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SYNTHESIS AND ANALYSIS OF REAL
SINGLE-SIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS
By
Leon Worthington Couch, II
A Dissertation Presented to the Graduate Council of
The University of Florida
in Partial Fulfillment of the Reauirements for the
Degree of Doctor of Philosophy
UNIVERSITY OF FLORIDA
1968
Copyright by
Leon Worthington Couch, II
1968
DEDICATION
The author proudly dedicates this dissertation to his parents,
Mrs, Leon Couch and the late Rev. Leon Couch, and to his wife, Margaret
Wheland Couch,
ACKNOWLEDGMENTS
The author wishes to express sincere thanks to some of the many
people who have contributed to his Ph.D program. In particular,
acknowledgment is made to his chairman, Professor T. S. George, for his
stimulating courses, sincere discussions, and his professional example.
The author also appreciates the help of the other members of his super-
visory committee. Thanks are expressed to Professor R. C. Johnson and
the other members of the staff of the Electronics Research Section,
Department of Electrical Engineering for their comments and suggestions.
The author is also grateful for the help of Miss Betty Jane Morgan who
typed the final draft and the final manuscript.
Special thanks are given to his wife, Margaret, for her inspi-
ration and encouragement.
The author is indebted to the Department of Electrical Engi-
neering for the teaching assistantship which enabled him to carry out
this study and also to Harry Diamond Laboratories which supported this
work in part under Contract DAAG39-67-C-0077, U. S. Army Materiel Com-
mand.
TABLE OF CONTENTS
ACKNOWLEDGMENTS . .
LIST OF FIGURES .
KEY TO SYMBOLS .
ABSTRACT . .
CHAPTER
I
I. INTRODUCTION . .
.1, MATHEMATICAL PRELIMINARIES .
I. SYNTHESIS OF SINGLE-SIDEBAND SIGNALS *
V. EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN
4.1. Example 1: Single-Sideband AM with
Suppressed-Carrier ....... *
4,2 Example 2: Single-Sideband PM *
4.3. Example 3: Single-Sideband FM *
4,4. Example 4: Single-Sideband a *
V. ANALYSIS OF SINGLE-SIDEBAND SIGNALS *
5. 1 Three Additional Equivalent Realiza
5.2, Suppressed-Carrier Signals *
5.3. Autocorrelation Functions *.
5.4. Bandwidth Considerations .. *.
5.4-1. Mean-type bandwidth *
5.4-2. RMS-type bandwidth .
5.4-3, Equivalent-noise bandwidth *
5 5, Efficiency . .
5.6, Peak-to-Average Power Ratio .* *
v
Page
* iv
* viii
* x
* xiii
* 1
4
* 9
* 18
tions
* *
. 18
19
28
* 30
. 3 42
* .* 43
*. 44
* 45
* .* 45
. 46
* *
* *
Page
VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS .. .48
6o1. Example 1: Single-Sideband AM With
Suppressed Carrier. ................ 48
62. Example 2: Single-Sideband PM . 51
6.3. Example 3: Single-Sideband FM * 68
6.4. Example 4: Single-Sideband a . 71
VII, COMPARISON OF SOME SYSTEMS . .. 75
7.1. Output Signal-to-Noise Ratios . 76
7 1-1 AM system . . 76
7.1-2. SSB-AM-SC system . 77
7.1-3. SSB-FM system . .. 78
7.1-4. FM system . . .. 84
7.1-5. Comparison of signal-to-noise ratios. 85
7.2, Energy-Per-Bit of Information .......... *89
7.2-1. AM system . . .. 93
7.2-2. SSB-AM-SC system . . 93
7.2-3. SSB-FM system . .. 94
7.2-4. FM system . . 94
7.2-5. Comparison of energy-per-bit for
various systems *. ..............** 95
7.3. System Efficiencies ................. 97
7,3-1. AM system . . .. 98
7.3-2c SSB-AM-SC system . . 98
7.3-3. SSB-FM system . .. 98
7.3-4. FM system . . .. 99
7.3-5. Comparison of system efficiencies 100
Page
VIII. SUMMARY . . .. 102
APPENDIX
I. PROOFS OF SEVERAL THEOREMS . .. .105
II. EVALUATION OF e j(x + jy) . . 119
REFERENCES . . . .. 121
BIOGRAPHICAL SKETCH . . . 124
LIST OF FIGURES
Figure
1. Voltage Spectrum of a Typical m(t) Waveform *.
2. Voltage Spectrum of the Analytic Signal Z(t) .*
3. Voltage Spectrum of an Entire Function of an Analytic
Signal .......................
4. Voltage Spectrum of the Positive Frequency-Shifted
Entire Function of the Analytic Signal *.
5. Voltage Spectrum of the Synthesized Upper Single-
Sideband Signal . . .
6. Voltage Spectrum of the Negative Frequency-Shifted
Entire Function of an Analytic Signal .
7. Voltage Spectrum of the Synthesized Lower Single-
Sideband Signal . .
8. Phasing Method for Generating USSB-AM-SC Signals -
9. USSB-PM Signal Exciter--Method I . .
10. USSB-FM Signal Exciter . .
11. Envelope-Detectable USSB Signal Exciter ...... .
12. Square-Law Detectable USSB Signal Exciter ......
13. USSB-PM Signal Exciter--Method 11 . .
14. USSB-PM Signal Exciter--Method III . .
15. Power Spectrum of a(t) *
16. AM Coherent Receiver . .
17. SSB-AM-SC Receiver . . .
18. SSB-FM Receiver . . .
19. Output to Input Signal-to-Noise Power Ratios for
Several Systems .* .
Page
9
11
* 16
* 20
. 22
. 24
* 26
. 27
* 53
. 54
* 67
* 76
* 78
* 78
* 86
vi11
* *
* *
Figure Page
20 Output Signal-to-Noise to Input Carrier-to-Noise Ratio
for Several Systems .* ..................... 87
21. Output Signal-to-Noise to input Signal-to-Normalized-
Noise Power Ratio for Various Systems .......... 90
22, Output Signal-to-Noise to Input Carrier-to-Normalized-
Noise Power Ratio for Various Systems . 91
23. Comparison of Energy-per-Bit for Various Systems .. .96
24. Efficiencies of Various Systems . 101
25. Contour of Integration . . .. 107
26. Contour of Integration .................. 115
KEY TO SYMBOLS
A0 = Amplitude Constant
AM = Amplitude-Modulation
b = Baseband Bandwidth (rad/s)
B = RF Signal Bandwidth (rad/s)
Cb = Baseband Channel Capacity
CB = RF Channel Capacity
Ci = Input Carrier Power
(C/N)i = Input Carrier-to-Noise Ratio
(C/N)I = Input Carrier-to-Normalized-Noise Ratio
D = Modulator Transducer Constant
FM = Frequency-Modulation
F(w) = Voltage Spectrum
F(-) = The Fourier Transform of (*)
g(W) = U(W) + jV(W) = An Entire Function
GN = Gaussian Noise
LSSB = Lower Single-Sideband
m(t) = Modulating Signal or a Real Function of the Modulating Signal
(see e(t) below)
M = Either Multiplex or Figure of Merit
Ni = Input Noise Power
NI = Normalized Input Noise Power
P()) = Power Spectral Density
PM = Phase-Modulation
R(,) ; Autocorrelation Function
Re(-) = Real Part of (*)
RF = Radio Freouency
Si = Input Signal Power
So: = Output Signal Power
(S/N)i = Input Signal-to-Noise Ratio
(S/N)I = Input Signal-to-Normalized-Noise Ratio
(S/N)o = Output Signal-to-Noise Ratio
SC = Suppressed-Carrier
USSB = Upper Single-Sideband
-(W) = The "Suppressed-Carrier" Function of U(W)
V(W) = The "Suppressed-Carrier" Function of V(W)
X(t) = A Real Modulated Signal
XL = Lower Single-Sideband Modulated Signal
XU = Upper Single-Sideband Modulated Signal
Z(t) = m(t) + jm(t) = The Analytic Signal of m(t)
a = Modulation (as defined in the text)
S= System Efficiency
6 = Modulation Index
n = Efficiency
e(t) = Modulating Signal (when m(t) is not the Modulating Signal)
o0 = Variance
om Average Power of m(t)
W = Angular Frequency
wrms = RMS-Type Bandwidth
AO = Eouivalent-Noise Bandwidth
S = Mean-Type Bandwidth
* = The Convolution Operator
(.)* = The Conjugate of (*)
(*) = The Hilbert Transform of (*)
(*) = The Averaging Operator
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
SYNTHESIS AND ANALYSIS OF REAL
SINGLE-SIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS
By
Leon Worthington Couch, II
June, 1968
Chairman: Professor T. S. George
Major Department: Electrical Engineering
A new approach to single-sideband (SSB) signal design and ana-
lysis for communications systems is developed. It is shown that SSB
signals may be synthesized by use of the conjugate functions of any
entire function where the arguments are the real modulating signal and
its Hilbert transform. Entire functions are displayed which give the
SSB amplitude-modulated (SSB-AM), SSB frequency-modulated (SSB-FM),
SSB envelope-detectable, and SSB square-law detectable signals. Both
upper and lower SSB signals are obtained by a simple sign change.
This entire generating function concept, along with analytic
signal theory, is used to obtain generalized formulae for the properties
of SSB signals, Formulae are obtained for (1) equivalent realizations
for a given SSB signal, (2) the condition for a suppressed-carrier SSB
signal, (3) autocorrelation function, (4) bandwidth (using various-de-
finitions), (5) efficiency of the SSB signal, and (6) peak-to-average
power ratio. The amplitude of the discrete carrier term is found to be
xiii
equal to the absolute value of the entire generating function evaluated
at the origin provided the modulating signal is AC coupled. Examples
of the use of these formulae are displayed where these properties are
evaluated for stochastic modulation.
The usefulness of a SSB signal depends not only on the pro-
perties of the signal but on the properties of the overall system as well.
Consequently, a comparison of AM, SSB-AM, SSB-FM, and FM systems is
made from the overall viewpoint of generation, transmission with additive
Gaussian noise, and detection. Three figures of merit are used in these
comparisons: (1) Output signal-to-noise ratios, (2) Energy-per-bit of
information, and (3) System efficiency.
In summary, the entire generating function concept is a new tool
for synthesis and analysis of single-sideband signals.
xiv
CHAPTER I
INTRODUCTION
In recent years the use of single-sideband modulation has become
more and more popular in communication systems. This is due to certain
advantages such as conservation of the frequency spectrum and larger post-
detection signal-to-noise ratios in suppressed carrier single-sideband
systems when comparison is made in terms of total transmitted power.
A single-sideband communication system is a system which generates
a real signal waveform from a real modulating signal such that the Fourier
transform, or voltage spectrum, of the generated signal is one-sided about
the carrier frequency of the transmitter. In conventional amplitude-modu-
lated systems the relationship between the real modulating waveform and
the real transmitted signal is given by the well-known formula:
XAM(t) = Ao [1 + m(t)] cos Wot m(t)|j 1 (1.1)
where Ao is the amplitude constant of the transmitter
m(t) is the modulating (real) waveform
Wu is the carrier frequency of the transmitter.
Likewise, frequency-modulated systems generate the transmitted waveform:
t
XFM(t) = Ao cos [Wot + D f m(t')dt'] (12)
where A0 is the amplitude constant of the transmitter
m(t) is the modulating (real) waveform
w0 is the frequency of the transmitter
D is the transducer constant of the modulator.
Now, what is the corresponding relationship for a single-sideband system?
Oswald, and Kuo and Freeny have given the relationship:
XSSB-AM(t) = Ao [m(t) cos "ot m(t) sin owt] (1.3)
where A0 is the amplitude constant of the transmitter
m(t) is the modulating signal
m(t) is the Hilbert transform of the modulating signal
(o is the frequency of the transmitter [1, 2].
This equation represents the conventional upper single-sideband suppressed-
carrier signal, which is now known as a single-sideband amplitude-modulated
suppressed-carrier signal (SSB-AM-SC). It will be shown here that this is
only one of an infinitely denumerable set of single-sideband signals. In-
deed, it will be shown that any member of the set can be represented by
XSSB(t) = Ao [U(m(t), m(t))cos wot T V(m(t), m(t)) sin wot] (1.4)
where Ao is the amplitude constant of the transmitter
U(x,y) and V(x,y) are the conjugate functions of any entire
function
m(t) is the modulating (real) waveform
m(t) is the Hilbert transform of m(t)
Wo is the transmitter frequency.
Various properties of these single-sideband signals will be analyzed in
3
general for the whole set, and some outstanding members of the set will
be chosen for examples to be examined in detail.
It should be noted that Bedrosian has classified various types of
modulation in a similar manner; however, he does not give a general repre-
sentation for single-sideband signals [3].
CHAPTER II
MATHEMATICAL PRELIMINARIES
Some properties of the Hilbert transform and the corresponding
analytic signal will be examined in this chapter. None of the material
presented in this chapter is new; in fact, it is essentially the same
as that given by Papoulis except for some changes in notation [4]. How-
ever, this background material will be very helpful in derivations pre-
sented in Chapter III and Chapter V
The Hilbert transform of m(t) is given by
1. m(x)dx 1
m(t) =- P m(t) -- (2,1)
St-x Ttt
where (*) is read "the Hilbert transforms of (*)"
P denotes the Cauchy principal value
indicates the convolution operation.
The inverse Hilbert transform is also defined by Ea (2.1) except that a
minus sign is placed in front of the right-hand side of the equation. It
is noted that these definitions differ from those used by the mathema-
ticians by a trivial minus sign. It can be shown, for example, that the
Hilbert transform of cos wot is sin mot when 0o > 0 and that the Hilbert
transform of a constant is zero. A list of Hilbert transforms has been
compiled and published under work done at the California Institute of Tech-
nology on the Bateman Manuscript Project [5].
The Fourier transform of m(t) is given by
Fm(w) = [-j sgn (w)] Fm(a) (2.2)
where
+ 1 W > 0
sgn (w) = 0 0 = (2.3)
1 W < 0
and Fm(w) is the Fourier transform of m(t). In other words, the Hilbert
transform operation is identical to that performed by a -900 all-pass
linear (ideally non-realizable) network.
From Eq. (2.2), it follows that
F^(w) = [-j sgn (w)]2 Fm(w) = -Fm(W) (2.4)
or
M(t) = -m(t). (2.5)
The (complex) analytic signal associated with the real signal
m(t) is defined by
Z(t) = m(t) + jm(t). (2.6)
The Fourier transform of Z(t) follows by the use of Ea. (2.2),
and it is
FZ(w) = Fm(w) + j[-j sgn (w)] Fm(w)
or
2Fm(w) >
FZ(w)= Fm(w) W = 0 (2.7)
L 0 W < j
Now suppose that m(t) is a stationary random process with auto-
correlation Rmm(t) and power spectrum Pmm(w)o Then the power spectrum
of m(t) is
Pim(w) = Pmm(w) 1-J sgn () = Pmm(w). (2.8)
This is readily seen by use of the transfer function of the Hilbert trans-
form operator given by Eq. (2.2). Then, by taking the inverse Fourier
transform of Eq. (2.8), it follows that
Rmf(-) = Rmm(r). (2.9)
The cross-correlation function is obtained as follows:
Rim(T) = m(t + T)m(t)
S-m(t + T A)m(t)dx
where () is the averaging operator. Thus
where (.) is the averaging operator. Thus
Rmm(i) = Rmm().
(2.10)
It follows that the spectrum of the cross-correlation function is given by
fmmif) [-j sgn (w)] Pmalnw)
(2,11)
it. is i..'fd that Pmm(() is a purely imaginary function since Pmm(w) is a
real function. Then
1
Rmm(L ) = I
[-j sgn (,)] Pmm(w) ejwT
which, for Pmm(w) a real even function, reduces to
" '0
Pinfri L ) sin t d.
Thus the cross-correlation function is an odd function of L:
Rnm(i) = -Rmm(-)
= -m(t [)m(t) = -m(t + 1)mr(t)
or
Rmm(' ) :. -Rm(-') = -Rmm(i)o
(2.12)
(2.13)
(2.14)
The autocui elation for the analytic signal is found as follows:
RZZ(i) = Z(tt+)Z*(t)
= [m(t+) + jm(t+r)] [m(t) jm(t)]
m(tr,)m(t) + m(t+T)m(t) + jm(t+. )m(t) jm(t+r)m(t)
= Rmm(') + RWm(T) + jRim(T) jRmm(i).
Using Eqs. (2.9), (2.10) and (2.14) we obtain
RZZ(T) = 2[Rmm(i) + jRm(t)] = 2[Rmmn() + 3Rmm(s)]. (2.15)
Thus (1/2)Rzz(T) is an analytic signal associated with Rmm(v). By use of
Eq. (2.7) it follows that
"4Pmm() W 0
PZZ(W) = 2Pmm(w) a 0. (2 16)
L 0 < O_
CHAPTER I ;
SYNTHESIS OF SINGLE: -SiUDEAIND GN'LS
Eq. (1.4), which specifies thc set of single-sideband signals
that can be generated from a given modulating waveform or process, will
be derived in this chapter. The equation must be a real function of a
real input waveform, m(t), since it represents the generating function
of a physically realizable system--the single-1sideband transmitter--and,
in general, it is non-linear. Analytic signal techniques will be used
in the derivation. It will be shown that if we have a complex function
k(z) where k(z) is analytic for z = x + jy in the upper half-plane (UHP),
then the voltage spectrum of k(x,O) k(t) is zero for < 0. In order
to synthesize real SSB signals from a real modulating waveform, an UHP
analytic generating function of the complex time real modulating process
must be found regardless of the particular (physically realizable) wave-
form that the process assumes.
Let m(t) be either the real baseband modulating signal or a reaL
function of the baseband modulating signal e(t), Then the amplitude of
the voltage spectrum of m(t) is double sided about the origin, for ex-
ample, as shown by Figure 1.
Fm( i)
Figure 1. Voltage Spectrum of a Typical m(t) Waveform
10
Since m(t) is generated by a physically realizable process, it con-
tains finite power for a finite time interval. This, of course, is equiva-
lent to saying that m(t) is a finite energy signal or, in mathematical
terms, it is a L2 function. By Theorem 91 of Titchmarsh, m(t) is also a
member of the L2 class of functions almost everywhere [6]. Now the complex
signal Z(t) is formed by
Z(t) = m(t) + jm(t). (3,1)
It is recalled that Z(t) is commonly called an analytic signal in the
literature. By Theorem 95 of Titchmarsh there exists an analytic func-
tion (regular for y > 0), Z (z), such that as y 0
Z1(x + jy) Z(t) = m(t) + jm(t) x t
for almost all t and, furthermore, Z(t) is a Ll (-, function [6]
It follows by Theorem 48 of Titchmarsh that the Fourier transform of Z(t)
exists [6].
Theorem I: If k(z) is analytic in the UHP then the
spectrum of k(t,O), denoted by Fk(w), is zero for
all w < 0, assuming that k(t,0) is Fourier trans-
formable,
For a proof of this theorem the reader is referred to Appendix I.
Thus the voltage spectrum of Z(t) is zero for w < 0 by Theorem I
since Z(t) takes on values of the analytic function Z7(z) almost every-
where along the x axis. Furthermore, since Z(t) is an analytic signal--
that is, it is defined by Eq. (3.1)--its voltage spectrum is given by
Eq. (2.7), which is
Fz(,) F j
where Fm(w) is the voltage spectrum of the signal m(t)
Figure 2 for our example used in Figure 1.
,> j F )
This is shown in
Figure 2 Voltage Spectrum of the Analytic Signal Z(t)
Figure 2. Voltage Spectrum of the Analytic Signal Z(t)
Now let a function g(W) be given such that
g(W) = U(ReW,ImW) + jV(ReW,ImW)
(3.3)
where g(W) is an entire function of the complex variable W.
Theorem. II: If Z(z) is an analytic function of z in
the UHP and if g(W) is an entire function of W, then
g[Z(z)] is an analytic function of z in the UH z-plane,
A proof of this theorem may be found in Appendix I.
Thus g[Z1(z)] is an analytic function of z in the UH z-plane, and
by Theorem I g[Z(t)] has a single-sided spectrum: the spectrum being
zero for w < 0. This is illustrated by Figure 3, where Fg(w) = F[g(Z(t))].
r1
(3 2)
SFga ()
Figure 3. Voltage Spectrum of an Entire Function
of an Analytic Signal
Now multiply the complex baseband signal g[Z(t)] by eJ"ut to
translate the signal up to the transmitting frequency, o,. It is noted
that g[Z,(z)] and ejWoz for mo > 0 are both analytic functions in the UH
z-plane, By the Lemma to Theorem I in Appendix I, g[Z,(z)]eJawz is ana-
lytic in the UH z-plane. Therefore, by Theorem I the voltage spectrum
of g[Z(t)]e JOt is one sided about the origin. Furthermore,
F[g(Z(t))eJoWt 1 TeF[g(Z(t))] F[eJ t]
= Fg(w) 6(w-Lo)
or
F[g(Z(t))ej ot] Fg(w-wo) 'o 0 (3.4)
This spectrum is illustrated in Figure 4.
Figure 4. Voltage Spectrum of the Positive Frequency-
Shifted Entire Function of the Analytic Signal
The real upper single-sideband signal can now be obtained from
the complex single-sideband signal, g[Z(t)]eJot, by taking the real
part, This is seen from Theorem III,
Theorem III. If h(z) is analytic for all z in the
UHP and F[h(x,O)] = Fh(w), then for wc > 0,
Fh(w-w)o a wo
F{Re[h(x,0)eJwox]} = 0 i W < mo
-Fi(46-w.) w s-w
This theorem is proved in Appendix I.
Thus the upper single-sideband signal for a given entire function is
XUSSB(t) = Re{g[Z(t)l]ejot}
= Re{[U(ReZ(t),ImZ(t)) + jV(ReZ(t),ImZ(t))]ej 't}
= Re{[U(m(t),m(t)) + jV(m(t),m(t))]ejot}
)ejwut i i
I XUSSB(t) = U(m(t),m(t)) cos t- V(m(t),m(t)) sin wmt I
where U(ReW,ImW) is the real part of the entire function g(W)
V(ReW,ImW) is the imaginary part of g(W)
m(t) is either the modulating signal or a real function of the
modulating signal e(t)
m(t) is the Hilbert transform of m(t),
Using Theorem III the voltage spectrum of XUSSB(t) is
FUX(w) = F[XUSSB(t)] =
Fg(- )
0
F*(-w-0)
g
, O -< coo
, ii < O 0
< --
(3.6)
This spectrum is illustrated by Figure 5.
IFUX( )i
-WO 0
Figure 5. Voltage Spectrum of the Synthesized
Upper Single-Sideband Signal
The lower single-sideband signal can be synthesized in a similar
manner from the complex baseband signal. Now we need to translate the
complex baseband signal down to the transmitting frequency instead of up,
(3.5)
15
as in the upper single-sideband synrthc:.. Then the Fourier transform of
the down-shifted complex baseband signal is
F[g(Z(t))e-Lot] [g ]
2;r
F[e-Jut]
Fg(v) I(ulJo)
F[g(Z(t))e-Jmot] Fg(w+u4)
, WU
(3,7)
This spectrum is illustrated in Figure 6.
IF[g(Z(t))e-jot]
Figure 6.
Shifted
W0 W4
Voltage Spectrum of the Negative Frequency-
Entire Function of an Analytic Signal
Theorem IV: If h(z) is analytic for all z in UHP and
F[h(x,O)] = Fh(w) where Fh(Q) = 0 for all > wo, then
for wo > 0
F{Re[h(x,O)e-JmWx]
0
Fh(w+wo)
, O < WO
, i > Wo
,0 > W ~-W
This theorem is proved in Appendix I.
Thus the real lower single-sideband signal for a given entire
function is
XLSSB(t) = Reg(Z(t))e-j 't
= Re{[U(m(t),m(t)) t jV(m(t),fi(t))]e-j o t
XLSSB(t) = U(m(t),m(t)) cos wot + V(m(t),m(t)) sin wt.
Using Theorem IV the voltage spectrum of XLSSB(t) is
(3.8)
FLX(M) = F[XLSSB(t)] =
F*(-wtwo)
0
Fg(W+Wo)
, 0 < < WO
, Ki > Wo
, O m u -W
It is noted that the requirements that Fg(w) be zero for w > wo is to
prevent spectral overlap at the origin. This requirement is satisfied
(for all practical purposes) for wo at radio frequencies.
The spectrum of FLX(M) is illustrated by Figure 7.
FLX(w)
(3.9)
-WO I O w
Figure 7. Voltage Spectrum of the Synthesized
Lower Single-Sideband Signal
To summarize, it has been shown that once an entire function 9gW)
is chosen, then an upper or lower single-sideband signal can be obtained
from the signal m(t). The signal m(t) is either the modulating signal
or a real function of the modulating signal b(t). The generalized ex-
pressions, which represent SSB signals, are given by Eq, (3 5) for the
USSB signal and by Eq. (3.8) for the LSSB signal. These expressions are
obviously the transfer functions that are implemented by the upper and
lower single-sideband transmitters respectively. Since there are dn in-
finitely denumerable number of entire functions, there are an inrinuiely
denumerable number of upper and lower single-sideband signals that can be
generated from any one modulation process, In Chapter IV some specific
entire functions will be chosen to illustrate some well-known single-
sideband signals.
CHAPTER IV
EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN
Specific examples of upper single-sideband signal design will now
be presented. Entire functions will be chosen to give signals which have
various distinct properties. In Chapter VI these properties will be ex-
amined in detail. Only upper sideband examples are presented here since
the corresponding lower sideband signals are given by the same equation
except for a sign change (Eq. (3.5) and Eqo (3.8)).
4.1. Example 1: Single-Sideband AM With Suppressed-Carrier
This is the conventional type of single-sideband signal that is
now widely used by the military, telephone companies, and amateur radio
operators. It will be denoted here by SSB-AM-SC.
Let the entire function be
g!(W) = W (4.1)
and let m(t) be the modulating signal. Then substituting the corresponding
analytic signal for W
g (Z(t)) = m(t) + jm(t)
or
UM(m(t),6(t)) = m(t) and Vj(m(t),m(t)) = m(t). (4.2 a,b)
Substituting Eqs. (4.2a) and (4o2b) into Eq, (3.5) we obtain the
upper single-sideband signal:
XUSSB-AM-SC(t) = m(t) cos wot m(t) sin wot (4.3)
where m(t) is the modulating audio or video signal and m(t) is the Hil-
bert transform of m(t). It is assumed that m(t) is AC coupled so that
it will have a zero mean.
The upper single-sideband transmitter corresponding to the gene-
rating function given by Eq. (4.3) is illustrated by the block diagram
in Figure 8. It is recalled that this is the well-known phasing method
for generating SSB-AM-SC signals [7, 8],,
4.2. Example 2: Single-Sideband PM
Single-sideband phase-modulation was described by Bedrosian in
1962 [3].
To synthesize this type of signal, denoted by SSB-PM, use the
entire function:
g,(W) = eJ (44)
Let m(t) be the modulating audio or video signal. Then
g2(Z(t)) = e(m(t) + j(t)) = e-(t) em(t)
or
U2(m(t),m(t)) = e-(t) cos m(t) (4.5a)
m(t) cos
Radio Frequency
Oscillator, wo
XUSSB-AM-SC(t)
+"^ -Modulated RF Output
Phasing Method for Generating USSB-AM-SC Signals
m(t)
Input
Hilbert Transform
{-90 Phase Shift
over Spectrum of
m(t)}
Figure 8.
and
V2(m(t),m(t)) = e-m(t) sin m(t). (4.5b)
Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper
single-sideband signal:
XUSSB-PM(t) = e-(t) cos m(t) cos wot e-m(t) sin m(t) sin wot
or
XUSSB-PM(t) = e-m(t) cos (cot + m(t)). (4.6)
It is again assumed that the modulation m(t) is AC coupled so that its
mean value is zero. The single-sideband exciter described by Eq. (4.6)
is shown in Figure 9.
4.3. Example 3: Single-Sideband FM
Single-sideband frequency-modulation is very similar to SSB-PM
in that they are both angle modulated single-sideband signals. In fact
the equations for SSB-FM are identical to those given in Section 4.2 ex-
cept that
t
m(t) = D f e(t)dt (47)
-00
where e(t) is now the modulating signal (instead of m(t)) and D is the
transducer constant.
Experiments with SSB-FM signals have been conducted by a number
of persons and are reported in the literature [9, 10].
m(t)
Modulating Input
XUSSB-PM(
Modulated RF Output
cos (mot+m(
Figure 9. USSB-PM Signal Exciter--Method I
Phase Modulator
at
Radio Frequency mo
Hilbert Transform
{-90 Phase Shift
over Spectrum of
m(t)}
The SSB-FM exciter as described by Eas. (4.6) and (4.7) is given
in Figure 10.
4.4. Example 4: Single-Sideband a
The term single-sideband a (SSB-a) will be used to denote a sub-
class of single-sideband signals which may be generated from a particular
entire function with a real parameter a. This notation was first used by
Bedrosian [3].
Let the entire function be
g3(W) = e"W (4.8)
where a is a real parameter, and let
m(t) = ln[l + e(t)] (4.9)
where e(t) is the video or audio modulation signal which is amplitude
limited such that le(t)| < 1. It is assumed that m(t) is AC coupled
(that is, it has a zero mean). Note that these assumptions are
usually met by communications systems since they are identical to the
restrictions in conventional AM modulations systems. Then
g3(Z(t)) = e[1m(t)+jm(t)]= e m(t) emja(t)
or
U3(m(t),6(t) = eam(t) cos (am(t)) (4.10a)
Figure 10.
USSB-FM Signal Exciter
V3 (m(t),m(t)) = em(.t) sin (am(t)). (410b)
Substituting Eqs. (4.10a) and (4.10b) into Eq. (3.5), the SSB-a signal is
XUSSB-a(t) = em"(t) cos (am(t)) cos wot
eam(t) sin (am(t)) sin wot
or
XUSSB-a(t) = eam(t) cos (mot + am(t)). (4.11)
In terms of the input audio waveform, Eq. (4.11) becomes
XUSSB(t) = ealn[l+e(t)] cos (wot + aln[l+e(t)])
or
XUSSB_-(t) = [1+e(t)] cos (Wot + aln[l+e(t)]). (4.12)
For a = 1 we have an envelope-detectable SSB signal, as is readily
seen from Ea. (4.12). Voelcker has recently published a paper demon-
strating the merits of the envelope-detectable SSB signal [11]. The real-
ization of Eq. (4.12) is shown in Figure 11.
For a = 1/2 we have a square-law detectable SSB signal. This type
of signal has been studied in detail by Powers [121. Figure 12 gives the
block-diagram realization for the square-law detectable SSB exciter.
e(t)
Modulating Input
DC Level
of +1
XUSSB-o = 1(t)
[1+e(t)]
Phase Modulator
at
Radio Frequency w0
Envelope-Detectable USSB Signal Exciter
Hilbert Transform
{-90 Phase Shift
over Spectrum of
In[1+e(t)]}
cos (w~t+i'[l +e(t) I
Figure 11.
e(t)
Modulating
Input
Positive Square I [l+e(t)]1
Root Circuit
Square-Law Detectable USSB Signal Exciter
Figure 12.
CHAPTER V
ANALYSIS OF SINGLE-SIDEBAND SIGNALS
The generalized SSB signal, that was developed in Chapter III,
will now be analyzed to determine such properties as equivalent gener-
alized SSB signals, presence or absence of a discrete carrier term,
autocorrelation functions, bandwidths, efficiency, and peak-to-averaige
power ratio. Some of these properties will depend only on the entire
function associated with the SSB signal, but most of the properties will
be a function of the statistics of the modulating signal as well.
5.1. Three Additional Equivalent Realizations
Three equivalent ways (in general) for generating an upper SSB
signal will now be found in addition to the realization given by Eq. (3.5).
Similar expressions will also be given for lower SSB signals which are
equivalent to Eq. (3.8). It is very desirable to know as many equivalent
realizations as possible since any ore of them might be the most econom-
ical to implement for particular SSB signal.
Theorem V: If h(x,y) = U(x,y) + jV(x,y) is
analytic in the UHP (including UH-) then
h(t,O) = U(t,O) + j[O(t,O)+k1] (5.1)
or
h(t,O) = [-(t,O)+k2] + jV(t,0) (5.2)
or
h(t,O) = [-V(t,O)+k2] + j[O(t,O)+k1] (5.3)
where
T-
k, = lim f V(R cos e,R sin e)de a real constant (5.4)
R** 0
k2 = lim U(R cos e,R sin e)de a real constant (5.5)
R-- o
A proof of this theorem is given in Appendix I.
Theorem V may be applied to ,the generalized SSB signal by letting
h(z) = g(Z1(z)) where g(.) is an entire function of (.), Z1(z) is analytic
in the UHP, and lim Z (z) = lim Z (t + jy) = m(t) + j6(t). Thus Theorem V
yO y+O
gives three additional equivalent expressions for g(Z(t)) in addition to
g(z(t)) = U(m(t),m(t)) + jV(m(t),M(t)) (5.6)
which was used in the derivation in Chapter III. Therefore, following
the same procedure as in Chapter III, equivalent upper SSB signals may be
found. Using Eq. (5.1) we have for the first equivalent representation
of Eq. (3.5):
XUSSB(t) = Re{g(Z(t))ejwOt}
= Re{g(m(t),m(t))eJwot}
= Re{[U(m(t),m(t) + jU(m(t),m(t)) + jkl]eJWOt}
or
XUSSB(t) = U(m(t),m(t)) cos wot [U(m(t),m(t))+ k,] sin o0t. (5.7)
Using Eq. (5.2) the second equivalent representation is
XUSSB(t) = [-V(m(t),m(t))+k2 cos mot V(m(t),m(t)) sin mot. (5.8)
Using Eq. (5.3) the third equivalent representation is
XUSSB(t) = [-V(m(t),m(t))+k2] cos mot [U(m(t),m(t))+k1],sin wot. (5.9)
Likewise the three lower SSB signals, which are equivalent to
Eq. (3.8), are
XLSSB(t) = U(m(t),m(t)) cos awt + [U(m(t),m(t))+k1] sin mot (5.10)
XLSSB(t) = [-V(m(t),m(t))+k2] cos mot + V(m(t),6(t)) sin wot (5.11)
and
XLSSB(t) = [-9(m(t),m(t))+k2] cos w0t + [U(m(t),i(t))+kj] sin wot.(5.12)
It should be noted, however, that if for a given entire function
k, and k2 are both zero, then all four representations for the USSB or
the LSSB signals are identical since by Theorem V, U = -V and V = U under
these conditions.
5.2. Suppressed-Carrier Signals
The presence of a discrete carrier term appears as impulses in
the (two-sided) spectrum of transmitted signal at frequencies wo and -woo
The impulses may have real, purely imaginary, or complex-valued weights
depending on whether the carrier term is cos wot, sin mot, or a com-
bination of the two. Thus the composite voltage spectrum of the modulated
signal consists of a continuous part due to the modulation plus impulse
functions at w0 and -mo if there is a discrete carrier term. As defined
here, the "continuous" part may contain impulse functions for some types
of modulation, but not at the carrier frequency. Taking the inverse
Fourier transform of the composite voltage spectrum it is seen that if
there is a discrete carrier term, the time waveform must be expressible
in the form:
X(t) = [f1(t)+c1] cos Wot [f2(t)+c2] sin mot (5.13)
where cI and c2 are due to the discrete carrier
f1(t) and f2(t) are due to the continuous part of the spectrum
and have zero mean values.
Thus Eq. (5.13) gives the condition that c2 and cl are not both zero if
there is a discrete carrier term.
To determine the condition for a discrete carrier in an upper
SSB signal, Eq. (5.13) will be identified with Eq. (5o9), which represents
the whole class of upper SSB signals. It is now argued that both U and V
have a zero mean value if the modulating process is stationary. This is
seen as follows:
U(m(t),m(t))= P U(m(t'),m(t')) dt' .
-00
But U[m(t'),m(t')] = c, a constant, since m(t) is wide-sense stationary.
Thus
U(m(t),m(t)) = IP c--dt' = 0.
-00
Likewise V has a zero mean value. Then, identifying Eq. (5.13) with
Eq. (5.9), it is seen that
fi(t) 4 -V(m(t),m(t)) (5.14a)
f2(t) = U(m(t),m(t)) (5.14b)
c, = kg and c2 k (5.14c,d)
Similarily, for lower SSB signals Eq. (5.13) can be identified
with Eq. (5.12).
Thus the SSB signal has a discrete carrier provided that k1 and k2
are not both zero.
As an aside, it is noted that the criterion for a discrete car-
rier, given by Eq. (5.13), is not limited to SSB signals; it holds for
alt modulated signals. For AM Eq. (5.13) can be identified with Eq. (1.1).
Here
f (t) Aom(t) (5.15a)
f2(t) = 0 (5.15b)
c, = Ao and c2 = 0 (5.15c,d)
because m(t) has a zero mean due to AC coupling in the modulator of the
transmitter. Thus for AM it is seen that there is a discrete carrier
term of amplitude c, = A which does not depend on the modulation. For FM
Eq. (1.2) can be expanded. Then, assuming sinusoidal modulation of fre-
quency wa, we obtain
XFM(t) = [Ao cos (- cos wat)] cos ot
wa
[A0 sin (-- cos wat)] sin wot. (5.16)
To identify Eq. (5.16) with Eq. (5.13) we have to find the DC terms of
f (t) + c A cos (- cos mat)
1 0 a
and
f (t) + c2 Ao sin (-cos wat).
These are
C, = A0 COS (L COS Wat)
wa
SAo T cos ( cos at)dt
= A ( D) (5.17a)
wa
and
c2 = A sin ( D cos wat)
Sa
T
= Ao0 sin (-k- cos wat)dt
T 0 wa
= 0 (5.17b)
Then for sinusoidal frequency-modulation it is seen that the discrete
carrier term has an amplitude of AoJo(D/ta) which may or may not be zero
depending on the modulation index D/wa. Consequently, for FM it is seen
that the discrete carrier term may or may not exist depending on the
modulation. Prof. T. S. George has given the discrete carrier condition
for the case of FM Gaussian noise [13].
Continuing with our SSB signals, it will now be shown that k, and
k2 depend only on the entire function associated with the SSB signal and
not on the modulation. From Theorem IV we have
k = I lim
R-*oo
k2 = lim
SRT
f V[ml(R cos e,R sin e) m (R cos e,R sin e)]de
0
U[m1(R coS e,R sin e) m(R cos e,R sin e)]de
0
where U and V are the real and imaginary parts of the entire function
Z1(z) = m1(z) + jm1(z) is the analytic function associated
with the analytic signal Z(t) of m(t).
It is seen that if
(5.18a)
lim m (R cos e,R sin e) = 0 0 e s
R+o1
lim m,(R cos e,R sin e) = 0
R-+o
(5.18b)
, 0 O
then k and k2 depend only on U and V of the entire function and not on
m. Thus we need to show that Ea. (5.18a) and (5.18b) are valid. By the
theory of Chapter III there exists a function Z1(z) = m (z) + j0(z)
which is analytic in the UHP such that (almost everywhere) lim Z1(t + jy)
y*
= Z(t) = m(t) + jm(t) where the Fourier transform of Z(t), F(o), is
L.2(-o, o). Then we have
F(w)ejzwdw.
It follows that
lim IZ (Rej')I2
R+.-
= lim (-)2
R -*O "
[F(w)][e-(R sin O)wej(R cos o)]dJ2
By use of Schwarz's inequality this becomes
lim 1Z1(Rej3) |2
R-o
12
f
IF(w)12d } {lim
R-xo
e-2(R sin O)adw,}
But F(M) e L ("', -) so that
f
0
IF(w) 12dw K.
Vim
R ~
e-(2R sin e)wdw = 0
e di=
, 0 < 0 < T.
Therefore we have
lim jZ(ReJ )I (-)2 K 0 = 0
Also
Z, (z) =
, 0 < e < n.
For e = 0 or e =
2(+) e = 0
lim Z I(ReJ)I = 0
R-o
since
Z(t) e L2(-, 2).
Then
lim jZi(ReJe)I = lim IZ (R cos e,R sin O)| = 0 0 s e s
R-co R-
which shows that Eqs. (5.18a) and (5.18b) are valid statements. Thus,
the presence (kI and k2 not both zero) or the absence (k, = k = 0) of
a discrete carrier depends only on the entire function associated with
the SSB signal and not on the modulation. Furthermore, it is seen that
the amplitude of the discrete carrier is given by the magnitude of the
entire function evaluated at the origin (of the W plane), and the power
in the discrete carrier is one-half the square of the magnitude.
For every generalized USSB signal represented by Eq. (3.5),
there exists a corresponding suppressed-carrier USSB signal:
XUSSB-SC(t) = '-(m(t),(t)) cos Wot *(m(t),m(t)) sin Wot
(5.19)
where the notation SC and denote the suppressed-carrier functions.
But what are these functions U and W? The condition for a suppressed
carrier is that ki = k2 = 0. By comparing Eq. (5.19) with Eq. (5.9) it
follows that th -V and V Furthermore by Theorem V of Section 5.1,
U = -V + kg and V = U + ki, Thus
U -V = U k2 (5,20)
and
V- U = V ki. (5,21)
It is also noted that 4 and + are a unique Hilbert transform pair. That
is, V- is the Hilbert transform of i, and U- is the inverse Hilbert trans-
form of -. This is readily shown by taking the Hilbert transform of
Eq. (5.20), comparing the result with Eq. (5.21), and by taking the in-
verse Hilbert transform of Eq. (5.21), comparing the result with Eq. (5.20).
Thus Eq. (5,19) may be re-written as
XUSSB-SC(t) = tt(m(t),'(t)) cos wot -(m(t),m(t)) sin o0t (5.22)
or
XUSSB-SC(t) = -V(m(t),m(t)) cos wot V(m(t),m(t)) sin wot (5.23)
where U and V-are given by Eq. (5.20) and Eq. (5.21).
It is interesting to note that the form of the USSB signal given
above checks with the expression given by Haber [14]. He indicates that
if a process n(t) has spectral components only for I)w > wo then n(t)
can be represented by
n(t) = s(t) cos wot 9(t) sin mot. (5.24)
Thus Eq. (5.22) checks with Eq. (5.24) where U = s(t), and Eq. (5.23)
checks also where -V E s(t).
The corresponding representations for LSSB suppressed-carrier
signals are given by
XLSSB-SC(t) = tJ(m(t),m(t)) cos wot + 4(m(t),m{t)) sin mot (5.25)
and
XLSSB-SC(t) = --(m(t),m(t)) cos wot + *(m(t),m(t)) sin mot (5.26)
where 4- and V are given by Eq. (5.20) and Eq. (5.21).
This representation also checks with that given by Haber for pro-
cesses with spectral components only for Iwl < wo which is
n(t) = s(t) cos wot + s(t) sin mot. (5o27)
5.3. Autocorrelation Functions
The autocorrelation function for the generalized SSB signal and
the corresponding suppressed-carrier SSB signal will now be derived.
Using the result of Chapter III, it is known that the generalized
upper SSB signal can be represented by
XUSSB(t) = Re{g(m(t),m(t))ej(ot+)} (5.28)
where a uniformly distributed phase angle > has been included to account
for the random start-up phase of the RF oscillator in the SSB exciter.
Then, using Middleton's result [15], the autocorrelation of the USSB sig-
nal is
RXU(t) = XUSSB(t+T)XUSSB(t) = Re{eJwoTRg(-)}
(5.29)
where
Rg(T) = g(M(t+T)mi(t+T))g*(m(t),rm(t))
(5.30)
and
g(m(t),m(t)) = U(m(t),m(t)) + jV(m(t),m(t)).
(5.31)
The subscript XU indicates the USSB signal. For the generalized LSSB
signal the corresponding formulae are
XLSSB(t) = Re{g(m(t),rn(t))eJj(bOt+f) }
and
IRXL(T) = Re{e-JWOTRg(T)}.
These equations can be simplified if we consider the autocorre-
lation for the continuous part of the spectrum of the SSB signal. The
suppressed DC carrier version of g, denoted by gSC, will first be found
in terms of g, and then the corresponding autocorrelation function Rg-sc()
(5.32)
(5.33)
will be determined in terms of Rg(T).
By examining Eqo (5o19) and comparing this equation to Eq. (35),
with the aid of Eq. (3.3) it is seen that the suppressed DC carrier
version of g is given by
gsc(m(t),m(t)) = f(m(t),m(t)) + jW(m(t),m(t)) (5.34)
where 4 and V are the suppressed-carrier functions defined by Eq. (5.20)
and Eq. (5.21). Then it follows that
g(m(t),m(t)) = gsc(m(t),m(t)) + [k2+jk,]o (5.35)
It is noted that the mean value of gSC is zero. This is readily seen via
Eqs. (5.34), (5.20), and (5.21) since it is recalled that the mean value
of U and V was shown to be zero in Section 5,2. Then, using Eq. (5.35),
the autocorrelation of g is obtained in terms of the autocorrelation of
9SC:
Rg(T) = Rg-SC(T) + (k 2+k ). (5.36)
Therefore the autocorrelation functions for the USSB signal,
Eq. (5.29), and the LSSB signal, Eq. (5.33), become
RXU(T) = Re{eJwo( 22+k22) + Rg-SC(r)]} (5o37)
and
RXL(') = Re{&WoT[ k2+k22) + Rg-SC(T)]}
(5.38)
It may be easier to calculate the autocorrelation for the USSB or
LSSB signal using this representation rather than that of Eq. (5.29) and
Eq. (5,33) since RgSC(T) may be easier to calculate than Rg(r). This
is shown below.
A simplified expression for Rg-SC(T) will now be derived. First,
it is recalled from Section 5.2 that and V- are a unique Hilbert trans-
form pair. Thus gSC, given by Eq. (5.34), can be expressed in terms of
two analytic signals:
gsc(m(t),m(t)) = t(m(t),m(t)) + j4(m(t),m(t)) (5.39)
and
gSC(m(t),m(t)) = -V(m(t),m(t)) + j-(m(t),m(t)) (5.40)
where Eq. (5.39) is the analytic signal associated with and Eq. (5.40)
is the analytic signal associated with -V-. Using Eq. (5.39) and Eq. (2.15),
the autocorrelation of gSC is given by
Rg-SC(T) = 2[R (T) + JR4.(T)] (5.41)
or by using Eqs. (5.40), (2.15),and (2.9)
Rg-SC(T) = 2[R.(T) + jR (T)]o (5.42)
Thus Rg-SC(T) may be easier to calculate than R (T) since only Ri,(T) or
R .(T) is needed. This, of course, is assuming that the Hilbert trans-
form is relatively easy to obtain. On the other hand Rg(T) may be calcu-
lated directly from g(m(t),m(t)) or indirectly by use of RUU.(T), Rvv(),
RUV(T), and RVU(T).
The autocorrelation functions for the generalized USSB and LSSB
signals having a suppressed-carrier are readily given by Eq. (5.37) and
Eq. (5.38) with k, = k2 = 0:
RXU-SC(i) = Re{eJmwTRg-SC(T)}
= CR. o) cos W"o Rtt() Sin wfo
= RV(7T) cos wo' RW.(T) sin moT
RXL-SC(t) = -Re{ eeoRg-SC()}
= R 1.(,) COs WOT + RiJ.(T) sin wor
= R ,(T) cos W0T + Rv.(T) sin wor.
(5.43a)
(5.43b)
(5.43c)
(5.44a)
(5.44b)
(5.44c)
It follows that the power spectral density of any of these SSB
signals may be obtained by taking the Fourier transform of the appro-
priate autocorrelation function presented above.
5.4. Bandwidth Considerations
The suppressed-carrier autocorrelation formulae developed above
will now be used to calculate bandwidths of SSB signals. It is noted
that the suppressed-carrier formulae are needed instead of the "total sig-
nal" formulae since, from the engineering point of view, the presence or
absence of a discrete carrier should not change the bandwidth of the sig-
nal. Various definitions of bandwidth will be used [16, 17]1
5o4-1. Mean-type bandwidth
Since the spectrum of a SSB signal is one-sided about the carrier
frequency, the average frequency as measured from the carrier frequency
is a measure of the bandwidth of the signal:
f WPg.SC(w)d Rg-Sc(O)
0 = ------- (5.45)
f Pg-SC(w)dw Rg-SC(O)
where Pg_SC(w) is the power spectral density of gSC(m(t),m(t))and the
prime indicates the derivative with respect to r. The relationship is
valid whenever R'_SC (0) and Rgsc(O) exist. Substituting Eg. (5.41)
into Eq. (5.45) we have
2[R (O) + jR (O)]
I 2[R*(0) + jRi(0O)]
But it recalled that Rju.(T) is an even function of T and, from Chapter II,
RUS.(T) is an odd function of T. Then R%,(0) = R(O) = 0 and it follows
that
R44u(O) R(0)
P"Ry(0) Rft(O)
(5,46)
I
It is noted that this formula is applicable whenever Ru.(0) and R. (O) or
R.(0O) and R,,(0) exist. That is, R,44(0), R 4(0), R (O), and R^_(0) may
or may not exist since Rg-SC(T) is analytic almost everywhere (Theorem 103
of Titchmarsh [6]).
5o4-2, RMS-type bandwidth
The rms bandwidth, wrms, may also be obtained.
CO
2 2 PgSC()d -Rg-SC(O)
(wrms)2 2 -0o
f Pg-SC(w)dw Rg-SC(O)
-Substituting Eq (5.41) once again, we have00
Substituting Eq. (5.41) once again, we have
(rms)2
(5.47)
-2[R~(0) + jR (0)]
2[R i (0) + JRi (O)]
Since RUU(T) is an odd function of T, R (O) = u(0) = 0, and we have
2 -R (0)
R-BB(0)
-RW(0)
R4.(0)
(5.48)
It is noted that this formula is applicable whenever R 1(0) and Ry,.(0)
or R,4(0) and RW (0) exist.
5,4-3. Equivalent-noise bandwidth
The equivalent-noise bandwidth, Aw, for the continuous part of
the power spectrum is defined by
(2Aw) Pg-SC(O)] 2= Pg-SC(w)d = Rg-SC(O)
-00
(5.49)
But
PgSC(O) = f Rg-SC(t)dT
COO
Thus
(AoW) =
1
Rg-SC(O)
f Rg-SC(r)dT
-00
Substituting for Rg-SC(T) by using Eq. (5.41) or Eq. (5.42) we
obtain (noting once again that R (T) is even and R (r) is odd)
TT i7
(Aw) (5.50)
1 f- R,,(T)dT R1 f Rv(T)dr
R(0) Rw() )
5.5. Efficiency
A commonly.Used definition of efficiency for modulated signals
is [18]
n = Sideband Power/Total Power.
I -
(5o51)
This definition will be used to obtain a formula expressing the efficiency
for the generalized SSB signal. Using Eq. (5.43) and Eq. (5.44) the side-
band power in either the USSB or LSSB signal is
RXU-SC(O) = RXL-SC(O) = RWo(O) = RW(O) (552)
It is also noted that Rg-SC(O) is not equal to the total power in the
real-signal sidebands since gSC is a complex (analytic) baseband signal;
instead, (l/2)Re[Rg_sc(O)] : Rwu(O) = Rwv(O) is the total real-signal
power. This is readily seen from Eq. (5.43a) and Eq. (5.44a).
Similarily the total power in either the USSB or LSSB signal is
obtained from Eq. (5.37) or Eq. (5.38):
Rxu(O) = RXL(O) = [kl2 + k2 + 2R4(0)]
= 1[k 2 + k22 + 2R (O)] (5.53)
Thus the efficiency of a SSB signal is
2R 4U(0) 2RV(0)
S= (5.54)
k 2 + k2 + 2R (0) k2 + k2 + 2R4.(0)
1 2 +12R2
5.6. Peak-to-Average Power Ratio
The ratio of the peak-average (over one cycle of the carrier-
frequency) to the average power for the SSB signal may also be obtained.
The expression for the peak-average power over one carrier-
frequency cycle of a SSB signal is easily obtained with the aid of
Eq. (3.5) and Eq. (3.8). Recalling that U and V are relatively slow
time-varying functions compared to cos mot and sin mot, we have for the
peak-average power:
P pv {[U(m(t),m(t))] + [V(m(t),m(t))]l}
pt i tpeak
(5.55)
where tpeak is the value of t which gives the maximum value for Eq. (5.55).
Using Eq. (5.20) and Eq. (5.21), Pp-Av can also be written as
Pp-Av -= {[ + k 2 + [+ k i]2 t = tpeak
2 2
= {[U-+ k2] + [*-+ kj] t tpeak
peak
= {[-V-+ k21 + [2 + kI t
peak .
The average power of the SSB signal was given previously by Eq, (5.53).
Thus the expression for peak-to-average power ratio for the generalized
SSB signal is
Pp-Av {[U(m(t),m(t))] + [V(m(t),m(t))] It = t (556a)
-- peak
PAv k 2 + k 2 + 2R (0)
2 o6 (2
{[U(m(t)m(t))] + [V(m(t),m(t))] t = tpeak (556b)
k2 + k2 + 2Rv (0)
{[U(m(t),m(t))+k2]2 + [4(m(t),m(t))+k,]2} it tpeak (5 56c)
k 2 + k22 + 2R.4(0)
{[-V-(m(t),m(t))+k2]2 + [V-(m(t),m(t))+k]21}
= ____________________t tpeak. (5.56d)
k, 2 + k22 + 2R .(0)
Several equivalent representations have been given for peak-to-average
power since one representation may be easier to use than another for a
particular SS1B signal.
CHAPTER VI
EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS
The examples of SSB signals that were presented in Chapter IV
will now be analyzed using the techniques which were developed in
Chapter V.
6.1. Example 1: Single-Sideband AM With Suppressed Carrier
The constants kI and k2 will first be determined to show that
indeed we have a suppressed carrier SSB signal. By substituting
Eq. (4.2b) into Eq. (5.4) we have
Tr
k = lim m(R cos e,R sin e)de
0
But from Eq. (5.18b) it follows that
lim m1(R cos e,R sin e) = 0 0 < e < .
R+->
Thus
k, = 0 (6.1)
Similarily substituting Eq. (4.2a) into Eq. (5.5) we have
k = lim P m(R cos e,R sin e)de = 0 (6.2)
2 R-wc J
since lim m(R cos e,R sin e) = 0 for 0 < e < x from Eq. (5.18a). Further-
R-x
more, since both k and k are zero, the equivalent realizations for the
SSB signals, as given by the equations in Section 5.1, reduce identically
to the phasing method of generating SSB-AM-SC signals (which was given
previously in Figure 8).
The autocorrelation for the SSB-AM-SC signal is readily given by
use of Eq. (4.2a) and Eq. (5.20). Thus
U-(m(t),m(t)) = m(t). (6.3)
Then the autocorrelation of the suppressed-carrier USSB-AM signal is
given via Eq. (5.43b), and it is
RXU-SC-SSB-AM() = Rmm () cos Wmo Rmm(T) sin wmo. (6.4)
Likewise, by use of Eq. (5o44b) the autocorrelation for the suppressed-
carrier LSSB-AM signal is
RL-SC-SSB-AM() = Rmm(T) cosw + mm(t) sin wor. (6.5)
From Eq. (6.4) it follows that the spectrum of the USSB-AM-SC
signal is just the positive-frequency spectrum of the modulation shifted
up to a0 and the negative-frequency spectrum of the modulation shifted
down to -wo. That is, there is a one-to-one correspondence between the
spectrum of this SSB signal and that of the modulation. This is due to
the fact that the corresponding entire function for the signal, g(W) = W,
is a linear function of W. Consequently, the bandwidths for this SSB
signal are identical to those for the modulation. This is readily shown
below.
The mean-type bandwidth
is given by use of Eq. (6.3) in
MSSB-AM
where 'm = Rmm(0), the power in
the rms bandwidth is
(when the numerator and denominator exist)
Eq. (5.46):
Rmm(O) Rmm(O)
(6.6)
Rmm(O) m
the modulating signal. By using Eq. (5.48)
M = (0) (6.7)
whenever Rmm(O) and 'm exist. By using Eq. (5.50) the equivalent-noise
bandwidth is
(Am)SSBAM = (6.8)
f- Rmm(r)dT
Thus the bandwidths of the SSB-AM-SC signal are identical to those of the
modulating process m(t).
The efficiency of the SSB-AM-SC signal is obtained by using
Eq. (5.54):
2Rmm(0)
nSC-SSB-AM -
2Rmm(0)
(6.9)
The peak-to-average power ratio for the SSB-AM-SC signal follows
from Eq. (5.56c), and it is
Pp-Av {[m(t)]2 + [m(t)]2}t = tpeak
PAV /SC-SSB-AM 2*m
(6.10)
6.2. Example 2: Single-Sideband PM
The SSB-PM signal has a discrete carrier term. This is shown by
calculating the constants k, and k2. Substituting Eq. (4.5b) into
Eq. (5.4) we have
k I lim e-m,(R cos e,R sin e)sin [m,(R cos e,R sin e)]de.
0
But from Eqs. (5.18a) and (5.18b) lim m1(R cos e,R sin e) = 0 for
R-o
0 _s e < T and lim m (R cos e, R sin e) = 0 for 0 e rr. Thus
R--o
ki = 0. (6.11)
Likewise, substituting Eq. (4,5a) into Eq. (5.5) we have
k = e-0 cos 0 de = 1. (6.12)
IT J
0
Thus the SSB-PM signal has a discrete carrier term since k2 = 1 f 0.
There are equivalent representations for the SSB-PM signal since
k and k2 are not both zero. For example, for the upper sideband signal,
equivalent representations are given by Egs. (5.7) and (5.8). It is
noted that Eq. (5.8) is identical to Eq. (5.9) for the USSB-PM signal
since k, = 0. Thus the two equivalent representations are:
XUSSB-PM(t) e= e(t)cos m(t)] cos Wot e acos m(t)] sin wot (6.13)
and
XUSSB-PM(t) =-(e-(t)sin m(t))+1]cos wot [e-m(t)sin m(t)]sin ot. (6.14)
The USSB-PM exciters corresponding to these equations are shown in Figure
13 and Figure 14. They may be compared to the first realization method
given in Figure 9.
The autocorrelation function for the SSB-PM signal will now be
examined. In Section 6.1 the autocorrelation for the SSB-AM-SC signal
was obtained in terms of the autocorrelation function of the modulation.
This was easy to obtain since 4 = m(t). However, for the SSB-PM case 4
and Vare non-linear functions of the modulation m(t). Consequently, the
density function for the modulation process will be needed in order to
obtain the autocorrelation of the SSB-FM signal in terms of Rmm(T).
To calculate the autocorrelation function for the SSB-PM signal,
first RV_(r) will be obtained in term of Rmm(T). Using kL = 0, Eq.(5.21),
and Eq. (4.5b) we have
V(m(t),m(t)) = V(m(t),m(t)) = e-m(t) sin m(t). (6Jo5)
Then
Rm jm(t) -jm(t) -Jm(t-)- jm(t-r)
R (1) = e ep-m(t-T) e-2j
W e 2, e-2/_
m(t)
Modulating Input
jcos -.t
XUSSB-PM(t)
OutDut
USSB-PM Signal Exciter--Method II
e4(t) cos m(t)
Figure 13.
m(t)
Modulating Input
e-m(t) sin m(t)
(e sin m(t)) Balanced [-(e tsin m(t})+1]cos wot
Modulator
+ -
DC Level
of +1
RF Oscillator
at mro
i sin wot
-90 Phase _
~ shift at wO
XUSSB-PM(t)
Modulated
RF Output
[e-&(t) sin m(t)]sin
Figure 14. USSB-PM Signal Exciter--Method III
or
R(r) ej[xi(tr)+jy(t)+jy )] ej[X2(tT)+jy(tT)]
eJ[x3(t,.)+jy(t,T)] + ej[x4(t,T)+jy(t,T)] (6.16)
where x1(t,r) = m(t) m(t-T.)
x2(t,T) m(t) + m(t-r)
x3(t,t) -m(t) m(t-T) = -x2(t,T)
x4(t,T) E -m(t) + m(t-t) = -x (t,T)
y(t,r) H i(t) + i(t-T).
Now Zet the modulation m(t) be a stationary Gaussian process with zero mean.
Then x1(t,T), X2(t,T), X3(t,T), x11(t,r), and y(t,T) are Gaussian processes
since they are obtained by linear operations on m(t). They are also stat-
ionary and have a zero mean value. It follows that x (t,T), y(t,r);
x2(t,i), y(t,7); x3(t,T), y(t,T); and x4(t,r), y(t,r) are jointly Gaussian
since the probability density of the input and output of a linear system
is jointly Gaussian when the input is Gaussian [15]. For example, to show
that x,(t,T) and y(t,r) are jointly Gaussian, a linear system with inputs
m(t) and m(t-r) can readily be found such that the output is y(t,r). Now
the averaging operation in Eq. (6.16) can be carried out by using the fol-
lowing formula which is derived in Appendix II:
eJ{x(t)+jy(t)} = e-{x +j2Pxy-ay2} (6.17)
where x(t) and y(t) are jointly Gaussian processes with zero mean,
x2 = x2(t)
y2 = 2(t)
uxy = x(t)y(t) .
Thus
2 2---- -7
O' [m(t)-m(t-t)] = 2[am -Rmm(T)]
ax r[m(t)+m(t-r)]2 2[om2+Rmm(T)]
2
OX3
[-m(t)-m(t-T)]= 2[ om2+Rmm( )]
x 2 = [-m(t)+m(t-t)]2= 2[am2-Rmm()]
and
y 2 [m(t)+m(t-1i)] = 2[cm2+Rmm([)] .
From Chapter II
-Rmm() so that
it is recalled that Rm(0) = 0 and Rm(i) = -Rlm() =
the p averages are
= [m(t)-m(t-r)][m(t)+m(rt-T)]
= -2Rmm()
[m(t-(tT( =
Sx3Y
and
X y = -[m(t)-m(t-T)][m(t)+m(t-r)] 2Rmm(T) .
Ix y
I
Ix2 =
Therefore, using Eq. (6.17), Eq. (6.16) becomes
R (T) = e-{2[am -Rmm(T)] + j2[-2Rmm(T)] 2[om2+Rmm(T)]}
e-2{2[m2 +Rmm(i)] + j2.0 2[am2+Rmm(T)]}
.- e-{2[m2+Rmm(,)] + j2 2[a m2+Rmm(T)]}
+ 1 e-{2[am2 -Rmm(T)] + j2[2Rmm(l)] 2[Om2+Rmm(T)]}
which reduces to
RVW-SSB-PM-GN(r) = {e2Rmm(T) cos (2Rmm(r)) 1} (6.18)
where W -SSB-PM-GN denotes the autocorrelation of the imaginary part of
the entire function which is associated with the suppressed-carrier SSB-
PM signal with Gaussian noise modulation.
It is noted that Eq. (6.18) is an even function of i, as it should
be, since it is the autocorrelation of the real function V(m(t),6(t)).
Furthermore R (0) is zero when Rmm(O) = 0, as it should be, since the
power in any suppressed-carrier signal should be zero when the modulating
power is zero.
The autocorrelation of the USSB-PM signal is now readily obtained
for the case of Gaussian noise modulation by substituting Eq. (6.18) into
Eq. (5.42) and using Eq. (5.37):
RXU-SSB-PM-GN(T) = Re eJ0 T{[e2Rmm() cos (2Rmm(())]
+ j[e Rmm T)cos (2Rmm(T)]}] (6.19)
Likewise, the autocorrelation of the LSSB-PM signal may be obtained by
using Eq. (5.38).
The autocorrelation of the suppressed-carrier USSB-PM signal with
Gaussian modulation is given by using Eq. (5.43a):
RXU-SC-SSB-PM-GN(t) = Re [eJ m([e2Rmm(t) cos (2Rmm(r)) 1]
+ j[e2Rmm( cos (2Rmm(r)]} (6.20)
Similarly, the autocorrelation of the suppressed-carrier LSSB-FM signal
may be obtained by using Eq. (5,44a).
The mean-type bandwidth will now be evaluated for the SSB-PM
signal assuming Gaussian noise modulation. From Eq. (6.18) we obtain
001 e2Rmm( ) cos (2Rmm(x)dA
P(tA)2
Then
1 e2Rmm() cos [2Rmm(x)]dx
R V_(O) P 2 (6o21)
and from Eq. (6.18)
R *(O) = [e2%m 1] (6.22)
where m = m2 is the average power of m(t). Substituting Eqs. (6.21)
and (6.22) into Eq,, (5.46) we have the mean-type bandwidth for the
Gaussian noise modulated SSB-PM signal:
17 P e2Rmm(A) cos[2Rmm(A)]dA
(W)SSB-PM-GN (6-23)
e29m -1
where m is the noise power of m(t), It is seen that Eq. (6.23) may or
may not exist depending on the autocorrelation of m(t).
The mis-type bandwidth can be obtained with the help of the second
derivative of Eq. (6.18):
R () {-e2Rmm()
+ {-e2Rmm(r)
+ {-e2Rmm ()
+ { e2Rmm ( )
+ {-e2Rmm(,)
+ { e2Rmm(t)
sin [2Rmm(T)]} 2[Rmm(T)]2
cos [2Rmm(T)]J 2[Rmm(T)l
sin [2Rmm(T)]}
Rmm ()
cos [2Rmm(I)]} 2[Rmm(i)]2
sin [2Rmm(T)]}
cos [2Rmm(T)]}
Imm
Rmm (T)
RW(O) = e2m {Rmm(0) 2[Rmm(0)]2}
Substituting Eq. (6.24) and Eq. (6.22) into Eq. (5.48) we have for the
rms-type bandwidth of the SSB-PM signal with Gaussian noise modulation:
/2(2[Rnm(O)12 Rm(O)}
6(; ?q)
v'rmsJssB-PM-GN 1 e-2m
This expression for the rms bandwidth may or may not exist depending on
the autocorrelation of m(t). It is interesting to note that Mazo and Salz
have obtained a formula for the rms bandwidth in terms of different para-
meters [19]. However both of these formulae give the same numerical re-
sults, as we shall demonstrate by Eqo (6.29).
Thus
(6.24)
2RMM(T)MM(T)
t \1 A
ll ?
The equivalent-noise bandwidth is obtained by substituting
Eq. (6.18) into Eq. (5.50):
(AW) =
1 e2Rmm() cos [2Rmm(T)] 1} dT
[e22m-l].
or
'T(e2m 1)
(Am)SSB-PM-GN = (6.26)
S{e2Rmm(T) cos [2Rmm(T)] l}dT
It is noted that the equivalent-noise bandwidth may exist when the formu-
lae for the other types of bandwidth are not valid because of the non-
existence of derivatives of Rmm(t) at T = 0.
It is obvious that the actual numerical values for the bandwidths
depend on the specific autocorrelation function of the Gaussian noise.
For example, the rms bandwidth of the SSB-PM signal will now be calculated
for the particular case of Gaussian modulation which also has a Gaussian
spectrum. Let
2
-w)
Pm(w) = e 232
where Pm(w) is the spectrum of m(t)
no = m is the total noise power in m(t)
2a is the "variance" of the spectrum.
The autocorrelation corresponding to this spectrum is
Rmm(1) = e 0, (6.27)
The Hilbert transform of Rmm(T) is also needed and is obtained by the
frequency domain approach. It is recalled from Chapter II that
P () =
mm
-j Pmm()
0
j Pmm
S> 0
Then
SRmm(T) ~ = T)
P .1
2i
00
f Pim(w) ejwt dw
0 -,2 0 -W
S e2 ej de J e2-' e2a ejTd]
f f
-0 0
which reduces to
w 2
integral is evaluated by using the
of the Bateman Manuscript Project,
1 [5]:
S 2 -
2V"
o
sin wT dm .
formula obtained from page 73,
Tables of Integral Transforms,
Erf Re a > 0
\2v/a /
Erf (x) e't dt.
O
This
#18,
vol.
where
a < 0
Thus
Rmm J) (- 0 e e-22T2) Erf (-4 2)
Rmm(T) = -j Rmm() Erf or .
\/j2 /
(6.28)
From Eao. (6.27) it follows that
Rmm(O) = -oo2
and from Ea. (6.28) we have
Rim(O) = 2
/27
Substituting these two equations into Ea. (6.25) we get
(Grms) 2, -e-2o--
Thus if m(t) has a Gaussian spectrum and if the modulation has a Gaussian
density function, the SSB-FM signal has the rms bandwidth:
20o2 [4( oo/n) + ]
("rms)SSB-PM-GN = l -2' -- (6.29)
where *o is the total noise power in m(t)
a'is the "variance" in the spectrum of m(t).
This has the same numerical value as that obtained from the result given
by Mazo and Salz [19]. The result may also be compared to that given by
Kahn and Thomas for the SSB-PM signal with sinusoidal modulation [20].
From Eq. (19) of their work
(wrms)ssB-PM-S = wa6 (6.30)
where ma is the frequency of the sinusoidal modulation and 6 is the modu-
lation index. For comparison purposes, equal power will be used for m(t)
in Eq. (6.30) as was used for m(t) in Eq. (6.29). Then Eq. (6,30) becomes
(wrms)SSB-PM-S = /2 wa 'u (6.31)
Thus it is seen that for Gaussian modulation the rms bandwidth is propor-
tional to the power in m(t) when the power is large (%o > > T/4), and for
sinusoidal modulation the rms bandwidth is proportional to the square root
of the power m(t).
The efficiency for the SSB-PM signal with Gaussian modulation
will now be obtained. Substituting En. (6.22) into Eq. (5.54) we have
e2m-_l
tSSB-PM-GN = + (e2m-1l)
or
SSB-PM-N e2m (6.32)
where 1m is the noise power of m(t).
The peak-average to average power ratio for Gaussian m(t) is given
by use of Eas. (4.5a), (4.5b), and (6.22) in Ea. (5.56b):
Pv
VAv
{[e-m(t) cos m(t)] [el(t) sin m(t)]2 t tpeak
1 + (e-2mm-l)
(6,33)
Note that m(t) may take on large negative values because it has a Gaussian
density function (since it was assumed at the outset that the modulation
was Gaussian), However, it is reasoned that for all practical purposes,
m(t) takes on maximum and minimum values of +3am and -3am volts where am
is the standard deviation of m(t). This approximation is useful only for
small values of am since e+2(3am) approximates the peak power only when the
exponential function does not increase too rapidly for larger values of am.
Thus the peak-to-average power ratio for the SSB-PM signal with Gaussian
noise modulation is
-Av
( v )SSB-PM-GN
Se6am e6/im-2 m
e -
when 'm is small.
It is noted that the efficiency and the peak-to-average power
ratio depend on the total power in the Gaussian modulation process and not
on the shape of the modulation spectrum. On the other hand the autocorre-
lation function and bandwidth for the SSB signal depend on the spectral
(6.34)
shape of the modulation as well.
The dependence of bandwidth on the spectrum of the Gaussian noise
modulation will be illustrated by another example. Consider the narrow-
band modulation process:
m(t) = a(t) cos (wat + 4) (6.35)
where a(t) is the (double-sideband) suppressed-subcarrier amplitude
modulation
ma is the frequency of the subcarrier
( is a uniformly distributed independent random phase due to
the subcarrier oscillator.
That is, we are considering a SSB signal which is phase modulated by the
m(t) given above. Then
Rmm(T) = Raa(f) cos wa{ (6.36)
where Raa(T) is the autocorrelation of the subcarrier modulation a(t).
Rmm(T) can be obtained from Eq. (6.36) by use of the product theorem [21].
Thus, assuming that the highest frequency in the power spectrum of a(t)
is less than wa,
Rmm(T) = Raa() sin aT (6.37)
Furthermore let a(t) be a Gaussian process; then m(t) is a
narrow-band Gaussian process. This is readily seen since Eq. (6.35) may
be expanded as follows:
m(t) = [a(t) cos (wat+q) a(t) sin (wat+f)]
+ [a(t) cos (a t+p) + a(t) sin (wat+4)] (6.38)
The terms in the brackets are the USSB and LSSB parts of the suppressed-
subcarrier signal m(t). But these USSB and LSSB parts are recognized
as the well-known representation for a narrow-band Gaussian process.
Thus m(t) is a narrow-band Gaussian process.
Now the previous expressions for bandwidth, which assume that
m(t) is Gaussian, may be used. The mean-type bandwidth for the multi-
plexed SSB-PM signal is then readily given via Eq. (6.23), and it is
00
eRaa(X) cos wax cos[Raa(x) sin wax]dA
-________ ______________ (6.39)
(u)M-SSB-PM-GN ~ e8a 1
where ag is the average power of the Gaussian distributed subcarrier
modulation a(t). Obtained in a similar manner, the rms bandwidth is
Wa2 ( a+1) Raa(0)
(wrms)M-SSB-PM-GN = e*a (6.40)
and the equivalent-noise bandwidth is
f[e2a-1]
(Aw)MSSB-PMN = 0 (6.41)
f eRaa(T) cos waT cos[Raa(r) sin waT]
-00
Thus, it is seen once again that the bandwidth depends on the spectrum of
the modulation, actually the subcarrier modulation a(t).
To obtain a numerical value for the rms bandwidth of the multi-
plexed SSB-PM signal assume that the spectrum of a(t) is flat over
a Wo < w"a
67
Pa()M
0 W W-+
Figure 15. Power Spectrum of a(t)
From Figure 15 we have
-Wo
or
NoWo sin Wo.
Raa(r) (6.42)
and
0a 0 (6.43)
IT
Then
Raa(O) (6.44)
31
Substituting the last two equations into Ea. (6.40) we obtain the rms
bandwidth for the SSB-PM multiplexed signal:
2 NOWO( N0W0 N oWo_
a -- -- +1 + -
(mrms)M-SSB-PM-GN = NoWo/f (6.45)
1 l -e-W/
where ma is the subcarrier frequency
No is the amplitude of the spectrum of the subcarrier Gaussian
noise modulation
Wo is the bandwidth of the subcarrier noise modulation.
Thus the rms bandwidth is proportional to the power in the subcarrier
modulation as No becomes large.
6.3. Example 3: Single-Sideband FM
As was indicated in Section 4.3. the representation for the SSB-FM
signal is very similar to that for the SSB-PM signal. In fact it will be
shown below that all the formulae for the properties of the SSB-PM signal
(which were obtained in the previous section) are directly applicable to
the SSB-FM signal.
The SSB-FM signal has a discrete carrier term since the entire
function for generating the SSB-FM signal is identical to that for the
SSB-PM signal, which has a discrete carrier term.
The other properties of the SSB-FM signal follow directly from
those of the SSB-PM signal if the autocorrelation of m(t) can be obtained
in terms of the spectrum for the frequency modulating signal e(t). It is
recalled from Eq. (4.7) that
t
m(t) = D ef (t')dt', (6.46)
-00
First, the question arises: Is m(t) stationary if e(t) is stationary?
The answer to this question has been given by Rowe; however, it is not
very satisfactory since he says that m(t) may or may not be stationary [22].
However, it will be shown that m(t), as given by Eq. (6.46), is stationary
in the strict sense if e(t) is stationary in the strict sense; and,
furthermore, m(t) is wide-sense stationary if e(t) is wide-sense stationary.
It is recalled that if
y(t) = L[x(t)]
where L is a linear time-invariant operator, then y(t) is strict-sense
stationary if x(t) is strict-sense stationary and that y(t) is wide-sense
stationary if x(t) is wide-sense stationary [4]. Since the integral is a
linear operator, we need to show only that it is time-invariant, that is
to show that
y(t+e) = L[x(t+E]
or
I (tl)dt f e(t2+e)dt2
e(tl)dt1
This is readily seen to be true by making a change in the variable,
letting tI = t2 + e. Thus, if 8(t) is stationary, then m(t) is stationary.
Moreover, in the same way it is seen that if m(t) had been defined by
t
m,(t) = D e,(t')dt' (6.47)
to
then m (t) is not necessarily stationary for e,(t) stationary since the
system is time-varying (i.e. it was turned on at to). But this should
not worry us because, as Middleton points out, aZZll physically realizable
systems have non-stationary outputs since no physical process could
have started out at t = -m and continued without some time variation in
the parameters 05]. However, after the "time-invariant" physical systems
have reached steady-state we may consider them to be stationary processes--
provided there is a steady state. Thus by letting t -m we are con-
sidering the steady-state process m(t) which we have shown to be stationary.
Now the autocorrelation of m(t) can be obtained by using power-spectrum
techniques since m(t) has been shown to be stationary. From Eq. (6.46)
we have
e(t) dm(t) (6.48)
Then in terms of power-spectrum densities
P ee() = W2Pmm(a) (6.49)
As Rowe points out, Pmm(w) must eventually fall off faster than k/w2,
where k is a constant, if e(t) is to contain finite power; and if Pmm(w) =
k/w2, Pee(w) will be flat and, consequently, white noise. Thus we have a
condition for the physical realizability of m(t): Pmm(w) falls off faster
than -6 db/octave at the high end. This condition is satisfied by physi-
cal systems since they do not have infinite frequency response. From
Eq. (6.49) we have
1
Pmm(W) PeO() (6650)
Immediately we see that if POe() takes on a constant value as IwI + 0
and at w = 0, m(t) will contain a large amount of power with spectral
components concentrated about the origin. In other words, m(t) has a
large block of power, located infinitely close to the origin which is
infinitely large. Thus m(t) contains a slowly varying "DC" term with a
period T and m2(t) -- -. By examining Eqo (6.46) we obtain the same
result from the time domain. That is, for Pee(w) equal to a constant,
e(t) contains a finite amount of power located infinitely close to the
origin which appears as a slowly varying finite "DC" term in e(t) such
that T -. Then by Eq. (6.46), m(t) has a infinite amplitude and,
consequently, infinite power, In other words, the system does not have a
steady-state output condition if the input has a power around w = 0. Thus,
this system is actually conditionally stable, the output being bounded
only if the input power spectrum has a slope greater than or equal to +6
db/octave near the origin (and, consequently, zero at the origin) as seen
from Eq. (6.50). It is interesting to note that for the case of FM, eJm(t)
is stationary regardless of the shape of the spectral density Pee(w). This
is due to the fact that ejm(t) is bounded regardless of whether m(t) is
bounded or not.
From Eq. (6.50) we can readily obtain Rmm(r) for any input process
e(t) which has a bounded output process m(t). Thus
00
Rmm()) : 2 eJWT dw (6.51)
-00
Furthermore, R"m(0), Rmm(M), and Rmm(O) may be obtained in terms of
Pee(w). By substituting for these quantities in the equations of Section
6.2, the properties of a SSB-FM signal can be obtained in terms of the
spectrum of the modulating process.
6.4. Example 4: Single-Sideband a
The SSB-a signal has a discrete carrier term. This is readily
shown by calculating the constants k, and k2. Substituting Eq. (4.10b)
into Eq. (5.4) we have
iT
ki = 'I lim em1(R cos eR sin e) sin am,(R cos e,R sin e)de .
SR_00 f
0
But lim m,(R cos e, R sin e) = 0, for 0 s e s 7 and lim m (R cos e,
R--o R4-
R sin e) = 0 for 0 s e To, Thus
k, = 0. (6.52)
Likewise, substituting Eq. (4.10a) into Eq. (5.5), we have
k2 = 1. (6.53)
Thus the SSB-a signal has a discrete carrier term.
It follows that equivalent representations for the SSB-a signal
are possible since k2 0. This is analogous to the discussion on equiva-
lent representations for SSB-PM signals (Section 6.2) so this subject will
not be pursued further.
The autocorrelation function for the SSB-a signal will now be ob-
tained in terms of Rmm(T)o Using Eq. (5.21) and Eq. (4.10b) we have
RW(T) = [eem(t) sin am(t)][eam(t-[) sin am(t-T)]
or
R (,) = k{ea[m(t)+m(t-T)]} eja[m(t)-m(t-T)] -eja[m(t)+m(t-T)]
+ {eam(t)+m(t-)]} {eJ[-m(t)-m(t-)] + ej[-m(t)+(t-T)}.
(6.54)
The density function of m(t) has to be specified in order to carry out
this average. It is recalled that m(t) is related to the modulating
signal e(t) by the equation:
m(t) = In [l+e(t)] o
Now assume that the density function of the modulation is chosen such
that m(t) is a Gaussian random process of all orders. Eq. (6.54) can
then be evaluated by the procedure that was used to evaluate Eq. (6.16).
Assuming a Gaussian m(t), Eq. (6.54) becomes
RW(T)SSB-a-GN = {e2a2Rmm(T) cos [2a2mm(T)] 1} o (6,55)
But this is identical to Eq. (6.18) except for the scale factor a2.
Thus for envelope detectable SSB (a = 1) with m(t) Gaussian, the auto-
correlation and spectral density functions are identical to those for
the SSB-PM signal with Gaussian m(t). Moreover, the properties are
identical for SSB-a and SSB-PM signals having Gaussian m(t) processes
such that (*m)SSB-PM = a2(m)SSB-a_
It is also seen that if |e(t)l < < 1 most of the time then
m(t) : e(t).
Thus, when e(t) is Gaussian with a small variance, m(t) is approximately
Gaussian most of the time. Then Eq. (6.55) becomes
RW(t)SSB-a-GN {e22Ree () cos [2a2Ree(f)] 1} (6.56)
when le(t)l < < 1 most of the time. Consequently, formulae for the auto-
correlation functions analogous to Eqs. (6.19) and (6.20), may be further
simplified to a function of Ree(i) instead of Rmm(t). Then the auto-
correlation functions for USSB-a and LSSB-a signals, assuming Gaussian
modulation e(t) with a small variance, are
RXU-SSB-a-GN(r) Re ejo'T {[e2a2Re(T) cos (2otee(T))]
+ j RLe2a2Ree() cos (2a2Reo())]} (6)57)
and
RXL-SSB-a-GN(,) Re e-jwo([e2a2Ree(T) cos (2c2(ee(T))]
+ j [e22Ree) cos (2W2 ee(r))]} (6.58)
The efficiency is readily obtained by substituting Eq. (6.56)
into Eq. (5.54):
nSSB-a-GN = 1 e"22m (659)
where Pm is the power in the Gaussian m(t) and le(t)| < < 1. This result
may be compared for a = 1 to that given by Voelcker for envelope-detectable
SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t);
and Eq. (6.59) becomes
nSSB-a-GN 1. 1 e-2e2 z 20e2, (6o60)
This agrees with Voelcker's result (his Eq. (38)) when the variance of the
modulation is small--the condition for Eqo (6o60) to be valid.
The expressions for the other properties of the SSB-a signal, such
as bandwidths and peak-to-average power ratio, will not be examined further
here since it was shown above that these properties are the same as those
obtained for the SSB-PM signal when (Wm)SSB-PM '= 2(m)SSB-a as long as
m(t) is Gaussian.
CHAPTER VII
COMPARISON OF SOME SYSTEMS
In the two preceding chapters properties of single-sideband sig-
nals have been studied. However, the choice of a particular modulation
scheme also depends on the properties of the receiver. For example,
the entire function g(W) = W2 can be used to generate a SSB signal, but
there is no easy way to detect this type of signal.
In this chapter a comparison of various types of modulated sig-
nals will be undertaken from the overall system viewpoint (i.e. generation,
transmission and detection). Systems will be compared in terms of the
degradation of the modulating signal which appears at the receiver out-
put when the modulated RF signal plus Gaussian noise is present at the
input. This degradation will be measured in terms of three figures of
merit:
1. The signal-to-noise ratio at the receiver output
2o The signal energy required at the receiver input for
a bit of information at the receiver output when com-
arison is made with the ideal system (Here the ideal
system is defined as a system which requires a minimum
amount of energy to transmit a bit of information as
predicted by Shannon's formula.)
3. The efficiency of the system as defined by the ratio
of the RF power required by an ideal system to the RF
power required by an actual system.(Here the ideal sys-
75
tem is taken to be a system which has optimum trade-off
between predetection signal bandwidth and postdetection
signal-to-noise ratio.)
Comparison of AM, FM, SSB-AM-SC, and SSB-FM systems will be made using
these three figures of merit. It is clear that these comparisons are
known to be valid only for the conditions specified; that is, for the
given modulation density function, and detection schemes which are used
in these comparisons.
7.1. Output Signal-to-Noise Ratios
7.1-1. AM system
Consider the coherent receiver as shown in Figure 16 where the
input AM signal plus narrow-band Gaussian noise is given by
X(t) + ni(t) = {Ao[l+6 sin wmt] cos wot}
+ {xc(t) cos )ot xs(t) sin wmot (7.1)
where X(t) is the input signal, ni(t) is the input noise with a flat spec-
trum over the bandwidth 2wm, and 6 is the modulation index.
AC Couple
X(t)+ni(t) Low Pass ___ Output
2k cos cot
Figure 16. AM Coherent Receiver
Then the output signal-to-noise power ratio, where Aok6 sin wmt is the
output signal, is given by
(S/N)o (S/N)i (7.2)
1 + _26
or
(S/N)o 62(C/N)i, (7.3)
where (S/N)i = The input signal-to-noise power ratio
(C/N)i = The input carrier-to-noise power ratio
and the spectrum of the noise is taken to be flat over the IF bandpass
which is 2wm(rad/s).
7.1-2, SSB-AM-SC system
Consider the coherent receiver (Figure 16) once again, where
the input is a SSB-AM-SC signal plus narrow-band Gaussian noise. Then
the input signal plus noise is
X(t) + ni(t) = {Ao[m(t) cos wot m(t) sin wot]}
+ [xc(t) cos wot Xs(t) sin mot] (7.4)
where
m(t) = 6 sin mt
and xs(t) = xc(t) if the IF passes only upper sideband components. The
input noise is assumed to have a flat spectrum over the bandwidth m.o
Then the output signal-to-noise power ratio, where Ak6 sin wmt is the
output signal, is given by [23]
(S/N)o = (S/N)i (7.5)
where the spectrum of the noise is taken to be flat over the IF bandpass
which is wm(rad/s)o
It is interesting to note that the same result is obtained from a
more complicated receiver as given in Figure 17. However, in some practi-
cal applications the receiver in Figure 17 may give much better perform-
ance due to better lower sideband noise rejection. That is, in Figure 17
the lower sideband noise is eliminated as the result of the approximate
Hilbert transform filter realized about w = 0; whereas, in Figure 16 the
lower sideband noise is rejected by the IF filter realized about w = mo.
Thus, in order to obtain equal lower sideband noise rejection in both
receivers, the IF bandpass for the receiver in Figure 16 would have to
have a very steep db/octave roll-off characteristic at w = wo.
Low Pass
Filter
X(t)+ni(t) + Output
2k cos wot
2k sin mot +
Low Pass Hilbert
Filter Filter
Figure 17. SSB-AM-SC Receiver
7.1-3. SSB-FM system
Now consider a FM receiver which is used to detect a SSB-FM sig-
nal plus narrow-band Gaussian noise as shown in Figure 18.
X(t)+ni(t) FM Output
Receiver
Figure 18. SSB-FM Receiver
The input signal plus noise is given by
X(t) + ni(t) = Aoe-(t) cos [wot + m(t)] + ni(t)
(7.6)
where Ao = The amplitude of carrier
o =s The radian frequency of the carrier
m(t) = D _t v(t) dt
m(t) = m(t) = The Hilbert transform of m(t)
ni(t) = Narrow-band Gaussian noise with power spectral density Fo
over the (one-sided spectral) IF band
and v(t) is the modulation on the upper SSB-FM signal. The independent
narrow-band Gaussian noise process may be represented by
ni(t) = R(t) cos [mot + (t)j = xc(t) cos wot xs(t) sin wot
where xs(t) = xc(t) since the IF passes only the frequencies on the upper
sideband of the carrier frequency.
Then the phase of the detector output is obtained from Eq. (7.6)
and is
p(t) = k tan-1
(7.7)
which reduces to
i(t) = km(t)
+ k tan- K
R(t) sin [p(t) m(t)]
(7,8)
where k is a constant due to the detector. The detector output voltage
is given by dt)- o Eq. (7.8) is identical to the phase output when the
input is conventional FM plus noise except for the factor e -(t)
For large input signal-to-noise ratios (i.e. Aoe-m(t) > > R(t)
most of the time), Eq. (7.8) becomes
kR(t)
p(t) = km(t) + ---- sin [p(t) m(t)] (7.9)
Aoe-m(t)
dno(t)
Then the noise output voltage is -d--- where
kem(t)
n (t) = R(t) sin [<(t) m(t)]. (7.10)
Ao
Now the phase p(t) is uniformly distributed over 0 to 2r since the input
noise is a narrow-band Gaussian process. Then for m(t) deterministic,
[<(t) m(t)] is distributed uniformly also. Furthermore, R(t) has a
Rayleigh density function. Then it follows that R(t) sin [q(t) m(t)]
is Gaussian (at least to the first order density) and, using Rice's
formulation [24, 25],
R(t) sin [p(t) m(t)] = xs(t) = l 2F(n) a sin [(Un- )t + en]
n=l 2i-T
where F(w) = F0 is the input noise spectrum and {en} are independent
random variables uniformly distributed over 0 to 2i. Actually it is
known that the presence of modulation produces some clicks in the out-
put [26], but this effect is not considered here. Eq. (7.10) then be-
comes
kem(t) _____
ke r
no(t) = 2F(wn) -w sin [(Pn-wo)t + On]
Ao n=l 2'
or
dno(t) kem ---
dt Ao nt 1 2F(tn) (wn-wo) cos [(n-mo)t+ On]
dt A0 n f n 2n
ke(t) dm(t) i F(n) Am n]
+ t-- 1 Fn) sin [(n-)t + n
Ao Ldt-] n=1 2Tn
Noting that {on} are independent as well as uniformly distributed and
that the noise spectrum is zero below the carrier frequency, the output
noise power is
dno(t) 2
Wm Wm
k k e2m(t) F + 2 e2m(t) 2dO(t)] id
^-dT2
Ao2 t F 2dw + e 2- Fd
k 2 (t F m3 k2 2m(t) dm(t) F0
2 [ -- 2 m (7.11)
Ao2 2 3 A0 dt 2n
where ,-) is the averaging operator and wm is the baseband bandwidth
(rad/s) o Now let v(t) = -Am cos Wmt then, averaging over t, we have
2e / 2
S2m(tj- wm e26 cos Wmt dt = Io(26) (7.12)
and
e2m(t) ^l(t)2 1 (m6)2 [io(26) 12(26)]
= m 26 I (26) (7.13)
2 m 1
where 6 = DAm/wm, the modulation index
Eq. (7.13) into Eq, (7,11) we obtain for the output noise power
k2Fo 0m3
N 0
0
1 (2
1a'
(7.14)
Referring to Eq. (7.9), the output signal power is
k2 2
= -T (DAm) .
2
dkm(t)
0 dt
Then the output signal-to-noise ratio is
k2
(DAm)2
S Ak2L
2TWn A 02 3
AO2 2
Fo
2--
21T
m I-o(26)
1
+ 61
2
1
+ -
2
i(26)
(7.16)
611(26)
Referring to Eq. (7.6), the signal power into the detector is
Si = Ao2 e2(t) cos2 [.ot + m(t)] = 1 A 2 e2(t)
=A2 e~~~(2
1 2
= Ao
2
Io(26).
Kahn and Thomas have given the ratio of the rms bandwidths (taken about
(7.15)
(S/N), =
(S/N)o =
(7.17)
Substituting Eq. (7.12) and
83
the mean of the one-sided spectrum) for a SSB-FM signal to a conventional
FM signal [20], and it is
BSSB-FM
BFM
(7.18)
I 1 2(26)
102(26)
It is known that the bandwidth (in rad/s) of a
FM signal is approxi-
mately
BFM = 2(6+1)wm.
(7,19)
Thus, to the first approximation, the SSB-FM bandwidth is
BSSB-FM 2
I 2(2 )
2 1 2=
(7.20)
(6+1)w .
m
Then, taking the IF bandwidth to be that of the SSB-FM signal, the input
noise power is
V 0
N B SSB-FM'
(7 21)
Consequently, the input signal-to-noise ratio is
Io(26)
(S/N)i =
(7,22)
mm (6+1) /2
I 2(26)
1- 2)
Io (26)
Fo
4 -
2-u
Using Eq. (7.16) and Eq. (7.22), we have
(7.23)
for the case of SSB-FM plus Gaussian noise into a FM detector.
The signal-to-noise output can also be obtained in terms of the
unmodulated-signal-to-noise ratio (i.e. the carrier-to-noise power at
the input). From Ea, (7.6) we obtain
(S/N)i lo(26) (C/N)i
and Eq. (7.23) becomes
iO2(26)
6 62(6+1) /2 1 ----
(S/N)o = 102(26) (C/N)i
Io(26) + 61,(26)
(7.24)
(7.25)
where (C/N)i is the carrier-to-noise power ratio.
7.1-4. FM system
The signal-to-noise ratio at the output of a FM receiver for a
FM signal plus narrow-band Gaussian noise at the input can be obtained
by the same procedure as used above for SSB-FM. The factor e (t)of
Eq. (7.6) is replaced by unity, and the bandwidth of the input noise
is given by Eqo (7.19). Then the output signal-to-noise ratio becomes
(S/N)o = 3 62(6+1) (S/N)i
6 62(6+1) /2 21 12(6)
(S/N)o = 2 (S/N)i
102(26) + T 61o(26)11(26)
(7.26)
when the input signal-to-noise ratio is large. It is also noted that
(S/N)i = (C/N)i. (7.27)
7.1-5. Comparison of signal-to-noise ratios
A comparison of the various modulation systems is now given by
plotting (S/N)o/(S/N)i as a function of the modulation index by use of
Eqs. (7.2), (7.5), (7.23), and (7.26). This plot is shown in Figure 19.
Likewise (S/N)o/(C/N)i as a function of the modulation index
is shown in Figure 20, where Eqs. (7.3), (7.25), (7.26), and (7.27) are
used. It is noted that in both of these figures the noise power band-
width was determined by the signal bandwidth.
When systems are compared in terms of signal-to-noise ratios, a
useful criterion is the output signal-to-noise ratio from the system
for a given RF signal power in the channel--that is, (S/N)o/Si. This
result can be obtained from (S/N)o/(S/N)i, which was obtained previously
for each system, if the input noise, Ni, is normalized to some convenient
constant. This is done, for example, by taking only the noise power in
the band 2mm (rad/s) for measurement purposes. (The actual input noise
power of each system is not changed, just the measurement of it.) Then
the normalized input noise power for all the systems is
F0
N = 2wm
2n
where the subscript I denotes the normalized power. Then the ratio
(S/N)o/(S/N), is identical to Ni[(S/N)o/Si] where NJ is the constant de-
fined above. Thus, to within the multiplicative constant NJ, comparison
of (S/N)o/(S/N)I for the various systems is a comparison of the output
3.5 ---- I
I
3.0 -
I
FM
(S/N)o f
(S/N)i
I
/ SSB-FM--FM Detection
1.5 5 --
SSB-AM-SC
1 ____ _
0.5 /
/ AM--Coherent
/ Detection
0 0.5 1.0 1.5 2.0 2.5
Modulation Index (6)
Figure 19. Output to Input Signal-to-Noise
Power Ratios for Several Systems
|
|
67
4;
Pa()
1
Wf
0)**
Figure 15. Power Spectrum of a(t)
From Figure 15 we have
Raa(T)
W
1
o
'aa^; 77 J -o'
~WA
eJTdw
or
Raa(T)
NqW0 /sin W t
* \ W0t
(6.42)
and
^ =
N W
oo
(6.43)
Then
ii ,
Raa(o)
-N0W0'
(6.44)
Substituting the last two equations into Eq. (6.40) we obtain the rms
bandwidth for the SSB-PM multiplexed signal:
rTTMoyw. i ^ + nw
^rms^M-SSB-PM-SN-
3tt
1 e-Nowo/lT
(6.45)
where u>a is the subcarrier frequency
N0 is the amplitude of the spectrum of the subcarrier Gaussian
noise modulation
WQ is the bandwidth of the subcarrier noise modulation.
106
and that these partial derivatives are continuous.
W = wrw2 = (Ux + j)(U2 + jV2)
= (U^- v1v2) + j(v1u2 + V2U1) = U + jV.
Then
and
aU =
u,
3Uz +
u,
Mi.
V,
av2
- v,
Ml
ax
1
ax
2
ax
1
ax
2
ax
av
Vi
3U9
u2
aVi
v2
aUi
+ u1
a V2
_ -
+
+
3y
sy
ay
ay
ay
By substituting Eqs. (I-1 a) and (I-lb) into Eq. (1-4),
3V1
aV aU2 all
= Un - + U0 i- + V,
ay
ax
ax
av2
ax
+ v.
ax
(1-3)
(1-4)
(1-5)
But Eq. (1-5) is identical to Eq. (1-3) and the partial derivatives are
continuous. Thus, the condition of Eq. (I-2a) is satisfied.
Also,
and
3V 3U2. + y Ml _l u Ml x 11 Ml
= V,
+ V,
+ u,
ax
ax
ax
ax
ax
Then
aU
3y
= u,
au2
ay
+ u.
alii
ay
- v,
av2
ay
- Vc
av1
ay
ay \ ax ax ax ax
(1-6)
(1-7)
and all the partial derivatives are continuous. By comparing Eq. (1-6)
with Eq. (1-7) it is seen that the condition of Eq. (I-2b) is satisfied.
Therefore W(z) is analytic in the UHP.
37
where the notation SC and denote the suppressed-carrier functions.
But what are these functions tt and ¥? The condition for a suppressed
carrier is that kx = k2 = 0. By comparing Eq. (5.19) with Eq. (5.9) it
follows that it = -V and if- = 0. Furthermore by Theorem V of Section 5.1,
U = -V + k2 and V = + kx. Thus
it = -V = U k2 (5.20)
and
Â¥ = U = V kr (5.21)
It is also noted that it and ¥ are a unique Hilbert transform pair. That
is, ¥ is the Hilbert transform of it, and it is the inverse Hilbert trans
form of 3t. This is readily shown by taking the Hilbert transform of
Eq. (5.20), comparing the result with Eq. (5.21), and by taking the in
verse Hilbert transform of Eq. (5.21), comparing the result with Eq. (5.20).
Thus Eq. (5.19) may be re-written as
XUSSB-SC^ = ,m(t)) cos w0t it(m(t),m(t)) sin wot
(5.22)
or
XUSSB-SC^ = "^(t) ,m(t)) cos w0t V
(5.23)
where it and ¥ are given by Eq. (5.20) and Eq. (5.21).
It is interesting to note that the form of the USSB signal given
_ *
above checks with the expression given by Haber [14]. He indicates that
if a process n(t) has spectral components only for |w| > wq then n(t)
can be represented by
Copyright by
Leon Worthington Couch, II
1968
116
or for h(z) = U(x,y) + jV(x,y)
0 = P
U(x,0) + jV(x,0)
x-t
dx Jtt[U(t,0) + jV(t,0)]
+ lim
R-+00
[U(R cos e,R sin 0) + jV(R cos 0,R sin
R(cos 0 + j sin 0) t
Aside: calculate the term:
t f R[U+jV][-sin 0 + j cos 0]
I im j --
R-x J (R cos e-t) + jR sin 0
d0
1 im ¡
R-* ~
R[U + jV][t sin 0 + j(R-t cos 0)]
(R cos 0-t)2 + R2 sin2 0
d0
lim
R400
{[Ut sin 0 + V(t cos 0-R)] + j[Vt sin 0 + U(R-t
R 2t cos 0 + t2/R
For finite t, lim (t cos 0 R) = -R, lim (R-t cos 0) = R, and
R-Kjo R--K
lim [R 2t cos 0 + t2/R] = R, Thus Eq. (1-30) becomes
R-x
1 im
R->oo
{[Ut sin 0 VR] + j[Vt sin 0 + UR]}
R
d0
1 im *
R-*
'Ut sin 0
R
- V
+ j
Vt sin 0
R
+ U
de
0)]RjeJed0
(1-29)
COS 0)]}
de.
(1-30)
(1-31)
Since U and V are real and imaginary parts of a function which is analytic
f//7aj
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APPENDIX II
EVALUATION
OF eo(x + J'y)
Assume: x and y are joint Gaussian random variables, bbth having zero
mean values.
To show: e^x ^ = e"^x +J2uxy~^y2l
when x and y are joint Gaussian random variables.
The joint density function is
1
p(x,y)
2lTaXCTy(l "P
T- e 2ax2-ay2^p2)
[ay2x2-2axavpXy+0x2y2]
Then
J[x+jy]
_2TTaxay(l-p2)^
00 00
r
,j(x+jy)e 2ax2ay2(1-p2)
[ay2x2*2ax0ypxy+ax2y2]
dxdy
oo oo
r -
2ax2(l-p2)
x2-2(^~ py+jax2(l-p2)j x+
CTy2
u
-oo co
y2+y2a/(l-p2))
dxdy
oo oo
r'
J
00 00
2ax2(l-p2)
[x-k(y|
[2 -
2ax2(l-p2) L | y
k^y)+^y2+y2ax2(l-p2l|
V dxdy
where k(y) = ~ py + jox2(l-p2)
ay
119
Page
VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS 48
6.1c Example 1: Single-Sideband AM With
Suppressed Carrier* ...... 48
6.2. Example 2: Single-Sideband PM 51
6.3. Example 3: Single-Sideband FM 68
6.4. Example 4: Single-Sideband a 71
VII. COMPARISON OF SOME SYSTEMS 75
7.1. Output Signal-to-Noise Ratios .... 76
7.1-1 AM system ....... ...... 76
7.1-2, SSB-AM-SC system ..... 77
7.1-3. SSB-FM system 78
7.1-4. FM system 84
7.1-5. Comparison of signal-to-noise ratios* 85
7.2. Energy-Per-Bit of Information 89
7.2-1. AM system 93
7.2-2. SSB-AM-SC system 93
7.2-3. SSB-FM system 94
7.2-4, FM system 94
7.2-5. Comparison of energy-per-bit for
various systems 95
7.3. System Efficiencies 97
7.3-1. AM system ..... ... 98
7.3-2. SSB-AM-SC system ...... 93
7.3-3. SSB-FM system ....... 93
7.3-4. FM system 99
7.3-5. Comparison of system efficiencies 100
vi
ACKNOWLEDGMENTS
The author wishes to express sincere thanks to some of the many
people who have contributed to his Ph.D program. In particular,
acknowledgment is made to his chairman, Professor T. S. George, for his
stimulating courses, sincere discussions, and his professional example.
The author also appreciates the help of the other members of his super
visory committee. Thanks are expressed to Professor R. C. Johnson and
the other members of the staff of the Electronics Research Section,
Department of Electrical Engineering for their comments and suggestions.
The author is also grateful for the help of Miss Betty Jane Morgan who
typed the final draft and the final manuscript.
Special thanks are given to his wife, Margaret, for her inspi
ration and encouragement.
The author is indebted to the Department of Electrical Engi
neering for the teaching assistantship which enabled him to carry out
this study and also to Harry Diamond Laboratories which supported this
work in part under Contract DAAG39-67-C-0077, U. S. Army Materiel Com
mand.
Figure 10. USSB-FM Signal Exciter
65
shape of the modulation as well.
The dependence of bandwidth on the spectrum of the Gaussian noise
modulation will be illustrated by another example. Consider the narrow-
band modulation process:
m(t) = a(t) cos (ojat + (f>) (6.35)
where a(t) is the (double-sideband) suppressed-subcarrier amplitude
modulation
a is the frequency of the subcarrier
<¡> is a uniformly distributed independent random phase due to
the subcarrier oscillator.
That is, we are considering a SSB signal which is phase modulated by the
m(t) given above. Then
Rmm(T) = *2 Raa(t) oos ^a1 (6.36)
where Raa(T) is the autocorrelation of the subcarrier modulation a(t).
Rmm(T) can obtained from Eq. (6.36) by use of the product theorem [21].
Thus, assuming that the highest frequency in the power spectrum of a(t)
is less than o>a,
Rmmi'O ~ Raa(x) sin a1 > (6.37)
Furthermore let a(t) be a Gaussian process; then m(t) is a
narrow-band Gaussian process. This is readily seen since Eq. (6.35) may
be expanded as follows:
m(t) = %[a(t) cos (wat+<|>) a(t) sin (cjat+c¡>)]
+ %[a(t) cos (cgt+tj)) + a(t) sin (ojat+<)>)] (6.38)
78
more complicated receiver as given in Figure 17 However, in some practi
cal applications the receiver in Figure 17 may give much better perform
ance due to better lower sideband noise rejection. That is, in Figure 17
the lower sideband noise is eliminated as the result of the approximate
Hilbert transform filter realized about oj = 0; whereas, in Figure 16 the
lower sideband noise is rejected by the IF filter realized about u = to0.
Thus, in order to obtain equal lower sideband noise rejection in both
receivers, the IF bandpass for the receiver in Figure 16 would have to
have a very steep db/octave roll-off characteristic at
Figure 17. SSB-AM-SC Receiver
7.1-3. SSB-FM system
Now consider a FM receiver which is used to detect a SSB-FM sig
nal plus narrow-band Gaussian noise as shown in Figure 18.
X{t)+n-j (t)
FM
Receiver
Output
Figure 18. SSB-FM Receiver
55
or
(6.16)
where XjU.r) = m(t) m(t-r)
x2(t,x) = m(t) + m(t-r)
x3(t,x) = -m(t) m(t-x) -x2(t,r)
X4(t,r) = -m(t) + m(t-r) = -x j(t jt)
y(t,r) = m(t) + m(t-x)
Now let the modulation m(t) be a stationary Gaussian process with zero mean.
Then x^t,/), x2(t,i), x3(t,T), x4(t,x), and y(t,x) are Gaussian processes
since they are obtained by linear operations on m(t). They are also stat
ionary and have a zero mean value. It follows that x (t,x), y(t,t);
x?_(t jt) j y(t,x); x 3 (t j i), y (t, T); and x4(t,T)s y(t,r) are jointly Gaussian
since the probability density of the input and output of a linear system
is jointly Gaussian when the input is Gaussian [15]. For example, to show
that Xj(t,x) and y(t,r) are jointly Gaussian, a linear system with inputs
m(t) and m(t-r) can readily be found such that the output is y(t,x). Now
the averaging operation in Eq. (6.16) can be carried out by using the fol
lowing formula which is derived in Appendix II:
ej{x(t)+jy(t)} = e-Js{ox2+j2yXy-ay2}
(6.17)
where x(t) and y(t) are jointly Gaussian processes with zero mean,
2 =
= X2(t)
oy2 = y2(t)
time-varying functions compared to cos ta0t and sin wot, we have for the
peak-average power:
47
Pp_Av = %{[U(m(t),m(t))]2 + [V(m(t),m(t))]/}I
P 't t,
(5.55)
where tpea(< is the value of t which gives the maximum value for Eq. (5.55).
Using Eq. (5.20) and Eq. (5.21), Pp_Av can also be written as
P Av [4M- k2]2 [* + k,]2) |
p peak
2 2
= ^(Du- + k9] + [ti- + k.] } L .
r Vak
= %{[4+ k]2 + |> + kL]2}11 t
1 xpeak .
The average power of the SSB signal was given previously by Eq. (5.53).
Thus the expression for peak-to-average power ratio for the generalized
SSB signal is
(5.56a)
(5.56b)
(5.56c)
(5.56d)
Several equivalent representations have been given for peak-to-average
power since one representation may be easier to use than another for a
particular SS>B signal.
Pn-Av f[U(m(t).ii(t))]2 + [V(m(t),m(t))]2) |t t
= _____ Lpeak
PAv k;i2 + k2 + 2R(JU(0)
{[U(m(t),m(t))]2 + [V(m(t),m(t))]2} L ,
k,2 + k22 + 2RW(0)
{[Wm(t).m(t))+k,]2 + [ |t t
= ^ 1 ~ Lpeak
kj2 + k22 + 2Rw(0)
{[-Â¥(m(t),m(t))+k2]2 + [Â¥-(m(t),m(t))+k1]2} |
= t ''peak.
k;i2 + k22 + 2RW(0)
68
Thus the niis bandwidth is proportional to the power in the subcarrier
modulation as N0 becomes large.
6.3/ Example 3: Single-Sideband FM
As was indicated in Section 4.3. the representation for the SSB-FM
signal is very similar to that for the SSB-PM signal. In fact it will be
shown below that all the formulae for the properties of the SSB-PM signal
(which were obtained in the previous section) are directly applicable to
the SSB-FM signal.
The SSB-FM signal has a discrete carrier term since the entire
function for generating the SSB-FM signal is identical to that for the
SSB-PM signal, which has a discrete carrier term.
The other properties of the SSB-FM signal follow directly from
those of the SSB-PM signal if the autocorrelation of m(t) can be obtained
in terms of the spectrum for the frequency modulating signal e(t). It is
recalled from Eq. (4.7) that
t
(6.46)
First, the question arises: Is m(t) stationary if e(t) is stationary?
The answer to this question has been given by Rowe; however, it is not
very satisfactory since he says that m(t) may or may not be stationary [22]
However, it will be shown that m(t), as given by Eq. (6.46), is stationary
in the strict sense if e(t) is stationary in the strict sense; and,
furthermore, m(t) is wide-sense stationary if e(t) is wide-sense stationary
It is recalled that if
y(t) L[x(t)l
109
in the UH z-plane and that they are continuous in the UHP also.
Now,
3V2 3V2 3Ux aV2 3V
(1-12)
3y 3U1 3y 3 V1 3y
Substituting Eqs. (1-9) and (1-10) into Eq. (1-12), Eq. (1-12) becomes
(1-13)
Thus the condition given by Eq. (I-11 a) is satisfied in the region where
these derivatives exist and are continuous.
Similarly,
3V2 3V2 311} 3V2 3Vi
(1-14)
3X 3 U x 3X 3 V i 3X
Substituting Eqs. (1-9) and (I-10) into Eq. (1-14), Eq. (1-14) becomes
Thus the condition given by Eq. (I-llb) is satisfied in the region
where these derivatives exist and are continuous.
It is now argued that the derivatives exist and are continuous
for z in the UHP. This is true because for any z in the UHP, including
UH , z(z) may take on any value in the finite W plane. Also, the
derivatives of Ux and V1 with respect to x and y exist and are continuous
for z in the UHP, and the derivatives of U2 and V2 with respect to U} and \1
exist and are continuous anywhere in the finite W plane. Thus the con
ditions given by Eqs. (I-11 a) and (I-llb) are satisfied, and g[Z(z)] is
analytic in the UH z-plane.
or
6
Fz(ui)'
2Fm(w) oj > 0
Fm(w) ai = 0
0 oj < Q
(2.7)
Now suppose that m(t) is a stationary random process with auto
correlation Rmm([) anc* power spectrum Pmm(u))o Then the power spectrum
of m(t) is
Fmm(w) = pmm(w) l~J s9n (w) I Pmmi^)- (2.8)
This is readily seen by use of the transfer function of the Hilbert trans
form operator given by Eq. (2.2). Then, by taking the inverse Fourier
transform of Eq. (2.8), it follows that
Rmm(T) ~ Rmm(1)
(2.9)
The cross-correlation function is obtained as follows:
Rmm(T) = (t + T)m(t)
m(t + i A)m(t)dA
TT A
Rmm(t-A)dA
where () is the averaging operator. Thus
Rmm(i~ Rmm(T)
(2.10)
73
Assuming a Gaussian m(t), Eg. (6.54) becomes
COS [2a2R|7im(x)] 1} (6.55)
But this is identical to Eg. (6.18) except for the scale factor a2.
Thus for envelope detectable SSB (a = 1) with m(t) Gaussian, the auto
correlation and spectral density functions are identical to those for
the SSB-PM signal with Gaussian m(t). Moreover, the properties are
identical for SSB-a and SSB-PM signals having Gaussian m(t) processes
such that (ipm)SSB-PM 0(2 ^m^sSB-a*
It is also seen that if |e(t)| < < 1 most of the time then
m(t) = e(t).
Thus, when e(t) is Gaussian with a small variance, m(t) is approximately
Gaussian most of the time. Then Eg. (6.55) becomes
2a2Ree(T) cos [2a2Ree(x)] 1}
(6.56)
RW(l)SSB-a-GN ~
when |e(t)| < < 1 most of the time. Conseguently, formulae for the auto
correlation functions analogous to Egs. (6.19) and (6.20), may be further
simplified to a function of Ree(x) instead of Rrnm(t). Then the auto
correlation functions for USSB-cx and LSSB-a signals, assuming Gaussian
modulation e(t) with a small variance, are
RXU-SSB-a-GN
(t) = H Re eJoT{te22Ree(T) cos (2<,2l?ee(t))]
j [e2"R86(T) cos (2c.^e(0)]>
+
(6,57)
88
signal-to-noise ratios for the systems for a given RF signal power. This
procedure is commonly used for system comparisons [23].
Likewise, a comparison of output signal-to-noise ratios for vari
ous systems for a given aarrier power can be carried out by comparing
(S/N)0/(C/N)j, where the subscript I denotes the normalized input noise
power once again.
The AM, SSB-AM-SC, SSB-FM, and FM systems will now be compared
by using this procedure.
For the AM system (S/N)i = (S/N)j so that Eq. (7.2) becomes
(S/N)0
1 +
(S/N)!
(7.28)
and, likewise, Eq. (7.3) becomes
(S/N)0 = 2 (C/N)].
For the SSB-AM system Eq. (7.5) becomes
(S/N)0 = 2(S/N)i.
(7.29)
(7.30)
For the SSB-FM system, Eq. (7.23) becomes
(7.31)
and Eq. (7.25) becomes
(S/N)0
662
10(2<$) + | 61,(26)
(C/N)j
(7.32)
41
It may be easier to calculate the autocorrelation for the USSB or
LSSB signal using this representation rather than that of Eq (5.29) and
Eq. (5.33) since Rg_$c(x) may be easier to calculate than Rg(x). This
is shown below.
A simplified expression for Rg_$c(T) will now be derived. First,
it is recalled from Section 5.2 that if and -V- are a unique Hilbert trans
form pair. Thus g$c given by Eq. (5.34), can be expressed in terms of
two analytic signals:
gsc(m(t),m(t)) = tf(m(t) ,m(t)) + jit(m(t) ,m(t)) (5.39)
and
gsc(m(t),m(t)) = -¥-(m(t) ,m(t)) + j¥-(m(t) ,m(t)) (5.40)
where Eq. (5.39) is the a^lytic signal associated with -H-and Eq. (5.40)
is the analytic signl associated with -Â¥. Using Eq. (5.39) and Eq. (2.15),
the autocorrelation of g5g is given by
Rg-SC(T) 2[:Rw(t) + jRw(r)] (5.41)
or by using Eqs. (5.40), (2.15), and (2.9)
Rg-SC(T) = 2[rw(t) + jRw(t)]. (5.42)
Thus Rg_$c(x) may be easier to calculate than Rg(x) since only Rw(x) or
RyyM is needed. This, of course, is assuming that the Hilbert trans-
by use of Eqs. (4.5a), (4.5b), and (6.22) in Eq. (5.56b):
64
i[e
m (t)
cos m(t)] + [e
rn(t)
sin m(t)]2}
1 + (e
1)
tpeak
or
e-2rn(t)
e
t = tpeak
Si'm
(6.33)
Note that m(t) may take on large negative values because it has a Gaussian
density function (since it was assumed at the outset that the modulation
was Gaussian). However, it is reasoned that for all practical purposes,
m(t) takes on maximum and minimum values of +3am and 40^ volts where cm
is the standard deviation of m(t). This approximation is useful only for
small values of crm since e+^m) approximates the peak power only when the
exponential function does not increase too rapidly for larger values of am.
Thus the peak-to-average power ratio for the SSB-PM signal with Gaussian
noise modulation is
SSB-PM-GN
(6.34)
when is small.
It is noted that the efficiency and the peak-to-average power
ratio depend on the total power in the Gaussian modulation process and not
on the shape of the modulation spectrum. On the other hand the autocorre
lation function and bandwidth for the SSB signal depend on the spectral
85
when the input signal-to-noise ratio is large. It is also noted that
(S/N)i = (C/N)j.
(7.27)
7.1-5. Comparison of signal-to-noise ratios
A comparison of the various modulation systems is now given by
plotting (S/N)0/(S/N)^ as a function of the modulation index by use of
Eqs. (7.2), (7.5), (7.23), and (7.26). This plot is shown in Figure 19.
Likewise (S/N)0/(C/N)j as a function of the modulation index
is shown in Figure 20, where Eqs. (7.3), (7.25), (7.26), and (7.27) are
used. It is noted that in both of these figures the noise power band
width was determined by the signal bandwidth.
When systems are compared in terms of signal-to-noise ratios, a
useful criterion is the output signal-to-noise ratio from the system
for a given RF signal power in the channel--that is, (S/N)0/S-¡, This
result can be obtained from (S/N)0/(S/N)-¡, which was obtained previously
for each system, if the input noise, N-¡, is normalized to some convenient
constant. This is done, for example, by taking only the noise power in
the band 2oim (rad/s) for measurement purposes. (The actual input noise
power of each system is not changed, just the measurement of it.) Then
the normalized input noise power for all the systems is
F0
[\| _ 2u)pi
2ir
where the subscript I denotes the normalized power. Then the ratio
(S/N)0/(S/N)j is identical to Nj[(S/N)0/Si] where Nj is the constant de
fined above. Thus, to within the multiplicative constant Nj, comparison
of (S/N)0/(S/N)j for the various systems is a comparison of the output
Using Eg. (2.2) and the difinition for the Fourier transform, we
F^(w) = FReh(a)) + j[-j sgn (w)] FReh(w) + j^irk^U)
or
FRehM
*sFh(w)
a) > 0
Ff-i(a))-j2-rrki6(ai) w s 0
Also, it is recalled that
FReh(-) = FReh()
This is seen from
Re[h(x
Thus from Eqs. (1-18) and (1-19) we obtain
,0)]eJtXdx
Re[h(x,0)]e"J)Xdx
FReh()
JgF^(a)) a) > 0
[F^uO-J^irk^w)] = [F^i-wJ+jZTrk^i-w)] w = 0
%Ffi(-o)) a) < 0
Proof of Theorem III:
By aid of Eg. (1-23) we have
111
obtain
(1-18)
(1-19)
h(x,0)ejox = {U(x,0)+j[U(x,0)+k1]}ejx .
94
7.2-3. SSB-FM system
To obtain M(6) for the SSB-FM system with (S/N)Q = 27.5 db, it
follows from Eq. (7.25) that (C/N)-f = 23.3 db for 6 = 1. Also, for the
SSB-FM system Eq. (7.38) becomes
M(6)
_ / I12(26)1
-I
2(6+1) o)m /2 /l
I (26)(C/N)i
y I02(26)
0.693 com log
6 62 (6+1) /2 /l -
1 +
T7(W
In2(2.)
-. (7.43)
10(26) + | 61^26)
(C/N)i
For (C/N)^ = 23.3 db, Eq. (7.43) reduces to
7.2-4. FM system
To obtain M() for the FM system, for (S/N)0 = 27.5 db, it follows
from Eqs. (7.26) and (7.27) that (C/N)-,* = 12 db (which is just above the
threshold) for 6=2. Also, for the FM system, Eq. (7.38) becomes
[2 (6+1) oom] (C/N)
M(6) = : .
0.693 aw log2 [1+3 62(6+1)(C/N)i]
(7.45)
112
Then
ReCh(x90)e^a>x3> = f (U(x,0) cos to0x [Uix.Oj+k^sin wox}eja)Xdx
U(x.O) >s(ej,oX + e1"x) + J[0{x,O)+k1]yejx-e-;ilX)>e-jxdx
00 CO
= h J U(x,0)e"J^"w^xdx + % f U(x,0)e"J'^a)+)^xdx
00 CO
00 00
+ 3h C U(x,O)e'J^"w0^xdx jig f U(x,0)e'^w+w^xdx
v/ J
00 oo
00 00
+ Jhk1 f ej("w+)o)xdx jj5k1 f ej(_)"a)o)xdx
and by using Eq. (2.2) and the Fourier transform of U(x,0),
'{Re[h(x,0)e,;)oX]}= FRe^()-)0) + % FR(ahU+o>0)
Reh'
+ 3h [-J sgn (o-jq)3 FReh(c-)0) j% [-j sgn (w+a)0)]FReh(cD+iD0)
+ 3h 2Trk1(-to+Jo) 3h 2-rrk16(-cj-u)o). (1-20)
Using Eq. (1-17) from the Lemma to Theorem III to evaluate FReh(*) in
Eq. (1-20), Eq. (1-20)becomes
%F^(t->o) w > wq
0 |o)| < uo
%F^(-o)-w0) w -w0
F{Re[h(x,0)eja)x]} =
Figure Page
20 Output Signal-to-Noise to Input Carrier-to-Noise Ratio
for Several Systems 87
21. Output Signal-to-Noise to Input Signal-to-Normalized-
Noise Power Ratio for Various Systems 90
22. Output Signal-to-Noise to Input Carrier-to-Normalized-
Noise Power Ratio for Various Systems 91
23 Comparison of Energy-per-Bit for Various Systems ....... 96
24,. Efficiencies of Various Systems 101
25. Contour of Integration 107
26o Contour of Integration 115
40
will be determined in terms of Rg(-r).
By examining Eq. (5.19) and comparing this equation to Eq. (3.5),
with the aid of Eq. (3.3) it is seen that the suppressed DC carrier
version of g is given by
gsc(m(t),m(t)) = if(m(t),m(t)) + j¥-(m(t) ,m(t)) (5.34)
where -0-and M- are the suppressed-carrier functions defined by Eq. (5.20)
and Eq. (5.21). Then it follows that
g(m(t),m(t)) = gsc(m(t) ,m(t)) + [k2+jk1]0 (5.35)
It is noted that the mean value of ggQ is zero. This is readily seen via
Eqs. (5.34), (5.20), and (5.21) since it is recalled that the mean value
of 0 and V was shown to be zero in Section 5.2. Then, using Eq. (5.35),
the autocorrelation of g is obtained in terms of the autocorrelation of
9sc;
Rg(0 = Rg-sc(x) + (kiZ+k22)
(5.36)
Therefore the autocorrelation functions for the USSB signal,
Eq. (5.29), and the LSSB signal, Eq. (5.33), become
Rxu(T) = ^(^[(k^2) + Rg-SC (T 1
and
Rxl(i) = %Re{eJwoT[(kl2+k22) + Rg_sc(x)]}o
(5.37)
(5.38)
i
14
or
XuSSB^t) = U(m(t) ,rn(t)) cos w0t V(m(t) ,m(t)) sin oj0t
(3.5)
where U(ReW,ImW) is the real part of the entire function g(W)
V(ReW,ImW) is the imaginary part of g(W)
m(t) is either the modulating signal or a real function of the
modulating signal e(t)
m(t) is the Hilbert transform of m(t).
Using Theorem III the voltage spectrum of Xu$$g(t) is
FUXU) = ^[XusSBt)]
%Fg(w-tQ) to >
, j co j < to.
lgFg(-w-aJ0) a) < -u0
(3.6)
This spectrum is illustrated by Figure 5.
The lower single-sideband signal can be synthesized in a similar
manner from the complex baseband signal Now we need to translate the
complex baseband signal down to the transmitting frequency instead of up,
KEY TO SYMBOLS
AM
b
B
Cb
Cb
Ci
(C/N)i
(C/N )i
D
FM
F(a>)
F(-)
g(w)
GN
LSSB
m(t)
M
Ni
P(u)
= Amplitude Constant
= Amplitude-Modulation
= Baseband Bandwidth (rad/s)
= RF Signal Bandwidth (rad/s)
= Baseband Channel Capacity
= RF Channel Capacity
= Input Carrier Power
= Input Carrier-to-Noise Ratio
= Input Carrier-to-Normalized-Noise Ratio
= Modulator Transducer Constant
= Frequency-Modulation
= Voltage Spectrum
= The Fourier Transform of (*)
= U(W) + jV(W) = An Entire Function
= Gaussian Noise
= Lower Single-Sideband
= Modulating Signal or a Real Function of the Modulating Signal
(see e(t) below)
= Either Multiplex or Figure of Merit
- Input Noise Power
= Normalized Input Noise Power
= Power Spectral Density
x
CHAPTER I
INTRODUCTION
In recent years the use of single-sideband modulation has become
more and more popular in communication systems. This is due to certain
advantages such as conservation of the frequency spectrum and larger post
detection signal-to-noise ratios in suppressed carrier single-sideband
systems when comparison is made in terms of total transmitted power,
A single-sideband communication system is a system which generates
a real signal waveform from a real modulating signal such that the Fourier
transform, or voltage spectrum, of the generated signal is one-sided about
the carrier frequency of the transmitter. In conventional amplitude-modu
lated systems the relationship between the real modulating waveform and
the real transmitted signal is given by the well-known formula:
XAM(t) = A0 H + m(t)] cos ojQt |m(t)| <1 (11)
where A() is the amplitude constant of the transmitter
m(t) is the modulating (real) waveform
oj0 is the carrier frequency of the transmitter.
Likewise, frequency-modulated systems generate the transmitted waveform:
Xp^(t) = A0 cos [w0t + D / tm(t")dt'] (1.2)
1
101
g(db)
Figure 24. Efficiencies of Various Systems
50
signal are identical to those for the modulation. This is readily shown
below.
The mean-type bandwidth (when the numerator and denominator exist)
is given by use of Eq. (6.3) in Eq. (5.46):
(6.6)
where = Rmm(0), the power in the modulating signal. By using Eq. (5.48)
the rms bandwidth is
(rms^SSB-AM
-tim(O)
I'm
(6.7)
whenever R^m(0) and exist. By using Eq. (5.50) the equivalent-noise
bandwidth is
ir
(A^SSB-AM = oo
4^ S Rmrn(T)dT
(6.8)
Thus the bandwidths of the SSB-AM-SC signal are identical to those of the
modulating process m(t).
The efficiency of the SSB-AM-SC signal is obtained by using
Eq. (5.54):
iSC-SSB-AM
2Rmm(0)
= 1 .
2Rmm(0)
(6.9)
CHAPTER II
MATHEMATICAL PRELIMINARIES
Some properties of the Hilbert transform and the corresponding
analytic signal will be examined in this chapter. None of the material
presented in this chapter is new; in fact, it is essentially the same
as that given by Papon1 is except for some changes in notation [4]. How
ever, this background material will be very helpful in derivations pre
sented in Chapter III and Chapter' V
The Hilbert transform of m(t) is given by
co
(2.1)
where () is read !lthe Hilbert transforms of ()"
P denotes the Cauchy principal value
* indicates the convolution operation.
The inverse Hilbert transform is also defined by Eo (2.1) except that a
minus sign is placed in front of the right-hand side of the equation. It
is noted that these definitions differ from those used by the mathema
ticians by a trivial minus sign. It can be shown, for example, that the
Hilbert transform of cos oo0t is sin to0t when w0 > 0 and that the Hilbert
transform of a constant is zero, A list of Hilbert transforms has been
compiled and published under work done at the California Institute of Tech
nology on the Bateman Manuscript Project [5],
4
31
signal consists of a continuous part due to the modulation plus impulse
functions at u0 and -o>0 if there is a discrete carrier term. As defined
here, the "continuous" part may contain impulse functions for some types
of modulation, but not at the carrier frequency. Taking the inverse
Fourier transform of the composite voltage spectrum it is seen that if
there is a discrete carrier term, the time waveform must be expressible
in the form:
X(t) = [f^tj+cj cos ai0t [f2(t)+c2] sin iOgt (5.13)
where c1 and c are due to the discrete carrier
f^t) and f2(t) are due to the continuous part of the spectrum
and have zero mean values.
Thus Eq. (5.13) gives the condition that c2 and Cj are not both zero if
there is a discrete carrier term.
To determine the condition for a discrete carrier in an upper
SSB signal, Eq. (5.13) will be identified with Eq. (5.9), which represents
/V A
the whole class of upper SSB signals. It is now argued that both U and V
have a zero mean value if the modulating process is stationary. This is
seen as follows:
U(m(t),£(t))-lp f df .
But U[m(t'),m(t')] = c, a constant, since m(t) is wide-sense stationary.
Thus
00
U(m(t) ,m(t)) = ^ P J'
00
dt' = 0.
Figure 8 Phasing Method for Generating USSB-AM-SC Signals
we have
70
(6.48)
Then in terms of power-spectrum densities
(6.49)
As Rowe points out, Pmm(oo) must eventually fall off faster than k/
where k is a constant, if e(t) is to contain finite power; and if Pmm(to) =
k/to2, Pqq(w) will be flat and, consequently, white noise. Thus we have a
condition for the physical realizability of m(t): Pmm(u)) falls off faster
than -6 db/octave at the high end. This condition is satisfied by physi
cal systems since they do not have infinite frequency response. From
Eq. (6.49) we have
(6.50)
Pmm() ~
C
Immediately we see that if P9g(to) takes on a constant value as |w| ->- 0
and at o> 0, m(t) will contain a large amount of power with spectral
components concentrated about the origin. In other words, m(t) has a
large block of power, located infinitely close to the origin which is
infinitely large. Thus m(t) contains a slowly varying "DC" term with a
period T and m2(t) - . By examining Eq. (6.46) we obtain the same
result from the time domain. That is, for Pe0(w) equal to a constant,
e(t) contains a finite amount of power located infinitely close to the
origin which appears as a slowly varying finite "DC" term in e(t) such
that T *
Then by Eq. (6.46), m(t) has a infinite amplitude and,
7
It follows that the spectrum of the cross-correlation function is given by
Pmm() ~ l-"j sgn (w)3 PfTirii(tlJ) (2.11)
it is noted that Pmm(w) is a purely imaginary function since Pmm(w) is a
real function. Then
00
Rmm(c) ~ T7 f (w)] Pmm(uduJ
l~ '* *00
which5 for Prnm(u)) a real even function, reduces to
Rmm( i) = J Pmm(^) s,n wt du) (2/12)
11 0
Thus the cross-correlation function is an odd function of r:
fynm(T) ~ Rmm(_T)
= -rh(t T)m(t) -m(t + i)m(t) (2/13)
or
wo = -W-O = -WO. (2.14)
The autocorrelation for the analytic signal is found as follows:
RZZ( ) = Z(tH)Z*(iy
= [m(t+-r) + jm(t+x)] [m(t) jm(tj]
= m(t+i)rn(t) + m(t+r)m(t) + jm(t+: )m(t) jm(t+r)m(t)
- Rmm() f Rmm(T) + JRmm(1) jRmm(TK
Figure 9. USSB-PM Signal Exciter--Method I
17
To summarize, it has been shown that once an entire function g(W)
is chosen, then an upper or lower single-sideband signal can be obtained
from the signal m(t) The signal m(t) is either the modulating signal
or a real function of the modulating signal e(t). The generalized ex
pressions, which represent SSB signals, are given by Eq (35) for the
USSB signal and by Eq, (3,8) for the LSSB signal. These expressions are
obviously the transfer functions that are implemented by the upper and
lower single-sideband transmitters respectively. Since there are an in
finitely denumerable number of entire functions, there are an infinitely
denumerable number of upper and lower single-sideband signals that can be
generated from any one modulation process, In Chapter IV some specific
entire functions will be chosen to illustrate some well-known single
sideband signals.
LIST OF FIGURES
Figure Page
1. Voltage Spectrum of a Typical m(t) Waveform 9
2. Voltage Spectrum of the Analytic Signal Z(t) 11
3. Voltage Spectrum of an Entire Function of an Analytic
Signal ]2
4. Voltage Spectrum of the Positive Frequency-Shifted
Entire Function of the Analytic Signal ..... 13
5. Voltage Spectrum of the Synthesized Upper Single-
Sideband Signal 14
6. Voltage Spectrum of the Negative Frequency-Shifted
Entire Function of an Analytic Signal 15
7. Voltage Spectrum of the Synthesized Lower Single-
Sideband Signal 16
8. Phasing Method for Generating USSB-AM-SC Signals 20
9. USSB-PM Signal Exciter--Method I .... 22
10. USSB-FM Signal Exciter 24
11. Envelope-Detectable USSB Signal Exciter 26
12. Square-Law Detectable USSB Signal Exciter 27
13. USSB-PM Signal Exciter--Method 11 ...... 53
14. USSB-PM Signal Exciter--Method III 54
15. Power Spectrum of a(t) 67
16. AM Coherent Receiver ....... 76
17. SSB-AM-SC Receiver 78
18. SSB-FM Receiver 78
19. Output to input Signal-to-Noise Power Ratios for
Several Systems ....... 86
vi i i
It follows that
35
CO
T-i (z) = '2~ ^ F())e^Zwd).
'00
lim |Z1(Rej0)|2
R-H
oo
= lim (-)2| ^ [FU)][e~(R sln 0)uej(R cos e)]du,
R -Xx>
A
By use of Schwarz's inequality this becomes
lim |Z (ReJ0) |2
R-*
(f)2c
'll
00 CO
r |F(a)) |2doj} {lim f e2(R sin e)Wh
0 R- 0
But F(to) e L2 (-5 ) so that
/
| F(to) 12dto < K.
Also
lim
R 7-00
e-(2R sin e)du)
0 < 0 < TT .
Therefore we have
lim 1Z, (ReJ6) | < {-f K 0 = 0 ,
R
0 < 0 < 1T .
23
The SSB-FM exciter as described by Eqs. (4.6) and (4.7) is given
in Figure 10.
4.4. Example 4; Singlet-Sideband a
The term single-sideband a (SSB-a) will be used to denote a sub
class of single-sideband signals which may be generated from a particular
entire function with a real parameter a. This notation was first used by
Bedrosian [3],
Let the entire function be
g3W eaW
(4.8)
where a is a real parameter, and let
m(t) = 1n[1 + e(t)]
(4.9)
where e(t) is the video or audio rnodulation signal which is amplitude
limited such that |e(t)| <1. It is assumed that m(t) is AC coupled
(that is, it has a zero mean). Note that these assumptions are
usually met by communications systems since they are identical to the
restrictions in conventional AM modulations systems. Then
g3(Z(t)) e'
1a[m(t)+jm(t)]= eam(t) ejam(t)
or
am(t) cos (am(t))
U3(m(t),m(t) = e
(4.10a)
SYNTHESIS AND ANALYSIS OF REAL
SINGLE-SIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS
By
Leon Worthington Couch, II
A Dissertation Presented to the Graduate Council of
The University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
UNIVERSITY OF FLORIDA
1968
83
the mean of the one-sided spectrum) for a SSB-FM signal to a conventional
FM signal [20], and it is
BSSB-FM
_____ a fz
bfm
I1z(26)
UHzT)
(7.18)
It is known that the bandwidth (in rad/s) of a FM signal is approxi
mately
Bfm = 2(+l (7.19)
Thus, to the first approximation, the SSB-FM bandwidth is
_ / 112(26) '
bssb-fm2 /2 y1 'S+1>V <7-20>
Then, taking the IF bandwidth to be that of the SSB-FM signal, the input
noise power is
i SSB-FM
n
(7.21)
Consequently, the input signal-to-noise ratio is
Ag2 Iq(26)
(S/N), -
Fo
4 a)m (6+1) /2
Ii2(26)
I02(26)
(7.22)
] 1
Eq (2.7), which is
Fz(uj)
2FmU)
03 >
F rn (0)
9 U3 =
0
5 0) <
(3 2)
where Fm(a)) is the voltage spectrum of the signal m(t), This is shown in
Figure 2 for our example used in Figure 1
Now let a function g(W) be given such that
g(W) i U(ReW,ImW) + jV(ReW,ImW)
(3.3)
where g(W) is an entire function of the complex variable W.
Theorem II: If Z(z) is an analytic function of z in
the UHP and if g(W) is an entire function of W, then
g[Z(z)] is an analytic function of z in the UH z-plane,
A proof of this theorem may be found in Appendix I.
Thus g[Z.(z)] is an analytic function of z in the UH z-plane, and
by Theorem I g[Z(t)] has a single-sided spectrum: the spectrum being
zero for w < 0. This is illustrated by Figure 3, where Fg(w) = F[g(Z(t))].
21
and
V2(m(t) ,m(t)) = e"rn(t) sin m(t).
(4.5b)
Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper
single-sideband signal:
or
XyssB-PM^) = e~m^ cos K1 +
(4.6)
It is again assumed that the modulation m(t) is AC coupled so that its
mean value is zero. The single-sideband exciter described by Eq. (4.6)
is shown in Figure 9.
4.3. Example 3: Single-Sideband FM
Single-sideband frequency-modulation is very similar to SSB-PM
in that they are both angle modulated single-sideband signals. In fact
the equations for SSB-FM are identical to those given in Section 4.2 ex
cept that
t
(4.7)
where e(t) is now the modulating signal (instead of m(t)) and D is the
transducer constant.
Experiments with SSB-FM signals have been conducted by a number
of persons and are reported in the literature [9, 10].
2
where A0 is the amplitude constant of the transmitter
m(t) is the modulating (real) waveform
u)0 is the frequency of the transmitter
D is the transducer constant of the modulator.
Now, what is the corresponding relationship for a single-sideband system?
Oswald, and Kuo and Freeny have given the relationship:
XsSB-AM^) = Ao Cm(t) cos <*>0t m(t) sin co01] (1.3)
where A0 is the amplitude constant of the transmitter
m(t) is the modulating signal
m(t) is the Hilbert transform of the modulating signal
co0 is the frequency of the transmitter [1, 2].
This equation represents the conventional upper single-sideband suppressed-
carrier signal, which is now known as a single-sideband amplitude-modulated
suppressed-carrier signal (SSB-AM-SC). It will be shown here that this is
only one of an infinitely denumerable set of single-sideband signals. In
deed, it will be shown that any member of the set can be represented by
xSSB(t) = Ao [u(mU)> m(t)) cos a)0t + V(m(t), m(t)) sin co0t] (1.4)
where A0 is the amplitude constant of the transmitter
U(x,y) and V(x,y) are the conjugate functions of any entire
function
m(t) is the modulating (real) waveform
m(t) is the Hilbert transform of m(t)
to0 is the transmitter frequency.
Various properties of these single-sideband signals will be analyzed in
44
It is noted that this formula is applicable whenever Ryy(O) and Ryy(O) or
Rw(0) and R^(0) exist. That is, RyyJO), RW(Q), Ry.(0), and R^_(0) may
or may not exist since ^g_sc^T) 1S ana^ytic almost everywhere (Theorem 103
of Titchmarsh [6]).
5.4-2.
RMS-type bandwidth
The rms bandwidth, wrms, may also be obtained.
oo
f pg-SC()d Rg-SC^0^
oo
(5.47)
Substituting Eq. (5.41) once again, we have
(rms)
-2[Ryy_(0) + jRyy.(0) ]
2[RW(0) + j%(0)]
Since Ryy(x) is an odd function of x, Ryy(O) = Ryy(O) =0, and we have
2
(rms)
-R^(0)
o
J
1
Rw(0)
c
o
w) j
-R^.(0)
rms J
c
o
i
c
o
(5.48)
It is noted that this formula is applicable whenever R^y.(0) and Rw(0)
or R^O) and R^(0) exist.
PM
RO
Re O
RF
Si
S0
(S/N)i
(S/N)j
(S/N)0
SC
USSB
(w)
-v-(w)
X(t)
XL
XU
Z(t)
a
5
6
n
e(t)
o2
0)
wrms
- Phase-Modulation
= Autocorrelation Function
= Real Part of (0
= Radio Freauency
-- Input Signal Power
* Output Signal Power
= Input Signal-to-Noise Ratio
= Input Signal-to-Normalized-Noise Ratio
= Output Signal-to-Noise Ratio
= Suppressed-Carrier
= Upper Single-Sideband
= The "Suppressed-Carrier" Function of U(W)
= The "Suppressed-Carrier" Function of V(W)
= A Real Modulated Signal
= Lower Single-Sideband Modulated Signal
= Upper Single-Sideband Modulated Signal
= m(t) + jm(t) = The Analytic Signal of m(t)
= a Modulation (as defined in the text)
= System Efficiency
= Modulation Index
= Efficiency
= Modulating Signal (when m(t) is not the Modulating Signal)
- Variance
- Average Power of m(t)
= Angular Frequency
= RMS-Type Bandwidth
xi
CHAPTER VII
COMPARISON OF SOME SYSTEMS
In the two preceding chapters properties of single-sideband sig
nals have been studied. However, the choice of a particular modulation
scheme also depends on the properties of the receiver. For example,
the entire function g(W) W can be used to generate a SSB signal, but
there is no easy way to detect this type of signal.
In this chapter a comparison of various types of modulated sig
nals will be undertaken from the overall system viewpoint {i.e* generation,
transmission and detection). Systems will be compared in terms of the
degradation of the modulating signal which appears at the receiver out
put when the modulated RF signal plus Gaussian noise is present at the
input. This degradation will be measured in terms of three figures of
merit:
1. The signal-to-noise ratio at the receiver output
2. The signal energy required at the receiver input for
a bit of information at the receiver output when com
an son is made with the ideal system (Here the ideal
system is defined as a system which requires a minimum
amount of energy to transmit a bit of information as
predicted by Shannon's formula.)
3. The efficiency of the system as defined by the ratio
of the RF power required by an ideal system to the RF
power required by an actual system,(Here the ideal sys-
75
29
or
h(t,0) = [-V(t,0)+k2] + J[U(t#0)+k1] (5.3)
where
. TV
k. = lim i V(R cos e,R sin e)de a real constant (5.4)
v R-* 0
TV
k2 = lim | U(R cos e,R sin e)de a real constant (5.5)
" R-X JQ
A proof of this theorem is given in Appendix I.
Theorem V may be applied to the generalized SSB signal by letting
h(z) = g(Zx(z)) where g(*) is an entire function of () ^(z) is analytic
in the UHP, and lim Zx(z) = lim Z (t + jy) = m(t) + jm(t). Thus Theorem V
y+0 y ">0
gives three additional equivalent expressions for g(Z(t)) in addition to
g(z(t)) = U(m(t),m(t)) + jV(m(t),m(t)) (5.6)
which was used in the derivation in Chapter III. Therefore, following
the same procedure as in Chapter III, equivalent upper SSB signals may be
found. Using Eq. (5.1) we have for the first equivalent representation
of Eq. (3.5):
xUS$B(t) = Re{g(Z(t))ej,Ot}
= Re{g(m(t),m(t))e^o^}
= Re [U(m(t) ,m(t) + jU(m(t) ,m(t)) + jk1]eJ'ot}
XyssB^) U(m(t),m(t)) cos w0t [U(m(t) ,m(t)) + kj sin taQt.
(5.7)
or
46
This definition will be used to obtain a formula expressing the efficiency
for the generalized SSB signal. Using Eq. (5.43) and Eq. (5.44) the side
band power in either the USSB or LSSB signal is
Rxu_sc(o) rxl-sc(0) = W) = y) (5.52)
It is also noted that Rg-SC(O) is not equal to the total power in the
real-signal sidebands since ggQ is a complex (analytic) baseband signal;
instead, (1 /2)Re[Rg_^(-.(0)3 Ryy(O) = Ryy(O) is the total real-signal
power. This is readily seen from Eq. (5.43a) and Eq. (5.44a).
Similarily the total power in either the USSB or LSSB signal is
obtained from Eq. (5.37) or Eq. (5.38):
Rxu(0) = rxl(0) = %[ki2 + k22 + 2W0)]
= Hlk* + k22 + 2RW(0)] (5.53)
Thus the efficiency of a SSB signal is
2Rnn(0) 2RW(0)
TTCT VTv"
n = ----- = 0 (5.54)
kx2 + k2 + 2Rm(0) k;|2 + k22 + 2RW(0)
5.6. Peak-to-Average Power Ratio
The ratio of the peak-average (over one cycle of the carrier-
frequency) to the average power for the SSB signal may also be obtained.
The expression for the peak-average power over one carrier-
frequency cycle of a SSB signal is easily obtained with the aid of
Eq. (3.5) and Eq. (3.8). Recalling that U and V are relatively slow
CHAPTER VI
EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS
The examples of SSB signals that were presented in Chapter IV
will now be analyzed using the techniques which were developed in
Chapter V.
6.1. Example 1: Single-Sideband AM With Suppressed Carrier
The constants k1 and k2 will first be determined to show that
indeed we have a suppressed carrier SSB signal. By substituting
Eq. (4.2b) into Eq. (5.4) we have
7T
0
But from Eq. (5.18b) it follows that
lim m^R cos e,R sin e) = 0 0 < e < tt ..
R-x
Thus
k = 0 .
(6.1)
Similarily substituting Eq. (4.2a) into Eq. (5.5) we have
IT
(6.2)
o
48
16
Thus the real lower single-sideband signal for a given entire
function is
XLSSB(t) = Re{g(Z(t))eJVc}
= Re [U(m(t) ,m(t)) + jV(m(t) ,rii(tj )]e"Ju3ot)
or
XLSSB^^ = U(m(t) ,m(t)) cos co0t + V(m(t) ,m(t)) sin u>0t.
(3.8)
Using Theorem IV the voltage spectrum of XL<^g(t) is
Flx(o>) i[XLSSB(t)]
JgFg (-(J+-a)0 ) 5 0 < 0) < COq
0 | o ¡ > Wq
hFq ()+)q ) 0 > a) > to
0
(3.9)
It is noted that the requirements that Fg(oj) be zero for m > w0 is to
prevent spectral overlap at the origin. This requirement is satisfied
(for all practical purposes) for ojq at radio frequencies.
The spectrum of FBx(^) is illustrated by Figure 7.
Figure 7. Voltage Spectrum of the Synthesized
Lower Single-Sideband Signal
8
Using Eqs. (2.9), (2.10) and (2.14) we obtain
Rzz(t) = 2[Rmm(t) + J Rmm(T)J = 2[Rmm(T) + jRmmU)]. (2.15)
Thus (1/2)R2(t) is an analytic signal associated with Rmm(-t). By use of
Eq. (2.7) it follows that
4Pmm(a>) w > 0
Pzz^)
2Pmm(a)) > w = 0
V
3
o
1
(2 16)
87
Figure 20. Output Signal-to-No1se to Input Carrler-
to-No1se Ratio for Several Systems
117
in the open UHP, U and V are finite as R for 0 < 0 < rr. Thus
1 im
R-x
Ut sin 6
R
0 and 1 im
R-*
Vt sin e
R
= 0
for 0 < e < tt, and Eq. (1-31) becomes
11m {-V + jU} de
R-" 0
(1-32)
Substituting the right side of Eq. (1-32) for the right-hand term on
the right side of Eq. (1-29), Eq. (1-29) becomes
O P / dx + j P
V(x,0) d..
x-t ax
- jirU(t.O)
+ irV(t,0) -
1im / Vde + j Tim I Ude
R-x J R-^> J
(1-33)
Setting the real and imaginary parts of Eq. (1-33) equal to zero we get
0 = P
U(x,0)
x-t
dx + ttV (t ,0)
1 im
R-*
Vde
(I-34a)
and
0 = P
V(x,0) d..
x-t ax
TT
TrU(t,0) + lim f Ude
R-* vi
(I-34b)
Thus
and
V(t,0) = U(t,0) + lim \ Vde
11 R-*= x
U(t,0) = -V(t,0) + lim f Ude
17 R-x J
(I-35a)
(I-35b)
0
115
and
k2 = 11m [ U(R cos 0,R sin e)de a real constant. (1-27)
^ R-x >
0
Proof of Theorem V;
By Cauchy's Theorem
h(x)
dz = 0
(1-28)
Jc z-t
for c as shown in Figure 26 since h(z) is analytic in the UHP, where t is
real and finite.
Figure 26. Contour of Integration
Thus for e > 0
0 = lim
£-0
t-e
h(x,0)
x-t
dx +
h(t+ee^e)ee'-l'ej
Je
eec
d9 + I dx
t+e
11. I h(Re^)RjeJe de
R-x
ReJ0-t
or
0 = P
J
rh(x-P>. dx jirh(t.O) + lim f MReJ' ^R-J-e~- de
X-t R^oo J DoJ0.
ReJ -t
89
For the FM system, Eo. (7.26) becomes
(S/N)0 = 3 62(S/N)j
and
(S/N)Q = 3 62(C/N)j.
(7.33)
(7.34)
A comparison of the output signal-to-noise ratios for the vari
ous modulation systems can be made now for a given input signal or car
rier power by using these equations. (S/N)0/(S/N)j as a function of
modulation index is plotted for various systems in Figure 21. Likewise
(S/N)0/(C/N)j is shown in Figure 22. From these figures, it is concluded
that FM gives the greatest signal-to-noise ratio at the detector output
for high index, followed by SSB-AM. For low index (6 < 1), SSB-AM is
best, followed by SSB-FM and FM which have about the same (S/N)0, and
AM gives the lowest (S/N)0.
7.2. Energy-Per-Bit of Information
The concept of RF energy required per bit of received information
is used by Raisbeck for comparing SSB-AM and FM systems [27]. This will
be extended to AM and SSB-FM systems in this section.
The (received) capacity of the system is given by [28]
Cb > (b/2ir) log2 [1 + (S/N)Q] (7.35)
where b is the baseband bandwidth (rad/s)
(S/N)0 is the output signal-to-noise power ratio.
Eq. (7.35) becomes an equality when the output noise is Gaussian.
123
30. W.G. Tuller, "Theoretical Limits on the Rate of Information,"
Proa, IREj vol. 37, 1949.
31. R.E.A.C. Paley and N. Wiener, "Fourier Transforms in the Complex
Domain," Am. Math, Soa. Colloq. Publ. vol. 10, 1934.
SYNTHESIS AND ANALYSIS OF REAL
SINGLE-SIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS
By
Leon Worthington Couch, II
A Dissertation Presented to the Graduate Council of
The University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
UNIVERSITY OF FLORIDA
1968
Copyright by
Leon Worthington Couch, II
1968
DEDICATION
The author proudly dedicates this dissertation to his parents,
Mrs,. Leon Couch and the late Rev, Leon Couch, and to his wife, Margaret
Wheland Couch,
ACKNOWLEDGMENTS
The author wishes to express sincere thanks to some of the many
people who have contributed to his Ph.D program. In particular,
acknowledgment is made to his chairman, Professor T. S. George, for his
stimulating courses, sincere discussions, and his professional example.
The author also appreciates the help of the other members of his super
visory committee. Thanks are expressed to Professor R. C. Johnson and
the other members of the staff of the Electronics Research Section,
Department of Electrical Engineering for their comments and suggestions.
The author is also grateful for the help of Miss Betty Jane Morgan who
typed the final draft and the final manuscript.
Special thanks are given to his wife, Margaret, for her inspi
ration and encouragement.
The author is indebted to the Department of Electrical Engi
neering for the teaching assistantship which enabled him to carry out
this study and also to Harry Diamond Laboratories which supported this
work in part under Contract DAAG39-67-C-0077, U. S. Army Materiel Com
mand.
table of contents
Page
ACKNOWLEDGMENTS iv
LIST OF FIGURES viii
KEY TO SYMBOLS x
ABSTRACT xiii
CHAPTER
I. INTRODUCTION 1
II. MATHEMATICAL PRELIMINARIES 4
III. SYNTHESIS OF SINGLE-SIDEBAND SIGNALS 9
IV, EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN 18
4.1. Example 1: Single-Sideband AM with
Suppressed-Carrier 18
4,2 Example 2: Single-Sideband PM . 19
4.3. Example 3: Single-Sideband FM 21
4.4. Example 4: Single-Sideband a 23
V.ANALYSIS OF SINGLE-SIDEBAND SIGNALS .... 28
51. Three Additional Equivalent Realizations 28
5.2. Suppressed-Carrier Signals 30
5.3. Autocorrelation Functions 38
5.4. Bandwidth Considerations 42
5.4-1. Mean-type bandwidth 43
5.4-2, RMS-type bandwidth 44
5.,4-3. Equivalent-noise bandwidth 45
5 5 Efficiency 45
5,6 Peak-to-Average Power Ratio 46
v
Page
VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS 48
6.1c Example 1: Single-Sideband AM With
Suppressed Carrier* ...... 48
6.2. Example 2: Single-Sideband PM 51
6.3. Example 3: Single-Sideband FM 68
6.4. Example 4: Single-Sideband a 71
VII. COMPARISON OF SOME SYSTEMS 75
7.1. Output Signal-to-Noise Ratios .... 76
7.1-1 AM system ....... ...... 76
7.1-2, SSB-AM-SC system ..... 77
7.1-3. SSB-FM system 78
7.1-4. FM system 84
7.1-5. Comparison of signal-to-noise ratios* 85
7.2. Energy-Per-Bit of Information 89
7.2-1. AM system 93
7.2-2. SSB-AM-SC system 93
7.2-3. SSB-FM system 94
7.2-4, FM system 94
7.2-5. Comparison of energy-per-bit for
various systems 95
7.3. System Efficiencies 97
7.3-1. AM system ..... ... 98
7.3-2. SSB-AM-SC system ...... 93
7.3-3. SSB-FM system ....... 93
7.3-4. FM system 99
7.3-5. Comparison of system efficiencies 100
vi
Page
VIII. SUMMARY 102
APPENDIX
L PROOFS OF SEVERAL THEOREMS 105
II. EVALUATION OF e j(x + Jy^ 119
REFERENCES 121
BIOGRAPHICAL SKETCH 124
vi i
LIST OF FIGURES
Figure Page
1. Voltage Spectrum of a Typical m(t) Waveform 9
2. Voltage Spectrum of the Analytic Signal Z(t) 11
3. Voltage Spectrum of an Entire Function of an Analytic
Signal ]2
4. Voltage Spectrum of the Positive Frequency-Shifted
Entire Function of the Analytic Signal ..... 13
5. Voltage Spectrum of the Synthesized Upper Single-
Sideband Signal 14
6. Voltage Spectrum of the Negative Frequency-Shifted
Entire Function of an Analytic Signal 15
7. Voltage Spectrum of the Synthesized Lower Single-
Sideband Signal 16
8. Phasing Method for Generating USSB-AM-SC Signals 20
9. USSB-PM Signal Exciter--Method I .... 22
10. USSB-FM Signal Exciter 24
11. Envelope-Detectable USSB Signal Exciter 26
12. Square-Law Detectable USSB Signal Exciter 27
13. USSB-PM Signal Exciter--Method 11 ...... 53
14. USSB-PM Signal Exciter--Method III 54
15. Power Spectrum of a(t) 67
16. AM Coherent Receiver ....... 76
17. SSB-AM-SC Receiver 78
18. SSB-FM Receiver 78
19. Output to input Signal-to-Noise Power Ratios for
Several Systems ....... 86
vi i i
Figure Page
20 Output Signal-to-Noise to Input Carrier-to-Noise Ratio
for Several Systems 87
21. Output Signal-to-Noise to Input Signal-to-Normalized-
Noise Power Ratio for Various Systems 90
22. Output Signal-to-Noise to Input Carrier-to-Normalized-
Noise Power Ratio for Various Systems 91
23 Comparison of Energy-per-Bit for Various Systems ....... 96
24,. Efficiencies of Various Systems 101
25. Contour of Integration 107
26o Contour of Integration 115
KEY TO SYMBOLS
AM
b
B
Cb
Cb
Ci
(C/N)i
(C/N )i
D
FM
F(a>)
F(-)
g(w)
GN
LSSB
m(t)
M
Ni
P(u)
= Amplitude Constant
= Amplitude-Modulation
= Baseband Bandwidth (rad/s)
= RF Signal Bandwidth (rad/s)
= Baseband Channel Capacity
= RF Channel Capacity
= Input Carrier Power
= Input Carrier-to-Noise Ratio
= Input Carrier-to-Normalized-Noise Ratio
= Modulator Transducer Constant
= Frequency-Modulation
= Voltage Spectrum
= The Fourier Transform of (*)
= U(W) + jV(W) = An Entire Function
= Gaussian Noise
= Lower Single-Sideband
= Modulating Signal or a Real Function of the Modulating Signal
(see e(t) below)
= Either Multiplex or Figure of Merit
- Input Noise Power
= Normalized Input Noise Power
= Power Spectral Density
x
PM
RO
Re O
RF
Si
S0
(S/N)i
(S/N)j
(S/N)0
SC
USSB
(w)
-v-(w)
X(t)
XL
XU
Z(t)
a
5
6
n
e(t)
o2
0)
wrms
- Phase-Modulation
= Autocorrelation Function
= Real Part of (0
= Radio Freauency
-- Input Signal Power
* Output Signal Power
= Input Signal-to-Noise Ratio
= Input Signal-to-Normalized-Noise Ratio
= Output Signal-to-Noise Ratio
= Suppressed-Carrier
= Upper Single-Sideband
= The "Suppressed-Carrier" Function of U(W)
= The "Suppressed-Carrier" Function of V(W)
= A Real Modulated Signal
= Lower Single-Sideband Modulated Signal
= Upper Single-Sideband Modulated Signal
= m(t) + jm(t) = The Analytic Signal of m(t)
= a Modulation (as defined in the text)
= System Efficiency
= Modulation Index
= Efficiency
= Modulating Signal (when m(t) is not the Modulating Signal)
- Variance
- Average Power of m(t)
= Angular Frequency
= RMS-Type Bandwidth
xi
Alo = Equivalent-Noise Bandwidth
nr = Mean-Type Bandwidth
* = The Convolution Operator
()* = The Conjugate of ()
() = The Hilbert Transform of ()
() = The Averaging Operator
XI 1
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
SYNTHESIS AND ANALYSIS OF REAL
SINGLE-SIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS
By
Leon Worthington Couch, II
June, 1968
Chairman: Professor T. S. George
Major Department: Electrical Engineering
A new approach to single-sideband (SSB) signal design and ana
lysis for communications systems is developed. It is shown that SSB
signals may be synthesized by use of the conjugate functions of any
entire function where the arguments are the real modulating signal and
its Hilbert transform. Entire functions are displayed which give the
SSB amplitude-modulated (SSB-AM), SSB frequency-modulated (SSB-FM),
SSB envelope-detectable, and SSB square-law detectable signals. Both
upper and lower SSB signals are obtained by a simple sign change.
This entire generating function concept, along with analytic
signal theory, is used to obtain generalized formulae for the properties
of SSB signals Formulae are obtained for (1) equivalent realizations
for a given SSB signal, (2) the condition for a suppressed-carrier SSB
signal, (3) autocorrelation function, (4) bandwidth (using various-de
finitions), (5) efficiency of the SSB signal, and (6) peak-to-average
power ratio. The amplitude of the discrete carrier term is found to be
xi i i
equal to the absolute value of the entire generating function evaluated
at the origin provided the modulating signal is AC coupled. Examples
of the use of these formulae are displayed where these properties are
evaluated for stochastic modulation.
The usefulness of a SSB signal depends not only on the pro
perties of the signal but on the properties of the overall system as well.
Consequently, a comparison of AM, SSB-AM, SSB-FM, and FM systems is
made from the overall viewpoint of generation, transmission with additive
Gaussian noise, and detection. Three figures of merit are used in these
comparisons: (1) Output signal-to-noise ratios, (2) Energy-per-bit of
information, and (3) System efficiency.
In summary, the entire generating function concept is a new tool
for synthesis and analysis of single-sideband signals.
xiv
CHAPTER I
INTRODUCTION
In recent years the use of single-sideband modulation has become
more and more popular in communication systems. This is due to certain
advantages such as conservation of the frequency spectrum and larger post
detection signal-to-noise ratios in suppressed carrier single-sideband
systems when comparison is made in terms of total transmitted power,
A single-sideband communication system is a system which generates
a real signal waveform from a real modulating signal such that the Fourier
transform, or voltage spectrum, of the generated signal is one-sided about
the carrier frequency of the transmitter. In conventional amplitude-modu
lated systems the relationship between the real modulating waveform and
the real transmitted signal is given by the well-known formula:
XAM(t) = A0 H + m(t)] cos ojQt |m(t)| <1 (11)
where A() is the amplitude constant of the transmitter
m(t) is the modulating (real) waveform
oj0 is the carrier frequency of the transmitter.
Likewise, frequency-modulated systems generate the transmitted waveform:
Xp^(t) = A0 cos [w0t + D / tm(t")dt'] (1.2)
1
2
where A0 is the amplitude constant of the transmitter
m(t) is the modulating (real) waveform
u)0 is the frequency of the transmitter
D is the transducer constant of the modulator.
Now, what is the corresponding relationship for a single-sideband system?
Oswald, and Kuo and Freeny have given the relationship:
XsSB-AM^) = Ao Cm(t) cos <*>0t m(t) sin co01] (1.3)
where A0 is the amplitude constant of the transmitter
m(t) is the modulating signal
m(t) is the Hilbert transform of the modulating signal
co0 is the frequency of the transmitter [1, 2].
This equation represents the conventional upper single-sideband suppressed-
carrier signal, which is now known as a single-sideband amplitude-modulated
suppressed-carrier signal (SSB-AM-SC). It will be shown here that this is
only one of an infinitely denumerable set of single-sideband signals. In
deed, it will be shown that any member of the set can be represented by
xSSB(t) = Ao [u(mU)> m(t)) cos a)0t + V(m(t), m(t)) sin co0t] (1.4)
where A0 is the amplitude constant of the transmitter
U(x,y) and V(x,y) are the conjugate functions of any entire
function
m(t) is the modulating (real) waveform
m(t) is the Hilbert transform of m(t)
to0 is the transmitter frequency.
Various properties of these single-sideband signals will be analyzed in
3
general for the whole set, and some outstanding members of the set will
be chosen for examples to be examined in detail.
It should be noted that Bedrosian has classified various types of
modulation in a similar manner; however, he does not give a general repre
sentation for single-sideband signals [3],
CHAPTER II
MATHEMATICAL PRELIMINARIES
Some properties of the Hilbert transform and the corresponding
analytic signal will be examined in this chapter. None of the material
presented in this chapter is new; in fact, it is essentially the same
as that given by Papon1 is except for some changes in notation [4]. How
ever, this background material will be very helpful in derivations pre
sented in Chapter III and Chapter' V
The Hilbert transform of m(t) is given by
co
(2.1)
where () is read !lthe Hilbert transforms of ()"
P denotes the Cauchy principal value
* indicates the convolution operation.
The inverse Hilbert transform is also defined by Eo (2.1) except that a
minus sign is placed in front of the right-hand side of the equation. It
is noted that these definitions differ from those used by the mathema
ticians by a trivial minus sign. It can be shown, for example, that the
Hilbert transform of cos oo0t is sin to0t when w0 > 0 and that the Hilbert
transform of a constant is zero, A list of Hilbert transforms has been
compiled and published under work done at the California Institute of Tech
nology on the Bateman Manuscript Project [5],
4
5
The Fourier transform of m(t) is given by
FfiU) = [-j sgn (o)] Fm(to)
(2.2)
where
sgn (u)
+ 1 u) > 0
0 a) = 0
- 1 to < 0_
(2.3)
and Fm(j) is the Fourier transform of m(t). In other words, the Hilbert
transform operation is identical to that performed by a -90 all-pass
linear (ideally non-realizable) network.
From Eq. (2.2), it follows that
F*(u>) = [-j sgn (w)]2 Fm(u>) -Fm(co) (2.4)
or
m(t) = -m(t).
(2.5)
The (complex) analytic signal associated with the real signal
m(t) is defined by
Z(t) = m(t) + jm(t).
(2.6)
The Fourier transform of Z(t) follows by the use of Eq. (2.2),
and it is
Fz(<>) = Fm(o>) + j[-j sgn (w)] Fm(w)
or
6
Fz(ui)'
2Fm(w) oj > 0
Fm(w) ai = 0
0 oj < Q
(2.7)
Now suppose that m(t) is a stationary random process with auto
correlation Rmm([) anc* power spectrum Pmm(u))o Then the power spectrum
of m(t) is
Fmm(w) = pmm(w) l~J s9n (w) I Pmmi^)- (2.8)
This is readily seen by use of the transfer function of the Hilbert trans
form operator given by Eq. (2.2). Then, by taking the inverse Fourier
transform of Eq. (2.8), it follows that
Rmm(T) ~ Rmm(1)
(2.9)
The cross-correlation function is obtained as follows:
Rmm(T) = (t + T)m(t)
m(t + i A)m(t)dA
TT A
Rmm(t-A)dA
where () is the averaging operator. Thus
Rmm(i~ Rmm(T)
(2.10)
7
It follows that the spectrum of the cross-correlation function is given by
Pmm() ~ l-"j sgn (w)3 PfTirii(tlJ) (2.11)
it is noted that Pmm(w) is a purely imaginary function since Pmm(w) is a
real function. Then
00
Rmm(c) ~ T7 f (w)] Pmm(uduJ
l~ '* *00
which5 for Prnm(u)) a real even function, reduces to
Rmm( i) = J Pmm(^) s,n wt du) (2/12)
11 0
Thus the cross-correlation function is an odd function of r:
fynm(T) ~ Rmm(_T)
= -rh(t T)m(t) -m(t + i)m(t) (2/13)
or
wo = -W-O = -WO. (2.14)
The autocorrelation for the analytic signal is found as follows:
RZZ( ) = Z(tH)Z*(iy
= [m(t+-r) + jm(t+x)] [m(t) jm(tj]
= m(t+i)rn(t) + m(t+r)m(t) + jm(t+: )m(t) jm(t+r)m(t)
- Rmm() f Rmm(T) + JRmm(1) jRmm(TK
8
Using Eqs. (2.9), (2.10) and (2.14) we obtain
Rzz(t) = 2[Rmm(t) + J Rmm(T)J = 2[Rmm(T) + jRmmU)]. (2.15)
Thus (1/2)R2(t) is an analytic signal associated with Rmm(-t). By use of
Eq. (2.7) it follows that
4Pmm(a>) w > 0
Pzz^)
2Pmm(a)) > w = 0
V
3
o
1
(2 16)
CHAPTER III
SYNTHESIS OF SINGLE-SIDEBAND SIGNALS
Eq. (1.4), which specifies the set of single-sideband signals
that can be generated from a given modulating waveform or process, will
be derived in this chapter. The equation must be a real function of a
real input waveform, m(t), since it represents the generating function
of a physically realizable system--the single-sideband transmitter--and,
in general, it is non-linear. Analytic signal techniques will be used
in the derivation. It will be shown that if we have a complex function
k(z) where k(z) is analytic for z = x + jy in the upper half-plane (UHP),
then the voltage spectrum of k(x,0) k(t) is zero for w < 0. In order
to synthesize real SSB signals from a real modulating waveform, an UHP
analytic generating function of the complex time veal modulating process
must be found regardless of the particular (physically realizable) wave
form that the process assumes.
Let m(t) be either the real baseband modulating signal or a veal
function of the baseband modulating signal e(t), Then the amplitude of
the voltage spectrum of m(t) is double sided about the origin, for ex
ample, as shown by Figure 1.
9
i o
Since m(t) is generated by a physically realizable process, it con
tains finite power for a finite time interval This, of course, is equiva
lent to saying that m(t) is a finite energy signal or, in mathematical
terms, it is a L2 function. By Theorem 91 of Titchmarsh, m(t) is also a
member of the L2 class of functions almost everywhere [6]. Now the complex
signal Z(t) is formed by
Z(t) = m(t) + jm(t). (3,1)
It is recalled that Z(t) is commonly called an analytic signal in the
literature. By Theorem 95 of Titchmarsh there exists an analytic func
tion (regular for y > 0), Z^z), such that as y -* 0
Zj(x + jy) Z(t) = m(t) + jm(t) x = t
for almost all t and, furthermore, Z(t) is a L2 (-*>, function [6]
It follows by Theorem 48 of Titchmarsh that the Fourier transform of Z(t)
exists [6],
Theorem I: If k(z) is analytic in the UHP then the
spectrum of k(t,0), denoted by F^oj), is zero for
all to < 0, assuming that k(t,0) is Fourier trans
formable.
For a proof of this theorem the reader is referred to Appendix I.
Thus the voltage spectrum of Z(t) is zero for ui < 0 by Theorem I
since Z(t) takes on values of the analytic function Z,(z) almost every
where along the x axis. Furthermore, since Z(t) is an analytic signal--
that is, it is defined by Eq. (3=1)its voltage spectrum is given by
] 1
Eq (2.7), which is
Fz(uj)
2FmU)
03 >
F rn (0)
9 U3 =
0
5 0) <
(3 2)
where Fm(a)) is the voltage spectrum of the signal m(t), This is shown in
Figure 2 for our example used in Figure 1
Now let a function g(W) be given such that
g(W) i U(ReW,ImW) + jV(ReW,ImW)
(3.3)
where g(W) is an entire function of the complex variable W.
Theorem II: If Z(z) is an analytic function of z in
the UHP and if g(W) is an entire function of W, then
g[Z(z)] is an analytic function of z in the UH z-plane,
A proof of this theorem may be found in Appendix I.
Thus g[Z.(z)] is an analytic function of z in the UH z-plane, and
by Theorem I g[Z(t)] has a single-sided spectrum: the spectrum being
zero for w < 0. This is illustrated by Figure 3, where Fg(w) = F[g(Z(t))].
12
Figure 3. Voltage Spectrum of an Entire Function
of an Analytic Signal
Now multiply the complex baseband signal g[Z(t)] by eJuJot to
translate the signal up to the transmitting frequency, oi0. It is noted
that g[Zj(z)] and eju)z for ojq > 0 are both analytic functions in the UH
z-plane By the Lemma to Theorem I in Appendix I, g[Z; (z)]ev,l'Uo2 is ana
lytic in the UH z-plane. Therefore, by Theorem I the voltage spectrum
of g[Z(t)]eJU)ot is one sided about the origin. Furthermore,
FCgUltiJeK4] = -LF[g(z(t))] *
(- 7T
= Fg (oo) 6(w-o)0)
or
F[gU(t))e^ot] = Fg(ld-u)q) o>0 > 0 (3.4)
This spectrum is illustrated in Figure 4.
13
\F[q(l(t))e^^} i
Figure 4 Voltage Spectrum of the Positive Frequency-
Shifted Entire Function of the Analytic Signal
The real upper single-sideband signal can now be obtained from
the complex single-sideband signal, g[Z(t)]e'-*a,ot, by taking the real
part. This is seen from Theorem III
Theorem III, If h(z) is analytic for all z in the
UHP and F[h(x,0)] e Fh(w), then for > 0,
F{Re[h(x,0)eJwox]}
^sFp-! (oCQ ) O) > 0)g
o
0) < 0),.
J O),.
This theorem is proved in Appendix I.
Thus the upper single-sideband signal for a given entire function is
XUSSB(t) = ReigCZUne^}
= Re{[U(ReZ(t) ,lmZ(t)) + jV^ReZ(t),lmZ(t))]eJwot}
= Re{[U(m(t) ,m(t)) + jV(mU) *m(t J)]eJa)^t}
14
or
XuSSB^t) = U(m(t) ,rn(t)) cos w0t V(m(t) ,m(t)) sin oj0t
(3.5)
where U(ReW,ImW) is the real part of the entire function g(W)
V(ReW,ImW) is the imaginary part of g(W)
m(t) is either the modulating signal or a real function of the
modulating signal e(t)
m(t) is the Hilbert transform of m(t).
Using Theorem III the voltage spectrum of Xu$$g(t) is
FUXU) = ^[XusSBt)]
%Fg(w-tQ) to >
, j co j < to.
lgFg(-w-aJ0) a) < -u0
(3.6)
This spectrum is illustrated by Figure 5.
The lower single-sideband signal can be synthesized in a similar
manner from the complex baseband signal Now we need to translate the
complex baseband signal down to the transmitting frequency instead of up,
15
as in the upper single-sideband synthesis. Then the Fourier transform of
the down-shifted complex baseband signal is
F[g(St))e-ot] F[g(Zt))] *
- Fn {W ) (u>+\on)
or
[g(Z(t))ej)ot] = Fg(u+u0) a), > 0. (3,7)
This spectrum is illustrated in Figure 6.
Figure 6. Voltage Spectrum of the Negative Frequency-
Shifted Entire Function of an Analytic Signal
Theorem IV: If h(z) is analytic for all z in UHP and
F[h(x,0)] e F^(oj) where Fh(fi) = 0 for all n > go0, then
for oj0 > 0
y{Re[h(x,0)e'JUJx]
^(-W+uJg)
0
0 < j < )q
M > uo
JgFh (tD+u)0}
0 > oo > -
This theorem is proved in Appendix I.
16
Thus the real lower single-sideband signal for a given entire
function is
XLSSB(t) = Re{g(Z(t))eJVc}
= Re [U(m(t) ,m(t)) + jV(m(t) ,rii(tj )]e"Ju3ot)
or
XLSSB^^ = U(m(t) ,m(t)) cos co0t + V(m(t) ,m(t)) sin u>0t.
(3.8)
Using Theorem IV the voltage spectrum of XL<^g(t) is
Flx(o>) i[XLSSB(t)]
JgFg (-(J+-a)0 ) 5 0 < 0) < COq
0 | o ¡ > Wq
hFq ()+)q ) 0 > a) > to
0
(3.9)
It is noted that the requirements that Fg(oj) be zero for m > w0 is to
prevent spectral overlap at the origin. This requirement is satisfied
(for all practical purposes) for ojq at radio frequencies.
The spectrum of FBx(^) is illustrated by Figure 7.
Figure 7. Voltage Spectrum of the Synthesized
Lower Single-Sideband Signal
17
To summarize, it has been shown that once an entire function g(W)
is chosen, then an upper or lower single-sideband signal can be obtained
from the signal m(t) The signal m(t) is either the modulating signal
or a real function of the modulating signal e(t). The generalized ex
pressions, which represent SSB signals, are given by Eq (35) for the
USSB signal and by Eq, (3,8) for the LSSB signal. These expressions are
obviously the transfer functions that are implemented by the upper and
lower single-sideband transmitters respectively. Since there are an in
finitely denumerable number of entire functions, there are an infinitely
denumerable number of upper and lower single-sideband signals that can be
generated from any one modulation process, In Chapter IV some specific
entire functions will be chosen to illustrate some well-known single
sideband signals.
CHAPTER IV
EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN
Specific examples of upper single-sideband signal design will now
be presented* Entire functions will be chosen to give signals which have
various distinct properties. In Chapter VI these properties will be ex
amined in detail. Only upper sideband examples are presented here since
the corresponding lower sideband signals are given by the same equation
except for a sign change (Eq. (3.5) and Eq. (3.8)).
4.1. Example 1: Single-Sideband AM With Suppressed-Carrier
This is the conventional type of single-sideband signal that is
now widely used by the military, telephone companies, and amateur radio
operators. It will be denoted here by SSB-AM-SC,
Let the entire function be
9l(W)=W (4.1)
and let m(t) be the modulating signal. Then substituting the corresponding
analytic signal for W
gx(Z(t)) = m(t) + jm(t)
or
(m(t),iii(t)) = m(t) and Vx(m(t),m(t)) = m(t). (4.2 a,b)
18
19
Substituting Eqs. (4.2a) and (4.2b) into Eq. (3.5) we obtain the
upper single-sideband signal:
XUSSB-AM-SC^ cos o1 m(t) sin Jot
(4.3)
where m(t) is the modulating audio or video signal and m(t) is the Hil
bert transform of m(t). It is assumed that m(t) is AC coupled so that
it will have a zero mean.
The upper single-sideband transmitter corresponding to the gene
rating function given by Eq, (4.3) is illustrated by the block diagram
in Figure 8. It is recalled that this is the well-known phasing method
for generating SSB-AM-SC signals [7, 8],
4.2. Example 2: Single-Sideband PM
Single-sideband phase-modulation was described by Bedrosian in
1962 [3].
To synthesize this type of signal, denoted by SSB-PM, use the
entire function:
(4.4)
Let m(t) be the modulating audio or video signal. Then
g2(Z(t)) = eJ(m^ + ^(t)) =
or
U2(m(t),iii(t)) = e-"i(t)
cos m(t)
(4.5a)
Figure 8 Phasing Method for Generating USSB-AM-SC Signals
21
and
V2(m(t) ,m(t)) = e"rn(t) sin m(t).
(4.5b)
Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper
single-sideband signal:
or
XyssB-PM^) = e~m^ cos K1 +
(4.6)
It is again assumed that the modulation m(t) is AC coupled so that its
mean value is zero. The single-sideband exciter described by Eq. (4.6)
is shown in Figure 9.
4.3. Example 3: Single-Sideband FM
Single-sideband frequency-modulation is very similar to SSB-PM
in that they are both angle modulated single-sideband signals. In fact
the equations for SSB-FM are identical to those given in Section 4.2 ex
cept that
t
(4.7)
where e(t) is now the modulating signal (instead of m(t)) and D is the
transducer constant.
Experiments with SSB-FM signals have been conducted by a number
of persons and are reported in the literature [9, 10].
Figure 9. USSB-PM Signal Exciter--Method I
23
The SSB-FM exciter as described by Eqs. (4.6) and (4.7) is given
in Figure 10.
4.4. Example 4; Singlet-Sideband a
The term single-sideband a (SSB-a) will be used to denote a sub
class of single-sideband signals which may be generated from a particular
entire function with a real parameter a. This notation was first used by
Bedrosian [3],
Let the entire function be
g3W eaW
(4.8)
where a is a real parameter, and let
m(t) = 1n[1 + e(t)]
(4.9)
where e(t) is the video or audio rnodulation signal which is amplitude
limited such that |e(t)| <1. It is assumed that m(t) is AC coupled
(that is, it has a zero mean). Note that these assumptions are
usually met by communications systems since they are identical to the
restrictions in conventional AM modulations systems. Then
g3(Z(t)) e'
1a[m(t)+jm(t)]= eam(t) ejam(t)
or
am(t) cos (am(t))
U3(m(t),m(t) = e
(4.10a)
Figure 10. USSB-FM Signal Exciter
25
and
V3(m(t)9m(t)) = em^ sin (am(t)). (4,10b)
Substituting Eqs. (4.10a) and (4.10b) into Eq. (3.5), the SSB-a signal is
XUSSB-a^ eam^ cos (om(t)) cos w0t
- eam^ sin .(am(t)) sin ajQt
or
XUSSB-a^ ~ cos (0t +
(4.11)
In terms of the input audio waveform, Eq. (4.11) becomes
XUSSB-a(t) = ealn[1+e(t)] cos ()Qt + an[l+e(t)])
or
XUSSB-a^ = [l+e(t)]a cos u0t + a1n[1+e(t)]).
(4.12)
For a = 1 we have an envelope-detectable SSB signal, as is readily
seen from Eq. (4.12). Voelcker has recently published a paper demon
strating the merits of the envelope-detectable SSB signal [11]. The real
ization of Eq. (4.12) is shown in Figure 11.
For a = 1/2 we have a square-law detectable SSB signal. This type
of signal has been studied in detail by Powers [12]. Figure 12 gives the
block-diagram realization for the square-law detectable SSB exciter.
Figure 11. Envelope-Detectable USSB Signal Exciter
e(t)
Figure 12. Square-Law Detectable USSB Signal Exciter
CHAPTER V
ANALYSIS OF SINGLE-SIDEBAND SIGNALS
The generalized SSB signal, that was developed in Chapter III,
will now be analyzed to determine such properties as equivalent gener
alized SSB signals, presence or absence of a discrete carrier term,
autocorrelation functions, bandwidths, efficiency, and peak-to-averajge
power ratio. Some of these properties will depend only on the entire
function associated with the SSB signal, but most of the properties will
be a function of the statistics of the modulating signal as well.
5.1. Three Additional Equivalent Realizations
Three equivalent ways (in general) for generating an upper SSB
signal will now be found in addition to the realization given by Eq. (3.5).
Similar expressions will also be given for lower SSB signals which are
equivalent to Eq. (3.8). It is very desirable to know as many equivalent
realizations as possible since ally orle of them might be the most econom
ical to implement for particular SSB signal.
Theorem V: If h(x,y) = U(x,y) + jV(x,y) is
analytic in the HP (including UH) then
h (t ,0) = U(t,0) + jiOU^+kj] (5.1)
or
h(t,0) = [4(t,0)+k2] + jV(t,0) (5.2)
28
29
or
h(t,0) = [-V(t,0)+k2] + J[U(t#0)+k1] (5.3)
where
. TV
k. = lim i V(R cos e,R sin e)de a real constant (5.4)
v R-* 0
TV
k2 = lim | U(R cos e,R sin e)de a real constant (5.5)
" R-X JQ
A proof of this theorem is given in Appendix I.
Theorem V may be applied to the generalized SSB signal by letting
h(z) = g(Zx(z)) where g(*) is an entire function of () ^(z) is analytic
in the UHP, and lim Zx(z) = lim Z (t + jy) = m(t) + jm(t). Thus Theorem V
y+0 y ">0
gives three additional equivalent expressions for g(Z(t)) in addition to
g(z(t)) = U(m(t),m(t)) + jV(m(t),m(t)) (5.6)
which was used in the derivation in Chapter III. Therefore, following
the same procedure as in Chapter III, equivalent upper SSB signals may be
found. Using Eq. (5.1) we have for the first equivalent representation
of Eq. (3.5):
xUS$B(t) = Re{g(Z(t))ej,Ot}
= Re{g(m(t),m(t))e^o^}
= Re [U(m(t) ,m(t) + jU(m(t) ,m(t)) + jk1]eJ'ot}
XyssB^) U(m(t),m(t)) cos w0t [U(m(t) ,m(t)) + kj sin taQt.
(5.7)
or
30
Using Eq. (5.2) the second equivalent representation is
XUSSB^ = t-V(m(t),m(t))+k2] cos w0t V(m(t),m(t)) sin w0t.
(5.8)
Using Eq. (5.3) the third equivalent representation is
XUSSB^ = iv(m(t;) iti(t))+k23 cos (o0t [U(m(t) ,m(t))+k1], sin u)Qt
(5.9)
Likewise the three lower SSB signals, which are equivalent to
Eq. (3.8), are
It should be noted, however, that if for a given entire function
k: and k2 are both zero, then all four representations for the USSB or
the LSSB signals are identical since by Theorem V, U = -V and V = 0 under
these conditions.
5.2. Suppressed-Carrier Signals
The presence of a discrete carrier term appears as impulses in
the (two-sided) spectrum of transmitted signal at frequencies w0 and -
The impulses may have real, purely imaginary, or complex-valued weights
depending on whether the carrier term is cos a)0t, sin u)0t, or a com
bination of the two. Thus the composite voltage spectrum of the modulated
31
signal consists of a continuous part due to the modulation plus impulse
functions at u0 and -o>0 if there is a discrete carrier term. As defined
here, the "continuous" part may contain impulse functions for some types
of modulation, but not at the carrier frequency. Taking the inverse
Fourier transform of the composite voltage spectrum it is seen that if
there is a discrete carrier term, the time waveform must be expressible
in the form:
X(t) = [f^tj+cj cos ai0t [f2(t)+c2] sin iOgt (5.13)
where c1 and c are due to the discrete carrier
f^t) and f2(t) are due to the continuous part of the spectrum
and have zero mean values.
Thus Eq. (5.13) gives the condition that c2 and Cj are not both zero if
there is a discrete carrier term.
To determine the condition for a discrete carrier in an upper
SSB signal, Eq. (5.13) will be identified with Eq. (5.9), which represents
/V A
the whole class of upper SSB signals. It is now argued that both U and V
have a zero mean value if the modulating process is stationary. This is
seen as follows:
U(m(t),£(t))-lp f df .
But U[m(t'),m(t')] = c, a constant, since m(t) is wide-sense stationary.
Thus
00
U(m(t) ,m(t)) = ^ P J'
00
dt' = 0.
32
Likewise V has a zero mean value. Then, identifying Eg. (5.13) with
Eq. (5.9), it is seen that
fx(t) S -V(m(t),m(t))
(5.14a)
f2(t) = U(m(t),m(t))
(5.14b)
1 = k2 and c2 = k .
(5.14c ,d)
Similarily, for lower SSB signals Eq. (5.13) can be identified
with Eq. (5.12).
Thus the SSB signal has a discrete carrier provided that kx and k2
are not both zero.
As an aside, it is noted that the criterion for a discrete car
rier, given by Eq. (5.13), is not limited to SSB signals; it holds for
all modulated signals. For AM Eq. (5.13) can be identified with Eq. (1.1)
Here
fj(t)
= A0m(t)
(5.15a)
f2(t)
5 0
(5.15b)
5 Ao
and c2 = 0
(5.15c,d)
because m(t) has a zero mean due to AC coupling in the modulator of the
transmitter. Thus for AM it is seen that there is a discrete carrier
term of amplitude c1 = AQ which does not depend on the modulation. For FM
Eq. (1.2) can be expanded. Then, assuming sinusoidal modulation of fre
quency wa, we obtain
33
XPM(t) = tAn cos (zr cos wat)] COS u)nt
'0 'Wa a*'-1 -0'
- [An sin (cos to t)] sin wnt.
(5.16)
To identify Eq. (5.16) with Eq (5.13) we have to find the DC terms of
and
f. (t) + c. e An cos (cos coat)
i i o wa a
and
f0(t) + c2 = A0 sin (- cos u)at).
wa
These are
c, = An cos ( cos w,t)
i u M- a
Ao
C D
/ cos ( cos wat)dt
v/ 0) a
Vo<£>
(5.17a)
c. = A sin ( cos wat)
2 o>a
_ A0
T o
0
J sin (^~ cos aiat)dt
(5.17b)
Then for sinusoidal frequency-modulation it is seen that the discrete
carrier term has an amplitude of AQJ0(D/ul^) which may or may not be zero
depending on the modulation index D/wa. Consequently, for FM it is seen
that the discrete carrier term may or may not exist depending on the
modulation. Prof. T. S. George has given the discrete carrier condition
34
for the case of FM Gaussian noise [13].
Continuing with our SSB signals, it will now be shown that kx and
k2 depend only on the entire function associated with the SSB signal and
not on the n]odulation. From Theorem IV we have
IT
k = lim V[m,(R cos e,R sin e) m. (R cos e,R sin e)]de
1 IT D J 1 i
R-* o
and
IT
k, = lim / U[m.,(R co$ e,R sin e) m(R cos e,R sin )]de
17 R^ o
where U and V are the real and imaginary parts of the entire function
l1{z) = m^z) + jm^z) is the analytic function associated
with the analytic signal Z(t) of m(t).
It is seen that if
lim m (R cos e,R sin e) = 0 0 < e < v (5.18a)
R-x
and
lim mJR cos e,R sin e) = 0 0 < e < tt (5.18b)
R-*
then kx and k2 depend only on U and V of the entire function and not on
m. Thus we need to show that Eq. (5.18a) and (5.18b) are valid. By the
theory of Chapter III there exists a function Z^z) = m^z) + jm^z)
which is analytic in the UHP such that (almost everywhere) 11^ Zx(t + jy)
y
= Z(t) = m(t) + jm(t) where the Fourier transform of Z(t), F(ai), is
l2(-, ). Then we have
It follows that
35
CO
T-i (z) = '2~ ^ F())e^Zwd).
'00
lim |Z1(Rej0)|2
R-H
oo
= lim (-)2| ^ [FU)][e~(R sln 0)uej(R cos e)]du,
R -Xx>
A
By use of Schwarz's inequality this becomes
lim |Z (ReJ0) |2
R-*
(f)2c
'll
00 CO
r |F(a)) |2doj} {lim f e2(R sin e)Wh
0 R- 0
But F(to) e L2 (-5 ) so that
/
| F(to) 12dto < K.
Also
lim
R 7-00
e-(2R sin e)du)
0 < 0 < TT .
Therefore we have
lim 1Z, (ReJ6) | < {-f K 0 = 0 ,
R
0 < 0 < 1T .
36
For e = 0 or 0 = tt
Z (+>) ,0 = 0
Tim ¡Z (Reje)
R->~
Z(-oo)
= 0
since
Z(t) e L (-*>, ).
Then
lim |Z1(ReJ0)| = 1 im |Z?L(R cos 0, R sin 0) | = 0 0 < 0 < rr
R-x Rx
which shows that Eqs. (5.18a) and (5.18b) are valid statements. Thus,
the presence (kx and k2 not both zero) or the absence (kx = k2 = 0) of
a discrete carrier depends only on the entire funtion associated with
the SSB signal and not on the modulation. Furthermore, it is seen that
the amplitude of the discrete carrier is given by the magnitude of the
entire function evaluated at the origin (of the W plane), and the power
in the discrete carrier is one-half the square of the magnitude.
For every generalized USSB signal represented by Eq. (3.5),
there exists a corresponding sppressed-carrier USSB signal:
XUSSB-SC^ = ,^Km('t) ,m(t)) cos w0t ¥-(m(t) ,m(t)) sih wot
(5.19)
37
where the notation SC and denote the suppressed-carrier functions.
But what are these functions tt and ¥? The condition for a suppressed
carrier is that kx = k2 = 0. By comparing Eq. (5.19) with Eq. (5.9) it
follows that it = -V and if- = 0. Furthermore by Theorem V of Section 5.1,
U = -V + k2 and V = + kx. Thus
it = -V = U k2 (5.20)
and
Â¥ = U = V kr (5.21)
It is also noted that it and ¥ are a unique Hilbert transform pair. That
is, ¥ is the Hilbert transform of it, and it is the inverse Hilbert trans
form of 3t. This is readily shown by taking the Hilbert transform of
Eq. (5.20), comparing the result with Eq. (5.21), and by taking the in
verse Hilbert transform of Eq. (5.21), comparing the result with Eq. (5.20).
Thus Eq. (5.19) may be re-written as
XUSSB-SC^ = ,m(t)) cos w0t it(m(t),m(t)) sin wot
(5.22)
or
XUSSB-SC^ = "^(t) ,m(t)) cos w0t V
(5.23)
where it and ¥ are given by Eq. (5.20) and Eq. (5.21).
It is interesting to note that the form of the USSB signal given
_ *
above checks with the expression given by Haber [14]. He indicates that
if a process n(t) has spectral components only for |w| > wq then n(t)
can be represented by
38
n(t) = s(t) cos wot s(t) sin wot. (5.24)
Thus Eq. (5.22) checks with Eq, (5.24) where it = s(t), and Eq. (5.23)
checks also where -V e s(t).
The corresponding representations for LSSB suppressed-carrier
signals are given by
XLSSB-SC^ = -y-(m(t),rn(t)) cos w0t + tt(m(t),m{t)) sin w0t
(5.25)
and
XLSSB-SC^ = "'^'(m(t) ,m(t)) cos w0t + ¥(m(t) ,m(t)) sin w0t
(5.26)
where it and-V-are given by Eq. (5.20) and Eq. (5.21).
This representation also checks with that given by Haber for pro
cesses with spectral components only for |w| < wo which is
n(t) = s(t) cos wot + §(t) sin w0t.
(5.27)
5.3. Autocorrelation Functions
The autocorrelation function for the generalized SSB signal and
the corresponding suppressed-carrier SSB signal will now be derived.
Using the result of Chapter III, it is known that the generalized
upper SSB signal can be represented by
XUSSB
j(w0t+)}
(t) = Re{g(m(t),m(t))e'
(5.28)
39
where a uniformly distributed phase angle 4> has been included to account
for the random start-up phase of the RF oscillator in the SSB exciter.
Then, using Middleton's result [15], the autocorrelation of the USSB sig
nal is
RXU^ XUSSB^t+T^XUSSB^t^ %Re{eJa)TRg(T)}
(5.29)
where
Rg(t) = g(m(t+x),m(t+T)) g*(m(t),m(t)) (5.30)
and
g(m(t),m(t)) = U(m(t),m(t)) + jV(m(t) ,m(t)). (5.31)
The subscript XU indicates the USSB signal. For the generalized LSSB
signal the corresponding formulae are
XL$SB(t) = Re{g(m(t),m(t))e"J^t+
(5.32)
and
Rxl(t) = %Re{e"J"wTRg(T)}.
(5.33)
These equations can be simplified if we consider the autocorre
lation for the continuous part of the spectrum of the SSB signal. The
suppressed DC carrier version of g, denoted by g$c> will first be found
in terms of g, and then the corresponding autocorrelation function Rg_sc(T)
40
will be determined in terms of Rg(-r).
By examining Eq. (5.19) and comparing this equation to Eq. (3.5),
with the aid of Eq. (3.3) it is seen that the suppressed DC carrier
version of g is given by
gsc(m(t),m(t)) = if(m(t),m(t)) + j¥-(m(t) ,m(t)) (5.34)
where -0-and M- are the suppressed-carrier functions defined by Eq. (5.20)
and Eq. (5.21). Then it follows that
g(m(t),m(t)) = gsc(m(t) ,m(t)) + [k2+jk1]0 (5.35)
It is noted that the mean value of ggQ is zero. This is readily seen via
Eqs. (5.34), (5.20), and (5.21) since it is recalled that the mean value
of 0 and V was shown to be zero in Section 5.2. Then, using Eq. (5.35),
the autocorrelation of g is obtained in terms of the autocorrelation of
9sc;
Rg(0 = Rg-sc(x) + (kiZ+k22)
(5.36)
Therefore the autocorrelation functions for the USSB signal,
Eq. (5.29), and the LSSB signal, Eq. (5.33), become
Rxu(T) = ^(^[(k^2) + Rg-SC (T 1
and
Rxl(i) = %Re{eJwoT[(kl2+k22) + Rg_sc(x)]}o
(5.37)
(5.38)
i
41
It may be easier to calculate the autocorrelation for the USSB or
LSSB signal using this representation rather than that of Eq (5.29) and
Eq. (5.33) since Rg_$c(x) may be easier to calculate than Rg(x). This
is shown below.
A simplified expression for Rg_$c(T) will now be derived. First,
it is recalled from Section 5.2 that if and -V- are a unique Hilbert trans
form pair. Thus g$c given by Eq. (5.34), can be expressed in terms of
two analytic signals:
gsc(m(t),m(t)) = tf(m(t) ,m(t)) + jit(m(t) ,m(t)) (5.39)
and
gsc(m(t),m(t)) = -¥-(m(t) ,m(t)) + j¥-(m(t) ,m(t)) (5.40)
where Eq. (5.39) is the a^lytic signal associated with -H-and Eq. (5.40)
is the analytic signl associated with -Â¥. Using Eq. (5.39) and Eq. (2.15),
the autocorrelation of g5g is given by
Rg-SC(T) 2[:Rw(t) + jRw(r)] (5.41)
or by using Eqs. (5.40), (2.15), and (2.9)
Rg-SC(T) = 2[rw(t) + jRw(t)]. (5.42)
Thus Rg_$c(x) may be easier to calculate than Rg(x) since only Rw(x) or
RyyM is needed. This, of course, is assuming that the Hilbert trans-
42
form is relatively easy to obtain On the other hand Rg(x) may be calcu
lated directly from g(m(t),m(t)) or indirectly by use of Ry^t), Rvv(x),
Ruv(t), and Rvu(t).
The autocorrelation functions for the generalized USSB and LSSB
signals having a suppressed-carrier are readily given by Eq. (5.37) and
Eg. (5.38) with kj = k, 0:
(5.43a)
(5.43b)
(5.43c)
and
(5.44a)
(5.44b)
(5.44c)
It follows that the power spectral density of any of these SSB
signals may be obtained by taking the Fourier transform of the appro
priate autocorrelation function presented above.
5.4. Bandwidth Considerations
The suppressed-carrier autocorrelation formulae developed above
will now be used to calculate bandwidths of SSB signals. It is noted
RXL-SCTRg.sc(r)}
Rw(t) cos O)0x + R^t) sin 000
W
Rw(t) cos UQT + Rw(x) sin CjOq '
RXU-SC^ %Re{eJTRg_sc(T)}
- Ryy.(t) cos o>qT Ryy( :) sin coot
w t) cos wot Ryy(x) sin u)0t
43
that the suppressed-carrier formulae are needed' instead of the "total sig
nal" formulae since, from the engineering point of view, the presence or
absence of a discrete carrier should not change the bandwidth of the sig
nal Various definitions of bandwidth will be used [16, 17],
5.4-1. Mean-type bandwidth
Since the spectrum of a SSB signal is one-sided about the carrier
frequency, the average frequency as measured from the carrier frequency
is a measure of the bandwidth of the signal:
f wPg_ScU)du j Rg_sc()
oo
- = *' (5 45)
CO v '
f Pg-SC^^ Rg-SC(O)
00
where Pg_^c(w) is the power spectral density of g$c(m(t) ,m(t))and the
prime indicates the derivative with respect to t. The relationship is
valid whenever Rg_sg(0) and Rg_$c^ exist. Substituting Eg. (5.41)
into Eq. (5.45) we have
f[R' (0) + JR' (0)]
j trtr trtr
" a'1 o
2[RW(0) + jRyyjO)]
But it recalled that Ryy(x) is an even function of t and, from Chapter II,
RyU(t) is an odd function of t. Then Ryy(O) = Ryy(O) = 0 and it follows
that
Ryy.(0) Rw(o)
Ryy-(o) Rw(o)
(5.46)
44
It is noted that this formula is applicable whenever Ryy(O) and Ryy(O) or
Rw(0) and R^(0) exist. That is, RyyJO), RW(Q), Ry.(0), and R^_(0) may
or may not exist since ^g_sc^T) 1S ana^ytic almost everywhere (Theorem 103
of Titchmarsh [6]).
5.4-2.
RMS-type bandwidth
The rms bandwidth, wrms, may also be obtained.
oo
f pg-SC()d Rg-SC^0^
oo
(5.47)
Substituting Eq. (5.41) once again, we have
(rms)
-2[Ryy_(0) + jRyy.(0) ]
2[RW(0) + j%(0)]
Since Ryy(x) is an odd function of x, Ryy(O) = Ryy(O) =0, and we have
2
(rms)
-R^(0)
o
J
1
Rw(0)
c
o
w) j
-R^.(0)
rms J
c
o
i
c
o
(5.48)
It is noted that this formula is applicable whenever R^y.(0) and Rw(0)
or R^O) and R^(0) exist.
45
5*4-3. Equivalent-noise bandwidth
the equivalent-noise bandwidth, Aw* for the continuous part of
the power spectrum is defined by
(2aw)
27 pg-scC0)
2tt
Pg-SC(w)du = Rg_sc()
(5.49)
But
Thus
00
Pg-Sc(O) = f Rg-Sc(T>dT
on 00
(Aco) =
g-sc
(0)
Rg-SC(-r)di
Substituting for Rg_sc(T) by using Eq. (5.41) or Eq. (5.42) we
obtain (noting once again that R^Ct) is even and Ryy(-r) is odd)
(5.50)
5.5. Efficiency
A commonly.Used definition of efficiency for modulated signals
is [18]
n = Sideband Power/Total Power.
5.51
46
This definition will be used to obtain a formula expressing the efficiency
for the generalized SSB signal. Using Eq. (5.43) and Eq. (5.44) the side
band power in either the USSB or LSSB signal is
Rxu_sc(o) rxl-sc(0) = W) = y) (5.52)
It is also noted that Rg-SC(O) is not equal to the total power in the
real-signal sidebands since ggQ is a complex (analytic) baseband signal;
instead, (1 /2)Re[Rg_^(-.(0)3 Ryy(O) = Ryy(O) is the total real-signal
power. This is readily seen from Eq. (5.43a) and Eq. (5.44a).
Similarily the total power in either the USSB or LSSB signal is
obtained from Eq. (5.37) or Eq. (5.38):
Rxu(0) = rxl(0) = %[ki2 + k22 + 2W0)]
= Hlk* + k22 + 2RW(0)] (5.53)
Thus the efficiency of a SSB signal is
2Rnn(0) 2RW(0)
TTCT VTv"
n = ----- = 0 (5.54)
kx2 + k2 + 2Rm(0) k;|2 + k22 + 2RW(0)
5.6. Peak-to-Average Power Ratio
The ratio of the peak-average (over one cycle of the carrier-
frequency) to the average power for the SSB signal may also be obtained.
The expression for the peak-average power over one carrier-
frequency cycle of a SSB signal is easily obtained with the aid of
Eq. (3.5) and Eq. (3.8). Recalling that U and V are relatively slow
time-varying functions compared to cos ta0t and sin wot, we have for the
peak-average power:
47
Pp_Av = %{[U(m(t),m(t))]2 + [V(m(t),m(t))]/}I
P 't t,
(5.55)
where tpea(< is the value of t which gives the maximum value for Eq. (5.55).
Using Eq. (5.20) and Eq. (5.21), Pp_Av can also be written as
P Av [4M- k2]2 [* + k,]2) |
p peak
2 2
= ^(Du- + k9] + [ti- + k.] } L .
r Vak
= %{[4+ k]2 + |> + kL]2}11 t
1 xpeak .
The average power of the SSB signal was given previously by Eq. (5.53).
Thus the expression for peak-to-average power ratio for the generalized
SSB signal is
(5.56a)
(5.56b)
(5.56c)
(5.56d)
Several equivalent representations have been given for peak-to-average
power since one representation may be easier to use than another for a
particular SS>B signal.
Pn-Av f[U(m(t).ii(t))]2 + [V(m(t),m(t))]2) |t t
= _____ Lpeak
PAv k;i2 + k2 + 2R(JU(0)
{[U(m(t),m(t))]2 + [V(m(t),m(t))]2} L ,
k,2 + k22 + 2RW(0)
{[Wm(t).m(t))+k,]2 + [ |t t
= ^ 1 ~ Lpeak
kj2 + k22 + 2Rw(0)
{[-Â¥(m(t),m(t))+k2]2 + [Â¥-(m(t),m(t))+k1]2} |
= t ''peak.
k;i2 + k22 + 2RW(0)
CHAPTER VI
EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS
The examples of SSB signals that were presented in Chapter IV
will now be analyzed using the techniques which were developed in
Chapter V.
6.1. Example 1: Single-Sideband AM With Suppressed Carrier
The constants k1 and k2 will first be determined to show that
indeed we have a suppressed carrier SSB signal. By substituting
Eq. (4.2b) into Eq. (5.4) we have
7T
0
But from Eq. (5.18b) it follows that
lim m^R cos e,R sin e) = 0 0 < e < tt ..
R-x
Thus
k = 0 .
(6.1)
Similarily substituting Eq. (4.2a) into Eq. (5.5) we have
IT
(6.2)
o
48
49
since 1 im m(R cos e,R sin e) = 0 for 0 < e < ir from Eq (5.18a). Further-
R-*
more, since both k and k2 are zero, the equivalent realizations for the
SSB signals, as given by the equations in Section 5.1, reduce identically
to the phasing method of generating SSB-AM-SC signals (which was given
previously in Figure 8).
The autocorrelation for the SSB-AM-SC signal is readily given by
use of Eq. (4.2a) and Eq (5.20). Thus
y-(m(t) ,m(t)) = m(t). (6.3)
Then the autocorrelation of the suppressed-carrier USSB-AM signal is
given via Eq. (5.43b), and it is
RXU-SC-SSB-AM^ Rmm^ cos WT Rmm^x^ sin oT*
(6.4)
Likewise, by use of Eq. (5.44b) the autocorrelation for the suppressed-
carrier LSSB-AM signal is
RXL-SC-SSB-AM^ WT)
cos
oT
r(t)
mm
sin
(6.5)
From Eq. (6.4) it follows that the spectrum of the USSB-AM-SC
signal is just the positive-frequency spectrum of the modulation shifted
up to on and the negative-frequency spectrum of the modulation shifted
down to oj0. That is, there is a one-to-one correspondence between the
spectrum of this SSB signal and that of the modulation. This is due to
the fact that the corresponding entire function for the signal, g(W) = W,
is a linear function of W. Consequently, the bandwidths for this SSB
50
signal are identical to those for the modulation. This is readily shown
below.
The mean-type bandwidth (when the numerator and denominator exist)
is given by use of Eq. (6.3) in Eq. (5.46):
(6.6)
where = Rmm(0), the power in the modulating signal. By using Eq. (5.48)
the rms bandwidth is
(rms^SSB-AM
-tim(O)
I'm
(6.7)
whenever R^m(0) and exist. By using Eq. (5.50) the equivalent-noise
bandwidth is
ir
(A^SSB-AM = oo
4^ S Rmrn(T)dT
(6.8)
Thus the bandwidths of the SSB-AM-SC signal are identical to those of the
modulating process m(t).
The efficiency of the SSB-AM-SC signal is obtained by using
Eq. (5.54):
iSC-SSB-AM
2Rmm(0)
= 1 .
2Rmm(0)
(6.9)
51
The peak-to-average power ratio for the SSB-AM-SC signal follows
from Eq. (5.56c), and it is
f[m(t)]2 + [ii(t)]2}
tpeak
SC-SSB-AM
2
(6.10)
m
6.2. Example 2: Single-Sideband PM
The SSB-PM signal has a discrete carrier term. This is shown by
calculating the constants k1 and k2. Substituting Eq. (4.5b) into
Eq. (5.4) we have
K .-Inin f cos e-R sin e)s1n [m,(R cos e.R sir, e)]de.
1 7r R-K. J 1
But from Eqs. (5.18a) and (5.18b) lim m^R cos e,R sin e) = 0 for
R-X
0 < e < Ti and lim m.(R cos e, R sin e) = 0 for 0 < e < tt. Thus
R-**>
kl = 0. (6.11)
Likewise, substituting Eq. (4.5a) into Eq. (5.5) we have
IT
k = C e" cos 0 de = 1. (6.12)
2 TT J
0
Thus the SSB-PM signal has a discrete carrier term since k2 = 1 f 0.
There are equivalent representations for the SSB-PM signal since
k and k are not both zero. For example, for the upper sideband signal,
1 2
equivalent representations are given by Egs. (5.7) and (5.8). It is
noted that Eq. (5.8) is identical to Eq. (5.9) for the USSB-PM signal
52
since k1 = 0. Thus the two equivalent representations are:
X
USSB PM^ = [e"m^cos m(t)] cos a)0t [_e '"'"'cos mQt;J sin w0t (6.13)
and
XUssB_p|v|(t) = [-^;sin m(t))+l]cos oi0t [e_r"^sin m(t)]sin w0t. (6.14)
The USSB-PM exciters corresponding to these equations are shown in Figure
13 and Figure 14. They may be compared to the first realization method
given in Figure 9.
The autocorrelation function for the SSB-PM signal will now be
examined. In Section 6.1 the autocorrelation for the SSB-AM-SC signal
was obtained in terms of the autocorrelation function of the modulation.
This was easy to obtain since 44 = m(t). However, for the SSB-PM case 44
and -V-are non-linear functions of the modulation m(t). Consequently, the
density function for the modulation process will be needed in order to
obtain the autocorrelation of the SSB-FM signal in terms of Rrnm(T)
To calculate the autocorrelation function for the SSB-PM signal,
first Ryy(t) will be obtained in term of R^fx). Using k;L = 0, Eq. (5.21),
and Eq. (4.5b) we have
V-(m(t) ,m(t)) a V(m(t),m(t)) = e_m(t) sin m(t).
(6.15)
Then
Figure 13* USSB-PM Signal ExciterMethod II
55
or
(6.16)
where XjU.r) = m(t) m(t-r)
x2(t,x) = m(t) + m(t-r)
x3(t,x) = -m(t) m(t-x) -x2(t,r)
X4(t,r) = -m(t) + m(t-r) = -x j(t jt)
y(t,r) = m(t) + m(t-x)
Now let the modulation m(t) be a stationary Gaussian process with zero mean.
Then x^t,/), x2(t,i), x3(t,T), x4(t,x), and y(t,x) are Gaussian processes
since they are obtained by linear operations on m(t). They are also stat
ionary and have a zero mean value. It follows that x (t,x), y(t,t);
x?_(t jt) j y(t,x); x 3 (t j i), y (t, T); and x4(t,T)s y(t,r) are jointly Gaussian
since the probability density of the input and output of a linear system
is jointly Gaussian when the input is Gaussian [15]. For example, to show
that Xj(t,x) and y(t,r) are jointly Gaussian, a linear system with inputs
m(t) and m(t-r) can readily be found such that the output is y(t,x). Now
the averaging operation in Eq. (6.16) can be carried out by using the fol
lowing formula which is derived in Appendix II:
ej{x(t)+jy(t)} = e-Js{ox2+j2yXy-ay2}
(6.17)
where x(t) and y(t) are jointly Gaussian processes with zero mean,
2 =
= X2(t)
oy2 = y2(t)
56
and
yxy 25 x(t)y(t) .
Thus
oxj ~ [m(t)-m(t-t )F = 2[am -R^ir)]
cx2 [m(t)+m(t-i)]T 2[orn2+Rmm( r)]
ox? [-m(t)-m(t-x )]2 = 2[am2+Rmm(x)]
axl C-nri(t)+m(t-r)]2- 2[am2-RtTim(x)]
and
a/ [l(t)+l(t-,)]2 2[am2+Rnlm(r)] .
From Chapter II it is recalled that Rm^(0) = 0 and R^r) = -R-m(x) -
-Rmm(x) so that the y averages are
yx v = [m(t)-m(t- r )][m(t)+m(t-i)] -2Rmm(t)
iy
yx v = [m(t)-m(t-x)][m(tT+m(t-x)] = 0
2J
Mv = -[m(t)+m(t-r)][m(t)+m(tX)] = 0
and
%y
= -[m(t)-m(t-x)][m(t)+m(t-x)] = 2Rmrtl(x) .
57
Therefore, using Eq. (6.17), Eq, (6.16) becomes
-%{2[am -Rmm(r)] + j2[-2Rmm(-t)] 2[am2+Rmm(T)]}
u e-^i2[om +Rmm(t)] + j20 2[am2+Rmm(T)]}
_ ^ e-%{2[am2+Rmt11(T)] + j20 2[am2+Rrnm(t)]}
+ ^ eJs{2[o|T| -Rmm(T)3 + j2[2Rmm(x)] 2[am2+Rmm(i:)]}
which reduces to
(6.18)
where W -SSB-PM-GN denotes the autocorrelation of the imaginary part of
the entire function which is associated with the suppressed-carrier SSB-
PM signal with Gaussian noise modulation.
It is noted that Eq. (6.18) is an even function of t, as it should
be, since it is the autocorrelation of the real function -V-(m(t) ,m(t))
Furthermore Ryy(O) is zero when Rmii)(0) 0, as it should be, since the
power in any suppressed-carrier signal should be zero when the modulating
power is zero.
The autocorrelation of the USSB-PM signal is now readily obtained
for the case of Gaussian noise modulation by substituting Eq. (6.18) into
Eq. (5.42) and using Eq. (537):
R
XU-SSB-PM-GN
(t) = % Re ejT{[e2R>(x) Cos (2Rmm(t))]
(6.19)
58
Likewise, the autocorrelation of the LSSB-PM signal may be obtained by
using Eq. (5.38).
The autocorrelation of the suppressed-carrier USSB-PM signal with
Gaussian modulation is given by using Eq. (543a):
RXU-SC-SSB-PM-GN^ = ** Re
e^oT{[e2Rmm(0 cos (2Rmm(T)) 1]
+ j[e2lWT) cos (2Rmm(T)]:
(6.20)
Similarly, the autocorrelation of the suppressed-carrier LSSB-FM signal
may be obtained by using Eq. (5.44a).
The mean-type bandwidth will now be evaluated for the SSB-PM
signal assuming Gaussian noise modulation. From Eq. (6.18) we obtain
R^(t) =
1 r e2R"(x) cos (2Rmm(i)di
2J (t-x)2
Then
Rvv(0)
2 it
,2Rmm(x)
cos [2Rmm(x)]d/
(6.21)
and from Eq. (6.18)
Rw(0) = %[e2^ l]
(6.22)
where = om2 is the average power of m(t). Substituting Eqs. (6.21)
and (6.22) into Eq. (5.46) we have the mean-type bandwidth for the
Gaussian noise modulated SSB-PM signal:
oo
P J~ ~T e2FWA) cos[2Rmm(x)]dx
A
()
SSB-PM-GN
32^m
(6.23)
59
where is the noise power of m(t) It is seen that Eq, (6,23) may or
may not exist depending on the autocorrelation of m(t).
The rms-type bandwidth can be obtained with the help of the second
derivative pf Eq. (6.18):
C(i) i-e2R(t) sin [2Rmm(T)]> 2[Rm(,)]2
*VV-
+ (-e2R(r) cos [2Rmm(T)]) 2[R^(t)]2
+ t-e2R(l) sin [2Rmm(T)]> R^(t)
+ t e2R(T> cos [2Rmnl(-r)]> 2[R(Of
+ <-e2R"(T) sin [2R(t)]} 2R^(T)t(,)
+ { e2R"
Thus
Rw() = e
2^m
{R¡>) 2C4(0)]2>
(6.24)
Substituting Eq. (6.24) and Eq. (6.22) into Eq. (5.48) we have for the
rms-type bandwidth of the SSB-PM signal with Gaussian noise modulation:
(wrms)
/2{2[RmlO)f R"(0)}
SSB-PM-GN
1 e
-2iPm
(6.25)
This expression for the rms bandwidth may or may not exist depending on
the autocorrelation of m(t). It is interesting to note that Mazo and Salz
have obtained a formula for the rms bandwidth in terms of different para
meters [19] However both of these formulae give the same numerical re
sults, as we shall demonstrate by Eq. (6.29).
60
The equivalent-noise bandwidth is obtained by substituting
Eq. (6.18) into Eq. (5.50):
(Ato)
%{e2Rmm(r) cos [26mrT1(T)] 1} dt
%[e^m-l] ^
or
(Aw)
;(e2^i 1)
SSB-PM-GN
{e2Rmm(T) co$ [2Rrnm(T)] 1 }di
(6.26)
It is noted that the equivalent-noise bandwidth may exist when the formu
lae for the other types of bandwidth are not valid because of the non
existence of derivatives of Rmm(i:) at t = 0.
It is obvious that the actual numerical values for the bandwidths
depend on the specific autocorrelation function of the Gaussian noise.
For example, the rms bandwidth of the SSB-PM signal will now be calculated
for the particular case of Gaussian modulation which also has a Gaussian
spectrum. Let
2
-0)
0
where Pm(u) is the spectrum of m(t)
4>0 = is the total noise power in m(t)
o2 is the "variance" of the spectrum.
The autocorrelation corresponding to this spectrum is
Rmm(r) ~ Vr
1 2 2
-Ho t
(6.27)
61
The Hilbert transform of Rmm(T) is also needed and is obtained by the
frequency domain approach. It is recalled from Chapter II that
P*(o)) =
mm
j PmmU) > w > 0
, cu 0
j Pmn ) < 0
Then
Wt) = ^im
W") eJT d
2tr
-U)2 t -00
iaiS ei#* ejwTdu> f e2^ eJ<*Tdw
which reduces to
fynm^)
/2rrcr
f
St ~coz ...
sin oox dw .
This integral 1s evaluated by using the formula obtained from page 73,
#18, of the 8ateman Manuscript Project, Tab1es of In teg ral fra ns forms,
vol. 1 [5]:
00
/
1 ..2
eaX sin xy dx e ^ Erf(~4=. y)
Z/a \2/a /
Re a > 0
where
Erf (x) *
Jr f
-t
e z dt.
62
Thus
Rmm(T) = "J i^o e 2 T j 0T
or
Rmm(T) ~ " Rmm(T) a n .
From Ea. (6.27) it follows that
Rmm() = -Vo2
and from Ea. (6.28) we have
Rmm(O) s ~pzr
/2tt
Substituting these two equations into Ea. (6.25) we get
(
ta
rms
2
1 e~Z>po
Thus if m(t) has a Gaussian spectrum and if the modulation has
density function, the SSB-FM signal has the rms bandwidth:
(wrms^sSB-PM-GN
2i|>0az[4(ip0/;ir) + 1]
1 e
where t|>0 is the total noise power in m(t)
a4is the "variance" in the spectrum of m(t).
(6.28)
Gaussian
(6.29)
63
This has the same numerical value as that obtained from the result given
by Mazo and Salz [19]. The result may also be compared to that given by
Kahn and Thomas for the SSB-PM signal with sinusoidal modulation [20].
From Eg. (19) of their work
^rms)$SB-,PM-S = ua^ (6.30)
where wa is the frequency of the sinusoidal modulation and 6 is the modu
lation index. For comparison purposes, equal power will be used for m(t)
in Eq. (6.30) as was used for m(t) in Eq. (6.29). Then Eq. (6.30) becomes
(k>rms)ssB-PM-S = ^ 3 *^o (6.31)
Thus it is seen that for Gaussian modulation the rms bandwidth is propor
tional to the power in m(t) when the power is large (ip0 > > tt/4), and for
sinusoidal modulation the rms bandwidth is proportional to the square root
of the power m(t).
The efficiency for the SSB-PM signal with Gaussian modulation
will now be obtained. Substituting Eo. (6.22) into Eq. (5.54) we have
e2+ra-i
"SSB-PM-GN +
or
- i p"2^m
nSSB-PM-GN
(6.32)
where is the noise power of m(t).
The peak-average to average power ratio for Gaussian m(t) is given
by use of Eqs. (4.5a), (4.5b), and (6.22) in Eq. (5.56b):
64
i[e
m (t)
cos m(t)] + [e
rn(t)
sin m(t)]2}
1 + (e
1)
tpeak
or
e-2rn(t)
e
t = tpeak
Si'm
(6.33)
Note that m(t) may take on large negative values because it has a Gaussian
density function (since it was assumed at the outset that the modulation
was Gaussian). However, it is reasoned that for all practical purposes,
m(t) takes on maximum and minimum values of +3am and 40^ volts where cm
is the standard deviation of m(t). This approximation is useful only for
small values of crm since e+^m) approximates the peak power only when the
exponential function does not increase too rapidly for larger values of am.
Thus the peak-to-average power ratio for the SSB-PM signal with Gaussian
noise modulation is
SSB-PM-GN
(6.34)
when is small.
It is noted that the efficiency and the peak-to-average power
ratio depend on the total power in the Gaussian modulation process and not
on the shape of the modulation spectrum. On the other hand the autocorre
lation function and bandwidth for the SSB signal depend on the spectral
65
shape of the modulation as well.
The dependence of bandwidth on the spectrum of the Gaussian noise
modulation will be illustrated by another example. Consider the narrow-
band modulation process:
m(t) = a(t) cos (ojat + (f>) (6.35)
where a(t) is the (double-sideband) suppressed-subcarrier amplitude
modulation
a is the frequency of the subcarrier
<¡> is a uniformly distributed independent random phase due to
the subcarrier oscillator.
That is, we are considering a SSB signal which is phase modulated by the
m(t) given above. Then
Rmm(T) = *2 Raa(t) oos ^a1 (6.36)
where Raa(T) is the autocorrelation of the subcarrier modulation a(t).
Rmm(T) can obtained from Eq. (6.36) by use of the product theorem [21].
Thus, assuming that the highest frequency in the power spectrum of a(t)
is less than o>a,
Rmmi'O ~ Raa(x) sin a1 > (6.37)
Furthermore let a(t) be a Gaussian process; then m(t) is a
narrow-band Gaussian process. This is readily seen since Eq. (6.35) may
be expanded as follows:
m(t) = %[a(t) cos (wat+<|>) a(t) sin (cjat+c¡>)]
+ %[a(t) cos (cgt+tj)) + a(t) sin (ojat+<)>)] (6.38)
66
The terms in the brackets are the USSB and LSSB parts of the suppressed-
subcarrier signal m(t). But these USSB and LSSB parts are recognized
as the well-known representation for a narrow-band Gaussian process.
Thus m(t) is a narrow-band Gaussian process.
Now the previous expressions for bandwidth, which assume that
m(t) is Gaussian, may be used. The mean-type bandwidth for the multi
plexed SSB-PM signal is then readily given via Eq. (6.23), and it is
(<*>)
M-SSB-PM-GN
oo
f eRaa(x) cos waA cos[RaaU) sin coaA]dA
00
e^a 1
(6.39)
where tpa is the average power of the Gaussian distributed subcarrier
modulation a(t). Obtained in a similar manner, the rms bandwidth is
(rms^.ssB.pM.
GN
a W1) Raa(0)
1 e^a
(6.40)
and the equivalent-noise bandwidth is
(aw)
r[e2*a-l]
M-SSB-PM-GN
J' eRaa^T^ cos aT Cos[Raa(x) sin toax]
(6.41)
Thus, it is seen once again that the bandwidth depends on the spectrum of
the modulation, actually the subcarrier modulation a(t).
To obtain a numerical value for the rms bandwidth of the multi
plexed SSB-PM signal assume that the spectrum of a(t) is flat over
|o)| < w0 < )a.
67
4;
Pa()
1
Wf
0)**
Figure 15. Power Spectrum of a(t)
From Figure 15 we have
Raa(T)
W
1
o
'aa^; 77 J -o'
~WA
eJTdw
or
Raa(T)
NqW0 /sin W t
* \ W0t
(6.42)
and
^ =
N W
oo
(6.43)
Then
ii ,
Raa(o)
-N0W0'
(6.44)
Substituting the last two equations into Eq. (6.40) we obtain the rms
bandwidth for the SSB-PM multiplexed signal:
rTTMoyw. i ^ + nw
^rms^M-SSB-PM-SN-
3tt
1 e-Nowo/lT
(6.45)
where u>a is the subcarrier frequency
N0 is the amplitude of the spectrum of the subcarrier Gaussian
noise modulation
WQ is the bandwidth of the subcarrier noise modulation.
68
Thus the niis bandwidth is proportional to the power in the subcarrier
modulation as N0 becomes large.
6.3/ Example 3: Single-Sideband FM
As was indicated in Section 4.3. the representation for the SSB-FM
signal is very similar to that for the SSB-PM signal. In fact it will be
shown below that all the formulae for the properties of the SSB-PM signal
(which were obtained in the previous section) are directly applicable to
the SSB-FM signal.
The SSB-FM signal has a discrete carrier term since the entire
function for generating the SSB-FM signal is identical to that for the
SSB-PM signal, which has a discrete carrier term.
The other properties of the SSB-FM signal follow directly from
those of the SSB-PM signal if the autocorrelation of m(t) can be obtained
in terms of the spectrum for the frequency modulating signal e(t). It is
recalled from Eq. (4.7) that
t
(6.46)
First, the question arises: Is m(t) stationary if e(t) is stationary?
The answer to this question has been given by Rowe; however, it is not
very satisfactory since he says that m(t) may or may not be stationary [22]
However, it will be shown that m(t), as given by Eq. (6.46), is stationary
in the strict sense if e(t) is stationary in the strict sense; and,
furthermore, m(t) is wide-sense stationary if e(t) is wide-sense stationary
It is recalled that if
y(t) L[x(t)l
69
where L is a linear time-invariant operator, then y(t) is strict-sense
stationary if x(t) is strict-sense stationary and that y(t) is wide-sense
stationary if x(t) is wide-sense stationary [4]. Since the integral is a
linear operator, we need to show only that it is time-invariant, that is
to show that
y(t+e) = L Jx(t+E)j
or
t+e
e(t1)dt1 = j e(t2+e)dtr
This is readily seen to be true by making a change in the variable,
letting t = t2 + e. Thus, if e(t) is stationary, then m(t) is stationary.
Moreover, in the same way it is seen that if m(t) had been defined by
t
mj (t) = D 1 (t')dt' (6.47)
tQ
then m (tj is not necessarily stationary for e^t) stationary since the
system is time-varying (i.e. it was turned on at tQ). But this should
not worry us because, as Middleton points out, all physically realizable
systems have non-stationary outputs since no physical process could
have started out at t = - and continued without some time variation in
the parameters D5]. However, after the "time-invariant" physical systems
have reached steady-state we may consider them to be stationary processes
provided there is a steady state. Thus by letting tQ -> - we are con
sidering the steady-state process m(t) which we have shown to be stationary
Now the autocorrelation of m(t) can be obtained by using power-spectrum
techniques since m(t) has been shown to be stationary. From Eq. (6.46)
we have
70
(6.48)
Then in terms of power-spectrum densities
(6.49)
As Rowe points out, Pmm(oo) must eventually fall off faster than k/
where k is a constant, if e(t) is to contain finite power; and if Pmm(to) =
k/to2, Pqq(w) will be flat and, consequently, white noise. Thus we have a
condition for the physical realizability of m(t): Pmm(u)) falls off faster
than -6 db/octave at the high end. This condition is satisfied by physi
cal systems since they do not have infinite frequency response. From
Eq. (6.49) we have
(6.50)
Pmm() ~
C
Immediately we see that if P9g(to) takes on a constant value as |w| ->- 0
and at o> 0, m(t) will contain a large amount of power with spectral
components concentrated about the origin. In other words, m(t) has a
large block of power, located infinitely close to the origin which is
infinitely large. Thus m(t) contains a slowly varying "DC" term with a
period T and m2(t) - . By examining Eq. (6.46) we obtain the same
result from the time domain. That is, for Pe0(w) equal to a constant,
e(t) contains a finite amount of power located infinitely close to the
origin which appears as a slowly varying finite "DC" term in e(t) such
that T *
Then by Eq. (6.46), m(t) has a infinite amplitude and,
71
consequently, infinite pwer. In other words, the system does not have a
steady-state output condition if the input has a power around oi =0. Thus,
this system ia actually conditionally stable, the output being bounded
only if the input power spectrum has a slope greater than or equal to +6
db/octave near the origin (and, consequently, zero at the origin) as seen
from Eq. (6,50). It is interesting to note that for the case of FM, ejm^
is stationary regardless of the shape of the spectral density Pqq(ua). This
is due to the fact that ejm^ is bounded regardless of whether m(t) is
bounded or not.
From Eq. (6.50) we can readily obtain Rmm(ir) for any input process
e(t) which has a bounded output process m(t). Thus
oo
WO = J- f ^-ej3T du (6.51)
IT J tl)
00
Furthermore, R^m(0), Rmm(T), and Rmm(0) may be obtained in terms of
Pee(w). By substituting for these quantities in the equations of Section
6.2, the properties of a SSB-FM signal can be obtained in terms of the
spectrum of the modulating process.
6.4. Example 4: Single-Sideband g
The SSB-a signal has a discrete carrier term. This is readily
shown by calculating the constants kT and k2. Substituting Eq. (4.10b)
into Eq. (5.4) we have
IC 1 11m f e"MR cos e*R s1n 6> sin am, (R cos 9.R sin 8)d8 .
1 n R-Mo J 1
.0
But lim m^R cos e, R sin e) = 0, for 0 < e s: tt and lim m^R cos e,
R-x R-x
R sin e) = 0 for 0 £ e $ it. Thus
kx =0. (6.52)
72
Likewise, substituting Eq. (4.10a) into Eq. (5.5), we have
k2 =1. (6.53)
Thus the SSB-a signal has a discrete carrier term.
It follows that equivalent representations for the SSB-a signal
are possible since k2 f 0. This is analogous to the discussion on equiva
lent representations for SSB-PM signals (Section 6.2) so this subject will
not be pursued further.
The autocorrelation function for the SSB-a signal will now be ob
tained in terms of Rmm(T) Using Eq. (5.21) and Eq. (4.10b) we have
Rw(t) = [eam(t) sin am(t)][eam(t"T) sin am(t-t)]
or
r (t) = 53{ea[m(t)+m(t-T)]} {eja[m(t)-m(t-T)] _eja[m(t)+m(t-T)]}
+ %{ea^(t)+m(t-r)]} {_eja[-m(t)-m(t-T)] + eja[-m(t)+m(t-r)]}.
(6.54)
The density function of m(t) has to be specified in order to carry out
this average. It is recalled that m(t) is related to the modulating
signal e(t) by the equation:
m(t) In [1+(t)H .
Now assume that the density function of the modulation is chosen such
that m(t) is a Gaussian random process of all orders. Eq. (6.54) can
then be evaluated by the procedure that was used to evaluate Eo. (6.16).
73
Assuming a Gaussian m(t), Eg. (6.54) becomes
COS [2a2R|7im(x)] 1} (6.55)
But this is identical to Eg. (6.18) except for the scale factor a2.
Thus for envelope detectable SSB (a = 1) with m(t) Gaussian, the auto
correlation and spectral density functions are identical to those for
the SSB-PM signal with Gaussian m(t). Moreover, the properties are
identical for SSB-a and SSB-PM signals having Gaussian m(t) processes
such that (ipm)SSB-PM 0(2 ^m^sSB-a*
It is also seen that if |e(t)| < < 1 most of the time then
m(t) = e(t).
Thus, when e(t) is Gaussian with a small variance, m(t) is approximately
Gaussian most of the time. Then Eg. (6.55) becomes
2a2Ree(T) cos [2a2Ree(x)] 1}
(6.56)
RW(l)SSB-a-GN ~
when |e(t)| < < 1 most of the time. Conseguently, formulae for the auto
correlation functions analogous to Egs. (6.19) and (6.20), may be further
simplified to a function of Ree(x) instead of Rrnm(t). Then the auto
correlation functions for USSB-cx and LSSB-a signals, assuming Gaussian
modulation e(t) with a small variance, are
RXU-SSB-a-GN
(t) = H Re eJoT{te22Ree(T) cos (2<,2l?ee(t))]
j [e2"R86(T) cos (2c.^e(0)]>
+
(6,57)
74
and
RXL-SSB-a-GN^ ~ Re
e"JuoT{[e2a2Ree(T) cos (2a2fr0e(r))]
+ J [e
2a2Ree(t)
cos (2a2tee(r))]}
(6.58)
The efficiency is readily obtained by substituting Eq. (6.56)
into Eq. (5.54):
'SSB-a-GN
= 1
5-2a2(J>m
(6.59)
where is the power in the Gaussian m(t) and |e(t)| < < 1. This result
may be compared for a = 1 to that given by Voelcker for envelope-detectable
SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t);
and Eq. (6.59) becomes
nSSB-oi-GN : 1 e 00 ~ 2o0/. (6.60)
This agrees with Voelcker"s result (his Eq. (38)) when the variance of the
modulation is small-~the condition for Eq. (6.60) to be valid.
The expressions for the other properties of the SSB-ct signal, such
as bandwidths and peak-to-average power ratio, will not be examined further
here since it was shown above that these properties are the same as those
obtained for the SSB-PM signal when Um)sSB-PM ~ a2(iJm^sSB-a
m(t) is Gaussian.
as long as
CHAPTER VII
COMPARISON OF SOME SYSTEMS
In the two preceding chapters properties of single-sideband sig
nals have been studied. However, the choice of a particular modulation
scheme also depends on the properties of the receiver. For example,
the entire function g(W) W can be used to generate a SSB signal, but
there is no easy way to detect this type of signal.
In this chapter a comparison of various types of modulated sig
nals will be undertaken from the overall system viewpoint {i.e* generation,
transmission and detection). Systems will be compared in terms of the
degradation of the modulating signal which appears at the receiver out
put when the modulated RF signal plus Gaussian noise is present at the
input. This degradation will be measured in terms of three figures of
merit:
1. The signal-to-noise ratio at the receiver output
2. The signal energy required at the receiver input for
a bit of information at the receiver output when com
an son is made with the ideal system (Here the ideal
system is defined as a system which requires a minimum
amount of energy to transmit a bit of information as
predicted by Shannon's formula.)
3. The efficiency of the system as defined by the ratio
of the RF power required by an ideal system to the RF
power required by an actual system,(Here the ideal sys-
75
76
tern is taken to be a system which has optimum trade-off
between predetection signal bandwidth and postdetection
signal-to-noise ratioo)
Comparison of AM, FM, SSB-AM-SC, and SSB-FM systems will be made using
these three figures of merit. It is clear that these comparisons are
known to be valid only for the conditions specified; that is, for the
given modulation density function, and detection schemes which are used
in these comparisons.
7.1, Output Signal-to-Noise Ratios
7.1-1. AM system
Consider the coherent receiver as shown in Figure 16 where the
input AM signal plus narrow-band Gaussian noise is given by
X(t) + n.¡(t) = {A0[l + 6 sin tmt] cos co0t}
+ ixc(t) cos )Qt xs(t) sin cdQt} (7.1)
where X(t) is the input signal, n-¡(t) is the input noise with a flat spec
trum over the bandwidth 2wm> and 6 is the modulation index.
X(t)+nj(t)
Low Pass
Fi 1 ter
AC Couple
2k cos wqL
Figure 16. AM Coherent Receiver
Output
Then the output signal-to-noise power ratio, where A0k6 sin ojmt is the
output signal, is given by
(S/N) 0 =
6 2
1 + %62
(S/N)1
(7.2)
77
or
(7.3)
where (S/N)-j The input signal-to-noise power ratio
(C/N)i = The input carrier-to-noise power ratio
and the spectrum of the noise is taken to be flat over the IF bandpass
which is 2a)m(rad/s).
7.1-2. SSB-AM-SC system
Consider the coherent receiver (Figure 16) once again, where
the input is a SSB-AM-SC signal plus narrow-band Gaussian noise. Then
the input signal plus noise is
X(t) + ni(t) = iA0[m(t) cos w0t m(t) sin o>0t]}
+ [xc(t) cos (O0t xs(t) sin oj0t]
(7.4)
where
m(t) = 6 sin wt
m
and xs(t) = xc(t) if the IF passes only upper sideband components. The
input noise is assumed to have a flat spectrum over the bandwidth )m.
Then the output signal-to-noise power ratio, where AQk6 sin wmt is the
output signal, is given by [23]
(7.5)
(S/N)Q = (S/N)i
where the spectrum of the noise is taken to be flat over the IF bandpass
which is cjjm(rad/s).
It is interesting to note that the same result is obtained from a
78
more complicated receiver as given in Figure 17 However, in some practi
cal applications the receiver in Figure 17 may give much better perform
ance due to better lower sideband noise rejection. That is, in Figure 17
the lower sideband noise is eliminated as the result of the approximate
Hilbert transform filter realized about oj = 0; whereas, in Figure 16 the
lower sideband noise is rejected by the IF filter realized about u = to0.
Thus, in order to obtain equal lower sideband noise rejection in both
receivers, the IF bandpass for the receiver in Figure 16 would have to
have a very steep db/octave roll-off characteristic at
Figure 17. SSB-AM-SC Receiver
7.1-3. SSB-FM system
Now consider a FM receiver which is used to detect a SSB-FM sig
nal plus narrow-band Gaussian noise as shown in Figure 18.
X{t)+n-j (t)
FM
Receiver
Output
Figure 18. SSB-FM Receiver
79
The input signal plus noise is given by
X(t) + n^(t) A0e"^^^ cos [o>0t + m(t)] + n-j(t) (7.6)
where A0 ~ The amplitude of carrier
u>o The radian frequency of the carrier
m(t) = D /t v(t) dt
m(t) = m(t) ~= The Hilbert transform of m(t)
nj(t) Narrow-band Gaussian noise with power spectral density F0
over the (one-sided spectral) IF band
and v(t) is the modulation on the upper SSB-FM signal. The independent
narrow-band Gaussian noise process may be represented by
n^(t) = R(t) cos [w0t + 4>(t)J = xc(t) cos w0t x$(t) sin w0t
where xs(t) = xc(t) since the IF passes only the frequencies on the upper
sideband of the carrier frequency.
Then the phase of the detector output is obtained from Eq. (7.6)
and is
;p(t) = k tan
which reduces to
A0e~^ sin m(t) + R(t) sin (t)
A0e^ cos m(t) + R(t) cos (t)
(7.7)
xp(t) km(t) + k tan^
R(t) sin [(t) m(t)]
A0e-^ + R(t) cos [m(t) (t)]
(7.8)
where k is a constant due to the detector. The detector output voltage
is given by Eq. (7.8) is identical to the phase output when the
input is conventional FM plus noise except for the factor e_n1' .
80
For large input signal-to-noise ratios {i.e. A0e-m^ > > R(t)
most of the time), Eq (7o8) becomes
kR(t)
ip(t) km(t) + sin [(t) m(t)]
(7.9)
dn0(t)
Then the noise output voltage is where
n (t) = R(t) sin [
A
(7.10)
Now the phase (t) is uniformly distributed over 0 to 2n since the input
noise is a narrow-band Gaussian process. Then for m(t) deterministic,
U(t) m(t)J is distributed uniformly also. Furthermore, R(t) has a
Rayleigh density function. Then it follows that R(t) sin [(t) m(t)]
is Gaussian (at least to the first order density) and, using Rice's
formulation [24, 25],
where F(u>) = F0 is the input noise spectrum and {n} are independent
random variables uniformly distributed over 0 to 2-rr. Actually it is
known that the presence of modulation produces some clicks in the out
put [26], but this effect is not considered here. Eq. (7.10) then be
comes
81
or
dn0(t)
dt
kem(t) % r___
~Yo J~2F(con) K-w0) COS [(wn-w0)t + 0n]
+
ke(t)
Ao
CO
z
n=l
^2F{n) 2. sin C(n-u.0)t + 8].
Noting that {en} are independent as well as uniformly distributed and
that the noise spectrum is zero below the carrier frequency, the output
noise power is
N
o
dn0(t)
dt
m
F^d. + -- e2S
Ao2
dm (t)
dt
2-it
^m
F0dw
o
k2
e2m(t)'
^ m3 + k2 e2m(t)
dm(t)
Ao2
J
2 rr 3 /\ 2
no
dt
- _
(7.11)
where () is the averaging operator and wm is the baseband bandwidth
(rad/s) Now let v(t) = -Am cos tomt then, averaging over t, we have
2m(t)
w,
2u/c|
m
2tt
m
e26 cos dt = In(26)
and
2m(t)
dm(t)
dt
I (m6)2 [10(26) Iz(2)]
1^6 1,(26)
(7.12)
(7.13)
82
where 6 DAm/tom, the modulation index. Substituting Eq, (7,12) and
Eq (7,13) into Eq, (7,11) we obtain for the output noise power
k2F0i%3
2ttA02
ll0(2i) +isl1(26)
Referring to Eq. (7.9), the output signal power is
2
= y
Then the output signal-to-noise ratio is
dkm(t)
dt
(7,14)
(7.15)
(S/N)0
1 1
- 1.(26) + 61,(26)
3 0 2 1
(S/N)n -
A0262
2 <%, 1 Io(26) + I 01,(26)1
2ir 3 2
(7.16)
Referring to Eq. (7.6), the signal power into the detector is
Si = A02 e2"1^ cos2 [o)0t + m(t)] = A02 e2^^^
jA02 I0(26)
(7.17)
Kahn and Thomas have given the ratio of the rms bandwidths (taken about
83
the mean of the one-sided spectrum) for a SSB-FM signal to a conventional
FM signal [20], and it is
BSSB-FM
_____ a fz
bfm
I1z(26)
UHzT)
(7.18)
It is known that the bandwidth (in rad/s) of a FM signal is approxi
mately
Bfm = 2(+l (7.19)
Thus, to the first approximation, the SSB-FM bandwidth is
_ / 112(26) '
bssb-fm2 /2 y1 'S+1>V <7-20>
Then, taking the IF bandwidth to be that of the SSB-FM signal, the input
noise power is
i SSB-FM
n
(7.21)
Consequently, the input signal-to-noise ratio is
Ag2 Iq(26)
(S/N), -
Fo
4 a)m (6+1) /2
Ii2(26)
I02(26)
(7.22)
84
Using Eo. (7.16) and Eq. (7.22), we have
6 6 2(6+1) /2 / 1
(S/N)0 =
Ij (26)
TTT
In2(2)+| 610(26)1,(26)
(S/N) -f
(7.23)
for the case of SSB-FM plus Gaussian noise into a FM detector.
The signal-to-noise output can also be obtained in terms of the
unmodulated-signal-to-noise ratio [i.e. the carrier-to-noise power at
the input). From Eq. (7.6) we obtain
(S/N)-f = 10(26) (C/N)i
(7.24)
and Eq. (7.23) becomes
(S/N)o =
Il2(26)
6 62(6+1) /2
1 _
10 2 (^ 6)
Iq(26) + | 6I1(26)
(C/N)i
where (C/N)^ is the carrier-to-noise power ratio.
(7.25)
7.1-4. FM system
The signal-to-noise ratio at the output of a FM receiver for a
FM signal plus narrow-band Gaussian noise at the input can be obtained
by the same procedure as used above for SSB-FM. The factor e_m^ of
Eq. (7.6) is replaced by unity, and the bandwidth of the input noise
is given by Eq. (7.19). Then the output signal-to-noise ratio becomes
(S/N)0 = 3 62(6+l) (S/N)i
(7.26)
85
when the input signal-to-noise ratio is large. It is also noted that
(S/N)i = (C/N)j.
(7.27)
7.1-5. Comparison of signal-to-noise ratios
A comparison of the various modulation systems is now given by
plotting (S/N)0/(S/N)^ as a function of the modulation index by use of
Eqs. (7.2), (7.5), (7.23), and (7.26). This plot is shown in Figure 19.
Likewise (S/N)0/(C/N)j as a function of the modulation index
is shown in Figure 20, where Eqs. (7.3), (7.25), (7.26), and (7.27) are
used. It is noted that in both of these figures the noise power band
width was determined by the signal bandwidth.
When systems are compared in terms of signal-to-noise ratios, a
useful criterion is the output signal-to-noise ratio from the system
for a given RF signal power in the channel--that is, (S/N)0/S-¡, This
result can be obtained from (S/N)0/(S/N)-¡, which was obtained previously
for each system, if the input noise, N-¡, is normalized to some convenient
constant. This is done, for example, by taking only the noise power in
the band 2oim (rad/s) for measurement purposes. (The actual input noise
power of each system is not changed, just the measurement of it.) Then
the normalized input noise power for all the systems is
F0
[\| _ 2u)pi
2ir
where the subscript I denotes the normalized power. Then the ratio
(S/N)0/(S/N)j is identical to Nj[(S/N)0/Si] where Nj is the constant de
fined above. Thus, to within the multiplicative constant Nj, comparison
of (S/N)0/(S/N)j for the various systems is a comparison of the output
86
Figure 19 Output to Input Signal-to-Noise
Power Ratios for Several Systems
87
Figure 20. Output Signal-to-No1se to Input Carrler-
to-No1se Ratio for Several Systems
88
signal-to-noise ratios for the systems for a given RF signal power. This
procedure is commonly used for system comparisons [23].
Likewise, a comparison of output signal-to-noise ratios for vari
ous systems for a given aarrier power can be carried out by comparing
(S/N)0/(C/N)j, where the subscript I denotes the normalized input noise
power once again.
The AM, SSB-AM-SC, SSB-FM, and FM systems will now be compared
by using this procedure.
For the AM system (S/N)i = (S/N)j so that Eq. (7.2) becomes
(S/N)0
1 +
(S/N)!
(7.28)
and, likewise, Eq. (7.3) becomes
(S/N)0 = 2 (C/N)].
For the SSB-AM system Eq. (7.5) becomes
(S/N)0 = 2(S/N)i.
(7.29)
(7.30)
For the SSB-FM system, Eq. (7.23) becomes
(7.31)
and Eq. (7.25) becomes
(S/N)0
662
10(2<$) + | 61,(26)
(C/N)j
(7.32)
89
For the FM system, Eo. (7.26) becomes
(S/N)0 = 3 62(S/N)j
and
(S/N)Q = 3 62(C/N)j.
(7.33)
(7.34)
A comparison of the output signal-to-noise ratios for the vari
ous modulation systems can be made now for a given input signal or car
rier power by using these equations. (S/N)0/(S/N)j as a function of
modulation index is plotted for various systems in Figure 21. Likewise
(S/N)0/(C/N)j is shown in Figure 22. From these figures, it is concluded
that FM gives the greatest signal-to-noise ratio at the detector output
for high index, followed by SSB-AM. For low index (6 < 1), SSB-AM is
best, followed by SSB-FM and FM which have about the same (S/N)0, and
AM gives the lowest (S/N)0.
7.2. Energy-Per-Bit of Information
The concept of RF energy required per bit of received information
is used by Raisbeck for comparing SSB-AM and FM systems [27]. This will
be extended to AM and SSB-FM systems in this section.
The (received) capacity of the system is given by [28]
Cb > (b/2ir) log2 [1 + (S/N)Q] (7.35)
where b is the baseband bandwidth (rad/s)
(S/N)0 is the output signal-to-noise power ratio.
Eq. (7.35) becomes an equality when the output noise is Gaussian.
90
Figure 21. Output SignaT-to-Noise to Input Signal-to-Normalized-
Noise Power Ratio for Various Systems
91
Figure 22. Output Signal-to-Noise to Input Carrier-to-Normalized-
Noise Power Ratio for Various Systems
92
Then the RF energy required per bit of received information is
Sl (F0B/2tt)(S/N)1 f0b (S/N)1
Cb S Cb S b l0g2 [1 + (S/N)q]
where F0 is the spectral density of the noise in the IF and B is the IF
bandwidth (rad/s). In an ideal system the capacity of the IF is eaual
to the capacity of the baseband even when (S/N).¡ -* 0. Therefore the
ideal system has an energy-per-bit given by
St Si
r r lim
Cb CB (S/N) -¡-K)
FnB (S/N)i
B log2 [1 + (S7N)i]
log2e
0.693 Fn. (7.37)
Then Eq. (7.36) may be written as
S
-L < (0.693 F0)
Cb
B (S/N)j
0.693 b Tog, [1 + (S/N)0]
_4
Now the figure of merit will be defined as
B (S/N)i
0.693 b log, [1 + (S/N)0]
(7.38)
which is the amount of energy required by the actual system over that of
the ideal system in order to receive a bit of information, provided that
the output noise is Gaussian. If the output noise is not Gaussian, the
value of M will be somewhat larger than the ratio, energy-per-bit for the
actual system to the energy-per-bit for the ideal system.
H as a function of modulation index will be derived below for com
parison of various systems.
93
7.2-1. AM system
We now want to find M(s) for the AM system, described in Section
7.1-1, such that we will have an output signal-to-noise ratio of 27.5 db
for 6=1. 27.5 db is an arbitrary value that is chosen here for com
parison of systems using M as a figure of merit. This value is repre
sentative of the (S/N)o requirement for actual communication systems.
From Eq. (7.3) it follows that (C/N)-j = 27.5 db for 6 = 1. Also, for the
AM system Eq. (7.38) becomes
2 )m [(1 + %62)(C/N)i3
M(6) = ; --
0.693 log2 [1 + 62('C/N)i]
For (C/N)j = 27.5 db, Eq. (7.39) reduces to
1620 (1 + %62)
M{6) =
log2 [1 +'560 62]
(7.39)
(7.40)
The values of M(6) for the AM system, as given by Eq. (7.40), will
be compared to those for other systems in Section 7.2-5.
7.2-2. SSB-AM-SC system
To obtain M(6) for the SSB-AM-SC system, (S/N)0 = 27.5 db will be
used once again. From Eq. (7.5) it follows that (S/N)j = 27.5 db. Also,
for the SSB-AM system Eq. (7.38) becomes
m (S/N)i
M(6) =
0.693 com log2 [1 + (S/N)0]
(7.41)
For (S/N) 0 = (S/N)-f = 27.5 db, Eq. (7.41) reduces to
M(6) = 19.5 db.
(7.42)
94
7.2-3. SSB-FM system
To obtain M(6) for the SSB-FM system with (S/N)Q = 27.5 db, it
follows from Eq. (7.25) that (C/N)-f = 23.3 db for 6 = 1. Also, for the
SSB-FM system Eq. (7.38) becomes
M(6)
_ / I12(26)1
-I
2(6+1) o)m /2 /l
I (26)(C/N)i
y I02(26)
0.693 com log
6 62 (6+1) /2 /l -
1 +
T7(W
In2(2.)
-. (7.43)
10(26) + | 61^26)
(C/N)i
For (C/N)^ = 23.3 db, Eq. (7.43) reduces to
7.2-4. FM system
To obtain M() for the FM system, for (S/N)0 = 27.5 db, it follows
from Eqs. (7.26) and (7.27) that (C/N)-,* = 12 db (which is just above the
threshold) for 6=2. Also, for the FM system, Eq. (7.38) becomes
[2 (6+1) oom] (C/N)
M(6) = : .
0.693 aw log2 [1+3 62(6+1)(C/N)i]
(7.45)
95
For (C/N)-j = 12 db, this reduces to
46 (6+1)
M(S)
log2 [1 + 48 2(+l)]
It is recalled that Raisbeck obtained this result [27].
(7.46)
7.2-5. Comparison of energy-per-bit for various systems
It is recalled that M(
bit for the actual system to the energy-per-bit for the ideal system
when the output system noise is Gaussian. The output noise is Gaussian
for the AM, SSB-AM-SC, and FM systems [for FM, (C/N)i = 12 db >> 0 db].
Also, from Eq. (7.10) it is seen that the noise out of the SSB-FM system
is Gaussian for small index (say 6 < 1). Thus, for the systems that are
analyzed above, M(6) represents the ratio of the energy-per-bit for the
actual system to the energy-per-bit for the ideal system. Then in db,
M() gives the energy-per-bit required above the ideal system.
The systems are compared in terms of energy-per-bit (db) above
the ideal system in Figure 23, where Eqs. (7.40), (7.42), (7.44), and
(7.46) have beqn plotted for the AM, SSB-AM-SC, SSB-FM, and FM systems.
From this figure it is seen that the FM system is best, followed by
SSB-AM-SC, SSB-FM and AM. Furthermore, the FM system is about 12 db
worse than the ideal system. These comparisons are valid for output
signal-to-noise ratios of about 25 db.
In addition, Figure 23 specifies the modulation index to use
for each type of system in order to minimize the energy required to
transmit one bit of information.
Energy (db) Required Above Ideal System
35
1.0 2.0
Modulation Index ()
3.0
4.0
Figure 23. Comparison of Energy-per-Bit for Various Systems
97
7.3. System Efficiencies
The third figure of merit which will be used to compare systems
is the system efficiency, defined by
Transmitted power required for an ideal system
6 =
Transmitted power required for an actual system
= sj/S-j (7.47)
where the ideal system is taken to be a system which has optimum trade
off between predetection signal bandwidth and postdetection signal-to-
noise ratio. This concept is used by Wright and doll iffe to compare
SSB-AM-SC and FM systems [29]. Here, it will be extended to AM and
SSB-FM systems.
The trade-off between predetection signal bandwidth and post
detection signal-to-noise ratio for an ideal system is obtained by
equating the predetection capacity to the postdetection capacity since
an ideal system does not lose information in the detection process [30].
Thus
(B/2it) log2 [1 + (S'/N)i] = (b/2ir) log [1 + (S'/N)0] (7.48)
where B is the IF bandwidth
b is the baseband bandwidth
(S'/N)-} is the input signal-to-noise ratio for the ideal system
(S'/N)0 is the output signal-to-noise ratio for the ideal system
The prime is used here to denote the ideal system. Eq. (7.48) reduces to
(s7n)0 = [1 + (S'/N)]Y 1 (7.49)
where y = B/b, the IF to baseband bandwidth ratio.
The efficiency, e will now be calculated for various types of
systems.
98
7.3-1, AM system
For the AM system y = B/b = 2. Then setting Eq. (7.49) equal to
Eq, (7.2) we have
[1 + (S/N)i32 -1
1 +
(S/N)i.
(7.50)
Substituting for S-¡ from Eq. (7.47), the efficiency for the AM system is
obtained, and it is
3 =
2
1
1 + ig2
_(S'/N)i + 2_
(7.51)
The AM efficiency will be compared to those for other systems in
Section 7.3-5 as a function of (S'/N)i with the modulation index as a
parameter.
7.3-2. SSB-AM-SC system
For the SSB-AM-SC system y = B/b = 1. Then, equating Eq. (7.49)
and Eq. (7.5), we have
(S'/NJi = (S/N) -f (7.52)
and substituting for Si using Eq. (7.47), the SSB-AM-SC efficiency is
(7.53)
7.3-3. SSB-FM system
For the SSB-FM system, using Eq. (7.20),
= = 2 (6+1) /2
m
' 1^(26)
1 Iq2(26)
Y
(7.54)
99
Note that for SSB-FM y and s are uniquely related to each other (by
Eq, (7.54)), unlike the AM and SSB-AM-SC cases. Equating Eq. (7.49) and
(7.23), we have
[i + (S7N),]Y l =
Then, substituting for Sn- from Eg. (7.47), the efficiency for SSB-FM is
32y
|_I02(2S) + | 6l0(2)I1(2)
(S'/N)i
[1 + (SVN)iF 1
(7.56)
362i
I02(26) + | 6I0(26)I1(26)
(S/N)-
(7.55)
where y and 6 are uniquely related by Eq. (7.54).
7.3-4. FM system
For the FM system, using Eq. (7.19),
Y = 7T = 2(6+1). (7.57)
m
Thus for FM, y and 6 are uniquely related, as was the case in SSB-FM.
Eauating Eq, (7.49) and Eq. (7.26) we have
[1 + (S'/N)i]T 1 = |y(J l)2 (S/N)-j. (7,58)
Then, substituting for S-j from Eg. (7.47), the efficiency for FM is
(7.59)
This is identical to the result obtained by Panter [23].
100
7,3-5. Comparison of system efficiencies
Eqs. (7.51), (7.53), (7.56), and (7.59) are plotted in Figure 24
in order to compare the efficiencies for the AM, SSB-AM-SC, SSB-FM, and
FM systems. The efficiency is given as a function of (S'/N)0 with the
modulation index, 6, as a parameter. For example, from the figure it
is seen that, for FM with 6=2 and (S'/N)i = 30 db, the FM system re
quires about 135 db more power than an ideal system with the same IF-to-
baseband bandwidth ratio and the same output signal-to-noise ratio.
From Figure 24, it is seen that SSB-AM-SC is an ideal system in
the sense of trading bandwidth for output signal-to-noise ratio. Also,
AM is the next best system, and SSB-FM and FM are the poorest systems
according to this criterion.
101
g(db)
Figure 24. Efficiencies of Various Systems
CHAPTER VIII
SUMMARY
In this work a new approach to SSB signal design and analysis
for communication systems has been presented. The key to this approach
is the philosophy of using a modulated-signal generating function--the
generating function bing any entire function.
It was hypothesized in Chapter I that SSB signals were of the
third basic modulation class, the first two being AM and FM.
In Chapter II a brief review of analytic signal theory was pre
sented, and this theory was used in successive chapters to facilitate
the derivations.
In Chapter III it was shown that signals of the SSB class could
be generated by use of entire generating functions and that these sig
nals were truly SSB signals regardless of the modulating process.
Generalized formulae were derived which may represent upper SSB or lower
SSB modulated signals. These formulae are analogous to those representing
AM and FM signals. However, it is noted that any SSB signal is a com
bination of AM and FM.
Chapter IV gave some examples of well-known SSB signals, using
the appropriate entire generating function to obtain their mathematical
representation and, consequently, their physical realization.
The generating function concept, along with analytic signal
theory, was used in Chapter V to obtain generalized formulae for the
properties of SSB signals. The properties that were studied were:
102
103
1. Equivalent realizations for a given SSB signal
2. The condition for a suppressed-carrier signal
3. Autocorrelation function
4. Bandwidth (using various definitions)
5. Efficiency
6. Peak-to-average power ratio.
The amplitude of the discrete carrier term was found to be equal to the
absolute value of the entire function (associated with a particular SSB
signal) evaluated at the origin and was not affected by the modulation.
Furthermore, for suppressed-carrier SSB signals, the real and imaginary
parts of the complex envelope are a unique Hilbert transform pair;
otherwise, they are a Hilbert transform pair to within an additive con
stant.
In Chapter VI the properties for examples of various SSB signals
were studied where stochastic modulation was assumed. The results were
compared with those published in the literature where possible.
In Chapter VII a comparison of AM, SSB-AM-SC, SSB-FM and FM
systems was carried out. This was a comparison of the various modu
lation schemes from the overall viewpoint of generation, transmission
with additive Gaussian noise, and detection. Three figures of merit
were used for comparison:
1. Output signal-to-noise ratios
2. Energy-per-bit of information
3. System efficiency.
It was found that, for a given RF signal power, FM has the greatest post
detection signal-to-noise ratio if the modulation index is large. For
small index SSB-AM-SC is best, with SSB-FM and FM second, and AM is
104
is poorest. For the lease energy-per-bit of information, FM is best,
followed by SSB-AM-SC, SSB-FM, and AM. When the systems are compared
in terms of optimum trade-off between predetection bandwidth and post
detection signal-to-noise ratio [i.e. system efficiency) SSB-AM-SC was
found to be ideal, with AM second best, followed by SSB-FM and FM.
In conclusion, the entire generating function concept should be
helpful in obtaining new types of SSB signals, and the corresponding
formulae for analyzing these signals will be helpful in classifying
these signals according to their properties. However, one should also
evaluate the overall system performance in the presence of noise to
determine the usefulness of these signals.
APPENDIX I
PROOFS OF SEVERAL THEOREMS
Theorem I
If k(z) is analytic in the UHP, then the spectrum of k(t,0),
denoted by F|<(w), is zero for all w < 0, assuming that k(t,0) is Fourier
transformable. (This result is included in Theorem 95 of Titchmarsh [6]
and in the work of Paley and Wiener [31].)
Lemma to Theorem I
If Wj(z) and W2(z) are analytic in the UHP, then W(z) W1(z)W2(z)
is analytic in the UHP.
Proof of the Lemma to Theorem I:
Assume that W1(z) and W2(z) are analytic in the UHP, which implies
that they are continuous. Then if W(z) satisifies the Cauchy-Riemann
(C-R) relation for all z in the UHP, W(z) is analytic in the UHP.
Given: Wx and W2 are analytic in UHP. Then
W
I = Uj + J'V1 =>
w2 = u2 + jV2 =>
9U,
8V1 .
3U
1^ _
1
in
the
UHP
(I-la)
3X
ay
3X
3y
8U2
3 V2
3V2
3U2
the
(I-lb)
=
-
in
UHP
3X
3X
3y
and these partial derivatives are continuous.
To show: W = U + jV is analytic for all z in the UHP by showing
in the UHP (I-2a)
3V __3U
3x --3y
in the UHP
(I-2b)
105
106
and that these partial derivatives are continuous.
W = wrw2 = (Ux + j)(U2 + jV2)
= (U^- v1v2) + j(v1u2 + V2U1) = U + jV.
Then
and
aU =
u,
3Uz +
u,
Mi.
V,
av2
- v,
Ml
ax
1
ax
2
ax
1
ax
2
ax
av
Vi
3U9
u2
aVi
v2
aUi
+ u1
a V2
_ -
+
+
3y
sy
ay
ay
ay
By substituting Eqs. (I-1 a) and (I-lb) into Eq. (1-4),
3V1
aV aU2 all
= Un - + U0 i- + V,
ay
ax
ax
av2
ax
+ v.
ax
(1-3)
(1-4)
(1-5)
But Eq. (1-5) is identical to Eq. (1-3) and the partial derivatives are
continuous. Thus, the condition of Eq. (I-2a) is satisfied.
Also,
and
3V 3U2. + y Ml _l u Ml x 11 Ml
= V,
+ V,
+ u,
ax
ax
ax
ax
ax
Then
aU
3y
= u,
au2
ay
+ u.
alii
ay
- v,
av2
ay
- Vc
av1
ay
ay \ ax ax ax ax
(1-6)
(1-7)
and all the partial derivatives are continuous. By comparing Eq. (1-6)
with Eq. (1-7) it is seen that the condition of Eq. (I-2b) is satisfied.
Therefore W(z) is analytic in the UHP.
107
Proof of Theorem I:
Given: k(z) is analytic in the UHP and eJ)Z is analytic in the
UHP for
To show
00
F(w) = | k(x,0)e"jwXdx = 0 ¥ u> < 0. (1-8)
By the Lemma k(z)eJuZ is analytic in the UHP for all
Cauchy's Theorem
k(z)e'ja)Zdz = 0
for c as shown in Figure 25 since k(z)e'J)Z is analytic in the UHP.
Thus
^ k(z)e"JCl)Zdz
k(x,0)eja)Xdx
+ lim f k(R sin e,R cos e)e
R-*
o
jReJ6 RJeJ'6d6 .
But for to < 0,
lira I f k eR s1n V>Rcs Wed6| < lim f |k|eR s1" eRde
R~**>
R->o
108
and |k| < M, a constant, since k is analytic in the UHP,
1 im
R-*
ke'jwReje Rjejede| < M 1im f eR sin eRde = 0
R-x 'ft
Therefore
F(oj) I k(x,0)e"JuXdx = 0 u> < 0
Theorem II
If Z(z) is an analytic function of z in the UHP and if g(W) is
an entire function of W, then g[Z(z)] is an analytic function of z in
the UH z-plane.
Proof of Theorem II:
The C-R relations will be used to show that g[Z(z)] is analytic
in the UH z-plane.
Given: Z(z) = Ux(x,y) + jV^x.y) is analytic in the UH z-plane.
This implies that
3U-,
- ?h.
. =
3 U,
3X
3y
3X
sy
(I-9a,b)
in UHP and these derivatives are continuous there.
g(W) U2(U1,V1) + JV2(Ux,Vx) is analytic in the finite W-plane.
This implies that
3U0 3 V
2 *2 .
3 U i 3V1
3V2 -3U2
3tj7 dV1
(I-10a,b)
in the finite W-plane and these derivative are continuous there,
To show: That
3U2 3V2 3V2 -3U2
ay
3x 3y
3X
(1-1la,b)
109
in the UH z-plane and that they are continuous in the UHP also.
Now,
3V2 3V2 3Ux aV2 3V
(1-12)
3y 3U1 3y 3 V1 3y
Substituting Eqs. (1-9) and (1-10) into Eq. (1-12), Eq. (1-12) becomes
(1-13)
Thus the condition given by Eq. (I-11 a) is satisfied in the region where
these derivatives exist and are continuous.
Similarly,
3V2 3V2 311} 3V2 3Vi
(1-14)
3X 3 U x 3X 3 V i 3X
Substituting Eqs. (1-9) and (I-10) into Eq. (1-14), Eq. (1-14) becomes
Thus the condition given by Eq. (I-llb) is satisfied in the region
where these derivatives exist and are continuous.
It is now argued that the derivatives exist and are continuous
for z in the UHP. This is true because for any z in the UHP, including
UH , z(z) may take on any value in the finite W plane. Also, the
derivatives of Ux and V1 with respect to x and y exist and are continuous
for z in the UHP, and the derivatives of U2 and V2 with respect to U} and \1
exist and are continuous anywhere in the finite W plane. Thus the con
ditions given by Eqs. (I-11 a) and (I-llb) are satisfied, and g[Z(z)] is
analytic in the UH z-plane.
no
Theorem III
If h(z) is analytic for all z in the UHP and if F[h(x,0)] = (oo),
then for u0 > 0
i{Re[h(xtO)eJwox]} .
h F^u-uq) u >
O) < Ur
jg Fh ( -u-Uq ) u < -uc
(1-16)
Lemma to Theorem III
If h(z) is analytic for all z in the UHP and if F[h(x,0)] = Fh(u),
then
FReh(u) = ^tRe[h(x,0)]}
% Fh(u)
u > 0
[F^uJ-j^TTk^u)! = [Fj!|(-u)+j2Trk1(-u)] u = 0
h Fh(-u) u < 0
(1-17)
Proof of Lemma to Theorem III:
From Eq. (1-23)
h(x,0) = U(x,0) + j[G(x,0)+k1].
Thus
Fh(u)
r
J
U(x,0) + j [ (x, 0) + kjle'^dx
U(x,0) e~JuXdx + j f U(xiO)e"JwXdx + jk, / e"j,xdx,
Using Eg. (2.2) and the difinition for the Fourier transform, we
F^(w) = FReh(a)) + j[-j sgn (w)] FReh(w) + j^irk^U)
or
FRehM
*sFh(w)
a) > 0
Ff-i(a))-j2-rrki6(ai) w s 0
Also, it is recalled that
FReh(-) = FReh()
This is seen from
Re[h(x
Thus from Eqs. (1-18) and (1-19) we obtain
,0)]eJtXdx
Re[h(x,0)]e"J)Xdx
FReh()
JgF^(a)) a) > 0
[F^uO-J^irk^w)] = [F^i-wJ+jZTrk^i-w)] w = 0
%Ffi(-o)) a) < 0
Proof of Theorem III:
By aid of Eg. (1-23) we have
111
obtain
(1-18)
(1-19)
h(x,0)ejox = {U(x,0)+j[U(x,0)+k1]}ejx .
112
Then
ReCh(x90)e^a>x3> = f (U(x,0) cos to0x [Uix.Oj+k^sin wox}eja)Xdx
U(x.O) >s(ej,oX + e1"x) + J[0{x,O)+k1]yejx-e-;ilX)>e-jxdx
00 CO
= h J U(x,0)e"J^"w^xdx + % f U(x,0)e"J'^a)+)^xdx
00 CO
00 00
+ 3h C U(x,O)e'J^"w0^xdx jig f U(x,0)e'^w+w^xdx
v/ J
00 oo
00 00
+ Jhk1 f ej("w+)o)xdx jj5k1 f ej(_)"a)o)xdx
and by using Eq. (2.2) and the Fourier transform of U(x,0),
'{Re[h(x,0)e,;)oX]}= FRe^()-)0) + % FR(ahU+o>0)
Reh'
+ 3h [-J sgn (o-jq)3 FReh(c-)0) j% [-j sgn (w+a)0)]FReh(cD+iD0)
+ 3h 2Trk1(-to+Jo) 3h 2-rrk16(-cj-u)o). (1-20)
Using Eq. (1-17) from the Lemma to Theorem III to evaluate FReh(*) in
Eq. (1-20), Eq. (1-20)becomes
%F^(t->o) w > wq
0 |o)| < uo
%F^(-o)-w0) w -w0
F{Re[h(x,0)eja)x]} =
113
Theorem IV
If h(z) is analytic for all z in the UHP and i[h(x,0)] 5 F^U)
where Fh(^) = 0 for all n ud, then for to0 > 0
F{Re[h(x50)ej)x]}
%Fp,(-a)+u0)
0
0 < ) < 0)Q
10)1 > coo
%Fh(co+co0)
0 > ) > aiQ
(1-21)
Proof of Theorem IV:
The proof for Theorem IV is very similar to that for Theorem III.
By the aid of Eq. (1-23) we have
h(x,0)e'ja)x = {U(x,0) + j[U(x.0)+k1]}eJwx .
Then
F{Re[h(x,0)eJuoX]} =
{U(x,0) cos co0x + [GU.CO+kJ sin
{U(x,0)%(ej)x
+ e~ja)x) jtUx.Oj+kjMe^-e"^)} e'j)Xdx
00
= %
00
U(xt0)e^u"u^xdx + %
00
00
U(x,O)e"^)+a>0^xdx
CO
U(x,0)e"^")xdx + jk J~ (xs0)e'^)+)^xdx
CO
- 3k
KieJ(-+o)xdx +
kjej(--.o)xdx
00
00
114
and by using Eq. (2.2) and the Fourier transform of U(x,0)
i,{Re[h(x,0)e Ja)ox]} = ^ FReh^-^o) + % F^U+wo)
- zh C-j sgn (w-wo)] FReh(oo-a)0) + zh [-j sgn(w+w0)]FRe(u)+u)0)
- j% 2ttk^ (~c+jo) + zh 2?rki6( uju)q) (1-22)
Using Eq. (1-17) from the Lemma to Theorem III to evaluate FRe^(*) in
Eq. (1-22) and noting that = 0 for Q > u0 Eq. (1-22) becomes
{Re[h(xs0)e"j)x]} =
%F^(-)+Wo) 9 0 < ) < )Q
^F^w+coo)
j |w| > u)g
0 > o) > JQ
Theorem V
If h(x,y) = U(x,y) + jV(x,y) is analytic in the UHP (including UH )
then
h(t,0) = U(t,0) + jHU.Oh^] (1-23)
or
h(t,0) = [-V(t,0)+k2] + jV(t,0) (1-24)
or
h(t,0) = [-V(t,0)+k2] + j[U(t,0)+k1]- (1-25)
where
TT
ki = -lim ( V(R cos e,R sin e)de a real constant (1-26)
n R^ J
o
115
and
k2 = 11m [ U(R cos 0,R sin e)de a real constant. (1-27)
^ R-x >
0
Proof of Theorem V;
By Cauchy's Theorem
h(x)
dz = 0
(1-28)
Jc z-t
for c as shown in Figure 26 since h(z) is analytic in the UHP, where t is
real and finite.
Figure 26. Contour of Integration
Thus for e > 0
0 = lim
£-0
t-e
h(x,0)
x-t
dx +
h(t+ee^e)ee'-l'ej
Je
eec
d9 + I dx
t+e
11. I h(Re^)RjeJe de
R-x
ReJ0-t
or
0 = P
J
rh(x-P>. dx jirh(t.O) + lim f MReJ' ^R-J-e~- de
X-t R^oo J DoJ0.
ReJ -t
116
or for h(z) = U(x,y) + jV(x,y)
0 = P
U(x,0) + jV(x,0)
x-t
dx Jtt[U(t,0) + jV(t,0)]
+ lim
R-+00
[U(R cos e,R sin 0) + jV(R cos 0,R sin
R(cos 0 + j sin 0) t
Aside: calculate the term:
t f R[U+jV][-sin 0 + j cos 0]
I im j --
R-x J (R cos e-t) + jR sin 0
d0
1 im ¡
R-* ~
R[U + jV][t sin 0 + j(R-t cos 0)]
(R cos 0-t)2 + R2 sin2 0
d0
lim
R400
{[Ut sin 0 + V(t cos 0-R)] + j[Vt sin 0 + U(R-t
R 2t cos 0 + t2/R
For finite t, lim (t cos 0 R) = -R, lim (R-t cos 0) = R, and
R-Kjo R--K
lim [R 2t cos 0 + t2/R] = R, Thus Eq. (1-30) becomes
R-x
1 im
R->oo
{[Ut sin 0 VR] + j[Vt sin 0 + UR]}
R
d0
1 im *
R-*
'Ut sin 0
R
- V
+ j
Vt sin 0
R
+ U
de
0)]RjeJed0
(1-29)
COS 0)]}
de.
(1-30)
(1-31)
Since U and V are real and imaginary parts of a function which is analytic
117
in the open UHP, U and V are finite as R for 0 < 0 < rr. Thus
1 im
R-x
Ut sin 6
R
0 and 1 im
R-*
Vt sin e
R
= 0
for 0 < e < tt, and Eq. (1-31) becomes
11m {-V + jU} de
R-" 0
(1-32)
Substituting the right side of Eq. (1-32) for the right-hand term on
the right side of Eq. (1-29), Eq. (1-29) becomes
O P / dx + j P
V(x,0) d..
x-t ax
- jirU(t.O)
+ irV(t,0) -
1im / Vde + j Tim I Ude
R-x J R-^> J
(1-33)
Setting the real and imaginary parts of Eq. (1-33) equal to zero we get
0 = P
U(x,0)
x-t
dx + ttV (t ,0)
1 im
R-*
Vde
(I-34a)
and
0 = P
V(x,0) d..
x-t ax
TT
TrU(t,0) + lim f Ude
R-* vi
(I-34b)
Thus
and
V(t,0) = U(t,0) + lim \ Vde
11 R-*= x
U(t,0) = -V(t,0) + lim f Ude
17 R-x J
(I-35a)
(I-35b)
0
118
Now show that the integrals in Eq, (1-35) are bounded. Using Schwarz's
inequality,
1 im
R-*
l
TT
IT
l2de
TT
f (lU)2"-7lvU>2
0
where |V|max = max 11 im V(R cos e,R sin e)| for 0 < e < ir, which is
R-x
finite since h(z) is analytic in the UHP.
Similarly
1 im
R-x
1
IT
TT
f U de
J
o
is bounded. Thus using Eqs. (I-35a) and (I-35b)
h(t,0) 5 U(t,0) + jV(t,0)
= U(t,0) + ¡[UU.O+k!]
= [-v(t,0)+k2] + jV(t,0)
= [-v(ts0)+k2] + jtuit.oj+kj
where
and
ki
1 im
R-*
r
J
0
V de
a finite real constant
k2
1
TT
1 im
R-**>
TT
de
o
a finite real constant.
APPENDIX II
EVALUATION
OF eo(x + J'y)
Assume: x and y are joint Gaussian random variables, bbth having zero
mean values.
To show: e^x ^ = e"^x +J2uxy~^y2l
when x and y are joint Gaussian random variables.
The joint density function is
1
p(x,y)
2lTaXCTy(l "P
T- e 2ax2-ay2^p2)
[ay2x2-2axavpXy+0x2y2]
Then
J[x+jy]
_2TTaxay(l-p2)^
00 00
r
,j(x+jy)e 2ax2ay2(1-p2)
[ay2x2*2ax0ypxy+ax2y2]
dxdy
oo oo
r -
2ax2(l-p2)
x2-2(^~ py+jax2(l-p2)j x+
CTy2
u
-oo co
y2+y2a/(l-p2))
dxdy
oo oo
r'
J
00 00
2ax2(l-p2)
[x-k(y|
[2 -
2ax2(l-p2) L | y
k^y)+^y2+y2ax2(l-p2l|
V dxdy
where k(y) = ~ py + jox2(l-p2)
ay
119
120
Then
J[x+jy]
27raxcjy(l-p2)^
[i/ZtT ax(l-p2)^] 2ax2(l-p2) *-
[-k2(y)+^2+y2
1 [y+L]2 E-L2+ay2ox2(l-p2)]
/2ir
e 2ay2 e 2cry2
dy
where L = av2(l-j p)
7y
Jy 0v
Thus
e J[x+jy] ___ (/27 oy) e 2oy2
/2t
CTy
or
eJ (x+jy) = e-Js(ax2+j 2vxy-oy2}
where
and
x2o-p2
yxy = xy .
REFERENCES
1. J.R.V. Oswald, "The Theory of Analytic Band Limited Signals Applied
to Carrier Systems," IRE' Trans. on Circuit Theory, vol. CT-3,
December 1956,
2. F.F. Kuo and S.L. Freeny, "Hilbert Transforms and Modulation Theory,"
Proa. NEC, vol. 18, 1962.
3. E. Bedrosian, "The Analytic Signal Representation of Modulated
Waveforms" Proa. IRE, vol. 50, October 1962.
4. A. Papoulis, Probability,Random Variables,and Stochastic Processes3
New York: McGraw-Hill, 1965.
5 Bateman Manuscript Project, Tables of Integral Transforms, vols, 1
and 2, New York: McGraw-Hill, 1954.
6. E.C. Titchmarsh, Theory of Fourier Integrals, Second Edition, London:
Oxford, 1948.
7. F.E. Terman, Electronic and Radio Engineering3 Fourth Edition, New
York: McGraw-Hill, 1955.
8. DoE. Norgoard, "The Phase-Shift Method of Single-Sideband Signal
Generation," Proc. ire, vol. 44, December 1956.
9. J.L. Dubois and J.S. Aagaard, "An Experimental SSB-FM System,"
IEEE Trans, on Communication Systems3 vol. CS-12, June 1964.
10. R.M. Glorioso and E.H. Brazeal, Jr., "Experiments in SSB-FM Communi
cation Systems, IEEE Trans, on Communication Technology, vol. COM-13,
March 1955.
11. H. Voelcker, "Demodulation of Single-Sideband Signals via Envelope
Detection," IEEE Trans, on Communication Technology3 vol. COM-14
February 1966.
12. K.H. Powers, "The Compatibility Problem in Single-Sideband Trans
mission," Proc. IRE, vol. 48, August 1960. Comment: L.R. Kahn,
same issue, p. 1504.
13. T.S. George, "Correlation Estimation in Noise-Modulation Systems
by Finite Time Averages," IEEE Trans, on Instrumentation and
Measurement3 vol. IM-14, March/June 1965.
14. F. Haber, "Signal Representation," IEEE Trans, on Communication
Technology, vol. COM-13, June 1965.
121
15.
122
D. Middleton, Introduction to Statistical Communication Theory3
New York: McGraw-Hill, 1960.
16. R.E, Kahn and J.B. Thomas, "Some Bandwidth Properties of Simultaneous
Amplitude and Angle Modulation," IEEE Trans, on Information Theory3
vol. IT-11, October 1965.
17. L. A. Wainstein and V.D. Zubakov, Extraction of Signals from Noise3
Englewood Cliffs, N.J.: Prentice-Hall, 1962.
18. H. Voelcker, "Toward a Unified Theory of Modulation-Part II: Zero
Manipulation," Proo. IEEE3 vol. 54, May 1966.
19. J.E. Mazo and J. Salz, "Spectral Properties of Single-Sideband Angle
Modulation," IEEE Trans, on Communication Technology3 vol. COM-16,
February 1968.
20. R.E. Kahn and J.B. Thomas, "Bandwidth Properties and Optimum Demodu
lation of Single-Sideband FM," IEEE Trans, on Communication Tech
nology 3 vol. COM-14, April 1966.
21. A.H Nuttall and E. Bedrosian, "On the Quadrature Approximation to
the Hilbert Transform of Modulated Signals," Proc. IEEE3 vol. 54,
October 1966.
22. H.E. Rowe, Signals and Noise in Communication Systems3 Princeton,
N. J.: Van Nostrand, 1965.
23. P.F. Panter, Uodulation3 Noise3 and Spectral Analysis3 New York:
McGraw-Hill, 1965.
24. S.O. Rice, "Mathematical Analysis of Random Noise," B.S.T.J.,
vols. 23 and 24, 1944 and 1945 (Reprinted in Selected Papers on
Noise and Stochastic Processes3 N. Wax, Dover Paperback, 1954).
25. S.O. Rice, "Statistical Properties of a Sine-Wave plus Random Noise,"
b.s.t.j.j vol. 27, January 1948.
26. S.O. Rice, "Noise in FM Receivers," Time Series Analysis3 M. Rosen
blatt, Editor, New York: Wiley, 1963.
27. G. Raisbeck, Information Theory3 Cambridge, Mass.: M.I.T. Press
(paperback), 1963.
28. C.E. Shannon, "The Mathematical Theory of Communciation," B.S.t.j.3
vol. 27, July and October 1948 (Also in paperback, Univ. of Ill.
Press, 1963).
29. N.L. Wright and S.A.W. Jolliffe, "Optimum System Engineering for
Satellite Communication Links with Special Reference to the Choice
of Modulation Method," J. Brit. IRE3 May 1962.
123
30. W.G. Tuller, "Theoretical Limits on the Rate of Information,"
Proa, IREj vol. 37, 1949.
31. R.E.A.C. Paley and N. Wiener, "Fourier Transforms in the Complex
Domain," Am. Math, Soa. Colloq. Publ. vol. 10, 1934.
BIOGRAPHICAL SKETCH
Leon Worthington Couch, II was born on July 6, 1941, in Durham,
North Carolina, In June, 1959, he was graduated from Goldsboro High
School, Goldsboro, North Carolina. The author received a degree of
Bachelor of Science in Electrical Engineering from Duke University,
Durham, North Carolina, in June, 1963; and in the following fall, he
entered the University of Florida where he received a degree of Master
of Engineering in August, 1964. In September, Mr. Couch continued his
studies at the University of Florida, Department of Electrical Engineering,
working toward the degree of Doctor of Philosophy. During his time of
study at the University of Florida, the author held a Graduate Teaching
Assistantship until August, 1966. At that time-he accepted a NASA
Traineeship which he resigned in January, 1967, to accept the position
of Research Associate in the Department of Electrical Engineering.
Mr. Couch is married to the former Margaret Elizabeth Wheland.
He is a member of Tau Beta Pi and Eta Kappa Nu and a student member
of the Institute of Electrical and Electronics Engineers. In addition,
the author holds a First Class Radiotelephone license and an Amateur
Radio license as issued by the Federal Communications Commission.
124
This dissertation was prepared under the direction of the chairman
of the candidate's supervisory committee and has been approved by all
members of that committee. It was submitted to the Dean of the College of
Engineering and to the Graduate Council, and was approved as partial ful
fillment of the requirements for the degree of Doctor of Philosophy.
June, 1968
Dean, College of Engineering
Dean, Graduate School
Supervisory Committee:
Chairman
f//7aj
Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Couch, Leon
TITLE: Synthesis and Analysis of Real Single Side Band...
PUBLICATION 1968
DATE:
I,
L^>ov\ CoucM
as copyright holder for the
aforementioned dissertation, hereby grant specific and limited archive and distribution rights to
the Board of Trustees-of the-University f-Elafida and its agents. I authorize the University of
Florida to digitize and distribute the dissertation described above for nonprofit, educational
purposes via the Internet or successive technologies.
This is a non-exclusive grant ol permissions for specific off-line and on-line uses for an
indefinite term. Off-line uses shall be llmittPtb those specifically allowed by "Fair Use" as
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84
Using Eo. (7.16) and Eq. (7.22), we have
6 6 2(6+1) /2 / 1
(S/N)0 =
Ij (26)
TTT
In2(2)+| 610(26)1,(26)
(S/N) -f
(7.23)
for the case of SSB-FM plus Gaussian noise into a FM detector.
The signal-to-noise output can also be obtained in terms of the
unmodulated-signal-to-noise ratio [i.e. the carrier-to-noise power at
the input). From Eq. (7.6) we obtain
(S/N)-f = 10(26) (C/N)i
(7.24)
and Eq. (7.23) becomes
(S/N)o =
Il2(26)
6 62(6+1) /2
1 _
10 2 (^ 6)
Iq(26) + | 6I1(26)
(C/N)i
where (C/N)^ is the carrier-to-noise power ratio.
(7.25)
7.1-4. FM system
The signal-to-noise ratio at the output of a FM receiver for a
FM signal plus narrow-band Gaussian noise at the input can be obtained
by the same procedure as used above for SSB-FM. The factor e_m^ of
Eq. (7.6) is replaced by unity, and the bandwidth of the input noise
is given by Eq. (7.19). Then the output signal-to-noise ratio becomes
(S/N)0 = 3 62(6+l) (S/N)i
(7.26)
15.
122
D. Middleton, Introduction to Statistical Communication Theory3
New York: McGraw-Hill, 1960.
16. R.E, Kahn and J.B. Thomas, "Some Bandwidth Properties of Simultaneous
Amplitude and Angle Modulation," IEEE Trans, on Information Theory3
vol. IT-11, October 1965.
17. L. A. Wainstein and V.D. Zubakov, Extraction of Signals from Noise3
Englewood Cliffs, N.J.: Prentice-Hall, 1962.
18. H. Voelcker, "Toward a Unified Theory of Modulation-Part II: Zero
Manipulation," Proo. IEEE3 vol. 54, May 1966.
19. J.E. Mazo and J. Salz, "Spectral Properties of Single-Sideband Angle
Modulation," IEEE Trans, on Communication Technology3 vol. COM-16,
February 1968.
20. R.E. Kahn and J.B. Thomas, "Bandwidth Properties and Optimum Demodu
lation of Single-Sideband FM," IEEE Trans, on Communication Tech
nology 3 vol. COM-14, April 1966.
21. A.H Nuttall and E. Bedrosian, "On the Quadrature Approximation to
the Hilbert Transform of Modulated Signals," Proc. IEEE3 vol. 54,
October 1966.
22. H.E. Rowe, Signals and Noise in Communication Systems3 Princeton,
N. J.: Van Nostrand, 1965.
23. P.F. Panter, Uodulation3 Noise3 and Spectral Analysis3 New York:
McGraw-Hill, 1965.
24. S.O. Rice, "Mathematical Analysis of Random Noise," B.S.T.J.,
vols. 23 and 24, 1944 and 1945 (Reprinted in Selected Papers on
Noise and Stochastic Processes3 N. Wax, Dover Paperback, 1954).
25. S.O. Rice, "Statistical Properties of a Sine-Wave plus Random Noise,"
b.s.t.j.j vol. 27, January 1948.
26. S.O. Rice, "Noise in FM Receivers," Time Series Analysis3 M. Rosen
blatt, Editor, New York: Wiley, 1963.
27. G. Raisbeck, Information Theory3 Cambridge, Mass.: M.I.T. Press
(paperback), 1963.
28. C.E. Shannon, "The Mathematical Theory of Communciation," B.S.t.j.3
vol. 27, July and October 1948 (Also in paperback, Univ. of Ill.
Press, 1963).
29. N.L. Wright and S.A.W. Jolliffe, "Optimum System Engineering for
Satellite Communication Links with Special Reference to the Choice
of Modulation Method," J. Brit. IRE3 May 1962.
45
5*4-3. Equivalent-noise bandwidth
the equivalent-noise bandwidth, Aw* for the continuous part of
the power spectrum is defined by
(2aw)
27 pg-scC0)
2tt
Pg-SC(w)du = Rg_sc()
(5.49)
But
Thus
00
Pg-Sc(O) = f Rg-Sc(T>dT
on 00
(Aco) =
g-sc
(0)
Rg-SC(-r)di
Substituting for Rg_sc(T) by using Eq. (5.41) or Eq. (5.42) we
obtain (noting once again that R^Ct) is even and Ryy(-r) is odd)
(5.50)
5.5. Efficiency
A commonly.Used definition of efficiency for modulated signals
is [18]
n = Sideband Power/Total Power.
5.51
table of contents
Page
ACKNOWLEDGMENTS iv
LIST OF FIGURES viii
KEY TO SYMBOLS x
ABSTRACT xiii
CHAPTER
I. INTRODUCTION 1
II. MATHEMATICAL PRELIMINARIES 4
III. SYNTHESIS OF SINGLE-SIDEBAND SIGNALS 9
IV, EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN 18
4.1. Example 1: Single-Sideband AM with
Suppressed-Carrier 18
4,2 Example 2: Single-Sideband PM . 19
4.3. Example 3: Single-Sideband FM 21
4.4. Example 4: Single-Sideband a 23
V.ANALYSIS OF SINGLE-SIDEBAND SIGNALS .... 28
51. Three Additional Equivalent Realizations 28
5.2. Suppressed-Carrier Signals 30
5.3. Autocorrelation Functions 38
5.4. Bandwidth Considerations 42
5.4-1. Mean-type bandwidth 43
5.4-2, RMS-type bandwidth 44
5.,4-3. Equivalent-noise bandwidth 45
5 5 Efficiency 45
5,6 Peak-to-Average Power Ratio 46
v
3
general for the whole set, and some outstanding members of the set will
be chosen for examples to be examined in detail.
It should be noted that Bedrosian has classified various types of
modulation in a similar manner; however, he does not give a general repre
sentation for single-sideband signals [3],
13
\F[q(l(t))e^^} i
Figure 4 Voltage Spectrum of the Positive Frequency-
Shifted Entire Function of the Analytic Signal
The real upper single-sideband signal can now be obtained from
the complex single-sideband signal, g[Z(t)]e'-*a,ot, by taking the real
part. This is seen from Theorem III
Theorem III, If h(z) is analytic for all z in the
UHP and F[h(x,0)] e Fh(w), then for > 0,
F{Re[h(x,0)eJwox]}
^sFp-! (oCQ ) O) > 0)g
o
0) < 0),.
J O),.
This theorem is proved in Appendix I.
Thus the upper single-sideband signal for a given entire function is
XUSSB(t) = ReigCZUne^}
= Re{[U(ReZ(t) ,lmZ(t)) + jV^ReZ(t),lmZ(t))]eJwot}
= Re{[U(m(t) ,m(t)) + jV(mU) *m(t J)]eJa)^t}
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
SYNTHESIS AND ANALYSIS OF REAL
SINGLE-SIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS
By
Leon Worthington Couch, II
June, 1968
Chairman: Professor T. S. George
Major Department: Electrical Engineering
A new approach to single-sideband (SSB) signal design and ana
lysis for communications systems is developed. It is shown that SSB
signals may be synthesized by use of the conjugate functions of any
entire function where the arguments are the real modulating signal and
its Hilbert transform. Entire functions are displayed which give the
SSB amplitude-modulated (SSB-AM), SSB frequency-modulated (SSB-FM),
SSB envelope-detectable, and SSB square-law detectable signals. Both
upper and lower SSB signals are obtained by a simple sign change.
This entire generating function concept, along with analytic
signal theory, is used to obtain generalized formulae for the properties
of SSB signals Formulae are obtained for (1) equivalent realizations
for a given SSB signal, (2) the condition for a suppressed-carrier SSB
signal, (3) autocorrelation function, (4) bandwidth (using various-de
finitions), (5) efficiency of the SSB signal, and (6) peak-to-average
power ratio. The amplitude of the discrete carrier term is found to be
xi i i
BIOGRAPHICAL SKETCH
Leon Worthington Couch, II was born on July 6, 1941, in Durham,
North Carolina, In June, 1959, he was graduated from Goldsboro High
School, Goldsboro, North Carolina. The author received a degree of
Bachelor of Science in Electrical Engineering from Duke University,
Durham, North Carolina, in June, 1963; and in the following fall, he
entered the University of Florida where he received a degree of Master
of Engineering in August, 1964. In September, Mr. Couch continued his
studies at the University of Florida, Department of Electrical Engineering,
working toward the degree of Doctor of Philosophy. During his time of
study at the University of Florida, the author held a Graduate Teaching
Assistantship until August, 1966. At that time-he accepted a NASA
Traineeship which he resigned in January, 1967, to accept the position
of Research Associate in the Department of Electrical Engineering.
Mr. Couch is married to the former Margaret Elizabeth Wheland.
He is a member of Tau Beta Pi and Eta Kappa Nu and a student member
of the Institute of Electrical and Electronics Engineers. In addition,
the author holds a First Class Radiotelephone license and an Amateur
Radio license as issued by the Federal Communications Commission.
124
34
for the case of FM Gaussian noise [13].
Continuing with our SSB signals, it will now be shown that kx and
k2 depend only on the entire function associated with the SSB signal and
not on the n]odulation. From Theorem IV we have
IT
k = lim V[m,(R cos e,R sin e) m. (R cos e,R sin e)]de
1 IT D J 1 i
R-* o
and
IT
k, = lim / U[m.,(R co$ e,R sin e) m(R cos e,R sin )]de
17 R^ o
where U and V are the real and imaginary parts of the entire function
l1{z) = m^z) + jm^z) is the analytic function associated
with the analytic signal Z(t) of m(t).
It is seen that if
lim m (R cos e,R sin e) = 0 0 < e < v (5.18a)
R-x
and
lim mJR cos e,R sin e) = 0 0 < e < tt (5.18b)
R-*
then kx and k2 depend only on U and V of the entire function and not on
m. Thus we need to show that Eq. (5.18a) and (5.18b) are valid. By the
theory of Chapter III there exists a function Z^z) = m^z) + jm^z)
which is analytic in the UHP such that (almost everywhere) 11^ Zx(t + jy)
y
= Z(t) = m(t) + jm(t) where the Fourier transform of Z(t), F(ai), is
l2(-, ). Then we have
51
The peak-to-average power ratio for the SSB-AM-SC signal follows
from Eq. (5.56c), and it is
f[m(t)]2 + [ii(t)]2}
tpeak
SC-SSB-AM
2
(6.10)
m
6.2. Example 2: Single-Sideband PM
The SSB-PM signal has a discrete carrier term. This is shown by
calculating the constants k1 and k2. Substituting Eq. (4.5b) into
Eq. (5.4) we have
K .-Inin f cos e-R sin e)s1n [m,(R cos e.R sir, e)]de.
1 7r R-K. J 1
But from Eqs. (5.18a) and (5.18b) lim m^R cos e,R sin e) = 0 for
R-X
0 < e < Ti and lim m.(R cos e, R sin e) = 0 for 0 < e < tt. Thus
R-**>
kl = 0. (6.11)
Likewise, substituting Eq. (4.5a) into Eq. (5.5) we have
IT
k = C e" cos 0 de = 1. (6.12)
2 TT J
0
Thus the SSB-PM signal has a discrete carrier term since k2 = 1 f 0.
There are equivalent representations for the SSB-PM signal since
k and k are not both zero. For example, for the upper sideband signal,
1 2
equivalent representations are given by Egs. (5.7) and (5.8). It is
noted that Eq. (5.8) is identical to Eq. (5.9) for the USSB-PM signal
49
since 1 im m(R cos e,R sin e) = 0 for 0 < e < ir from Eq (5.18a). Further-
R-*
more, since both k and k2 are zero, the equivalent realizations for the
SSB signals, as given by the equations in Section 5.1, reduce identically
to the phasing method of generating SSB-AM-SC signals (which was given
previously in Figure 8).
The autocorrelation for the SSB-AM-SC signal is readily given by
use of Eq. (4.2a) and Eq (5.20). Thus
y-(m(t) ,m(t)) = m(t). (6.3)
Then the autocorrelation of the suppressed-carrier USSB-AM signal is
given via Eq. (5.43b), and it is
RXU-SC-SSB-AM^ Rmm^ cos WT Rmm^x^ sin oT*
(6.4)
Likewise, by use of Eq. (5.44b) the autocorrelation for the suppressed-
carrier LSSB-AM signal is
RXL-SC-SSB-AM^ WT)
cos
oT
r(t)
mm
sin
(6.5)
From Eq. (6.4) it follows that the spectrum of the USSB-AM-SC
signal is just the positive-frequency spectrum of the modulation shifted
up to on and the negative-frequency spectrum of the modulation shifted
down to oj0. That is, there is a one-to-one correspondence between the
spectrum of this SSB signal and that of the modulation. This is due to
the fact that the corresponding entire function for the signal, g(W) = W,
is a linear function of W. Consequently, the bandwidths for this SSB
36
For e = 0 or 0 = tt
Z (+>) ,0 = 0
Tim ¡Z (Reje)
R->~
Z(-oo)
= 0
since
Z(t) e L (-*>, ).
Then
lim |Z1(ReJ0)| = 1 im |Z?L(R cos 0, R sin 0) | = 0 0 < 0 < rr
R-x Rx
which shows that Eqs. (5.18a) and (5.18b) are valid statements. Thus,
the presence (kx and k2 not both zero) or the absence (kx = k2 = 0) of
a discrete carrier depends only on the entire funtion associated with
the SSB signal and not on the modulation. Furthermore, it is seen that
the amplitude of the discrete carrier is given by the magnitude of the
entire function evaluated at the origin (of the W plane), and the power
in the discrete carrier is one-half the square of the magnitude.
For every generalized USSB signal represented by Eq. (3.5),
there exists a corresponding sppressed-carrier USSB signal:
XUSSB-SC^ = ,^Km('t) ,m(t)) cos w0t ¥-(m(t) ,m(t)) sih wot
(5.19)
43
that the suppressed-carrier formulae are needed' instead of the "total sig
nal" formulae since, from the engineering point of view, the presence or
absence of a discrete carrier should not change the bandwidth of the sig
nal Various definitions of bandwidth will be used [16, 17],
5.4-1. Mean-type bandwidth
Since the spectrum of a SSB signal is one-sided about the carrier
frequency, the average frequency as measured from the carrier frequency
is a measure of the bandwidth of the signal:
f wPg_ScU)du j Rg_sc()
oo
- = *' (5 45)
CO v '
f Pg-SC^^ Rg-SC(O)
00
where Pg_^c(w) is the power spectral density of g$c(m(t) ,m(t))and the
prime indicates the derivative with respect to t. The relationship is
valid whenever Rg_sg(0) and Rg_$c^ exist. Substituting Eg. (5.41)
into Eq. (5.45) we have
f[R' (0) + JR' (0)]
j trtr trtr
" a'1 o
2[RW(0) + jRyyjO)]
But it recalled that Ryy(x) is an even function of t and, from Chapter II,
RyU(t) is an odd function of t. Then Ryy(O) = Ryy(O) = 0 and it follows
that
Ryy.(0) Rw(o)
Ryy-(o) Rw(o)
(5.46)
72
Likewise, substituting Eq. (4.10a) into Eq. (5.5), we have
k2 =1. (6.53)
Thus the SSB-a signal has a discrete carrier term.
It follows that equivalent representations for the SSB-a signal
are possible since k2 f 0. This is analogous to the discussion on equiva
lent representations for SSB-PM signals (Section 6.2) so this subject will
not be pursued further.
The autocorrelation function for the SSB-a signal will now be ob
tained in terms of Rmm(T) Using Eq. (5.21) and Eq. (4.10b) we have
Rw(t) = [eam(t) sin am(t)][eam(t"T) sin am(t-t)]
or
r (t) = 53{ea[m(t)+m(t-T)]} {eja[m(t)-m(t-T)] _eja[m(t)+m(t-T)]}
+ %{ea^(t)+m(t-r)]} {_eja[-m(t)-m(t-T)] + eja[-m(t)+m(t-r)]}.
(6.54)
The density function of m(t) has to be specified in order to carry out
this average. It is recalled that m(t) is related to the modulating
signal e(t) by the equation:
m(t) In [1+(t)H .
Now assume that the density function of the modulation is chosen such
that m(t) is a Gaussian random process of all orders. Eq. (6.54) can
then be evaluated by the procedure that was used to evaluate Eo. (6.16).
30
Using Eq. (5.2) the second equivalent representation is
XUSSB^ = t-V(m(t),m(t))+k2] cos w0t V(m(t),m(t)) sin w0t.
(5.8)
Using Eq. (5.3) the third equivalent representation is
XUSSB^ = iv(m(t;) iti(t))+k23 cos (o0t [U(m(t) ,m(t))+k1], sin u)Qt
(5.9)
Likewise the three lower SSB signals, which are equivalent to
Eq. (3.8), are
It should be noted, however, that if for a given entire function
k: and k2 are both zero, then all four representations for the USSB or
the LSSB signals are identical since by Theorem V, U = -V and V = 0 under
these conditions.
5.2. Suppressed-Carrier Signals
The presence of a discrete carrier term appears as impulses in
the (two-sided) spectrum of transmitted signal at frequencies w0 and -
The impulses may have real, purely imaginary, or complex-valued weights
depending on whether the carrier term is cos a)0t, sin u)0t, or a com
bination of the two. Thus the composite voltage spectrum of the modulated
63
This has the same numerical value as that obtained from the result given
by Mazo and Salz [19]. The result may also be compared to that given by
Kahn and Thomas for the SSB-PM signal with sinusoidal modulation [20].
From Eg. (19) of their work
^rms)$SB-,PM-S = ua^ (6.30)
where wa is the frequency of the sinusoidal modulation and 6 is the modu
lation index. For comparison purposes, equal power will be used for m(t)
in Eq. (6.30) as was used for m(t) in Eq. (6.29). Then Eq. (6.30) becomes
(k>rms)ssB-PM-S = ^ 3 *^o (6.31)
Thus it is seen that for Gaussian modulation the rms bandwidth is propor
tional to the power in m(t) when the power is large (ip0 > > tt/4), and for
sinusoidal modulation the rms bandwidth is proportional to the square root
of the power m(t).
The efficiency for the SSB-PM signal with Gaussian modulation
will now be obtained. Substituting Eo. (6.22) into Eq. (5.54) we have
e2+ra-i
"SSB-PM-GN +
or
- i p"2^m
nSSB-PM-GN
(6.32)
where is the noise power of m(t).
The peak-average to average power ratio for Gaussian m(t) is given
25
and
V3(m(t)9m(t)) = em^ sin (am(t)). (4,10b)
Substituting Eqs. (4.10a) and (4.10b) into Eq. (3.5), the SSB-a signal is
XUSSB-a^ eam^ cos (om(t)) cos w0t
- eam^ sin .(am(t)) sin ajQt
or
XUSSB-a^ ~ cos (0t +
(4.11)
In terms of the input audio waveform, Eq. (4.11) becomes
XUSSB-a(t) = ealn[1+e(t)] cos ()Qt + an[l+e(t)])
or
XUSSB-a^ = [l+e(t)]a cos u0t + a1n[1+e(t)]).
(4.12)
For a = 1 we have an envelope-detectable SSB signal, as is readily
seen from Eq. (4.12). Voelcker has recently published a paper demon
strating the merits of the envelope-detectable SSB signal [11]. The real
ization of Eq. (4.12) is shown in Figure 11.
For a = 1/2 we have a square-law detectable SSB signal. This type
of signal has been studied in detail by Powers [12]. Figure 12 gives the
block-diagram realization for the square-law detectable SSB exciter.
71
consequently, infinite pwer. In other words, the system does not have a
steady-state output condition if the input has a power around oi =0. Thus,
this system ia actually conditionally stable, the output being bounded
only if the input power spectrum has a slope greater than or equal to +6
db/octave near the origin (and, consequently, zero at the origin) as seen
from Eq. (6,50). It is interesting to note that for the case of FM, ejm^
is stationary regardless of the shape of the spectral density Pqq(ua). This
is due to the fact that ejm^ is bounded regardless of whether m(t) is
bounded or not.
From Eq. (6.50) we can readily obtain Rmm(ir) for any input process
e(t) which has a bounded output process m(t). Thus
oo
WO = J- f ^-ej3T du (6.51)
IT J tl)
00
Furthermore, R^m(0), Rmm(T), and Rmm(0) may be obtained in terms of
Pee(w). By substituting for these quantities in the equations of Section
6.2, the properties of a SSB-FM signal can be obtained in terms of the
spectrum of the modulating process.
6.4. Example 4: Single-Sideband g
The SSB-a signal has a discrete carrier term. This is readily
shown by calculating the constants kT and k2. Substituting Eq. (4.10b)
into Eq. (5.4) we have
IC 1 11m f e"MR cos e*R s1n 6> sin am, (R cos 9.R sin 8)d8 .
1 n R-Mo J 1
.0
But lim m^R cos e, R sin e) = 0, for 0 < e s: tt and lim m^R cos e,
R-x R-x
R sin e) = 0 for 0 £ e $ it. Thus
kx =0. (6.52)
CHAPTER VIII
SUMMARY
In this work a new approach to SSB signal design and analysis
for communication systems has been presented. The key to this approach
is the philosophy of using a modulated-signal generating function--the
generating function bing any entire function.
It was hypothesized in Chapter I that SSB signals were of the
third basic modulation class, the first two being AM and FM.
In Chapter II a brief review of analytic signal theory was pre
sented, and this theory was used in successive chapters to facilitate
the derivations.
In Chapter III it was shown that signals of the SSB class could
be generated by use of entire generating functions and that these sig
nals were truly SSB signals regardless of the modulating process.
Generalized formulae were derived which may represent upper SSB or lower
SSB modulated signals. These formulae are analogous to those representing
AM and FM signals. However, it is noted that any SSB signal is a com
bination of AM and FM.
Chapter IV gave some examples of well-known SSB signals, using
the appropriate entire generating function to obtain their mathematical
representation and, consequently, their physical realization.
The generating function concept, along with analytic signal
theory, was used in Chapter V to obtain generalized formulae for the
properties of SSB signals. The properties that were studied were:
102
107
Proof of Theorem I:
Given: k(z) is analytic in the UHP and eJ)Z is analytic in the
UHP for
To show
00
F(w) = | k(x,0)e"jwXdx = 0 ¥ u> < 0. (1-8)
By the Lemma k(z)eJuZ is analytic in the UHP for all
Cauchy's Theorem
k(z)e'ja)Zdz = 0
for c as shown in Figure 25 since k(z)e'J)Z is analytic in the UHP.
Thus
^ k(z)e"JCl)Zdz
k(x,0)eja)Xdx
+ lim f k(R sin e,R cos e)e
R-*
o
jReJ6 RJeJ'6d6 .
But for to < 0,
lira I f k eR s1n V>Rcs Wed6| < lim f |k|eR s1" eRde
R~**>
R->o
CHAPTER IV
EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN
Specific examples of upper single-sideband signal design will now
be presented* Entire functions will be chosen to give signals which have
various distinct properties. In Chapter VI these properties will be ex
amined in detail. Only upper sideband examples are presented here since
the corresponding lower sideband signals are given by the same equation
except for a sign change (Eq. (3.5) and Eq. (3.8)).
4.1. Example 1: Single-Sideband AM With Suppressed-Carrier
This is the conventional type of single-sideband signal that is
now widely used by the military, telephone companies, and amateur radio
operators. It will be denoted here by SSB-AM-SC,
Let the entire function be
9l(W)=W (4.1)
and let m(t) be the modulating signal. Then substituting the corresponding
analytic signal for W
gx(Z(t)) = m(t) + jm(t)
or
(m(t),iii(t)) = m(t) and Vx(m(t),m(t)) = m(t). (4.2 a,b)
18
100
7,3-5. Comparison of system efficiencies
Eqs. (7.51), (7.53), (7.56), and (7.59) are plotted in Figure 24
in order to compare the efficiencies for the AM, SSB-AM-SC, SSB-FM, and
FM systems. The efficiency is given as a function of (S'/N)0 with the
modulation index, 6, as a parameter. For example, from the figure it
is seen that, for FM with 6=2 and (S'/N)i = 30 db, the FM system re
quires about 135 db more power than an ideal system with the same IF-to-
baseband bandwidth ratio and the same output signal-to-noise ratio.
From Figure 24, it is seen that SSB-AM-SC is an ideal system in
the sense of trading bandwidth for output signal-to-noise ratio. Also,
AM is the next best system, and SSB-FM and FM are the poorest systems
according to this criterion.
e(t)
Figure 12. Square-Law Detectable USSB Signal Exciter
15
as in the upper single-sideband synthesis. Then the Fourier transform of
the down-shifted complex baseband signal is
F[g(St))e-ot] F[g(Zt))] *
- Fn {W ) (u>+\on)
or
[g(Z(t))ej)ot] = Fg(u+u0) a), > 0. (3,7)
This spectrum is illustrated in Figure 6.
Figure 6. Voltage Spectrum of the Negative Frequency-
Shifted Entire Function of an Analytic Signal
Theorem IV: If h(z) is analytic for all z in UHP and
F[h(x,0)] e F^(oj) where Fh(fi) = 0 for all n > go0, then
for oj0 > 0
y{Re[h(x,0)e'JUJx]
^(-W+uJg)
0
0 < j < )q
M > uo
JgFh (tD+u)0}
0 > oo > -
This theorem is proved in Appendix I.
76
tern is taken to be a system which has optimum trade-off
between predetection signal bandwidth and postdetection
signal-to-noise ratioo)
Comparison of AM, FM, SSB-AM-SC, and SSB-FM systems will be made using
these three figures of merit. It is clear that these comparisons are
known to be valid only for the conditions specified; that is, for the
given modulation density function, and detection schemes which are used
in these comparisons.
7.1, Output Signal-to-Noise Ratios
7.1-1. AM system
Consider the coherent receiver as shown in Figure 16 where the
input AM signal plus narrow-band Gaussian noise is given by
X(t) + n.¡(t) = {A0[l + 6 sin tmt] cos co0t}
+ ixc(t) cos )Qt xs(t) sin cdQt} (7.1)
where X(t) is the input signal, n-¡(t) is the input noise with a flat spec
trum over the bandwidth 2wm> and 6 is the modulation index.
X(t)+nj(t)
Low Pass
Fi 1 ter
AC Couple
2k cos wqL
Figure 16. AM Coherent Receiver
Output
Then the output signal-to-noise power ratio, where A0k6 sin ojmt is the
output signal, is given by
(S/N) 0 =
6 2
1 + %62
(S/N)1
(7.2)
79
The input signal plus noise is given by
X(t) + n^(t) A0e"^^^ cos [o>0t + m(t)] + n-j(t) (7.6)
where A0 ~ The amplitude of carrier
u>o The radian frequency of the carrier
m(t) = D /t v(t) dt
m(t) = m(t) ~= The Hilbert transform of m(t)
nj(t) Narrow-band Gaussian noise with power spectral density F0
over the (one-sided spectral) IF band
and v(t) is the modulation on the upper SSB-FM signal. The independent
narrow-band Gaussian noise process may be represented by
n^(t) = R(t) cos [w0t + 4>(t)J = xc(t) cos w0t x$(t) sin w0t
where xs(t) = xc(t) since the IF passes only the frequencies on the upper
sideband of the carrier frequency.
Then the phase of the detector output is obtained from Eq. (7.6)
and is
;p(t) = k tan
which reduces to
A0e~^ sin m(t) + R(t) sin (t)
A0e^ cos m(t) + R(t) cos (t)
(7.7)
xp(t) km(t) + k tan^
R(t) sin [(t) m(t)]
A0e-^ + R(t) cos [m(t) (t)]
(7.8)
where k is a constant due to the detector. The detector output voltage
is given by Eq. (7.8) is identical to the phase output when the
input is conventional FM plus noise except for the factor e_n1' .
38
n(t) = s(t) cos wot s(t) sin wot. (5.24)
Thus Eq. (5.22) checks with Eq, (5.24) where it = s(t), and Eq. (5.23)
checks also where -V e s(t).
The corresponding representations for LSSB suppressed-carrier
signals are given by
XLSSB-SC^ = -y-(m(t),rn(t)) cos w0t + tt(m(t),m{t)) sin w0t
(5.25)
and
XLSSB-SC^ = "'^'(m(t) ,m(t)) cos w0t + ¥(m(t) ,m(t)) sin w0t
(5.26)
where it and-V-are given by Eq. (5.20) and Eq. (5.21).
This representation also checks with that given by Haber for pro
cesses with spectral components only for |w| < wo which is
n(t) = s(t) cos wot + §(t) sin w0t.
(5.27)
5.3. Autocorrelation Functions
The autocorrelation function for the generalized SSB signal and
the corresponding suppressed-carrier SSB signal will now be derived.
Using the result of Chapter III, it is known that the generalized
upper SSB signal can be represented by
XUSSB
j(w0t+)}
(t) = Re{g(m(t),m(t))e'
(5.28)
103
1. Equivalent realizations for a given SSB signal
2. The condition for a suppressed-carrier signal
3. Autocorrelation function
4. Bandwidth (using various definitions)
5. Efficiency
6. Peak-to-average power ratio.
The amplitude of the discrete carrier term was found to be equal to the
absolute value of the entire function (associated with a particular SSB
signal) evaluated at the origin and was not affected by the modulation.
Furthermore, for suppressed-carrier SSB signals, the real and imaginary
parts of the complex envelope are a unique Hilbert transform pair;
otherwise, they are a Hilbert transform pair to within an additive con
stant.
In Chapter VI the properties for examples of various SSB signals
were studied where stochastic modulation was assumed. The results were
compared with those published in the literature where possible.
In Chapter VII a comparison of AM, SSB-AM-SC, SSB-FM and FM
systems was carried out. This was a comparison of the various modu
lation schemes from the overall viewpoint of generation, transmission
with additive Gaussian noise, and detection. Three figures of merit
were used for comparison:
1. Output signal-to-noise ratios
2. Energy-per-bit of information
3. System efficiency.
It was found that, for a given RF signal power, FM has the greatest post
detection signal-to-noise ratio if the modulation index is large. For
small index SSB-AM-SC is best, with SSB-FM and FM second, and AM is
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62
Thus
Rmm(T) = "J i^o e 2 T j 0T
or
Rmm(T) ~ " Rmm(T) a n .
From Ea. (6.27) it follows that
Rmm() = -Vo2
and from Ea. (6.28) we have
Rmm(O) s ~pzr
/2tt
Substituting these two equations into Ea. (6.25) we get
(
ta
rms
2
1 e~Z>po
Thus if m(t) has a Gaussian spectrum and if the modulation has
density function, the SSB-FM signal has the rms bandwidth:
(wrms^sSB-PM-GN
2i|>0az[4(ip0/;ir) + 1]
1 e
where t|>0 is the total noise power in m(t)
a4is the "variance" in the spectrum of m(t).
(6.28)
Gaussian
(6.29)
Page
VIII. SUMMARY 102
APPENDIX
L PROOFS OF SEVERAL THEOREMS 105
II. EVALUATION OF e j(x + Jy^ 119
REFERENCES 121
BIOGRAPHICAL SKETCH 124
vi i
97
7.3. System Efficiencies
The third figure of merit which will be used to compare systems
is the system efficiency, defined by
Transmitted power required for an ideal system
6 =
Transmitted power required for an actual system
= sj/S-j (7.47)
where the ideal system is taken to be a system which has optimum trade
off between predetection signal bandwidth and postdetection signal-to-
noise ratio. This concept is used by Wright and doll iffe to compare
SSB-AM-SC and FM systems [29]. Here, it will be extended to AM and
SSB-FM systems.
The trade-off between predetection signal bandwidth and post
detection signal-to-noise ratio for an ideal system is obtained by
equating the predetection capacity to the postdetection capacity since
an ideal system does not lose information in the detection process [30].
Thus
(B/2it) log2 [1 + (S'/N)i] = (b/2ir) log [1 + (S'/N)0] (7.48)
where B is the IF bandwidth
b is the baseband bandwidth
(S'/N)-} is the input signal-to-noise ratio for the ideal system
(S'/N)0 is the output signal-to-noise ratio for the ideal system
The prime is used here to denote the ideal system. Eq. (7.48) reduces to
(s7n)0 = [1 + (S'/N)]Y 1 (7.49)
where y = B/b, the IF to baseband bandwidth ratio.
The efficiency, e will now be calculated for various types of
systems.
114
and by using Eq. (2.2) and the Fourier transform of U(x,0)
i,{Re[h(x,0)e Ja)ox]} = ^ FReh^-^o) + % F^U+wo)
- zh C-j sgn (w-wo)] FReh(oo-a)0) + zh [-j sgn(w+w0)]FRe(u)+u)0)
- j% 2ttk^ (~c+jo) + zh 2?rki6( uju)q) (1-22)
Using Eq. (1-17) from the Lemma to Theorem III to evaluate FRe^(*) in
Eq. (1-22) and noting that = 0 for Q > u0 Eq. (1-22) becomes
{Re[h(xs0)e"j)x]} =
%F^(-)+Wo) 9 0 < ) < )Q
^F^w+coo)
j |w| > u)g
0 > o) > JQ
Theorem V
If h(x,y) = U(x,y) + jV(x,y) is analytic in the UHP (including UH )
then
h(t,0) = U(t,0) + jHU.Oh^] (1-23)
or
h(t,0) = [-V(t,0)+k2] + jV(t,0) (1-24)
or
h(t,0) = [-V(t,0)+k2] + j[U(t,0)+k1]- (1-25)
where
TT
ki = -lim ( V(R cos e,R sin e)de a real constant (1-26)
n R^ J
o
i o
Since m(t) is generated by a physically realizable process, it con
tains finite power for a finite time interval This, of course, is equiva
lent to saying that m(t) is a finite energy signal or, in mathematical
terms, it is a L2 function. By Theorem 91 of Titchmarsh, m(t) is also a
member of the L2 class of functions almost everywhere [6]. Now the complex
signal Z(t) is formed by
Z(t) = m(t) + jm(t). (3,1)
It is recalled that Z(t) is commonly called an analytic signal in the
literature. By Theorem 95 of Titchmarsh there exists an analytic func
tion (regular for y > 0), Z^z), such that as y -* 0
Zj(x + jy) Z(t) = m(t) + jm(t) x = t
for almost all t and, furthermore, Z(t) is a L2 (-*>, function [6]
It follows by Theorem 48 of Titchmarsh that the Fourier transform of Z(t)
exists [6],
Theorem I: If k(z) is analytic in the UHP then the
spectrum of k(t,0), denoted by F^oj), is zero for
all to < 0, assuming that k(t,0) is Fourier trans
formable.
For a proof of this theorem the reader is referred to Appendix I.
Thus the voltage spectrum of Z(t) is zero for ui < 0 by Theorem I
since Z(t) takes on values of the analytic function Z,(z) almost every
where along the x axis. Furthermore, since Z(t) is an analytic signal--
that is, it is defined by Eq. (3=1)its voltage spectrum is given by
CHAPTER III
SYNTHESIS OF SINGLE-SIDEBAND SIGNALS
Eq. (1.4), which specifies the set of single-sideband signals
that can be generated from a given modulating waveform or process, will
be derived in this chapter. The equation must be a real function of a
real input waveform, m(t), since it represents the generating function
of a physically realizable system--the single-sideband transmitter--and,
in general, it is non-linear. Analytic signal techniques will be used
in the derivation. It will be shown that if we have a complex function
k(z) where k(z) is analytic for z = x + jy in the upper half-plane (UHP),
then the voltage spectrum of k(x,0) k(t) is zero for w < 0. In order
to synthesize real SSB signals from a real modulating waveform, an UHP
analytic generating function of the complex time veal modulating process
must be found regardless of the particular (physically realizable) wave
form that the process assumes.
Let m(t) be either the real baseband modulating signal or a veal
function of the baseband modulating signal e(t), Then the amplitude of
the voltage spectrum of m(t) is double sided about the origin, for ex
ample, as shown by Figure 1.
9
60
The equivalent-noise bandwidth is obtained by substituting
Eq. (6.18) into Eq. (5.50):
(Ato)
%{e2Rmm(r) cos [26mrT1(T)] 1} dt
%[e^m-l] ^
or
(Aw)
;(e2^i 1)
SSB-PM-GN
{e2Rmm(T) co$ [2Rrnm(T)] 1 }di
(6.26)
It is noted that the equivalent-noise bandwidth may exist when the formu
lae for the other types of bandwidth are not valid because of the non
existence of derivatives of Rmm(i:) at t = 0.
It is obvious that the actual numerical values for the bandwidths
depend on the specific autocorrelation function of the Gaussian noise.
For example, the rms bandwidth of the SSB-PM signal will now be calculated
for the particular case of Gaussian modulation which also has a Gaussian
spectrum. Let
2
-0)
0
where Pm(u) is the spectrum of m(t)
4>0 = is the total noise power in m(t)
o2 is the "variance" of the spectrum.
The autocorrelation corresponding to this spectrum is
Rmm(r) ~ Vr
1 2 2
-Ho t
(6.27)
77
or
(7.3)
where (S/N)-j The input signal-to-noise power ratio
(C/N)i = The input carrier-to-noise power ratio
and the spectrum of the noise is taken to be flat over the IF bandpass
which is 2a)m(rad/s).
7.1-2. SSB-AM-SC system
Consider the coherent receiver (Figure 16) once again, where
the input is a SSB-AM-SC signal plus narrow-band Gaussian noise. Then
the input signal plus noise is
X(t) + ni(t) = iA0[m(t) cos w0t m(t) sin o>0t]}
+ [xc(t) cos (O0t xs(t) sin oj0t]
(7.4)
where
m(t) = 6 sin wt
m
and xs(t) = xc(t) if the IF passes only upper sideband components. The
input noise is assumed to have a flat spectrum over the bandwidth )m.
Then the output signal-to-noise power ratio, where AQk6 sin wmt is the
output signal, is given by [23]
(7.5)
(S/N)Q = (S/N)i
where the spectrum of the noise is taken to be flat over the IF bandpass
which is cjjm(rad/s).
It is interesting to note that the same result is obtained from a
61
The Hilbert transform of Rmm(T) is also needed and is obtained by the
frequency domain approach. It is recalled from Chapter II that
P*(o)) =
mm
j PmmU) > w > 0
, cu 0
j Pmn ) < 0
Then
Wt) = ^im
W") eJT d
2tr
-U)2 t -00
iaiS ei#* ejwTdu> f e2^ eJ<*Tdw
which reduces to
fynm^)
/2rrcr
f
St ~coz ...
sin oox dw .
This integral 1s evaluated by using the formula obtained from page 73,
#18, of the 8ateman Manuscript Project, Tab1es of In teg ral fra ns forms,
vol. 1 [5]:
00
/
1 ..2
eaX sin xy dx e ^ Erf(~4=. y)
Z/a \2/a /
Re a > 0
where
Erf (x) *
Jr f
-t
e z dt.
equal to the absolute value of the entire generating function evaluated
at the origin provided the modulating signal is AC coupled. Examples
of the use of these formulae are displayed where these properties are
evaluated for stochastic modulation.
The usefulness of a SSB signal depends not only on the pro
perties of the signal but on the properties of the overall system as well.
Consequently, a comparison of AM, SSB-AM, SSB-FM, and FM systems is
made from the overall viewpoint of generation, transmission with additive
Gaussian noise, and detection. Three figures of merit are used in these
comparisons: (1) Output signal-to-noise ratios, (2) Energy-per-bit of
information, and (3) System efficiency.
In summary, the entire generating function concept is a new tool
for synthesis and analysis of single-sideband signals.
xiv
12
Figure 3. Voltage Spectrum of an Entire Function
of an Analytic Signal
Now multiply the complex baseband signal g[Z(t)] by eJuJot to
translate the signal up to the transmitting frequency, oi0. It is noted
that g[Zj(z)] and eju)z for ojq > 0 are both analytic functions in the UH
z-plane By the Lemma to Theorem I in Appendix I, g[Z; (z)]ev,l'Uo2 is ana
lytic in the UH z-plane. Therefore, by Theorem I the voltage spectrum
of g[Z(t)]eJU)ot is one sided about the origin. Furthermore,
FCgUltiJeK4] = -LF[g(z(t))] *
(- 7T
= Fg (oo) 6(w-o)0)
or
F[gU(t))e^ot] = Fg(ld-u)q) o>0 > 0 (3.4)
This spectrum is illustrated in Figure 4.
CHAPTER V
ANALYSIS OF SINGLE-SIDEBAND SIGNALS
The generalized SSB signal, that was developed in Chapter III,
will now be analyzed to determine such properties as equivalent gener
alized SSB signals, presence or absence of a discrete carrier term,
autocorrelation functions, bandwidths, efficiency, and peak-to-averajge
power ratio. Some of these properties will depend only on the entire
function associated with the SSB signal, but most of the properties will
be a function of the statistics of the modulating signal as well.
5.1. Three Additional Equivalent Realizations
Three equivalent ways (in general) for generating an upper SSB
signal will now be found in addition to the realization given by Eq. (3.5).
Similar expressions will also be given for lower SSB signals which are
equivalent to Eq. (3.8). It is very desirable to know as many equivalent
realizations as possible since ally orle of them might be the most econom
ical to implement for particular SSB signal.
Theorem V: If h(x,y) = U(x,y) + jV(x,y) is
analytic in the HP (including UH) then
h (t ,0) = U(t,0) + jiOU^+kj] (5.1)
or
h(t,0) = [4(t,0)+k2] + jV(t,0) (5.2)
28
39
where a uniformly distributed phase angle 4> has been included to account
for the random start-up phase of the RF oscillator in the SSB exciter.
Then, using Middleton's result [15], the autocorrelation of the USSB sig
nal is
RXU^ XUSSB^t+T^XUSSB^t^ %Re{eJa)TRg(T)}
(5.29)
where
Rg(t) = g(m(t+x),m(t+T)) g*(m(t),m(t)) (5.30)
and
g(m(t),m(t)) = U(m(t),m(t)) + jV(m(t) ,m(t)). (5.31)
The subscript XU indicates the USSB signal. For the generalized LSSB
signal the corresponding formulae are
XL$SB(t) = Re{g(m(t),m(t))e"J^t+
(5.32)
and
Rxl(t) = %Re{e"J"wTRg(T)}.
(5.33)
These equations can be simplified if we consider the autocorre
lation for the continuous part of the spectrum of the SSB signal. The
suppressed DC carrier version of g, denoted by g$c> will first be found
in terms of g, and then the corresponding autocorrelation function Rg_sc(T)
APPENDIX I
PROOFS OF SEVERAL THEOREMS
Theorem I
If k(z) is analytic in the UHP, then the spectrum of k(t,0),
denoted by F|<(w), is zero for all w < 0, assuming that k(t,0) is Fourier
transformable. (This result is included in Theorem 95 of Titchmarsh [6]
and in the work of Paley and Wiener [31].)
Lemma to Theorem I
If Wj(z) and W2(z) are analytic in the UHP, then W(z) W1(z)W2(z)
is analytic in the UHP.
Proof of the Lemma to Theorem I:
Assume that W1(z) and W2(z) are analytic in the UHP, which implies
that they are continuous. Then if W(z) satisifies the Cauchy-Riemann
(C-R) relation for all z in the UHP, W(z) is analytic in the UHP.
Given: Wx and W2 are analytic in UHP. Then
W
I = Uj + J'V1 =>
w2 = u2 + jV2 =>
9U,
8V1 .
3U
1^ _
1
in
the
UHP
(I-la)
3X
ay
3X
3y
8U2
3 V2
3V2
3U2
the
(I-lb)
=
-
in
UHP
3X
3X
3y
and these partial derivatives are continuous.
To show: W = U + jV is analytic for all z in the UHP by showing
in the UHP (I-2a)
3V __3U
3x --3y
in the UHP
(I-2b)
105
95
For (C/N)-j = 12 db, this reduces to
46 (6+1)
M(S)
log2 [1 + 48 2(+l)]
It is recalled that Raisbeck obtained this result [27].
(7.46)
7.2-5. Comparison of energy-per-bit for various systems
It is recalled that M(
bit for the actual system to the energy-per-bit for the ideal system
when the output system noise is Gaussian. The output noise is Gaussian
for the AM, SSB-AM-SC, and FM systems [for FM, (C/N)i = 12 db >> 0 db].
Also, from Eq. (7.10) it is seen that the noise out of the SSB-FM system
is Gaussian for small index (say 6 < 1). Thus, for the systems that are
analyzed above, M(6) represents the ratio of the energy-per-bit for the
actual system to the energy-per-bit for the ideal system. Then in db,
M() gives the energy-per-bit required above the ideal system.
The systems are compared in terms of energy-per-bit (db) above
the ideal system in Figure 23, where Eqs. (7.40), (7.42), (7.44), and
(7.46) have beqn plotted for the AM, SSB-AM-SC, SSB-FM, and FM systems.
From this figure it is seen that the FM system is best, followed by
SSB-AM-SC, SSB-FM and AM. Furthermore, the FM system is about 12 db
worse than the ideal system. These comparisons are valid for output
signal-to-noise ratios of about 25 db.
In addition, Figure 23 specifies the modulation index to use
for each type of system in order to minimize the energy required to
transmit one bit of information.
Figure 11. Envelope-Detectable USSB Signal Exciter
no
Theorem III
If h(z) is analytic for all z in the UHP and if F[h(x,0)] = (oo),
then for u0 > 0
i{Re[h(xtO)eJwox]} .
h F^u-uq) u >
O) < Ur
jg Fh ( -u-Uq ) u < -uc
(1-16)
Lemma to Theorem III
If h(z) is analytic for all z in the UHP and if F[h(x,0)] = Fh(u),
then
FReh(u) = ^tRe[h(x,0)]}
% Fh(u)
u > 0
[F^uJ-j^TTk^u)! = [Fj!|(-u)+j2Trk1(-u)] u = 0
h Fh(-u) u < 0
(1-17)
Proof of Lemma to Theorem III:
From Eq. (1-23)
h(x,0) = U(x,0) + j[G(x,0)+k1].
Thus
Fh(u)
r
J
U(x,0) + j [ (x, 0) + kjle'^dx
U(x,0) e~JuXdx + j f U(xiO)e"JwXdx + jk, / e"j,xdx,
33
XPM(t) = tAn cos (zr cos wat)] COS u)nt
'0 'Wa a*'-1 -0'
- [An sin (cos to t)] sin wnt.
(5.16)
To identify Eq. (5.16) with Eq (5.13) we have to find the DC terms of
and
f. (t) + c. e An cos (cos coat)
i i o wa a
and
f0(t) + c2 = A0 sin (- cos u)at).
wa
These are
c, = An cos ( cos w,t)
i u M- a
Ao
C D
/ cos ( cos wat)dt
v/ 0) a
Vo<£>
(5.17a)
c. = A sin ( cos wat)
2 o>a
_ A0
T o
0
J sin (^~ cos aiat)dt
(5.17b)
Then for sinusoidal frequency-modulation it is seen that the discrete
carrier term has an amplitude of AQJ0(D/ul^) which may or may not be zero
depending on the modulation index D/wa. Consequently, for FM it is seen
that the discrete carrier term may or may not exist depending on the
modulation. Prof. T. S. George has given the discrete carrier condition
69
where L is a linear time-invariant operator, then y(t) is strict-sense
stationary if x(t) is strict-sense stationary and that y(t) is wide-sense
stationary if x(t) is wide-sense stationary [4]. Since the integral is a
linear operator, we need to show only that it is time-invariant, that is
to show that
y(t+e) = L Jx(t+E)j
or
t+e
e(t1)dt1 = j e(t2+e)dtr
This is readily seen to be true by making a change in the variable,
letting t = t2 + e. Thus, if e(t) is stationary, then m(t) is stationary.
Moreover, in the same way it is seen that if m(t) had been defined by
t
mj (t) = D 1 (t')dt' (6.47)
tQ
then m (tj is not necessarily stationary for e^t) stationary since the
system is time-varying (i.e. it was turned on at tQ). But this should
not worry us because, as Middleton points out, all physically realizable
systems have non-stationary outputs since no physical process could
have started out at t = - and continued without some time variation in
the parameters D5]. However, after the "time-invariant" physical systems
have reached steady-state we may consider them to be stationary processes
provided there is a steady state. Thus by letting tQ -> - we are con
sidering the steady-state process m(t) which we have shown to be stationary
Now the autocorrelation of m(t) can be obtained by using power-spectrum
techniques since m(t) has been shown to be stationary. From Eq. (6.46)
118
Now show that the integrals in Eq, (1-35) are bounded. Using Schwarz's
inequality,
1 im
R-*
l
TT
IT
l2de
TT
f (lU)2"-7lvU>2
0
where |V|max = max 11 im V(R cos e,R sin e)| for 0 < e < ir, which is
R-x
finite since h(z) is analytic in the UHP.
Similarly
1 im
R-x
1
IT
TT
f U de
J
o
is bounded. Thus using Eqs. (I-35a) and (I-35b)
h(t,0) 5 U(t,0) + jV(t,0)
= U(t,0) + ¡[UU.O+k!]
= [-v(t,0)+k2] + jV(t,0)
= [-v(ts0)+k2] + jtuit.oj+kj
where
and
ki
1 im
R-*
r
J
0
V de
a finite real constant
k2
1
TT
1 im
R-**>
TT
de
o
a finite real constant.
56
and
yxy 25 x(t)y(t) .
Thus
oxj ~ [m(t)-m(t-t )F = 2[am -R^ir)]
cx2 [m(t)+m(t-i)]T 2[orn2+Rmm( r)]
ox? [-m(t)-m(t-x )]2 = 2[am2+Rmm(x)]
axl C-nri(t)+m(t-r)]2- 2[am2-RtTim(x)]
and
a/ [l(t)+l(t-,)]2 2[am2+Rnlm(r)] .
From Chapter II it is recalled that Rm^(0) = 0 and R^r) = -R-m(x) -
-Rmm(x) so that the y averages are
yx v = [m(t)-m(t- r )][m(t)+m(t-i)] -2Rmm(t)
iy
yx v = [m(t)-m(t-x)][m(tT+m(t-x)] = 0
2J
Mv = -[m(t)+m(t-r)][m(t)+m(tX)] = 0
and
%y
= -[m(t)-m(t-x)][m(t)+m(t-x)] = 2Rmrtl(x) .
104
is poorest. For the lease energy-per-bit of information, FM is best,
followed by SSB-AM-SC, SSB-FM, and AM. When the systems are compared
in terms of optimum trade-off between predetection bandwidth and post
detection signal-to-noise ratio [i.e. system efficiency) SSB-AM-SC was
found to be ideal, with AM second best, followed by SSB-FM and FM.
In conclusion, the entire generating function concept should be
helpful in obtaining new types of SSB signals, and the corresponding
formulae for analyzing these signals will be helpful in classifying
these signals according to their properties. However, one should also
evaluate the overall system performance in the presence of noise to
determine the usefulness of these signals.
59
where is the noise power of m(t) It is seen that Eq, (6,23) may or
may not exist depending on the autocorrelation of m(t).
The rms-type bandwidth can be obtained with the help of the second
derivative pf Eq. (6.18):
C(i) i-e2R(t) sin [2Rmm(T)]> 2[Rm(,)]2
*VV-
+ (-e2R(r) cos [2Rmm(T)]) 2[R^(t)]2
+ t-e2R(l) sin [2Rmm(T)]> R^(t)
+ t e2R(T> cos [2Rmnl(-r)]> 2[R(Of
+ <-e2R"(T) sin [2R(t)]} 2R^(T)t(,)
+ { e2R"
Thus
Rw() = e
2^m
{R¡>) 2C4(0)]2>
(6.24)
Substituting Eq. (6.24) and Eq. (6.22) into Eq. (5.48) we have for the
rms-type bandwidth of the SSB-PM signal with Gaussian noise modulation:
(wrms)
/2{2[RmlO)f R"(0)}
SSB-PM-GN
1 e
-2iPm
(6.25)
This expression for the rms bandwidth may or may not exist depending on
the autocorrelation of m(t). It is interesting to note that Mazo and Salz
have obtained a formula for the rms bandwidth in terms of different para
meters [19] However both of these formulae give the same numerical re
sults, as we shall demonstrate by Eq. (6.29).
98
7.3-1, AM system
For the AM system y = B/b = 2. Then setting Eq. (7.49) equal to
Eq, (7.2) we have
[1 + (S/N)i32 -1
1 +
(S/N)i.
(7.50)
Substituting for S-¡ from Eq. (7.47), the efficiency for the AM system is
obtained, and it is
3 =
2
1
1 + ig2
_(S'/N)i + 2_
(7.51)
The AM efficiency will be compared to those for other systems in
Section 7.3-5 as a function of (S'/N)i with the modulation index as a
parameter.
7.3-2. SSB-AM-SC system
For the SSB-AM-SC system y = B/b = 1. Then, equating Eq. (7.49)
and Eq. (7.5), we have
(S'/NJi = (S/N) -f (7.52)
and substituting for Si using Eq. (7.47), the SSB-AM-SC efficiency is
(7.53)
7.3-3. SSB-FM system
For the SSB-FM system, using Eq. (7.20),
= = 2 (6+1) /2
m
' 1^(26)
1 Iq2(26)
Y
(7.54)
Figure 13* USSB-PM Signal ExciterMethod II
Alo = Equivalent-Noise Bandwidth
nr = Mean-Type Bandwidth
* = The Convolution Operator
()* = The Conjugate of ()
() = The Hilbert Transform of ()
() = The Averaging Operator
XI 1
REFERENCES
1. J.R.V. Oswald, "The Theory of Analytic Band Limited Signals Applied
to Carrier Systems," IRE' Trans. on Circuit Theory, vol. CT-3,
December 1956,
2. F.F. Kuo and S.L. Freeny, "Hilbert Transforms and Modulation Theory,"
Proa. NEC, vol. 18, 1962.
3. E. Bedrosian, "The Analytic Signal Representation of Modulated
Waveforms" Proa. IRE, vol. 50, October 1962.
4. A. Papoulis, Probability,Random Variables,and Stochastic Processes3
New York: McGraw-Hill, 1965.
5 Bateman Manuscript Project, Tables of Integral Transforms, vols, 1
and 2, New York: McGraw-Hill, 1954.
6. E.C. Titchmarsh, Theory of Fourier Integrals, Second Edition, London:
Oxford, 1948.
7. F.E. Terman, Electronic and Radio Engineering3 Fourth Edition, New
York: McGraw-Hill, 1955.
8. DoE. Norgoard, "The Phase-Shift Method of Single-Sideband Signal
Generation," Proc. ire, vol. 44, December 1956.
9. J.L. Dubois and J.S. Aagaard, "An Experimental SSB-FM System,"
IEEE Trans, on Communication Systems3 vol. CS-12, June 1964.
10. R.M. Glorioso and E.H. Brazeal, Jr., "Experiments in SSB-FM Communi
cation Systems, IEEE Trans, on Communication Technology, vol. COM-13,
March 1955.
11. H. Voelcker, "Demodulation of Single-Sideband Signals via Envelope
Detection," IEEE Trans, on Communication Technology3 vol. COM-14
February 1966.
12. K.H. Powers, "The Compatibility Problem in Single-Sideband Trans
mission," Proc. IRE, vol. 48, August 1960. Comment: L.R. Kahn,
same issue, p. 1504.
13. T.S. George, "Correlation Estimation in Noise-Modulation Systems
by Finite Time Averages," IEEE Trans, on Instrumentation and
Measurement3 vol. IM-14, March/June 1965.
14. F. Haber, "Signal Representation," IEEE Trans, on Communication
Technology, vol. COM-13, June 1965.
121
57
Therefore, using Eq. (6.17), Eq, (6.16) becomes
-%{2[am -Rmm(r)] + j2[-2Rmm(-t)] 2[am2+Rmm(T)]}
u e-^i2[om +Rmm(t)] + j20 2[am2+Rmm(T)]}
_ ^ e-%{2[am2+Rmt11(T)] + j20 2[am2+Rrnm(t)]}
+ ^ eJs{2[o|T| -Rmm(T)3 + j2[2Rmm(x)] 2[am2+Rmm(i:)]}
which reduces to
(6.18)
where W -SSB-PM-GN denotes the autocorrelation of the imaginary part of
the entire function which is associated with the suppressed-carrier SSB-
PM signal with Gaussian noise modulation.
It is noted that Eq. (6.18) is an even function of t, as it should
be, since it is the autocorrelation of the real function -V-(m(t) ,m(t))
Furthermore Ryy(O) is zero when Rmii)(0) 0, as it should be, since the
power in any suppressed-carrier signal should be zero when the modulating
power is zero.
The autocorrelation of the USSB-PM signal is now readily obtained
for the case of Gaussian noise modulation by substituting Eq. (6.18) into
Eq. (5.42) and using Eq. (537):
R
XU-SSB-PM-GN
(t) = % Re ejT{[e2R>(x) Cos (2Rmm(t))]
(6.19)
42
form is relatively easy to obtain On the other hand Rg(x) may be calcu
lated directly from g(m(t),m(t)) or indirectly by use of Ry^t), Rvv(x),
Ruv(t), and Rvu(t).
The autocorrelation functions for the generalized USSB and LSSB
signals having a suppressed-carrier are readily given by Eq. (5.37) and
Eg. (5.38) with kj = k, 0:
(5.43a)
(5.43b)
(5.43c)
and
(5.44a)
(5.44b)
(5.44c)
It follows that the power spectral density of any of these SSB
signals may be obtained by taking the Fourier transform of the appro
priate autocorrelation function presented above.
5.4. Bandwidth Considerations
The suppressed-carrier autocorrelation formulae developed above
will now be used to calculate bandwidths of SSB signals. It is noted
RXL-SCTRg.sc(r)}
Rw(t) cos O)0x + R^t) sin 000
W
Rw(t) cos UQT + Rw(x) sin CjOq '
RXU-SC^ %Re{eJTRg_sc(T)}
- Ryy.(t) cos o>qT Ryy( :) sin coot
w t) cos wot Ryy(x) sin u)0t
81
or
dn0(t)
dt
kem(t) % r___
~Yo J~2F(con) K-w0) COS [(wn-w0)t + 0n]
+
ke(t)
Ao
CO
z
n=l
^2F{n) 2. sin C(n-u.0)t + 8].
Noting that {en} are independent as well as uniformly distributed and
that the noise spectrum is zero below the carrier frequency, the output
noise power is
N
o
dn0(t)
dt
m
F^d. + -- e2S
Ao2
dm (t)
dt
2-it
^m
F0dw
o
k2
e2m(t)'
^ m3 + k2 e2m(t)
dm(t)
Ao2
J
2 rr 3 /\ 2
no
dt
- _
(7.11)
where () is the averaging operator and wm is the baseband bandwidth
(rad/s) Now let v(t) = -Am cos tomt then, averaging over t, we have
2m(t)
w,
2u/c|
m
2tt
m
e26 cos dt = In(26)
and
2m(t)
dm(t)
dt
I (m6)2 [10(26) Iz(2)]
1^6 1,(26)
(7.12)
(7.13)
90
Figure 21. Output SignaT-to-Noise to Input Signal-to-Normalized-
Noise Power Ratio for Various Systems
DEDICATION
The author proudly dedicates this dissertation to his parents,
Mrs,. Leon Couch and the late Rev, Leon Couch, and to his wife, Margaret
Wheland Couch,
66
The terms in the brackets are the USSB and LSSB parts of the suppressed-
subcarrier signal m(t). But these USSB and LSSB parts are recognized
as the well-known representation for a narrow-band Gaussian process.
Thus m(t) is a narrow-band Gaussian process.
Now the previous expressions for bandwidth, which assume that
m(t) is Gaussian, may be used. The mean-type bandwidth for the multi
plexed SSB-PM signal is then readily given via Eq. (6.23), and it is
(<*>)
M-SSB-PM-GN
oo
f eRaa(x) cos waA cos[RaaU) sin coaA]dA
00
e^a 1
(6.39)
where tpa is the average power of the Gaussian distributed subcarrier
modulation a(t). Obtained in a similar manner, the rms bandwidth is
(rms^.ssB.pM.
GN
a W1) Raa(0)
1 e^a
(6.40)
and the equivalent-noise bandwidth is
(aw)
r[e2*a-l]
M-SSB-PM-GN
J' eRaa^T^ cos aT Cos[Raa(x) sin toax]
(6.41)
Thus, it is seen once again that the bandwidth depends on the spectrum of
the modulation, actually the subcarrier modulation a(t).
To obtain a numerical value for the rms bandwidth of the multi
plexed SSB-PM signal assume that the spectrum of a(t) is flat over
|o)| < w0 < )a.
86
Figure 19 Output to Input Signal-to-Noise
Power Ratios for Several Systems
99
Note that for SSB-FM y and s are uniquely related to each other (by
Eq, (7.54)), unlike the AM and SSB-AM-SC cases. Equating Eq. (7.49) and
(7.23), we have
[i + (S7N),]Y l =
Then, substituting for Sn- from Eg. (7.47), the efficiency for SSB-FM is
32y
|_I02(2S) + | 6l0(2)I1(2)
(S'/N)i
[1 + (SVN)iF 1
(7.56)
362i
I02(26) + | 6I0(26)I1(26)
(S/N)-
(7.55)
where y and 6 are uniquely related by Eq. (7.54).
7.3-4. FM system
For the FM system, using Eq. (7.19),
Y = 7T = 2(6+1). (7.57)
m
Thus for FM, y and 6 are uniquely related, as was the case in SSB-FM.
Eauating Eq, (7.49) and Eq. (7.26) we have
[1 + (S'/N)i]T 1 = |y(J l)2 (S/N)-j. (7,58)
Then, substituting for S-j from Eg. (7.47), the efficiency for FM is
(7.59)
This is identical to the result obtained by Panter [23].
19
Substituting Eqs. (4.2a) and (4.2b) into Eq. (3.5) we obtain the
upper single-sideband signal:
XUSSB-AM-SC^ cos o1 m(t) sin Jot
(4.3)
where m(t) is the modulating audio or video signal and m(t) is the Hil
bert transform of m(t). It is assumed that m(t) is AC coupled so that
it will have a zero mean.
The upper single-sideband transmitter corresponding to the gene
rating function given by Eq, (4.3) is illustrated by the block diagram
in Figure 8. It is recalled that this is the well-known phasing method
for generating SSB-AM-SC signals [7, 8],
4.2. Example 2: Single-Sideband PM
Single-sideband phase-modulation was described by Bedrosian in
1962 [3].
To synthesize this type of signal, denoted by SSB-PM, use the
entire function:
(4.4)
Let m(t) be the modulating audio or video signal. Then
g2(Z(t)) = eJ(m^ + ^(t)) =
or
U2(m(t),iii(t)) = e-"i(t)
cos m(t)
(4.5a)
113
Theorem IV
If h(z) is analytic for all z in the UHP and i[h(x,0)] 5 F^U)
where Fh(^) = 0 for all n ud, then for to0 > 0
F{Re[h(x50)ej)x]}
%Fp,(-a)+u0)
0
0 < ) < 0)Q
10)1 > coo
%Fh(co+co0)
0 > ) > aiQ
(1-21)
Proof of Theorem IV:
The proof for Theorem IV is very similar to that for Theorem III.
By the aid of Eq. (1-23) we have
h(x,0)e'ja)x = {U(x,0) + j[U(x.0)+k1]}eJwx .
Then
F{Re[h(x,0)eJuoX]} =
{U(x,0) cos co0x + [GU.CO+kJ sin
{U(x,0)%(ej)x
+ e~ja)x) jtUx.Oj+kjMe^-e"^)} e'j)Xdx
00
= %
00
U(xt0)e^u"u^xdx + %
00
00
U(x,O)e"^)+a>0^xdx
CO
U(x,0)e"^")xdx + jk J~ (xs0)e'^)+)^xdx
CO
- 3k
KieJ(-+o)xdx +
kjej(--.o)xdx
00
00
Energy (db) Required Above Ideal System
35
1.0 2.0
Modulation Index ()
3.0
4.0
Figure 23. Comparison of Energy-per-Bit for Various Systems
108
and |k| < M, a constant, since k is analytic in the UHP,
1 im
R-*
ke'jwReje Rjejede| < M 1im f eR sin eRde = 0
R-x 'ft
Therefore
F(oj) I k(x,0)e"JuXdx = 0 u> < 0
Theorem II
If Z(z) is an analytic function of z in the UHP and if g(W) is
an entire function of W, then g[Z(z)] is an analytic function of z in
the UH z-plane.
Proof of Theorem II:
The C-R relations will be used to show that g[Z(z)] is analytic
in the UH z-plane.
Given: Z(z) = Ux(x,y) + jV^x.y) is analytic in the UH z-plane.
This implies that
3U-,
- ?h.
. =
3 U,
3X
3y
3X
sy
(I-9a,b)
in UHP and these derivatives are continuous there.
g(W) U2(U1,V1) + JV2(Ux,Vx) is analytic in the finite W-plane.
This implies that
3U0 3 V
2 *2 .
3 U i 3V1
3V2 -3U2
3tj7 dV1
(I-10a,b)
in the finite W-plane and these derivative are continuous there,
To show: That
3U2 3V2 3V2 -3U2
ay
3x 3y
3X
(1-1la,b)
58
Likewise, the autocorrelation of the LSSB-PM signal may be obtained by
using Eq. (5.38).
The autocorrelation of the suppressed-carrier USSB-PM signal with
Gaussian modulation is given by using Eq. (543a):
RXU-SC-SSB-PM-GN^ = ** Re
e^oT{[e2Rmm(0 cos (2Rmm(T)) 1]
+ j[e2lWT) cos (2Rmm(T)]:
(6.20)
Similarly, the autocorrelation of the suppressed-carrier LSSB-FM signal
may be obtained by using Eq. (5.44a).
The mean-type bandwidth will now be evaluated for the SSB-PM
signal assuming Gaussian noise modulation. From Eq. (6.18) we obtain
R^(t) =
1 r e2R"(x) cos (2Rmm(i)di
2J (t-x)2
Then
Rvv(0)
2 it
,2Rmm(x)
cos [2Rmm(x)]d/
(6.21)
and from Eq. (6.18)
Rw(0) = %[e2^ l]
(6.22)
where = om2 is the average power of m(t). Substituting Eqs. (6.21)
and (6.22) into Eq. (5.46) we have the mean-type bandwidth for the
Gaussian noise modulated SSB-PM signal:
oo
P J~ ~T e2FWA) cos[2Rmm(x)]dx
A
()
SSB-PM-GN
32^m
(6.23)
120
Then
J[x+jy]
27raxcjy(l-p2)^
[i/ZtT ax(l-p2)^] 2ax2(l-p2) *-
[-k2(y)+^2+y2
1 [y+L]2 E-L2+ay2ox2(l-p2)]
/2ir
e 2ay2 e 2cry2
dy
where L = av2(l-j p)
7y
Jy 0v
Thus
e J[x+jy] ___ (/27 oy) e 2oy2
/2t
CTy
or
eJ (x+jy) = e-Js(ax2+j 2vxy-oy2}
where
and
x2o-p2
yxy = xy .
This dissertation was prepared under the direction of the chairman
of the candidate's supervisory committee and has been approved by all
members of that committee. It was submitted to the Dean of the College of
Engineering and to the Graduate Council, and was approved as partial ful
fillment of the requirements for the degree of Doctor of Philosophy.
June, 1968
Dean, College of Engineering
Dean, Graduate School
Supervisory Committee:
Chairman
74
and
RXL-SSB-a-GN^ ~ Re
e"JuoT{[e2a2Ree(T) cos (2a2fr0e(r))]
+ J [e
2a2Ree(t)
cos (2a2tee(r))]}
(6.58)
The efficiency is readily obtained by substituting Eq. (6.56)
into Eq. (5.54):
'SSB-a-GN
= 1
5-2a2(J>m
(6.59)
where is the power in the Gaussian m(t) and |e(t)| < < 1. This result
may be compared for a = 1 to that given by Voelcker for envelope-detectable
SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t);
and Eq. (6.59) becomes
nSSB-oi-GN : 1 e 00 ~ 2o0/. (6.60)
This agrees with Voelcker"s result (his Eq. (38)) when the variance of the
modulation is small-~the condition for Eq. (6.60) to be valid.
The expressions for the other properties of the SSB-ct signal, such
as bandwidths and peak-to-average power ratio, will not be examined further
here since it was shown above that these properties are the same as those
obtained for the SSB-PM signal when Um)sSB-PM ~ a2(iJm^sSB-a
m(t) is Gaussian.
as long as
82
where 6 DAm/tom, the modulation index. Substituting Eq, (7,12) and
Eq (7,13) into Eq, (7,11) we obtain for the output noise power
k2F0i%3
2ttA02
ll0(2i) +isl1(26)
Referring to Eq. (7.9), the output signal power is
2
= y
Then the output signal-to-noise ratio is
dkm(t)
dt
(7,14)
(7.15)
(S/N)0
1 1
- 1.(26) + 61,(26)
3 0 2 1
(S/N)n -
A0262
2 <%, 1 Io(26) + I 01,(26)1
2ir 3 2
(7.16)
Referring to Eq. (7.6), the signal power into the detector is
Si = A02 e2"1^ cos2 [o)0t + m(t)] = A02 e2^^^
jA02 I0(26)
(7.17)
Kahn and Thomas have given the ratio of the rms bandwidths (taken about
80
For large input signal-to-noise ratios {i.e. A0e-m^ > > R(t)
most of the time), Eq (7o8) becomes
kR(t)
ip(t) km(t) + sin [(t) m(t)]
(7.9)
dn0(t)
Then the noise output voltage is where
n (t) = R(t) sin [
A
(7.10)
Now the phase (t) is uniformly distributed over 0 to 2n since the input
noise is a narrow-band Gaussian process. Then for m(t) deterministic,
U(t) m(t)J is distributed uniformly also. Furthermore, R(t) has a
Rayleigh density function. Then it follows that R(t) sin [(t) m(t)]
is Gaussian (at least to the first order density) and, using Rice's
formulation [24, 25],
where F(u>) = F0 is the input noise spectrum and {n} are independent
random variables uniformly distributed over 0 to 2-rr. Actually it is
known that the presence of modulation produces some clicks in the out
put [26], but this effect is not considered here. Eq. (7.10) then be
comes
32
Likewise V has a zero mean value. Then, identifying Eg. (5.13) with
Eq. (5.9), it is seen that
fx(t) S -V(m(t),m(t))
(5.14a)
f2(t) = U(m(t),m(t))
(5.14b)
1 = k2 and c2 = k .
(5.14c ,d)
Similarily, for lower SSB signals Eq. (5.13) can be identified
with Eq. (5.12).
Thus the SSB signal has a discrete carrier provided that kx and k2
are not both zero.
As an aside, it is noted that the criterion for a discrete car
rier, given by Eq. (5.13), is not limited to SSB signals; it holds for
all modulated signals. For AM Eq. (5.13) can be identified with Eq. (1.1)
Here
fj(t)
= A0m(t)
(5.15a)
f2(t)
5 0
(5.15b)
5 Ao
and c2 = 0
(5.15c,d)
because m(t) has a zero mean due to AC coupling in the modulator of the
transmitter. Thus for AM it is seen that there is a discrete carrier
term of amplitude c1 = AQ which does not depend on the modulation. For FM
Eq. (1.2) can be expanded. Then, assuming sinusoidal modulation of fre
quency wa, we obtain
5
The Fourier transform of m(t) is given by
FfiU) = [-j sgn (o)] Fm(to)
(2.2)
where
sgn (u)
+ 1 u) > 0
0 a) = 0
- 1 to < 0_
(2.3)
and Fm(j) is the Fourier transform of m(t). In other words, the Hilbert
transform operation is identical to that performed by a -90 all-pass
linear (ideally non-realizable) network.
From Eq. (2.2), it follows that
F*(u>) = [-j sgn (w)]2 Fm(u>) -Fm(co) (2.4)
or
m(t) = -m(t).
(2.5)
The (complex) analytic signal associated with the real signal
m(t) is defined by
Z(t) = m(t) + jm(t).
(2.6)
The Fourier transform of Z(t) follows by the use of Eq. (2.2),
and it is
Fz(<>) = Fm(o>) + j[-j sgn (w)] Fm(w)
52
since k1 = 0. Thus the two equivalent representations are:
X
USSB PM^ = [e"m^cos m(t)] cos a)0t [_e '"'"'cos mQt;J sin w0t (6.13)
and
XUssB_p|v|(t) = [-^;sin m(t))+l]cos oi0t [e_r"^sin m(t)]sin w0t. (6.14)
The USSB-PM exciters corresponding to these equations are shown in Figure
13 and Figure 14. They may be compared to the first realization method
given in Figure 9.
The autocorrelation function for the SSB-PM signal will now be
examined. In Section 6.1 the autocorrelation for the SSB-AM-SC signal
was obtained in terms of the autocorrelation function of the modulation.
This was easy to obtain since 44 = m(t). However, for the SSB-PM case 44
and -V-are non-linear functions of the modulation m(t). Consequently, the
density function for the modulation process will be needed in order to
obtain the autocorrelation of the SSB-FM signal in terms of Rrnm(T)
To calculate the autocorrelation function for the SSB-PM signal,
first Ryy(t) will be obtained in term of R^fx). Using k;L = 0, Eq. (5.21),
and Eq. (4.5b) we have
V-(m(t) ,m(t)) a V(m(t),m(t)) = e_m(t) sin m(t).
(6.15)
Then
91
Figure 22. Output Signal-to-Noise to Input Carrier-to-Normalized-
Noise Power Ratio for Various Systems
92
Then the RF energy required per bit of received information is
Sl (F0B/2tt)(S/N)1 f0b (S/N)1
Cb S Cb S b l0g2 [1 + (S/N)q]
where F0 is the spectral density of the noise in the IF and B is the IF
bandwidth (rad/s). In an ideal system the capacity of the IF is eaual
to the capacity of the baseband even when (S/N).¡ -* 0. Therefore the
ideal system has an energy-per-bit given by
St Si
r r lim
Cb CB (S/N) -¡-K)
FnB (S/N)i
B log2 [1 + (S7N)i]
log2e
0.693 Fn. (7.37)
Then Eq. (7.36) may be written as
S
-L < (0.693 F0)
Cb
B (S/N)j
0.693 b Tog, [1 + (S/N)0]
_4
Now the figure of merit will be defined as
B (S/N)i
0.693 b log, [1 + (S/N)0]
(7.38)
which is the amount of energy required by the actual system over that of
the ideal system in order to receive a bit of information, provided that
the output noise is Gaussian. If the output noise is not Gaussian, the
value of M will be somewhat larger than the ratio, energy-per-bit for the
actual system to the energy-per-bit for the ideal system.
H as a function of modulation index will be derived below for com
parison of various systems.
93
7.2-1. AM system
We now want to find M(s) for the AM system, described in Section
7.1-1, such that we will have an output signal-to-noise ratio of 27.5 db
for 6=1. 27.5 db is an arbitrary value that is chosen here for com
parison of systems using M as a figure of merit. This value is repre
sentative of the (S/N)o requirement for actual communication systems.
From Eq. (7.3) it follows that (C/N)-j = 27.5 db for 6 = 1. Also, for the
AM system Eq. (7.38) becomes
2 )m [(1 + %62)(C/N)i3
M(6) = ; --
0.693 log2 [1 + 62('C/N)i]
For (C/N)j = 27.5 db, Eq. (7.39) reduces to
1620 (1 + %62)
M{6) =
log2 [1 +'560 62]
(7.39)
(7.40)
The values of M(6) for the AM system, as given by Eq. (7.40), will
be compared to those for other systems in Section 7.2-5.
7.2-2. SSB-AM-SC system
To obtain M(6) for the SSB-AM-SC system, (S/N)0 = 27.5 db will be
used once again. From Eq. (7.5) it follows that (S/N)j = 27.5 db. Also,
for the SSB-AM system Eq. (7.38) becomes
m (S/N)i
M(6) =
0.693 com log2 [1 + (S/N)0]
(7.41)
For (S/N) 0 = (S/N)-f = 27.5 db, Eq. (7.41) reduces to
M(6) = 19.5 db.
(7.42)
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