musical instruments, it is reasonable to assume that these results are applicable to more ge neral waveforms, in particular, to sawtooth waveforms. To avoid the interaction between the nu mber of cycles and pitch, for purposes of the algorithm, we set the minimum number of cycles necessary to determine pitch to four, the maximum among the minimum number of cycl es required over all frequencies. Since SWIPE and SWIPE are designed to produce maximu m pitch strength for a sawtooth waveform4 and zero pitch strength for a flat spectrum5, a natural choice to decide whether a sound has pitch is to use as thre shold half the pitch st rength of a sawtooth waveform. (In Section 3.9.1 it was found that the pitch strength of a sawtooth waveform is about 0.93 for SWIPE and between 0.83 and 0.93 for SWIPE .) To make these algorithms produce maximum pitch strength, 4 In fact, SWIPE produces maximum pitch strength for sawtoo th waveforms with the non-prime harmonics removed (except the first one), but we believe this type of signal is unlikely to occur in nature. 5 The pitch strength of a flat spectrum is in fact negative because of the decaying kernel envelope.

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73 a perfect match between the kernel and the spec trum of the signal is necessary, which requires that the window contains eigh t cycles of the sawtooth waveform, when using a Hann window. If the signal contains exactly eight cycles (i.e., if it is zero outside th e window) and is shifted slightly with respect to the window, the pitch strength decreases, and it reaches a limit of zero when the signal gets completely out of the wi ndow. Although hard to show analytically, it is easy to show numerically that that the relation be tween the shift and pitch strength is linear. Therefore, if the window contains four or mo re cycles of the sawtooth waveform, the pitch strength is at least half the maximum attainable pitch strength (i.e., the one achieved when the window is full of the sawtooth waveform), and if the window contains less than four cycles of the sawtooth waveform, the pitc h strength is less than half the maximum attainable pitch strength. Figure 3-16. Windows overlapping. Object 3-2. Four cycles of a 100 Hz sawtooth waveform (WAV file, 2 KB)

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74 Therefore, if we determine the existence of pitch based on a pitch strength threshold equal to half the maximum attainable pitch strength, to determine as pitched a signal consisting of four cycles of a sawtooth waveform, we need to ensu re that there exists at least one window whose coverage includes the whole signal. It is straightforward to show that to achieve this goal, we need to distribute the windows such that their separation in no larger than four cycles of the pitch period of the signal. In other words, th e windows must overlap by at least 50%. This situation is illustrated in Figure 3-16, which shows a signal cons isting of four cycles of a sawtooth waveform (listen to Object 3-2) and two Hann windows cen tered at the beginning and the end of the signal. The windows are separate d at a distance of four cycles, and the support of each of them overlaps with the whole signal, making it possible for each window to reach the pitch strength threshold. If the si gnal is slightly shifted in any di rection, one of the windows will cover less than four periods, but th e other will cover the four periods. This would not be true if the separation of the windows is larg er than four cycles. If the support of one of the windows overlaps complete ly with the signal but the separation of the windows is larger than four cycles, the other wi ndow will not cover the signal completely, and therefore a small shift of the signal towards th e latter window would not necessarily put the whole signal inside the window, making it impossible for any of the windows to produce a pitch strength larger than the threshold. 3.12.1.2 Using only power-of-two window sizes There is a problem with the optimal window size (O-WS) proposed in Section 3.7: each pitch candidate has its own, which means that a different STFT must be computed for each candidate. If we separate the candidates at a di stance of 1/8 semitone over a range of 5 octaves (appropriate for music, for exampl e), we will need to compute 8*12*5 = 480 STFTs for each

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75 pitch estimate. Not only that, for some WSs it may be inefficient to use an FFT (recall that the FFT is more efficient for windows sizes that are powers of two). To alleviate this problem, we propose to s ubstitute the O-WS with the power-of-two (P2) WS that produces the maximum K+-NIP between the square-root of the main lobe of the spectrum and the cosine kernel. To find such a WS, it is convenient to have a closed-form formula for the K+-NIP of these functions, but this invol ves integrating the product of a cosine and the square-root of the sum of three sinc functions, which is anal ytically intractable. As an alternative, we approximate the square -root of the spectral lobe with an idealized spectral lobe (ISL) consisting of the function it approximates: a positive cosine lobe. Figure 3-17 shows a K+-normalized cosine whose pos itive part has a width of f /2 (i.e., the cosine template used by an f Hz pitch candidate), and tw o normalized ISLs whose widths are half and twice the width of the positive part of the cosine. Since th e cosine and the ISLs are symmetric around zero, the K+-NIP can be computed using only th e positive frequencies. Hence, the K+-NIP Figure 3-17. Idealized spectral lobes.

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76 of the central positive lobe of a cosine with period rf (the ISL) and a cosine with period f (the template) can be computed as 2 / 1 4 / 0 2 2 / 1 4 / 0 2 4 / 0' ) / 2 ( cos ) / 2 ( cos ) / 2 cos( ) / 2 cos( ) ( f r f r fdf f f df f rf df f f f rf r P 8 / 8 / / ) 1 ( 2 cos / ) 1 ( 2 cos 2 12 / 1 2 / 1 4 / 0f r f df f f r f f rr f r f f r r f f r rr f f f 1 / ) 1 ( 2 sin 1 / ) 1 ( 2 sin 24 / 0 1 2 / ) 1 ( sin 1 2 / ) 1 ( sin 2 r r r r r r r (3-16) It is convenient to transform the input of this function to a base-2 logarithmic scale, = log2( r ), and then redefine the function as 2 1 2 / ) 1 2 ( sin 2 1 2 / ) 1 2 ( sin 2 ) (2 / 1 (3-17) Figure 3-18A shows () for between -1 and 1 (i.e., r = 2 between 1/2 and 2). As departs from zero, () departs from 1, as expected. However, the distribution is not symmetric: a decrease in has a larger effect on () than an increase in This make sense since a decrease in corresponds to a widening of the ISL, which puts part of it in the region where the cosine template is negative (see wider ISL in Fi gure 3-1), producing a large decrease in (). On the other hand, narrowing the ISL keeps it in the pos itive region of the cosine template, producing a smaller decrease in ().

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77 Figure 3-18. K+-normalized inner product between temp late and idealized spectral lobes. Figure 3-18A can be helpful in finding the P2-WS that produces the largest K+-NIP between the ISL and the template. If the O-WS for the template is T* seconds and the sampling rate is fs, then the O-WS in samples is N* = T fs, which correspond to = 0 in the figure. Smaller ’s correspond to smaller WSs, and larger ’s correspond to la rger WSs. In general, the WS in number of samples, denoted N and are related through the equation N = 2N*. It is straightforward to show that the two ’s that correspond to th e two closest P2-WSs to the optimal must be between -1 and 1, and not on ly that, their difference must be 1. Figure 3-18B shows the difference between () and ( 1) as a function of for between 0 and 1. From the figure we can infer that, for ’s between 0 and 0.56, we should use the larger P2-WS, and for between 0.56 and 1, we should use the smaller P2-WS. However, Figure 3-18B shows also that there is not much loss in the K+-NIP by choosing 0.5 as threshold rather than 0.56. Therefore, to simplify the algorithm, we decided to set the thres hold at 0.5. In other words, to determine the

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78 Figure 3-19. Individu al and combined pitch strength curves. P2-WS to use for a pitch candidate, we transform the O-WS and the P2-WSs to a logarithmic scale, and choose the P2-W S closest to the optimal. Unfortunately, this approach pr oduces discontinuities in the p itch strength (PS) curves, as illustrated in Figure 3-19A. The PS values marked with a plus sign were produced using a WS larger than the WS than the ones marked with a circle. To emphasize the e ffect, the pitch of the signal (220 Hz) was chosen to ma tch the point at which the cha nge of WS occurs. Since the PS values produced by the larger window in the neighborhood of the pitch are larger than the ones produced by the smaller window, the pitch co uld be biased toward a lower value. Although an effort was made to find an approp riate value for the threshold, it was based on an idealized spectrum, which does not have the side lobes found in real spectra. This problem can be alleviated by using a threshold larger than 0.56, determined through trial and error, but we

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79 found a better solution: to compute the PS as a lin ear combination of the PS values produced by the two closest P2-WSs, where th e coefficients of the combinat ion are proportional to the logdistance between the P2-WSs and the O-WS. Concretely, to determine the P2-WSs used to compute the PS of a candidate with frequency f Hz, the O-WS is written as a power of two, N* = 2L + where L is an integer and 0 < 1. Then, the PS values S0( f ) and S1( f ) are computed using windows of size 2L and 2L +1, respectively. Finally, these PSs are combined into a single one to produce the final PS ) ( ) ( ) 1 ( ) (1 0f S f S f S (3-18) Figure 3-19B shows how this combination of PS curves smoothes the discontinuity found in Figure 3-19A. It would be interesting to know how much is lost in PS by using the formula proposed in Equation 3-18, when the O-WS is not a power of two. This lost can be approximated by finding Figure 3-20. Pitch strength loss when using suboptimal window sizes.

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80 the minimum of the linear combination (1-) () + (-1) for 0 < < 1, which is plotted in Figure 3-20. It can be seen that it has a minimum of 0.93 at around = 0.4. Therefore, the maximum loss when computing PS using the two closest P2-WSs is 7%. Since the minimum PS of a sawtooth waveform when using an O-WS is about 0.92 for SWIPE and 0.83 for SWIPE (see Figure 3-15), the minimum pitch strength of a sawtooth waveform when using the two closest P2-WSs is about 0.86 for SWIPE and 0.77 for SWIPE Besides using an optimal window size for th e FFT computation, the approximation of O-WSs using P2-WSs has another advantage that is probably more important: the same FFT can be shared by several pitch candidates, more prec isely, by all the candidates within an octave of the optimal pitch for that FFT. Going back to the example that starte d this section, the replacement of the O-WS with the closest P2-WSs reduces the number of FFTs required to estimate the pitch from 480 to just 5: a huge save in computation. Using this approach, and translating the algor ithm to a discrete-time domain (necessary to compute an FFT), we can write the SWIPE estimate of the pitch at the discrete-time index as ) ( ) ( ) ( )) ( 1 ( max arg ] [1 ) ( ) (f S f f S f pf L f L f (3-19) where ) ( ) ( ) (*f L f L f (3-20) ) ( ) (*f L f L (3-21) ) / 4 ( log ) (2 *f kf f Ls (3-22)

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81 2 / 1 ) ( ERBs 0 2 2 / 1 ) ( ERBs 0 ) ( ERBs 0 2 / 1 2 2 / 1max max max| )] ( [ ˆ | )) ( ( ) ( 1 | )] ( [ ˆ | )) ( ( ) ( 1 ) ( f m f m f m Lm X m f K m m X m f K m f SL L,(3-23) s N Nf N f N X N I f X / }], 1 ,..., 0 { [ }, 1 ,..., 0 { ] [ ˆ (3-24) N j N Ne x w X/ 2'] [ ] [ ] [, (3-25) is the ERB scale step size (0.1 gives good enough resolution), I ( ,) is an interpolating function that uses the functional relations k = F ( k) to predict the value of F ( ), and XN[,] ( = 0, 1,…, N 1) is the discrete Fourier transform (c omputed via FFT) of the discrete signal x [ ], multiplied by the sizeN windowing function wN[ ], centered at The other variables, constants, and functions are define d as before (see Section 3.8). A Matlab implementation of this algorithm is given in Appendix A. 3.10.2 Reducing the Number of Spectral Integral Transforms The pitch resolution of SWIPE and SWIPE depends on the granularity of the pitch candidates. Therefore, to achieve high pitch reso lution, a large number of pitch candidates must be used, and since the pitch strength of each candidate is determined by computing a K+-NIP between its kernel and the spec trum, the computational cost of the algorithm would increase enormously. To avoid this situ ation, we propose to compute K+-NIPs only for certa in candidates, and then use interpolation to estimate th e pitch strength of the other candidates. As noted by de Cheveign (2002), the AC of a signal is the Fourier tr ansform of its power spectrum, and therefore the AC is a sum of cosines that can be approximated around zero by using a Taylor series expansion with even powers. If the signal is periodic, its AC is also

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82 periodic, and therefore the shape of the AC around the pitch period is the same as the shape around zero, and therefore it can also be approximate d by the same Taylor series, centered at the pitch period. If the width of th e spectral lobes is narrow and the energy of the high frequency components is small, the terms of order 4 in th e series vanish as the independent variable approaches the pitch period, and therefore the series can be approximated using a parabola. Since SWIPE perform an inner product between the spectrum and a kernel consisting of cosine lobes, a similar argument can be applied to the pitch strength curves produce by SWIPE. However, the quality of the fit of a parabola is not guaranteed for two reasons: first, the width of the spectral lobes produced by SWIP E are not narrow, in fact, they are as wide as the positive lobes of the cosine; and second, the use of the square-root of th e spectrum rather than its energy makes the contribution of the high frequency components large, vi olating the requirement of low contribution of high frequency components. Nevertheless, para bolic interpolation produces a good fit to the pitch strength curve in the neighb orhood of the SWIPE peaks, as we will proceed to show. Let’s derive an approximation to the pitch strength curve ( t ) produced by SWIPE for a sawtooth waveform with fundamental frequency f0 = 1/ T0 Hz in the neighborhood of the pitch period T0. To simplify the equations, let’s de fine the scaling transformations = 2f and = 2t / T0. To make the calculations tractable, let’s us e idealized spectral lobes (i.e. cosine lobes) and let’s ignore the normalization factors and the change of widt h of the spectral lobe with change of window size caused by a change of pitch candidate. Le t’s also repla ce the continuous decaying envelope of the kernel with a decayi ng step function that gives a weight of 1/ k to the k -th harmonic. With all this simplifications, the pitch strength of a candida te with scaled pitch

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83 period in the neighborhood of 2 (i.e., when the non-scaled pitch period t is in the neighborhood of T0) can be approximated as n k k 1) ( ) ( (3-26) where 4 / 1 4 / 1) 2 cos( ) cos( 1 ) (k k kd w k t 4 / 1 4 / 1) 2 ( cos ) 2 ( cos 2 1k kd t t k 2 ) 2 ( sin 2 ) 2 ( sin 2 14 / 1 4 / 1 t t t t kk k 2 ) 2 )( 4 / 1 ( sin ) 2 )( 4 / 1 ( sin 2 1 t t k t k k 2 ) 2 )( 4 / 1 ( sin ) 2 )( 4 / 1 ( sin t t k t k (3-27) Since we are interested in approximati ng this function in the neighborhood of 2, we can equivalently shift the function 2 units to the left by defining k ( ) = k ( +2 ), and then approximate k ( ) in the neighborhood of zero. Since sin( x ) / x = 1 x2/3! + x4/5! O ( x6) in the neighborhood of zero, it is useful to express k ( ) as ) 4 / 1 ( ) 4 / 1 ( sin 2 4 / 1 ) 4 / 1 ( ) 4 / 1 ( sin 2 4 / 1 ) ( k k k k k k k kk 4 ) 4 / 1 ( sin ) 4 / 1 ( sin 2 1 k k k (3-28) which has the Taylor series expansion

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84 6 4 4 2 2! 5 ) 4 / 1 ( 3 ) 4 / 1 ( 1 2 4 / 1 ) ( O k k k kk 6 4 4 2 2! 5 ) 4 / 1 ( 3 ) 4 / 1 ( 1 2 4 / 1 O k k k k 5 3 3! 3 ) 4 / 1 ( ) 4 / 1 ( ) 4 ( 2 1 O k k k 5 3 3! 3 ) 4 / 1 ( ) 4 / 1 ( O k k (3-29) in the neighborhood of zero. Finally, the approximati on of the pitch strength curve in the shiftedtime domain is n k kO a a a a a1 5 4 4 3 3 2 2 1 0) ( ) ( (3-30) Figure 3-21. Coefficients of the pitc h strength interpolation polynomial.

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85 Figure 3-21 shows the relative valu e of the coefficients of the expansion as a function of the number of harmonics in the signal. As the number of harmonics increases, the relative weight of the order-4 coefficient increases. However, as approaches zero, its fourth power becomes so small that its overall contribution to the sum is sm all compared to the contribution of the order-2 term. This effect is clear in Figure 3-22, which shows ( ) for a sawtooth waveform with 15 harmonics using polynomials of orde r 2 and order 4 in the range +/ 0.045, which corresponds to +/ 1/8 semitones. The curve has been scaled to have a maximum of 1. The large circles correspond to candidates separated by 1/8 semit ones, which is the in terval used in our implementation of SWIPE and SWIPE for the distance between p itch candidates for which the pitch strength is computed directly. The ot her markers correspond to candidates separated by 1/64 semitones, which is the resolution used to fine tune the pitch strength curve based on the pitch strength of the candidates fo r which the pitch strength is co mputed directly. As observed in Figure 3-22. Interpolated pitch strength.

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86 the figure, for such small values of the pitch strength values obtained with an order 2 polynomial (squares) are indistinguishable from the ones obtained with an order 4 polynomial (diamonds). Hence, a parabola is good enough to estimate the pitch streng th between candidates separated at distances as small as 1/8 semitones. 3.11 Summary This chapter described the SWIPE algorithm and its variation SWIPE The initial approach of the algorithm was the search for th e frequency that maximizes the average peak-tovalley distance at harmonic locati ons. Several modifications to this idea were applied to improve its performance: the locations of the harmonics were blurred, the spectral amplitude and the frequency scale were warped, an appropria te window type and si ze were chosen, and simplifications to reduce computational cost were introduced. After these modifications, SWIPE estimates the pitch as the fundamental frequenc y of the sawtooth waveform whose spectrum best matches the spectrum of the input signal. Its variation, SWIPE uses only the first and prime harmonics of the signal.

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87 CHAPTER 4 EVALUATION To asses the relevance of SWIPE and SWIPE they were compared against other algorithms using two speech databases and a musical instruments database. This chapter presents a brief description of these algorithms, database s, and the evaluation process. A more detailed description is given in Appendix B. 4.1 Algorithms The algorithms with which SWIPE and SWIPE were compared were the following: AC-P: This algorithm (Boersma, 1993) comput es the autocorrelati on of the signal and divides it by the autocorrelati on of the window used to anal yze the signal. It uses postprocessing to reduce discontinuities in the pitch trace. It is available with the Praat System at http://www.fon.hum.uva.nl/praat The name of the function is ac AC-S: This algorithm uses the autocorrelation of the cubed signal. It is available with the Speech Filing System at http://www.phon.ucl.ac.uk/resource/sfs The name of the function is fxac ANAL: This algorithm (Secrest and Doddington, 1983) uses autocorrelation to estimate the pitch, and dynamic programming to remove di scontinuities in the pitch trace. It is available with the Speech Filing System at http://www.phon.ucl.ac.uk/resource/sfs The name of the function is fxanal CATE: This algorithm uses a quasi autocorrela tion function of the speech excitation signal to estimate the pitch. We implemented it ba sed on its original description (Di Martino, 1999). The dynamic programming component used to remove discontinuities in the pitch trace was not implemented. CC: This algorithm uses crosscorrelation to estimate the pitch and post-processing to remove discontinuities in the pitch trace. It is available with the Praat System at http://www.fon.hum.uva.nl/praat The name of the function is cc CEP: This algorithm (Noll, 1967) uses the cepstru m of the signal and is available with the Speech Filing System at http://www.phon.ucl.ac.uk/resource/sfs The name of the function is fxcep ESRPD: This algorithm (Bagshaw, 1993; Medan, 1991) uses a normalized cross-correlation to estimate the pitch, a nd post-processing to remove discontinuities in the pitch trace. It is available with the Festival Speech Filing System at http://www.cstr.ed.ac.uk/projects/festival The name of the function is pda

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88 RAPT: This algorithm (Secrest and Doddingt on, 1983) uses a normalized crosscorrelation to estimate the pitch, and dynami c programming to rem ove discontinuities in the pitch trace. It is availabl e with the Speech Filing System at http://www.phon.ucl.ac.uk/resource/sfs The name of the function is fxrapt SHS: This algorithm (Hermes, 1988) uses subharm onic summation. It is available with the Praat System at http://www.fon.hum.uva.nl/praat The name of the function is shs SHR: This algorithm (Sun, 2000) uses the subharm onic-to-harmonic ratio. It is available at Matlab Central http://www.mathworks.com/matlabcentral under the title “Pitch Determination Algorithm”. The name of the function is shrp TEMPO: This algorithm (Kawahara et al., 1999) uses the instantaneous frequency of the outputs of a filterbank. It is available with the STRAIGHT System at its author web page http://www.wakayama-u.ac.jp/~kawahara The name of the function is exstraightsource YIN: This algorithm (de Cheveign and Kawaha ra, 2002) uses a modified version of the average squared difference function. It is available from its author web page at http://www.ircam.fr/pcm/cheveign/sw/yin.zip The name of the function is yin 4.2 Databases The databases used to test th e algorithms were the following: DVD: Disordered Voice Database This database contains 657 samples of sustained vowels produced by persons with disordered voice. It can be bought from Kay Pentax http://www.kayelemetrics.com KPD: Keele Pitch Database This speech database was collect ed by Plante et. al (1995) at Keele University with the purpose of evalua ting pitch estimation algorithms. It contains about 8 minutes of speech spoken by five ma les and five females. Laryngograph data was recorded simultaneously with speech, a nd was used to produce estimates of the fundamental frequency. It is publicly available at ftp://ftp.cs.keele.ac.uk/pub/pitch MIS: Musical Instruments Samples This database contains more than 150 minutes of sound produced by 20 different musical instruments. It was collected at the University of Iowa Electronic Music Studios, directed by Lawrence Fritts, a nd is publicly available at http://theremin.music.uiowa.edu PBD: Paul Bagshaw’s Database for evalua ting pitch determination algorithms This database contains about 8 minutes of speech spoken by one male and one female. Laryngograph data was recorded simultaneously with speech, and was used to produce estimates of the fundamental frequency. It was collected by Pa ul Bagshaw at the University of Edinburg (Bagshaw et. al 1993; Bagshaw 1994), and is publicly available at http://www.cstr.ed.ac.uk/research/projects/fda

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89 4.3 Methodology The algorithms were asked to produce a pitch estimate every millisecond. The search range was set to 40-800 Hz for speech and 30-1666 Hz fo r musical instruments. The algorithms were given the freedom to decide if the sound was pitc hed or not. However, to compute our statistics, we considered only the time instants at which all the algorithms agreed that the sound was pitched. Special care was taken to account for time mi salignments. Specifically, the pitch estimates were associated to the time corresponding to the center of their respective analysis windows, and when the ground truth pitch varied over time (i.e ., for PBD and KPD), the estimated pitch time series were shifted within a range of 100 ms to find the best alignment with the ground truth. The performance measure used to compare the algorithms was the gross error rate (GER). A gross error occurs when the estimated pitch is off from the reference pitch by more than 20%. At first glance this margin of error seems too larg e, but considering that mo st of the errors pitch estimation algorithms produce are octave errors (i .e., halving or doubling the pitch), this is a reasonable metric. On the other hand, this tolera nce gives room for deal ing with misalignments. The GER measure has been used previously to test PEAs by other rese archers (Bagshaw, 1993; Di Martino, 1999; de Chev eigne and Kawahara, 2002). 4.4 Results Table 4-1 shows the GERs for each of the algorithms over each of the speech databases. Both the rows and the columns are sorted by average GER: the best algorithms are at the top, and the more difficult databases are at the right. The best algorithm overall is SWIPE followed by SHS and SWIPE. Although on average SHS performs better than SWIPE, the only database in which SHS beats SWIPE is in the disordered voice database, which indicates that SWIPE performs better than SHS on normal speech.

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90 Table 4-1. Gross error rates for speech* Gross error (%) Algorithm PBD KPD DVD Average SWIPE 0.130.830.63 0.53 SHS 0.151.001.10 0.75 SWIPE 0.150.871.70 0.91 RAPT 0.751.002.40 1.40 TEMPO 0.321.902.00 1.40 YIN 0.331.404.50 2.10 SHR 0.691.505.10 3.50 ESRPD 1.403.904.60 5.00 CEP 6.104.2014.00 5.90 AC-P 0.732.9016.00 6.70 CATE 2.6010.007.20 6.60 CC 0.483.605.00 2.40 ANAL 0.832.0035.00 13.00 AC-S 8.807.0040.00 19.00 Average 1.703.009.90 4.90 Values computed using two significant digits. Table 4-2. Proportion of overestimation e rrors relative to total gross errors* Proportion of overestimations Algorithm DVD PBD KPD Average CC 0.00.00.1 0.0 SHS 0.00.00.3 0.1 RAPT 0.00.10.5 0.2 SHR 0.00.40.3 0.2 AC 0.00.40.2 0.2 AC 0.00.20.3 0.2 ANAL 0.00.50.4 0.3 CEP 0.40.50.4 0.4 SWIPE 0.00.60.7 0.4 SWIPE 0.10.60.7 0.4 YIN 0.10.90.5 0.5 TEMPO 0.10.80.9 0.6 CATE 0.50.50.8 0.6 ESRPD 0.50.70.9 0.7 Average 0.10.40.5 0.3 Values computed using one significant digit. Table 4-2 shows the proportion of GEs caused by overestimations of the pitch with respect to the total number of GEs. Th e proportion of GEs caused by underes timation of the pitch is just

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91 Table 4-3. Gross error rates by gender* Gross error (%) Algorithm Male Female Average SWIPE 0.362.401.4 SHS 0.552.501.5 SWIPE 0.492.701.6 RAPT 0.422.901.7 TEMPO 0.673.101.9 SHR 0.613.602.1 YIN 1.103.202.2 AC-P 2.103.602.9 CEP 1.804.203.0 CC 2.404.503.5 ESRPD 3.103.903.5 ANAL 1.305.903.6 AC-S 3.2010.006.6 CATE 11.004.207.6 Average 2.104.003.1 Values computed using two significant digits. one minus the values shown in the table. Algorith ms at the top have a tendency to underestimate the pitch while algorithms at the bottom have a te ndency to overestimate it. Most algorithms tend to underestimate the pitch in the di sordered voice database while th e errors are more balanced in the normal speech databases. Table 4-3 shows the pitch estimation perfor mance as a function of gender for the two databases for which we had access to this information: PVD and KPD. The error rates are on average larger for female speech than for male speech. Table 4-4 shows the GERs for the musical inst ruments database. Some of the algorithms were not evaluated on this database because they did not provide a mechanism to set the search range, and the range they covered was smaller that the pitch range spanned by the database. The two algorithms that perfor med the best were SWIPE and SWIPE.

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92 Table 4-4. Gross error rates for musical instruments* Gross error (%) Algorithm Underestimates Overestimates Total SWIPE 1.000.101.10 SWIPE 1.300.021.30 SHS 0.881.001.90 TEMPO 0.291.702.00 YIN 1.600.832.40 AC-P 3.200.003.20 CC 3.600.003.60 ESRPD 5.301.506.80 SHR 15.005.3020.00 Average 3.601.204.70 Values computed using two significant digits. Table 4-5. Gross error ra tes by instrument family* Gross error (%) Algorithm Brass Bowed Strings WoodwindsPiano Plucked Strings Average SWIPE' 0.01 0.190.142.208.80 2.30 SWIPE 0.00 0.220.230.0211.00 2.30 TEMPO 0.00 2.601.407.304.00 3.10 YIN 0.03 1.101.500.3614.00 3.40 SHS 0.02 1.500.7212.008.10 4.50 AC-P 0.03 0.560.800.3626.00 5.60 CC 0.07 0.831.000.3628.00 6.00 ESRPD 4.00 6.907.106.0011.00 7.00 SHR 22.00 25.0038.0026.0015.00 25.00 Average 2.90 4.305.606.1014.00 6.60 Values computed using two significant digits. Brass: French horn, bass/tenor trombones, trumpet, and tuba. Bowed strings: double bass, cello, viola, and violin. W oodwinds: flute, bass/alto flutes, bass/Bb/Eb clarinets, alto/soprano saxes. Plucked strings: cello and violin. Table 4-5 shows the GERs by instrument fa mily. The two best algorithms are SWIPE and SWIPE. SWIPE tends to perform better than SWIPE except for the piano, for which SWIPE produces almost no error. On the other hand, SWIPE performance on piano is relatively bad compared to correlation based algorithms. The fam ily for which fewer errors were obtained was the brass family; many algorithms achieved almost perfect performance for this family. The

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93 Table 4-6. Gross error rates fo r musical instruments by octave* Gross error (%) Algorithm 46.2 Hz +/1/2 oct. 92.5 Hz +/1/2 oct. 185 Hz +/1/2 oct. 370 Hz +/1/2 oct. 740 Hz +/1/2 oct. 1480 Hz +/1/2 oct. Average SWIPE' 1.20 1.00 2.300.890.130.29 0.97 SWIPE 0.08 1.20 3.001.000.250.38 0.99 YIN 3.20 0.95 5.301.800.690.96 2.20 AC-P 0.24 2.00 7.802.500.710.30 2.30 SHS 7.80 2.60 3.201.200.230.14 2.50 CC 0.26 2.60 8.202.700.930.40 2.50 TEMPO 15.00 2.80 2.001.100.520.31 3.60 ESRPD 7.90 2.60 4.804.2012.0032.00 11.00 SHR 37.00 0.60 1.8027.0070.0081.00 36.00 Average 8.10 1.80 4.304.709.5013.00 6.90 Values computed using two significant digits. family for which more errors were produced was the strings family playing pizzicato i.e., by plucking the strings. Indeed, pi zzicato sounds were the ones for which the performers produced more errors and the ones that were hard est for us to labe l (see Appendix B). Table 4-6 shows the GERs as a function of octave. The best perf ormance on average was achieved by SWIPE and SWIPE. The results of the algorit hms with an average GER less than 0.0 0.1 1.0 10.0 100.0 46.292.51853707401480 Pitch (Hz)GER (%) SWIPE' SWIPE YIN AC-P SHS CC TEMPO Figure 4-1. Gross error rates for musical instruments as a function of pitch.

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94 Table 4-7. Gross error rates fo r musical instruments by dynamic* Gross error (%) Algorithm pp mf ff Average SWIPE' 1.301.200.92 1.10 SWIPE 1.401.401.20 1.30 SHS 1.502.302.00 1.90 TEMPO 2.001.902.00 2.00 YIN 2.202.502.40 2.40 AC-P 3.303.203.30 3.30 CC 3.603.303.80 3.60 ESRPD 5.707.107.60 6.80 SHR 27.0029.0029.00 28.00 Average 5.305.805.80 5.60 Values computed using two significant digits. 10% is reproduced in Figure 4-1. All algorithms have approximately the same tendency, except at the lowest octave, where a larger va riance in the GERs can be observed. Table 4-7 shows the GERs as a function of dyna mic (i.e., loudness). In general, there is no significant variation of GERs with changes in loudness, although SWIPE has a tendency to reduce the GER as loudness increases [i.e., as the dynamic moves from pianissimo ( pp ) to fortissimo ( ff ) ]. As a final test, we wanted to validate the choi ces we made in Chapter 3, i.e., shape of the kernel, warping of the spectrum, weighting of th e harmonics, warping of the frequency scale, and selection of window type and size. Fo r this purpose, we evaluated SWIPE replacing every time one of its features with a more standard featur e, i.e., smooth vs. pulsed kernels, square-root vs. raw spectrum, decaying vs. flat kernel envelo pe, ERB vs. Hertz frequency scale, and pitchoptimized vs. fixed window size. We varied each of these variables in dependently and obtained the results shown in Table 4-8. Although some of the variations made SWIPE improve in some of the databases, overall SWIPE worked better with the features we proposed in Chapter 3.

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95 Table 4-8. Gross error rate s for variations of SWIPE Gross error (%) Variation PBD KPD DVD MIS Average Original 0.13 0.830.631.10 0.67 Flat envelope 0.16 1.001.400.60 0.79 Hertz scale1 0.23 1.701.400.37 0.93 Pulsed kernel 0.21 0.843.002.60 1.70 Raw spectrum2 0.25 2.101.604.90 2.20 Fixed WS3 0.15 0.771.709.10 2.90 Values computed using two significant digits. 1 FFTs were computed using optimal window sizes and the spectrum was inter/extrapolated to frequency bins separated at 5 Hz.2 The use of the raw spectrum rather than the square root of the spectrum implies the us e of a kernel whose envelope decays as 1/ f rather than 1/ f to match the spectral envelope of a sawtooth waveform.3 The power-of-two window size whose optimal pitch was closest to the geometric mean pitch of the database was used in each case. A window of size 1024 samples was used for the speech databases and a window of size of 256 samples was used for the musical instruments database. 4.5 Discussion SWIPE showed the best performance in all cat egories. SWIPE was the second best ranked for musical instruments and normal speech bu t not for disordered speech, for which SHS performed better (see Table 4-1). One possible reas on is that it is common for disordered voices to have energy at multiples of subharmonics of the pitch, and therefore algorithms that apply negative weights to the spectral regions between harmonics (e.g., SWIPE, SWIPE and all autocorrelation based algorithms) are prone to produce low scores for the pitch. Although SWIPE is among this group, its use of only the firs t and prime harmonics, reduces substantially the score subharmonics of the pitc h, producing most of the time a la rger score for the pitch than for its subharmonics. The rankings of the algorithms are relatively st able in all the tables except for SHR, which showed a good performance for speech but not fo r musical instruments. We believe this is caused by the wide pitch range spanned by the musi cal instruments. This is suggested by the results in Table 4-6, which show that SHR pe rforms well in the octaves around 92.5 Hz and 185

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96 Hz, which corresponds to the pitch region of spe ech, but performs very bad as the pitch moves from this region. Figure 4-1 shows that the relati ve trend on performance with pitch for musical instruments is about the same for all the algorithms except in the lowest region, where a large variance in performance was observed. However, this vari ance may be caused by a significant reduction in the numbers of samples in this region (about 4% of the data). The figure also shows an overall increase in GER in the octave around 185 Hz. We believe this is caused by the presence of a set of difficult sounds in the database wi th pitches in that region, since it is hard to be lieve that there is an inherent difficulty of the algorit hms to recognize pitc h in that region.

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97 CHAPTER 5 CONCLUSION The SWIPE pitch estimator has been develo ped. SWIPE estimates the pitch as the fundamental frequency of the sawtooth waveform whose spectrum best matches the spectrum of the input signal. The schematic description of the algorithm is the following: 1. For each pitch candidate f within a pitch range fminfmax, compute its pitch strength as follows: a. Compute the square-root of the spectrum of the signal. b. Normalize the square-root of the spectrum and apply an integral transform using a normalized cosine kernel whose envelope decays as 1/ f 2. Estimate the pitch as the candidate with highest strength. An implicit objective of the algorithm was to find the frequency for which the average peak-to-valley distance at its harm onics is maximized. To achieve th is, the kernel was set to zero below the first negative lobe and above the last negative lobe, and to avoid bias, the magnitude of these two lobes was halved. To make the contribution of each harmonic of the sawtooth waveform proportional to its amplitude and not to the square of its amplit ude, the square-root of the spectrum was taken before applying the in tegral transform. To make the kernel match the normalized squa re-root spectrum of the sawtooth waveform, a 1/ f envelope was applied to the kernel. The kernel was nor malized using only its positive part. To maximize the similarity betw een the kernel and the square -root of the input spectrum, each pitch candidate required its own window size, which in general is not a power of two, and therefore not ideal to compute an FFT. To re duce computational cost, the two closest power-oftwo window sizes were used, and their results are combined to produce a single pitch strength value. This had the extra advantage of allowing an FFT to be shared by many pitch candidates.

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98 Another technique used to reduce computational cost was to compute a coarse pitch strength curve and then fine tune it by using parabolic in terpolation. A last technique used to reduce computational cost was to reduce the amount of window overlap while allowing the pitch of a signal as short as four cycles to be recognized. The ERB frequency scale was used to compute the spectral integral transform since the density of this scale decrease s almost proportionally to fre quency, which avoids wasting computation in regions where there little spectral ener gy is expected. SWIPE a variation of SWIPE, uses only the fi rst and prime harmonics of the signal, producing a large reduction in subharmonic errors by reducing si gnificantly the scores of subharmonics of the pitch. Except for the obvious architectural decisions that must be taken when creating an algorithm (e.g., selection of the kernel), ther e are no free parameters in SWIPE and SWIPE at least in terms of “magic numbers”. SWIPE and SWIPE were tested using speech and musical instruments databases and their performance was compared against twelve othe r algorithms which have been cited in the literature and for which free implementations exist. SWIPE was shown to outperform all the algorithms on all the databases. SWIPE was ranked second in the normal speech and musical instruments databases, and was ranked th ird in the disordered speech database.

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99 APPENDIX A MATLAB IMPLEMENTATION OF SWIPE This is a Matlab implementation of SWIPE To convert it into SWIPE just replace [ 1 primes(n) ] in the for loop of the function pitchStrengthOneCandidate with [ 1:n ]. function [p,t,s] = swipep(x,fs,plim,dt,dlog2p,dERBs,sTHR) % SWIPEP Pitch estimation using SWIPE'. % P = SWIPEP(X,Fs,[PMIN PMAX],DT,DLOG2F,DERBS,STHR) estimates the pitch of % the vector signal X with sampling frequency Fs (in Hertz) every DT % seconds. The pitch is estimated by sampling the spectrum in the ERB scale % using a step of size DERBS ERBs. The pitch is searched within the range % [PMIN PMAX] (in Hertz) sampled every DLOG2P units in a base-2 logarithmic % scale of Hertz. The pitch is fine tuned by using parabolic interpolation % with a resolution of 1/64 of semitone (approx. 1.6 cents). Pitches with a % strength lower than STHR are treated as undefined. % % [P,T,S] = SWIPEP(X,Fs,[PMIN PMAX],DT,DLOG2P,DERBS,S/thr) returns the times % T at which the pitch was estimated and their corresponding pitch strength. % % P = SWIPEP(X,Fs) estimates the pitch using the default settings PMIN = % 30 Hz, PMAX = 5000 Hz, DT = 0.01 s, DLOG2P = 1/96 (96 steps per octave), % DERBS = 0.1 ERBs, and STHR = -Inf. % % P = SWIPEP(X,Fs,...[],...) uses the default setting for the parameter % replaced with the placeholder []. % % EXAMPLE: Estimate the pitch of the signal X every 10 ms within the % range 75-500 Hz using the default resolution (i.e., 96 steps per % octave), sampling the spectrum every 1/20th of ERB, and discarding % samples with pitch strength lower than 0.4. Plot the pitch trace. % [x,Fs] = wavread(filename); % [p,t,s] = swipep(x,Fs,[75 500],0.01,[],1/20,0.4); % plot(1000*t,p) % xlabel('Time (ms)') % ylabel('Pitch (Hz)')if ~ exist( 'plim' ) | isempty(plim), plim = [30 5000]; end if ~ exist( 'dt' ) | isempty(dt), dt = 0.01; end if ~ exist( 'dlog2f' ) | isempty(dlog2f), dlog2f = 1/96; end if ~ exist( 'dERBs' ) | isempty(dERBs), dERBs = 0.1; end if ~ exist( 'sTHR' ) | isempty(sTHR), sTHR = -Inf; end t = [ 0: dt: length(x)/fs ]'; % Times dc = 4; % Hop size (in cycles) K = 2; % Parameter k for Hann window % Define pitch candidates log2pc = [ log2(plim(1)): dlog2f: log2(plim(end)) ]'; pc = 2 .^ log2pc; S = zeros( length(pc), length(t) ); % Pitch strength matrix % Determine P2-WSs logWs = round( log2( 4*K fs ./ plim ) ); ws = 2.^[ logWs(1): -1: logWs(2) ]; % P2-WSs pO = 4*K fs ./ ws; % Optimal pitches for P2-WSs % Determine window sizes used by each pitch candidate d = 1 + log2pc log2( 4*K*fs./ws(1) );

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100 % Create ERBs spaced frequencies (in Hertz) fERBs = erbs2hz([ hz2erbs(pc(1)/4): dERBs: hz2erbs(fs/2) ]'); for i = 1 : length(ws) dn = round( dc fs / pO(i) ); % Hop size (in samples) % Zero pad signal xzp = [ zeros( ws(i)/2, 1 ); x(:); zeros( dn + ws(i)/2, 1 ) ]; % Compute spectrum w = hanning( ws(i) ); % Hann window o = max( 0, round( ws(i) dn ) ); % Window overlap [ X, f, ti ] = specgram( xzp, ws(i), fs, w, o ); % Interpolate at equidistant ERBs steps M = max( 0, interp1( f, abs(X), fERBs, 'spline', 0) ); % Magnitude L = sqrt( M ); % Loudness % Select candidates that use this window size if i==length(ws); j=find(d-i>-1); k=find(d(j)-i<0); elseif i==1; j=find(d-i<1); k=find(d(j)-i>0); else j=find(abs(d-i)<1); k=1:length(j); end Si = pitchStrengthAllCandidates( fERBs, L, pc(j) ); % Interpolate at desired times if size(Si,2) > 1 Si = interp1( ti, Si', t, 'linear', NaN )'; else Si = repmat( NaN, length(Si), length(t) ); end lambda = d( j(k) ) i; mu = ones( size(j) ); mu(k) = 1 abs( lambda ); S(j,:) = S(j,:) + repmat(mu,1,size(Si,2)) .* Si; end % Fine-tune the pitch using parabolic interpolation p = repmat( NaN, size(S,2), 1 ); s = repmat( NaN, size(S,2), 1 ); for j = 1 : size(S,2) [ s(j), i ] = max( S(:,j) ); if s(j) < sTHR continue, end if i==1, p(j)=pc(1); elseif i==length(pc), p(j)=pc(1); else I = i-1 : i+1; tc = 1 ./ pc(I); ntc = ( tc/tc(2) 1 ) 2*pi; c = polyfit( ntc, S(I,j), 2 ); ftc = 1 ./ 2.^[ log2(pc(I(1))): 1/12/64: log2(pc(I(3))) ]; nftc = ( ftc/tc(2) 1 ) 2*pi; [s(j) k] = max( polyval( c, nftc ) ); p(j) = 2 ^ ( log2(pc(I(1))) + (k-1)/12/64 ); end end function S = pitchStrengthAllCandidates( f, L, pc ) % Normalize loudness warning off MATLAB:divideByZero L = L ./ repmat( sqrt( sum(L.*L) ), size(L,1), 1 ); warning on MATLAB:divideByZero % Create pitch salience matrix S = zeros( length(pc), size(L,2) ); for j = 1 : length(pc) S(j,:) = pitchStrengthOneCandidate( f, L, pc(j) );

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101 end function S = pitchStrengthOneCandidate( f, L, pc ) n = fix( f(end)/pc 0.75 ); % Number of harmonics k = zeros( size(f) ); % Kernel q = f / pc; % Normalize frequency w.r.t. candidate for i = [ 1 primes(n) ] a = abs( q i ); % Peak's weigth p = a < .25; k(p) = cos( 2*pi q(p) ); % Valleys' weights v = .25 < a & a < .75; k(v) = k(v) + cos( 2*pi q(v) ) / 2; end % Apply envelope k = k .* sqrt( 1./f ); % K+-normalize kernel k = k / norm( k(k>0) ); % Compute pitch strength S = k' L; function erbs = hz2erbs(hz) erbs = 21.4 log10( 1 + hz/229 ); function hz = erbs2hz(erbs) hz = ( 10 .^ (erbs./21.4) 1 ) 229;

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102 APPENDIX B DETAILS OF THE EVALUATION B.1 Databases All the databases used in this work are fr ee and publicly available on the Internet, except the disordered voice database. Besides speech recordings, the speech databases contain simultaneous recordings of laryngograph data, which facilitates the computation of the fundamental frequency. The authors of these data bases used them to produce ground truth pitch values, which are also included in the databases. The disordered voice database includes fundamental frequency estimates, but as it will be explained later, a di fferent ground truth data set was used. The musical instruments database c ontains the names of the notes in the names of the files. B.1.1 Paul Bagshaw’s Database Paul Bagshaw’s database (PBD) for evalua tion of pitch determination algorithms (Bagshaw et. al 1993; Bagshaw 1994) was collected at th e University of Edinburgh, and is available at http://www.cstr.ed.ac.uk/research/projects/fda The speech and laryngograph signals of this database were sampled at 20 kHz. The ground truth fundamental frequency was computed by estimating the location of the glotta l pulses in the laryngogra ph data and taking the inverse of the distance between each pair of consecutive pulses. Each fundamental frequency estimate is associated to the time instant in the middle between the pair of pulses used to derive the estimate. B.1.2 Keele Pitch Database The Keele Pitch Database (KPD) (Plante et. al 1995) was created at Keele University and is available at ftp://ftp.cs.keele.ac.uk/pub/pitch The speech and laryngograph signals were sampled at 20 kHz. The fundament al frequency was estimated by using autocorre lation over a

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103 26.5 ms window shifted at interv als of 10 ms. Windows where the pitch is unclear are marked with special codes. Both of these speech databases PBD and KPD have been reported to contain errors (de Cheveigne, 2002), especially at the end of sentences, where the energy of speech decays and malformed pulses may occur. We will explai n later how we deal with this problem. B.1.3 Disordered Voice Database The disordered voice database (DVD) was collected by Kay Pentax http://www.kayelemetrics.com It includes 657 disordered voi ce samples of the sustained vowel “ah” sampled at 25 kH, and some few at 50 kHz. The database includes samples from patients with a wide variety of organic, neurological, traumatic psychogenic, and other voice disorders. The database includes fundamental frequenc y estimates, but by definition, they do not necessarily match their pitch. Therefore we estimated the pitch by ourselves by listening to the samples through earphones, and matching the pitch to the closest note, using as reference a synthesizer playing sawtooth waveforms. Assuming that we chose one of the two closest notes every time, this procedure should introduce an erro r no larger than 6%, which is smaller than the 20% necessary to produce a GE (see Chapter 4). There were some samples for which the pitch ra nged over a perfect fourth or more (i.e., the higher pitch was more than 33% higher than the lo wer pitch). Since this range is large compared to the permissible 20%, these samples were excl uded. Samples for which the range did not span more than a major third (i.e., the higher pitch was no more than 26% higher than the lower pitch) were preserved, and they were assigned the note corresponding to the median of the range. If the median was between two notes, it was assigned to any of them. This should introduce an error no

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104 larger than two semitones (12%), which is a bout half the maximum permissible error of 20%. There were 30 samples for which we could not perc eive with confidence a pitch, so they were excluded as well. Since the ground truth data was based on the percep tion of only one listener (the author), it could be argued that this data has low validity. To alleviate this, we excluded the samples for which the minimum error produced by a ny algorithm was larger than 50%. After excluding the non-pitch, variable pi tch, and samples at which the algorithms disagreed with the ground truth, we ended up with 612 samples out of the original 657. Appendix C shows the ground truth used for each of these 612 samples. B.1.4 Musical Instruments Database The musical instruments samples database was co llected at the University of Iowa, and is available at http://theremin.music.uiowa.edu The recordings were made using CD quality sampling at a rate of 44,100 kHz, but we downsampled them to 10 kHz in order to reduce computational cost. No noticeable change of per ceptual pitch was perceived by doing this, even for the highest pitch sounds. This database contains recordings of 20 instru ments, for a total of more than 150 minutes and 4,000 notes. The no tes are played in sequ ence using a chromatic scale with silences in between. E ach file usually spans one octave and is labeled with the name of the initial and final notes, plus the name of the instrument, a nd other details (e.g., Violin.pizz.mf.sulG.C4B4.aiff). In order to test the algorithms, the files were split into separate files containing each of them a single note with no leading or trailing silence. This proce ss was done in a semi-automatic way by using a power-based segmentation method, and then checking visually and auditively the quality of the segmentation.

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105 While doing this task it was discovered th at some of the note la bels were wrong. The intervals produced by the performers were sometim es larger than a semitone, and therefore the names of the files did not corres pond to the notes that were in fact played. This situation was common with string instruments, especially when playing in pizzicato Therefore, after splitting the files, we listen ed to each of them, and manually corrected the wrong names by using as reference an electr onic keyboard. This procedure sometimes introduced name conflicts (i.e., there were repeat ed notes played by the same instrument, same dynamic, etc.), and when this occurred, we remove d the repeated notes trying to keep the closest note to the target. When the conflicting notes were equally close to the target, the “best quality” sound was preserved. This removal of files was done to avoid the overh ead of having to add extra symbols to the file names to allow for repetitions, which would have complicated the generation of scripts to test the algorithms. Since this process of manually correcting the names of the notes was very tedious, especially for the pizzicato sounds, after fixing a ll the pizzicato bass and violin notes, the process was abandoned and the cello and viola pizzicato sounds were ex cluded from our evaluation. Arguably, except for the bass, pi zzicato sounds are not very co mmon in music, and therefore leaving the cello and viola pizzicato sounds out did not affect the repr esentativeness of the sample significantly. B.2 Evaluation Using Speech Whenever possible, each of the algorithms was asked to give a pitch estimate every millisecond within the range 40-800 Hz, using the default settings of the algorithm (an exception was made for ESRPD: instead of using the defau lt settings in the Festival implementation, the recommendations suggested by the author of the algorithm were followed). The range 40-800 was used to make the results comparable to the results published by de Cheveigne (2002).

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106 However, a full comparison is not possible since so me other variables were treated differently in that study. The commands issued for each of the algorithms were the following6: AC-P: To Pitch (ac)... 0.001 40 15 no 0.03 0.45 0.01 0.35 0.14 800 AC-S: fxac input_file ANAL: fxanal input_file CC: To Pitch (cc)... 0.001 40 15 no 0.03 0.45 0.01 0.35 0.14 800 CEP: fxcep input_file ESRPD: pda input_file -o output_file -L -d 1 -shift 0.001 -length 0.0384 -fmax 800 -fmin 40 -lpfilter 600 RAPT: fxrapt input_file SHS: To Pitch (shs)... 0.001 40 15 1250 15 0.84 800 48 SHR: [ t, p ] =shrp( x, fs, [40 800], 40, 1, 0.4, 1250, 0, 0 ); SWIPE: [ p, t ] = swipe( x, fs, [40 800], 0.001, 1/96, 0.1, -Inf ); SWIPE : [ p, t ] = swipep( x, fs, [40 800], 0.001, 1/96, 0.1, -Inf ); TEMPO: f0raw = exstraightsource( x, fs ); YIN: p.minf0 = 40; p.maxf0 = 800; p.hop = 20; p.sr = fs; r = yin( x, p ); where x is the input signal and fs is the sampling rate in Hertz. An important issue that had to be consider ed was the time associated to each pitch estimate. Since all algorithms use symmetric window s, a reasonable choice was to associate each estimate to the time at the center of the windo w. For CATE, ESRPD, and SHR, the user is allowed to determine the size of the window, so we followed the recommendation of their authors and we set the window sizes to 51.2, 38.4, and 40 ms, respectively. YIN uses a different window size for each pitch candidate, but the window s are always centered at the same time instant, and the largest window si ze is two periods of the larges t expected pitch period. For the Praat’s algorithms AC-P, CC, and SHS, through trial and error we found that they use windows of size 3, 1, and 2 times the la rgest expected pitch period, resp ectively. For AC-S, ANAL, CEP, RAPT, and TEMPO, the user is not allowed to se t up the window size, but the algorithms output 6 The command for CATE is not reported because we used our own implementation.

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107 the time instants associated to each pitch esti mate, so we used these times hoping that they correspond to the centers of the analysis windows used to determine the pitch. The times associated to the pitch ground trut h series are explicitly given in the PBD database, but not in the KPD database. For KPD, each pitch value was associated to the center of the window. Therefore, since th e ground truth pitch values were computed using 26.5 msec windows separated at a distance of 10 msec, the first pitch estim ate was assigned a time of 13.25 msec, and the time associated to each successive pitch estimate added 10 msec to the time of the previous estimate. For the DVD databases, each vow el was assumed to have a constant pitch, so the ground truth pitch time series was assumed to be constant. The purpose of the evaluation was to compare th e pitch estimates of the algorithms, but not their ability to distinguish the existence of pitc h. Therefore, we include d in the evaluation only the regions of the signal at wh ich all algorithms and the ground tr uth data agreed that pitch existed. To achieve this, we took the time in stants of the ground trut h values and the time instants produced by all the algorithms that es timated the pitch every millisecond (9 out of 13 algorithms), rounded them to the closest multiple of 1 millisecond, and took the intersection. This intersection would form the set of times at which all the algorithms would be evaluated. The algorithms that produced pitch estimates at a rate lower than 1,000 per second were not considered for finding the inters ection because that would reduce the time granularity of our evaluation, which was desire d to be one millisecond. As suggested in the previous paragraph, so me algorithms do not necessarily produce pitch estimates at times that are multip les of one millisecond, i.e., th ey may produce the estimates at the times t + t ms, where t is an integer and | t | < 1. Thus, to evaluate them at multiples of one millisecond, the pitch values at the desired times we re inter/extrapolated in a logarithmic scale.

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108 In other words, we took the logarithm of the estim ated pitches, inter/extrapolated them to the desired times, and took the exponential of the inter/ extrapolated pitches. Inter/extrapolation in the logarithmic domain was preferre d because we believe this is th e natural scale for pitch. This is what allows us to recognize a song ev en if it is sung by a male or a female. An important issue that must be considered when using simultaneous recordings of the laryngograph and speech signals is th at the latter are typically dela yed with respect to the former. An attempt to correct this misalignment was repo rted by the authors of KPD, but the success was not warranted. No attempt of correction was re ported for PBD. Since pitch in speech is timevarying, such misalignment could increase the es timation error significantly. To alleviate this problem, each pitch time series produced by each algorithm was delayed or advanced, in steps of 1 msec, and up to 100 msec, in order to find th e best match with the ground truth data. B.3 Evaluation Using Musical Instruments Considering that many algorithms were design ed for speech, the pitch range of the MIS database is probably too large for them to handle. To alleviate this, we excluded the samples that were outside the range 30-1666 Hz, which is nevert heless large, compared to the pitch range of speech. Since the range 30-1666 Hz was found to be too large for the Speech Filing System algorithms (AC-S, ANAL, CEP, and RAPT) thes e algorithms were not evaluated on the MIS database. The commands issued for each of the algorithms were the following: AC-P: To Pitch (ac)... 0.001 30 15 no 0.03 0.45 0.01 0.35 0.14 1666 CC: To Pitch (cc)... 0.001 30 15 no 0.03 0.45 0.01 0.35 0.14 1666 ESRPD: pda input_file -o output_file -P -d 1 -shift 0.001 -length 0.0384 -fmax 1666 -fmin 30 -n 0 -m 0 SHS: To Pitch (shs)... 0.001 30 15 5000 15 0.84 1666 48 SHR: [ t, p ] = shrp( x, fs, [30 1666], 40, 1, 0.4, 5000, 0, 0 ); SWIPE: [ p, t ] = swipe( x, fs, [30 1666], 0.001, 1/96, 0.1, -Inf ); SWIPE : [ p, t ] = swipep( x, fs, [30 1666], 0.001, 1/96, 0.1, -Inf ); YIN: p.minf0 = 30; p.maxf0 = 1666; p.hop = 10; p.sr = 10000; r = yin(x,p);

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109 Besides the widening of the pi tch range, the only difference w ith respect to the commands used for the speech databases were for ESRPD a nd SHS. For both of them, the low-pass filtering was removed in order to use as much informa tion from the spectrum as possible. This was convenient because the sounds were already low-pa ss filtered at 5 kHz, a nd therefore the highest pitch sounds (around 1666 Hz) had no more than th ree harmonics in the spectrum. The second change was the use of the ESRPD peak-tracker ( option -P) as an attempt to make the algorithm improve upon its results with speech. The evaluation process was very similar to th e one followed for speech: the time instants of the ground truth and the pitch estimates were rounded to the closest millisecond, the intersection of all the times was taken, and the stat istics were computed only at the times of this intersection. However, there was an issue that was necessary consider in this database. Some instruments played much longer notes than others The range of durations goes from tenths of second for strings playing in pizzicato to several seconds for some notes of the piano. If the overall error is computed without taking this into account, the results will be highly biased toward the performance produced with the in struments that play the largest notes. To account for this, the GER was computed independently for each sample, and then averaged over all the samples. However, this introduced an undesired effect: some samples had very few pitch estimates (only one estimate in so me cases), and therefor e this procedure would give them too much weight, whic h potentially would introduce noi se in our results. Therefore, we discarded the samples for which the time instants at which the algorithms were evaluated were less than half the durati on of the sample (in milliseconds). This discarded 164 samples, resulting in a total of 3459 samples, which wa s nevertheless a significant amount of data to quantify the performan ce of the algorithms.

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110 APPENDIX C GROUND TRUTH PITCH FOR THE DISORDERED VOICE DATABASE Table C-1. Ground truth pitch values for the disordered voice database AAK02 220.0 AAS16 123.5 ABB09 246.9 AB G04 116.5 ACG13 207.7 ACG20 164.8 ACH16 185.0 ADM14 138.6 ADP02 155.6 ADP11 116.5 AEA03 220.0 AFR17 246.9 AHK02 110.0 AHS20 196.0 AJF12 110.0 AJ M05 138.6 AJM29 123.5 AJP25 233.1 ALB18 123.5 ALW27 174.6 ALW28 220.0 AMB22 146.8 AMC14 92.5 AMC16 146.8 AMC23 196.0 AMD07 130.8 AMJ23 123.5 AMK25 77.8 AMP12 220.0 AMT11 246.9 AMV23 185.0 ANA15 155.6 ANA20 155.6 ANB28 196.0 AOS21 110.0 ASK21 116.5 ASR20 92.5 ASR23 130.8 AWE04 155. 6 AXD11 174.6 AXD19 196.0 AXL04 196.0 AXL22 196.0 AXS08 155.6 AXT11 185.0 AXT13 196.0 BAH13 98.0 BAS19 293.7 BAT19 185.0 BBR24 164.8 BCM08 233.1 BEF05 185.0 BGS05 246.9 BJH05 174.6 BJK16 174.6 BJK29 103.8 BKB13 87.3 BLB03 110.0 BMK05 246.9 BMM09 233.1 BPF03 116.5 BRT18 311.1 BSD30 130.8 BS G13 174.6 BXD17 138.6 CAC10 185.0 CAH02 196.0 CAK25 196.0 CAL12 92.5 CAL28 261.6 CAR10 196.0 CBD17 164.8 CBD19 174.6 CBD21 207.7 CBR29 174.6 CCM15 110.0 CDW03 146.8 CEN21 92.5 CER16 185.0 CER30 174.6 CFW04 155.6 CJB27 116.5 CJP10 98.0 CLE29 116.5 CLS31 185.0 CMA06 123.5 CMA22 103.8 CMR01 185.0 CMR06 110.0 CMR26 174.6 CMS10 196.0 CMS25 185.0 CNP07 196.0 CNR01 185.0 CPK19 155.6 CPK21 174.6 CPW28 220.0 CRM12 185.0 CSJ16 233.1 CSY01 110.0 CTB30 146.8 CTY03 130.8 CXL08 174.6 CXM07 130.8 CXM14 220.0 CXM18 146.8 CXP02 207.7 CXR13 146.8 CXT08 155.6 DAC26 155.6 DAG01 185. 0 DAM08 174.6 DAP17 130.8 DAS10 146.8 DAS24 146.8 DAS30 87.3 DAS40 77.8 DB A02 220.0 DBF18 155.6 DBG14 103.8 DFB09 233.1 DFS23 293.7 DFS24 293. 7 DGL30 207.7 DGO03 110.0 DHD08 123.5 DJF23 146.8 DJM14 130.8 DJM28 185.0 DJP04 110.0 DLB25 261.6 DLL25 174.6 DLT09 207.7 DLW04 130.8 DMC03 185.0 DMF11 293.7 DMG07 146.8 DMG24 196.0 DMG27 155.6 DMP04 123.5 DMR27 233.1 DMS01 146.8 DOA27 92.5 DRC15 196.0 DRG19 116.5 DSC25 277.2 DSW14 138. 6 DVD19 164.8 DWK04 130.8 DXS20 123.5 EAB27 164.8 EAL06 207.7 EAS11 110.0 EAS15 138.6 EAW21 207.7 EBJ03 146.8 EDG19 196.0 EEB24 164.8 EEC04 196.0 EED 07 554.4 EFC08 130.8 EGK30 196.0 EGT03 138.6 EGW23 220.0 EJB01 92.5 EJM04 123.5 ELL04 116.5 EMD08 82.4 EML18 370.0 EMP27 174.6 EOW04 164.8 EPW04 164.8 EPW07 123.5 ERS07 185.0 ESL28 207.7 ESM05 138.6 ESP04 138.6 ESS05 174.6 ESS24 220.0 EWW05 174.6 EXE06 146.8 EXH21 185.0 EXI04 110.0 EXI05 116.5 EXS07 207.7 EXW12 164.8 FAH01 164.8 FGR15 130.8 FJL23 116.5 FLL27 207.7 FLW13 207.7 FMC08 196.0 FMM21 207.7 FMM29 207.7 FMQ20 155.6 FMR17 116.5 FRH18 146.8 FSP13 155.6 FXC12 110.0 FXE24 196.0 FXI23 103.8 GCU31 123.5 GEA24 130.8 GEK02 138.6 GJW09 174.6 GLB01 77.8 GLB22 98.0 GMM06 196.0 GMM07 207.7 GMS03 110.0 GMS05 261.6 GMW18 146.8 GRS20 110.0 GSB11 164.8 GSL04 116.5 GTN21 130.8 GXL21 196.0 GXT10 155.6 GXX13 164.8 HB S12 196.0 HED26 123.5 HJH07 130.8 HLC16 110.0 HLK01 116.5 HLK15 130.8 HLM24 138.6 HMG03 185.0 HML26 207.7 HWR04 164.8 HXB20 196.0 HXI29 82.4 HXL58 116.5 HXR23 116.5 IGD08 196.0 IGD16 174.6 JAB08 130.8 JAB30 164.8 JAF15 146.8 JAJ10 207.7 JAJ22 155.6 JAJ31 155.6 JAL05 174.6 JAM01 207.7 9-Jan 130.8 JAP02 138.6 JAP17 174.6 JAP25 174.6 JBP14 98.0 JBR26 110.0 JBS17 82.4 JBW14 130.8 JCC08 164.8 JCC10 207.7 JCH13 110.0 JCH21 116.5 JCL12 174.6 JCL20 146.8 JCR01 233.1 JDM04 110.0 JEG29 246.9 JES29 123.5 JFC28 82.4 JFG08 138.6 JFG26 138.6 JFM24 174.6 JFN11 110.0 JFN21 116.5 JHW29 146.8 JIJ30 146.8 JJD06 174.6 JJD11 185.0 JJD29 138.6 JJI03 110.0 JJ M28 220.0 JLC08 185.0 JLD24 233.1 JLH03 174.6 JLM18 207.7 JLM27 123.5 JLS11 130.8 JLS18 138.6 JMC18 138.6 JME23 164.8 JMH22 155.6 JMJ04 207.7 JMZ16 196.0 JOP07 130.8 JPB07 98.0 JPB17 164.8 JPB30 98.0 JPM25 110.0 JPP27 207.7 JRF30 123.5 JRP20 110.0 JSG18 207.7 JTM05 87.3 JTS02 103.8 JWE23 185.0 JWK27 98.0 JWM15 116.5 JXB16 110.0 JXB26 116.5 JXC21 220.0 JXD01 138.6 JXD08 138.6 JXD30 123.5 JXF11 246.9 JXF29 103.8 JXG05 138.6 JXM30 146.8 JXS09 110.0 JXS14 146.8 JXS23 98.0 JXS39 146.8 JXZ11 123.5 KAB03 185.0 KAC07 246.9 KAO09 261.6 KAS09 233.1 KAS14 220.0 KCG23 246.9 KC G25 220.0 KDB23 220.0 KEP27 87.3

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111 Table C-1. Continued KEW22 220.0 KGM22 220.0 KJB19 164.8 KJI23 138.6 KJI24 130.8 KJL11 116.5 KJM08 130.8 KJS28 207.7 KJW07 103.8 KLC06 207.7 KLD26 164.8 KMC19 207.7 KMC22 207.7 KMC27 207.7 KMS29 155.6 KMW05 311.1 KPS25 103.8 KTJ26 220.0 KWD22 185.0 KXA21 164.8 KXB17 246.9 KXH19 246.9 LAC02 164.8 LAD13 130.8 LAI04 174.6 LAP05 116.5 LAR05 116.5 LBA24 220.0 LCW30 196.0 LDJ11 82.4 LGK25 110.0 LGM01 185.0 LHL08 207.7 LJH06 207.7 LJM24 196.0 LJS31 220.0 LLM22 277.2 LMB18 116.5 LMM04 185.0 L MM17 196.0 LMP12 196.0 LNC11 98.0 LPN14 146.8 LRD21 116.5 LRM03 293.7 LSB18 174.6 LVD28 261.6 LWR18 220.0 LXC01 207.7 LXC11 207.7 LXC28 207.7 LXD22 207.7 LXG17 116.5 LXR15 103.8 LXS05 196.0 MAB06 196.0 MAB11 146.8 MAC03 185.0 MAM08 207.7 MAM21 220.0 MAT26 261.6 MAT28 233.1 MBM05 155.6 MBM21 196.0 MBM25 185.0 MCA07 164.8 MCB20 174.6 MCW14 277.2 MCW21 196.0 MEC06 196.0 MEC28 174.6 MEH26 196.0 MEW15 246.9 MFC20 123.5 MGM28 220.0 MGV01 103.8 MHL19 138.6 MID08 174.6 MJL02 130.8 MJM04 207.7 MJZ 18 196.0 MKL31 123.5 ML B16 196.0 MLC08 233.1 MLC23 174.6 MLF13 196.0 MLG10 233.1 MMD01 233.1 MMD15 233.1 MMG27 246.9 MMM12 246.9 MMR01 138.6 MMS29 130.8 MNH04 207.7 MNH14 261.6 MPB23 103.8 MPC21 207.7 MPF25 110.0 MPH12 220.0 MPS09 246.9 MPS21 233.1 MPS23 311.1 MPS26 220.0 MRB11 98.0 MRB25 98.0 MRB30 92.5 MRC20 174.6 MRM16 155.6 MRR22 174.6 MSM20 77.8 MWD28 110.0 MXC10 233.1 MXN24 233.1 MXS06 246.9 MXS10 233.1 MYW04 220.0 MYW14 207.7 NAC21 98.0 NAP26 92.5 NFG08 207.7 NGA16 116.5 NJS06 207.7 NLC08 185.0 NM B28 185.0 NMC22 233.1 NMF04 164.8 NML15 196.0 NMR29 123.5 NMV07 207.7 NXM18 185.0 NXR08 185.0 OAB28 69.3 ORS18 98.0 OWH04 233.1 OWP02 246.9 PAM01 92.5 PAT10 110.0 PCL24 110.0 PDO11 110.0 PEE09 185.0 PFM03 103.8 PG B16 110.0 PJM12 98.0 PLW14 207.7 PMC26 92.5 PMD25 130.8 PMF03 233.1 PSA21 155.6 PTO18 98.0 PTO22 98.0 PTS01 130.8 RAB08 185.0 RAB22 196.0 RAE12 110.0 RAM30 261.6 RAN30 261.6 RBC09 155.6 RBD03 155.6 RCC11 233.1 REC19 233.1 REW16 110.0 RFC19 233.1 RFC28 116.5 RFH18 155.6 RFH19 130.8 RGE19 82.4 RHG07 220.0 RHP12 196.0 RJC24 98.0 RJF14 164.8 RJF22 174.6 RJL28 92.5 RJR15 110.0 RJR29 116.5 RJZ16 185.0 RLM21 123.5 RMB07 98.0 RMC07 155.6 RMC18 196.0 RMF14 196.0 RML13 233.1 RMM13 246.9 RPC14 174.6 RPJ15 116.5 RPQ20 103.8 RSM20 130.8 RTH15 87.3 RTL17 87.3 RWC23 98.0 RWF06 146.8 RWR14 110.0 RWR16 116.5 RXG29 98.0 RXM15 110.0 RXP02 138.6 RXS13 130.8 SAC10 103.8 SAE01 164.8 SAM25 138.6 SAR14 207.7 SAV18 277.2 SBF11 207.7 SBF24 207.7 SCC15 138.6 SCH15 207.7 SEC02 196.0 SEF10 98.0 SEG18 130.8 SEH26 174.6 SEH28 246.9 SEK06 164.8 SEM27 116.5 SFD17 116.5 SFD23 87.3 SFM22 92.5 SGN18 138.6 SHC07 164.8 SHD17 220.0 SHT20 138.6 SJD28 123.5 SLC23 220.0 SLG05 196.0 SLM27 87.3 SMD22 207.7 SMK04 370.0 SMK23 146.8 SMW17 77.8 SPM26 92.5 SRB31 174.6 SRR24 130.8 SWB14 123.5 SWS04 155.6 SXC02 146.8 SXG23 174.6 SXH10 185.0 SXM27 196.0 SXS16 220.0 SXZ01 87.3 TAB21 174.6 TAC22 207.7 TAR18 155.6 TCD26 138.6 TES03 220.0 TLP13 233.1 TLS08 185.0 TMK04 261.6 TNC14 207.7 TPM04 155.6 TPP11 220.0 TPP24 185.0 TPS16 116.5 TRF06 116.5 TRF21 98.0 TRS28 185.0 VFM11 220.0 VJV02 130.8 VJV09 110.0 VMB18 174.6 VMS04 277.2 VMS05 246.9 VRS01 164.8 WBR12 277.2 WCB24 174.6 WDK04 110.0 WDK13 220.0 WDK17 130.8 WDK47 146.8 WFC07 116.5 WJB06 233.1 WJB12 110.0 WJF15 174.6 WJP20 123.5 WPB30 123.5 WPK11 110.0 WSB06 110.0 WST20 87.3 WTG07 130.8 WXE04 123.5 WXH02 103.8 WXS21 110.0 LME07 659.3 EAM05 146.8 JEC18 196.0 TMD12 349.2 SMA08 220. 0 SHD04 349.2 KXH30 174.6 VAW07 174.6

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116 BIOGRAPHICAL SKETCH Arturo Camacho was born in San Jose, Co sta Rica, on October 21, 1972. He did his elementary school at Centro Educativo Roberto Cantillano Vindas and his high school at Liceo Salvador Umaa Castro. After th at, he studied Music at the Un iversidad Nacional, and at the same time he performed as pianis t in some of the most popular Co sta Rican Latin music bands in the 1990’s. He also studied Computer and Informa tion Science at the Univ ersidad de Costa Rica, where he obtained his B.S. degree in 2001. He worked for a short time as a software engineer in Banco Central de Costa Rica during that year, but soon he moved to the United States to pursue graduate studies in Computer Engineering at the University of Florida. He received his M.S. and Ph.D. degrees in 2003 and 2007, respectively. Arturo’s research interests span all areas of automatic music analysis, from the lowest level tasks like pitch estimation and timbre identification, to the highest levels tasks like analysis of harmony and gender. His dream is to have one da y a computer program that allows him (and everyone) to analyze music as well or bett er than a well-traine d musician would do. Currently, Arturo lives happily with his loved wife Alexandra, who is another Ph. D. gator in Computer Engineering and who he married in 2002, and their love d daughter Melissa, who was born in 2006.


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