SYNTHESIS AND ANALYSIS OF REAL
SINGLESIDEBAND 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 DAAG3967C0077, 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 SINGLESIDEBAND SIGNALS *
V. EXAMPLES OF SINGLESIDEBAND SIGNAL DESIGN
4.1. Example 1: SingleSideband AM with
SuppressedCarrier ....... *
4,2 Example 2: SingleSideband PM *
4.3. Example 3: SingleSideband FM *
4,4. Example 4: SingleSideband a *
V. ANALYSIS OF SINGLESIDEBAND SIGNALS *
5. 1 Three Additional Equivalent Realiza
5.2, SuppressedCarrier Signals *
5.3. Autocorrelation Functions * *.
5.4. Bandwidth Considerations .. *.
5.41. Meantype bandwidth *
5.42. RMStype bandwidth . .
5.43, Equivalentnoise bandwidth *
5 5, Efficiency . . .
5.6, PeaktoAverage 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 SINGLESIDEBAND SIGNAL ANALYSIS . .. .48
6o1. Example 1: SingleSideband AM With
Suppressed Carrier. . ................ 48
62. Example 2: SingleSideband PM . . . 51
6.3. Example 3: SingleSideband FM * * * 68
6.4. Example 4: SingleSideband a . . . 71
VII, COMPARISON OF SOME SYSTEMS . . . .. 75
7.1. Output SignaltoNoise Ratios . . . 76
7 11 AM system . . . . 76
7.12. SSBAMSC system . . . 77
7.13. SSBFM system . . . .. 78
7.14. FM system . . . . .. 84
7.15. Comparison of signaltonoise ratios. * 85
7.2, EnergyPerBit of Information .......... *89
7.21. AM system . . . . .. 93
7.22. SSBAMSC system . . . . 93
7.23. SSBFM system . . . .. 94
7.24. FM system . . . . 94
7.25. Comparison of energyperbit for
various systems *. ..............** 95
7.3. System Efficiencies ................. 97
7,31. AM system . . . . .. 98
7.32c SSBAMSC system . . . . 98
7.33. SSBFM system . . . .. 98
7.34. FM system . . . . .. 99
7.35. 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 FrequencyShifted
Entire Function of the Analytic Signal * *.
5. Voltage Spectrum of the Synthesized Upper Single
Sideband Signal . . . . .
6. Voltage Spectrum of the Negative FrequencyShifted
Entire Function of an Analytic Signal . .
7. Voltage Spectrum of the Synthesized Lower Single
Sideband Signal . . . .
8. Phasing Method for Generating USSBAMSC Signals 
9. USSBPM Signal ExciterMethod I . . .
10. USSBFM Signal Exciter . . .
11. EnvelopeDetectable USSB Signal Exciter ...... .
12. SquareLaw Detectable USSB Signal Exciter ......
13. USSBPM Signal ExciterMethod 11 . . .
14. USSBPM Signal ExciterMethod III . . .
15. Power Spectrum of a(t) . *
16. AM Coherent Receiver . . . .
17. SSBAMSC Receiver . . . . .
18. SSBFM Receiver . . . . .
19. Output to Input SignaltoNoise 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 SignaltoNoise to Input CarriertoNoise Ratio
for Several Systems * .* ..................... 87
21. Output SignaltoNoise to input SignaltoNormalized
Noise Power Ratio for Various Systems .......... * 90
22, Output SignaltoNoise to Input CarriertoNormalized
Noise Power Ratio for Various Systems . . . 91
23. Comparison of EnergyperBit 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 = AmplitudeModulation
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 CarriertoNoise Ratio
(C/N)I = Input CarriertoNormalizedNoise Ratio
D = Modulator Transducer Constant
FM = FrequencyModulation
F(w) = Voltage Spectrum
F() = The Fourier Transform of (*)
g(W) = U(W) + jV(W) = An Entire Function
GN = Gaussian Noise
LSSB = Lower SingleSideband
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 = PhaseModulation
R(,) ; Autocorrelation Function
Re() = Real Part of (*)
RF = Radio Freouency
Si = Input Signal Power
So: = Output Signal Power
(S/N)i = Input SignaltoNoise Ratio
(S/N)I = Input SignaltoNormalizedNoise Ratio
(S/N)o = Output SignaltoNoise Ratio
SC = SuppressedCarrier
USSB = Upper SingleSideband
(W) = The "SuppressedCarrier" Function of U(W)
V(W) = The "SuppressedCarrier" Function of V(W)
X(t) = A Real Modulated Signal
XL = Lower SingleSideband Modulated Signal
XU = Upper SingleSideband 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 = RMSType Bandwidth
AO = EouivalentNoise Bandwidth
S = MeanType 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
SINGLESIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS
By
Leon Worthington Couch, II
June, 1968
Chairman: Professor T. S. George
Major Department: Electrical Engineering
A new approach to singlesideband (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 amplitudemodulated (SSBAM), SSB frequencymodulated (SSBFM),
SSB envelopedetectable, and SSB squarelaw 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 suppressedcarrier SSB
signal, (3) autocorrelation function, (4) bandwidth (using variousde
finitions), (5) efficiency of the SSB signal, and (6) peaktoaverage
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, SSBAM, SSBFM, 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 signaltonoise ratios, (2) Energyperbit of
information, and (3) System efficiency.
In summary, the entire generating function concept is a new tool
for synthesis and analysis of singlesideband signals.
xiv
CHAPTER I
INTRODUCTION
In recent years the use of singlesideband 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 signaltonoise ratios in suppressed carrier singlesideband
systems when comparison is made in terms of total transmitted power.
A singlesideband 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 onesided about
the carrier frequency of the transmitter. In conventional amplitudemodu
lated systems the relationship between the real modulating waveform and
the real transmitted signal is given by the wellknown 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, frequencymodulated 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 singlesideband system?
Oswald, and Kuo and Freeny have given the relationship:
XSSBAM(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 singlesideband suppressed
carrier signal, which is now known as a singlesideband amplitudemodulated
suppressedcarrier signal (SSBAMSC). It will be shown here that this is
only one of an infinitely denumerable set of singlesideband 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 singlesideband 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 singlesideband 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)
Stx 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 righthand 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 allpass
linear (ideally nonrealizable) 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) 1J 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 crosscorrelation function is obtained as follows:
Rim(T) = m(t + T)m(t)
Sm(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 crosscorrelation 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 crosscorrelation 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 singlesideband 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 systemthe single1sideband transmitterand,
in general, it is nonlinear. 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 halfplane (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 zplane,
A proof of this theorem may be found in Appendix I.
Thus g[Z1(z)] is an analytic function of z in the UH zplane, and
by Theorem I g[Z(t)] has a singlesided 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
zplane, By the Lemma to Theorem I in Appendix I, g[Z,(z)]eJawz is ana
lytic in the UH zplane. 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(wLo)
or
F[g(Z(t))ej ot] Fg(wwo) '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 singlesideband signal can now be obtained from
the complex singlesideband 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(ww)o a wo
F{Re[h(x,0)eJwox]} = 0 i W < mo
Fi(46w.) w sw
This theorem is proved in Appendix I.
Thus the upper singlesideband 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*(w0)
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 SingleSideband Signal
The lower singlesideband 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 singlesideband synrthc:.. Then the Fourier transform of
the downshifted complex baseband signal is
F[g(Z(t))eLot] [g ]
2;r
F[eJut]
Fg(v) I(ulJo)
F[g(Z(t))eJmot] Fg(w+u4)
, WU
(3,7)
This spectrum is illustrated in Figure 6.
IF[g(Z(t))ejot]
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)eJmWx]
0
Fh(w+wo)
, O < WO
, i > Wo
,0 > W ~W
This theorem is proved in Appendix I.
Thus the real lower singlesideband signal for a given entire
function is
XLSSB(t) = Reg(Z(t))ej 't
= Re{[U(m(t),m(t)) t jV(m(t),fi(t))]ej 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 SingleSideband Signal
To summarize, it has been shown that once an entire function 9gW)
is chosen, then an upper or lower singlesideband 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 singlesideband transmitters respectively. Since there are dn in
finitely denumerable number of entire functions, there are an inrinuiely
denumerable number of upper and lower singlesideband signals that can be
generated from any one modulation process, In Chapter IV some specific
entire functions will be chosen to illustrate some wellknown single
sideband signals.
CHAPTER IV
EXAMPLES OF SINGLESIDEBAND SIGNAL DESIGN
Specific examples of upper singlesideband 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: SingleSideband AM With SuppressedCarrier
This is the conventional type of singlesideband signal that is
now widely used by the military, telephone companies, and amateur radio
operators. It will be denoted here by SSBAMSC.
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 singlesideband signal:
XUSSBAMSC(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 singlesideband 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 wellknown phasing method
for generating SSBAMSC signals [7, 8],,
4.2. Example 2: SingleSideband PM
Singlesideband phasemodulation was described by Bedrosian in
1962 [3].
To synthesize this type of signal, denoted by SSBPM, 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
XUSSBAMSC(t)
+"^ Modulated RF Output
Phasing Method for Generating USSBAMSC Signals
m(t)
Input
Hilbert Transform
{90 Phase Shift
over Spectrum of
m(t)}
Figure 8.
and
V2(m(t),m(t)) = em(t) sin m(t). (4.5b)
Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper
singlesideband signal:
XUSSBPM(t) = e(t) cos m(t) cos wot em(t) sin m(t) sin wot
or
XUSSBPM(t) = em(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 singlesideband exciter described by Eq. (4.6)
is shown in Figure 9.
4.3. Example 3: SingleSideband FM
Singlesideband frequencymodulation is very similar to SSBPM
in that they are both angle modulated singlesideband signals. In fact
the equations for SSBFM 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 SSBFM signals have been conducted by a number
of persons and are reported in the literature [9, 10].
m(t)
Modulating Input
XUSSBPM(
Modulated RF Output
cos (mot+m(
Figure 9. USSBPM Signal ExciterMethod I
Phase Modulator
at
Radio Frequency mo
Hilbert Transform
{90 Phase Shift
over Spectrum of
m(t)}
The SSBFM exciter as described by Eas. (4.6) and (4.7) is given
in Figure 10.
4.4. Example 4: SingleSideband a
The term singlesideband a (SSBa) will be used to denote a sub
class of singlesideband 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.
USSBFM 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 SSBa signal is
XUSSBa(t) = em"(t) cos (am(t)) cos wot
eam(t) sin (am(t)) sin wot
or
XUSSBa(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 envelopedetectable SSB signal, as is readily
seen from Ea. (4.12). Voelcker has recently published a paper demon
strating the merits of the envelopedetectable SSB signal [11]. The real
ization of Eq. (4.12) is shown in Figure 11.
For a = 1/2 we have a squarelaw detectable SSB signal. This type
of signal has been studied in detail by Powers [121. Figure 12 gives the
blockdiagram realization for the squarelaw detectable SSB exciter.
e(t)
Modulating Input
DC Level
of +1
XUSSBo = 1(t)
[1+e(t)]
Phase Modulator
at
Radio Frequency w0
EnvelopeDetectable 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
SquareLaw Detectable USSB Signal Exciter
Figure 12.
CHAPTER V
ANALYSIS OF SINGLESIDEBAND 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 peaktoaveraige
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. SuppressedCarrier Signals
The presence of a discrete carrier term appears as impulses in
the (twosided) spectrum of transmitted signal at frequencies wo and woo
The impulses may have real, purely imaginary, or complexvalued 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 widesense stationary.
Thus
U(m(t),m(t)) = IP cdt' = 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 frequencymodulation 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
Ro
12
f
IF(w)12d } {lim
Rxo
e2(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
Ro
since
Z(t) e L2(, 2).
Then
lim jZi(ReJe)I = lim IZ (R cos e,R sin O) = 0 0 s e s
Rco 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 onehalf the square of the magnitude.
For every generalized USSB signal represented by Eq. (3.5),
there exists a corresponding suppressedcarrier USSB signal:
XUSSBSC(t) = '(m(t),(t)) cos Wot *(m(t),m(t)) sin Wot
(5.19)
where the notation SC and denote the suppressedcarrier 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 rewritten as
XUSSBSC(t) = tt(m(t),'(t)) cos wot (m(t),m(t)) sin o0t (5.22)
or
XUSSBSC(t) = V(m(t),m(t)) cos wot V(m(t),m(t)) sin wot (5.23)
where U and Vare 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 suppressedcarrier
signals are given by
XLSSBSC(t) = tJ(m(t),m(t)) cos wot + 4(m(t),m{t)) sin mot (5.25)
and
XLSSBSC(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 suppressedcarrier 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 startup 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{eJWOTRg(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 Rgsc()
(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 suppressedcarrier 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) = RgSC(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) + RgSC(r)]} (5o37)
and
RXL(') = Re{&WoT[ k2+k22) + RgSC(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 RgSC(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
RgSC(T) = 2[R (T) + JR4.(T)] (5.41)
or by using Eqs. (5.40), (2.15),and (2.9)
RgSC(T) = 2[R.(T) + jR (T)]o (5.42)
Thus RgSC(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 suppressedcarrier are readily given by Eq. (5.37) and
Eq. (5.38) with k, = k2 = 0:
RXUSC(i) = Re{eJmwTRgSC(T)}
= CR. o) cos W"o Rtt() Sin wfo
= RV(7T) cos wo' RW.(T) sin moT
RXLSC(t) = Re{ eeoRgSC()}
= 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 suppressedcarrier autocorrelation formulae developed above
will now be used to calculate bandwidths of SSB signals. It is noted
that the suppressedcarrier 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
5o41. Meantype bandwidth
Since the spectrum of a SSB signal is onesided 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 RgSc(O)
0 =  (5.45)
f PgSC(w)dw RgSC(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 RgSC(T) is analytic almost everywhere (Theorem 103
of Titchmarsh [6]).
5o42, RMStype bandwidth
The rms bandwidth, wrms, may also be obtained.
CO
2 2 PgSC()d RgSC(O)
(wrms)2 2 0o
f PgSC(w)dw RgSC(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)
RBB(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.
5,43. Equivalentnoise bandwidth
The equivalentnoise bandwidth, Aw, for the continuous part of
the power spectrum is defined by
(2Aw) PgSC(O)] 2= PgSC(w)d = RgSC(O)
00
(5.49)
But
PgSC(O) = f RgSC(t)dT
COO
Thus
(AoW) =
1
RgSC(O)
f RgSC(r)dT
00
Substituting for RgSC(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
RXUSC(O) = RXLSC(O) = RWo(O) = RW(O) (552)
It is also noted that RgSC(O) is not equal to the total power in the
realsignal sidebands since gSC is a complex (analytic) baseband signal;
instead, (l/2)Re[Rg_sc(O)] : Rwu(O) = Rwv(O) is the total realsignal
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. PeaktoAverage Power Ratio
The ratio of the peakaverage (over one cycle of the carrier
frequency) to the average power for the SSB signal may also be obtained.
The expression for the peakaverage 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
timevarying functions compared to cos mot and sin mot, we have for the
peakaverage 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), PpAv can also be written as
PpAv = {[ + 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 peaktoaverage power ratio for the generalized
SSB signal is
PpAv {[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 peaktoaverage
power since one representation may be easier to use than another for a
particular SS1B signal.
CHAPTER VI
EXAMPLES OF SINGLESIDEBAND 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: SingleSideband 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 Rwc J
since lim m(R cos e,R sin e) = 0 for 0 < e < x from Eq. (5.18a). Further
Rx
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 SSBAMSC signals (which was given
previously in Figure 8).
The autocorrelation for the SSBAMSC 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 suppressedcarrier USSBAM signal is
given via Eq. (5.43b), and it is
RXUSCSSBAM() = Rmm () cos Wmo Rmm(T) sin wmo. (6.4)
Likewise, by use of Eq. (5o44b) the autocorrelation for the suppressed
carrier LSSBAM signal is
RLSCSSBAM() = Rmm(T) cosw + mm(t) sin wor. (6.5)
From Eq. (6.4) it follows that the spectrum of the USSBAMSC
signal is just the positivefrequency spectrum of the modulation shifted
up to a0 and the negativefrequency spectrum of the modulation shifted
down to wo. That is, there is a onetoone 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 meantype bandwidth
is given by use of Eq. (6.3) in
MSSBAM
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 equivalentnoise
bandwidth is
(Am)SSBAM = (6.8)
f Rmm(r)dT
Thus the bandwidths of the SSBAMSC signal are identical to those of the
modulating process m(t).
The efficiency of the SSBAMSC signal is obtained by using
Eq. (5.54):
2Rmm(0)
nSCSSBAM 
2Rmm(0)
(6.9)
The peaktoaverage power ratio for the SSBAMSC signal follows
from Eq. (5.56c), and it is
PpAv {[m(t)]2 + [m(t)]2}t = tpeak
PAV /SCSSBAM 2*m
(6.10)
6.2. Example 2: SingleSideband PM
The SSBPM 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 em,(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
Ro
0 _s e < T and lim m (R cos e, R sin e) = 0 for 0 e rr. Thus
Ro
ki = 0. (6.11)
Likewise, substituting Eq. (4,5a) into Eq. (5.5) we have
k = e0 cos 0 de = 1. (6.12)
IT J
0
Thus the SSBPM signal has a discrete carrier term since k2 = 1 f 0.
There are equivalent representations for the SSBPM 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 USSBPM signal
since k, = 0. Thus the two equivalent representations are:
XUSSBPM(t) e= e(t)cos m(t)] cos Wot e acos m(t)] sin wot (6.13)
and
XUSSBPM(t) =(e(t)sin m(t))+1]cos wot [em(t)sin m(t)]sin ot. (6.14)
The USSBPM 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 SSBPM signal will now be
examined. In Section 6.1 the autocorrelation for the SSBAMSC signal
was obtained in terms of the autocorrelation function of the modulation.
This was easy to obtain since 4 = m(t). However, for the SSBPM case 4
and Vare nonlinear 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 SSBFM signal in terms of Rmm(T).
To calculate the autocorrelation function for the SSBPM 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)) = em(t) sin m(t). (6Jo5)
Then
Rm jm(t) jm(t) Jm(t) jm(tr)
R (1) = e epm(tT) e2j
W e 2, e2/_
m(t)
Modulating Input
jcos .t
XUSSBPM(t)
OutDut
USSBPM Signal ExciterMethod II
e4(t) cos m(t)
Figure 13.
m(t)
Modulating Input
em(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
XUSSBPM(t)
Modulated
RF Output
[e&(t) sin m(t)]sin
Figure 14. USSBPM Signal ExciterMethod 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(tT.)
x2(t,T) m(t) + m(tr)
x3(t,t) m(t) m(tT) = x2(t,T)
x4(t,T) E m(t) + m(tt) = x (t,T)
y(t,r) H i(t) + i(tT).
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(tr) 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 +j2Pxyay2} (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(tt)] = 2[am Rmm(T)]
ax r[m(t)+m(tr)]2 2[om2+Rmm(T)]
2
OX3
[m(t)m(tT)]= 2[ om2+Rmm( )]
x 2 = [m(t)+m(tt)]2= 2[am2Rmm()]
and
y 2 [m(t)+m(t1i)] = 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(tr)][m(t)+m(rtT)]
= 2Rmm()
[m(t(tT( =
Sx3Y
and
X y = [m(t)m(tT)][m(t)+m(tr)] 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)]}
e2{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
RVWSSBPMGN(r) = {e2Rmm(T) cos (2Rmm(r)) 1} (6.18)
where W SSBPMGN denotes the autocorrelation of the imaginary part of
the entire function which is associated with the suppressedcarrier 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 suppressedcarrier signal should be zero when the modulating
power is zero.
The autocorrelation of the USSBPM 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):
RXUSSBPMGN(T) = Re eJ0 T{[e2Rmm() cos (2Rmm(())]
+ j[e Rmm T)cos (2Rmm(T)]}] (6.19)
Likewise, the autocorrelation of the LSSBPM signal may be obtained by
using Eq. (5.38).
The autocorrelation of the suppressedcarrier USSBPM signal with
Gaussian modulation is given by using Eq. (5.43a):
RXUSCSSBPMGN(t) = Re [eJ m([e2Rmm(t) cos (2Rmm(r)) 1]
+ j[e2Rmm( cos (2Rmm(r)]} (6.20)
Similarly, the autocorrelation of the suppressedcarrier LSSBFM signal
may be obtained by using Eq. (5,44a).
The meantype bandwidth will now be evaluated for the SSBPM
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 meantype bandwidth for the
Gaussian noise modulated SSBPM signal:
17 P e2Rmm(A) cos[2Rmm(A)]dA
(W)SSBPMGN (623)
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 mistype 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
rmstype bandwidth of the SSBPM signal with Gaussian noise modulation:
/2(2[Rnm(O)12 Rm(O)}
6(; ?q)
v'rmsJssBPMGN 1 e2m
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 equivalentnoise bandwidth is obtained by substituting
Eq. (6.18) into Eq. (5.50):
(AW) =
1 e2Rmm() cos [2Rmm(T)] 1} dT
[e22ml].
or
'T(e2m 1)
(Am)SSBPMGN = (6.26)
S{e2Rmm(T) cos [2Rmm(T)] l}dT
It is noted that the equivalentnoise 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 SSBPM 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 e22T2) 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, e2o
Thus if m(t) has a Gaussian spectrum and if the modulation has a Gaussian
density function, the SSBFM signal has the rms bandwidth:
20o2 [4( oo/n) + ]
("rms)SSBPMGN = 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 SSBPM signal with sinusoidal modulation [20].
From Eq. (19) of their work
(wrms)ssBPMS = 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)SSBPMS = /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 SSBPM signal with Gaussian modulation
will now be obtained. Substituting En. (6.22) into Eq. (5.54) we have
e2m_l
tSSBPMGN = + (e2m1l)
or
SSBPMN e2m (6.32)
where 1m is the noise power of m(t).
The peakaverage 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
{[em(t) cos m(t)] [el(t) sin m(t)]2 t tpeak
1 + (e2mml)
(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 peaktoaverage power ratio for the SSBPM signal with Gaussian
noise modulation is
Av
( v )SSBPMGN
Se6am e6/im2 m
e 
when 'm is small.
It is noted that the efficiency and the peaktoaverage 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 (doublesideband) suppressedsubcarrier 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
narrowband 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 wellknown representation for a narrowband Gaussian process.
Thus m(t) is a narrowband Gaussian process.
Now the previous expressions for bandwidth, which assume that
m(t) is Gaussian, may be used. The meantype bandwidth for the multi
plexed SSBPM 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)MSSBPMGN ~ 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)MSSBPMGN = e*a (6.40)
and the equivalentnoise bandwidth is
f[e2a1]
(Aw)MSSBPMN = 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 SSBPM 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 SSBPM multiplexed signal:
2 NOWO( N0W0 N oWo_
a   +1 + 
(mrms)MSSBPMGN = NoWo/f (6.45)
1 l eW/
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: SingleSideband FM
As was indicated in Section 4.3. the representation for the SSBFM
signal is very similar to that for the SSBPM signal. In fact it will be
shown below that all the formulae for the properties of the SSBPM signal
(which were obtained in the previous section) are directly applicable to
the SSBFM signal.
The SSBFM signal has a discrete carrier term since the entire
function for generating the SSBFM signal is identical to that for the
SSBPM signal, which has a discrete carrier term.
The other properties of the SSBFM signal follow directly from
those of the SSBPM 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 widesense stationary if e(t) is widesense stationary.
It is recalled that if
y(t) = L[x(t)]
where L is a linear timeinvariant operator, then y(t) is strictsense
stationary if x(t) is strictsense stationary and that y(t) is widesense
stationary if x(t) is widesense stationary [4]. Since the integral is a
linear operator, we need to show only that it is timeinvariant, 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 timevarying (i.e. it was turned on at to). But this should
not worry us because, as Middleton points out, aZZll physically realizable
systems have nonstationary 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 "timeinvariant" physical systems
have reached steadystate we may consider them to be stationary processes
provided there is a steady state. Thus by letting t m we are con
sidering the steadystate process m(t) which we have shown to be stationary.
Now the autocorrelation of m(t) can be obtained by using powerspectrum
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 powerspectrum 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
steadystate 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 SSBFM signal can be obtained in terms of the
spectrum of the modulating process.
6.4. Example 4: SingleSideband a
The SSBa 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,
Ro 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 SSBa signal has a discrete carrier term.
It follows that equivalent representations for the SSBa signal
are possible since k2 0. This is analogous to the discussion on equiva
lent representations for SSBPM signals (Section 6.2) so this subject will
not be pursued further.
The autocorrelation function for the SSBa 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(tT)]
or
R (,) = k{ea[m(t)+m(tT)]} eja[m(t)m(tT)] eja[m(t)+m(tT)]
+ {eam(t)+m(t)]} {eJ[m(t)m(t)] + ej[m(t)+(tT)}.
(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)SSBaGN = {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 SSBPM signal with Gaussian m(t). Moreover, the properties are
identical for SSBa and SSBPM signals having Gaussian m(t) processes
such that (*m)SSBPM = a2(m)SSBa_
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)SSBaGN {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 USSBa and LSSBa signals, assuming Gaussian
modulation e(t) with a small variance, are
RXUSSBaGN(r) Re ejo'T {[e2a2Re(T) cos (2otee(T))]
+ j RLe2a2Ree() cos (2a2Reo())]} (6)57)
and
RXLSSBaGN(,) Re ejwo([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):
nSSBaGN = 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 envelopedetectable
SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t);
and Eq. (6.59) becomes
nSSBaGN 1. 1 e2e2 z 20e2, (6o60)
This agrees with Voelcker's result (his Eq. (38)) when the variance of the
modulation is smallthe condition for Eqo (6o60) to be valid.
The expressions for the other properties of the SSBa signal, such
as bandwidths and peaktoaverage power ratio, will not be examined further
here since it was shown above that these properties are the same as those
obtained for the SSBPM signal when (Wm)SSBPM '= 2(m)SSBa as long as
m(t) is Gaussian.
CHAPTER VII
COMPARISON OF SOME SYSTEMS
In the two preceding chapters properties of singlesideband 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 signaltonoise 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 tradeoff
between predetection signal bandwidth and postdetection
signaltonoise ratio.)
Comparison of AM, FM, SSBAMSC, and SSBFM 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 SignaltoNoise Ratios
7.11. AM system
Consider the coherent receiver as shown in Figure 16 where the
input AM signal plus narrowband 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 signaltonoise 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 signaltonoise power ratio
(C/N)i = The input carriertonoise power ratio
and the spectrum of the noise is taken to be flat over the IF bandpass
which is 2wm(rad/s).
7.12, SSBAMSC system
Consider the coherent receiver (Figure 16) once again, where
the input is a SSBAMSC signal plus narrowband 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 signaltonoise 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 rolloff 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. SSBAMSC Receiver
7.13. SSBFM system
Now consider a FM receiver which is used to detect a SSBFM sig
nal plus narrowband Gaussian noise as shown in Figure 18.
X(t)+ni(t) FM Output
Receiver
Figure 18. SSBFM 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) = Narrowband Gaussian noise with power spectral density Fo
over the (onesided spectral) IF band
and v(t) is the modulation on the upper SSBFM signal. The independent
narrowband 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 tan1
(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 signaltonoise ratios (i.e. Aoem(t) > > R(t)
most of the time), Eq. (7.8) becomes
kR(t)
p(t) = km(t) +  sin [p(t) m(t)] (7.9)
Aoem(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 narrowband 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 2iT
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 [(Pnwo)t + On]
Ao n=l 2'
or
dno(t) kem 
dt Ao nt 1 2F(tn) (wnwo) cos [(nmo)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 signaltonoise ratio is
k2
(DAm)2
S Ak2L
2TWn A 02 3
AO2 2
Fo
2
21T
m Io(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 onesided spectrum) for a SSBFM signal to a conventional
FM signal [20], and it is
BSSBFM
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 SSBFM bandwidth is
BSSBFM 2
I 2(2 )
2 1 2=
(7.20)
(6+1)w .
m
Then, taking the IF bandwidth to be that of the SSBFM signal, the input
noise power is
V 0
N B SSBFM'
(7 21)
Consequently, the input signaltonoise ratio is
Io(26)
(S/N)i =
(7,22)
mm (6+1) /2
I 2(26)
1 2)
Io (26)
Fo
4 
2u
Using Eq. (7.16) and Eq. (7.22), we have
(7.23)
for the case of SSBFM plus Gaussian noise into a FM detector.
The signaltonoise output can also be obtained in terms of the
unmodulatedsignaltonoise ratio (i.e. the carriertonoise 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 carriertonoise power ratio.
7.14. FM system
The signaltonoise ratio at the output of a FM receiver for a
FM signal plus narrowband Gaussian noise at the input can be obtained
by the same procedure as used above for SSBFM. 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 signaltonoise 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 signaltonoise ratio is large. It is also noted that
(S/N)i = (C/N)i. (7.27)
7.15. Comparison of signaltonoise 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 signaltonoise ratios, a
useful criterion is the output signaltonoise ratio from the system
for a given RF signal power in the channelthat 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
/ SSBFMFM Detection
1.5 5 
SSBAMSC
1 ____ _
0.5 /
/ AMCoherent
/ Detection
0 0.5 1.0 1.5 2.0 2.5
Modulation Index (6)
Figure 19. Output to Input SignaltoNoise
Power Ratios for Several Systems
