- 4

any conventional statistical signal processing

shared by m y methods treat random signals a s if they were sta-
tistically stationary, in which case the parameters
are described: (1)
of the underlying physical mechanism that generates the
enables generation signal would not vary with time. But for most manmade
signals encountered in communication, telemetry, radar,
and sonar systems, some parameters do vary periodically
with time. In some cases even multiple incommlensurate
(not harmonically related) periodicities are involhed. Ex-
amples include sinusoidal carriers in amplitude, phase,
and frequency modulation systems, periodic keying of the
onproperty for signal compon- amplitude, phase, or frequency in digital modulation sys-
in distinct spectral bands. It is tems, and periodic scanning in television, facsirnile, and
some radar systems. Although in some cases these peri-
n that these three properties are
odicities can be ignored by signal processors, such a s re-
nt martijestationsof a single at-
ceivers which must detect the presence of signals of in-
terest, estimate their parameters, and/or extr.act their
attribute is studied, messages, in many cases there can be much to gain in
waysfor exploiting it terms of improvements in performance of these signal
1 processing tasks in- processors by recognizing and exploiting underlying pe-
tion and estimation of riodicity. This typically requires that the random signal
rmpted manmade signals are be modeled a s cyclostationary, in which case t!he statis-
tical parameters vary in time with single or multiple pe-
riodicities.
This article explains that the cyclostationarity attri-
bute, a s it is reflected in the periodicities of (second-order)
moments of the signal, can be interpreted in terms of the
property that enables generation of spectral lines from the
0 1991 E
l053-58~8/91/0400-00I4sr.oo I EE

This material is based upon work supported by the National Science

Foundation under Grant No, MIP-88-12902.

14 IEEESPMAGAZINE APRIL 1991

signal by putting it through a (quadratic)nonlinear trans-
formation. It also explains the fundamental link between
the spectral-line generation property and the statistical
property called spectral correlation, which corresponds
to the correlation that exists between the random fluc-
tuations of components of the signal residing in distinct
spectral bands. The article goes on to explain the effects
on the spectral-correlation characteristics of some basic
signal processing operations, such as filtering, product
modulation, and time sampling. It is shown how to use
these rl-sults to derive the spectral-correlation character-
istics for various types of manmade signals.
Some examples of signals that can be appropriately
modeled a s cyclostationary can be interpreted a s the re-
sponse of a linear or nonlinear system with some period-
ically varying parameters to stationary random excita-
tion. Specific examples include stationary random
modulation of the amplitude, phase, or frequency of a
sinewave: stationary random modulation of the ampli-
tudes, widths, or positions of pulses in an otherwise pe-
riodic pulse train; periodically varying Doppler effect on
a stationary random wave: and periodic sampling, mul-
tiplexing, or coding of stationary random data. In addi-
tion to )theseexamples of manmade signals, some natural
signals also exhibit cyclostationarity due, for example, to
seasonal effects in time-series data sets obtained in me-
teorology, climatology, atmospheric science, oceanogra-
phy, arid hydrology, as well a s astronomy. Numerous ex-
amples and references are given in [ 1, Chapters 12 and
141, [2, Chapter 121.
Finally, and most importantly, this article describes
some ways of exploiting the inherent spectral redun-
dancy associated with spectral correlation to perform
various signal processing tasks. These include detecting
the presence of signals buried in noise andlor severely
masked by interference: recognizing such corrupted sig-
nals according to modulation type: estimating parame-
ters such a s time-difference-of-arrival a t two reception
platforms and direction of arrival a t a reception array on
a single platform: blind-adaptive spatial filtering of sig-
nals impinging on a reception array: reduction of signal
corrupl ion due to cochannel interference andlor channel
fading for single-receiver systems: linear periodically
time-variant prediction; and identification of linear and
nonlinear systems from input and output measurements.
The descriptions include brief explanations of how and
why the signal processors that exploit spectral redun-
dancy (can outperform their more conventional counter-
parts that ignore spectral redundancy or, equivalently,
ignore cyclostationarity.
In the next section, the possibility of generating spectral
lines by simply squaring the signal is illustrated for two
types of signals: the random-amplitude modulated sine
wave and the random-amplitude modulated periodic
pulse train. Then it is explained that the property that

APRIL 1991 IEEESP MAGAZINE 15

enables spectral-line generation with some type of ference, and distortion,
quadratic time-invariant transformation is called 5) A prediction of a future value of a random signal,
cyclostatfonarity and is characterized by the cyclic and
autocorrelationfunction, which is a generalization of the 6) An estimate of the input-output relation of a linear
conventional autocorrelation function. Following this, or nonlinear system based on measurements of
it is shown that a signal exhibits cyclostationarity if a n d the system's response to random excitation.
only if the signal is correlated with certain frequency-
shifted versions of itself. The article concludes with a brief section indicating
In the third section, the correlation of frequency- how the theory of second-order cyclostationarity sur-
shifted versions of a signal is localized in the frequency veyed in the preceding sections can be generalized to
domain and this leads to the definition of a spectral- higher-order cyclostationarity, which corresponds to
correlattm density function (SCD). It is then ex- the property that enables spectral-line generation
plained that this function is the Fourier transform of using higher-order nonlinearities, such as cubic, quar-
the cyclic autocorrelation function. This Fourier- tic, and so on. References to more in-depth treatments
transform relation between these two functions in- of the theory and its applications are given throughout
cludes as a special case the well-known Wiener rela- the article.
tion between the power spectral density function and
the conventional autocorrelation function. A normal-
ization of the spectral-correlation density function that
converts it into a spectral correlation coefficient, whose
magnitude is between zero and unity, is then intro-
duced as a convenient measure of the degree of spec-
A signal x(t) contains a finite-strength additive sine-
tral redundancy in a signal.
wave component (an ac component) with frequency a ,
Continuing in this SCD section, the effects on the
spectral-correlation density function of several signal say
processing operations are described. These include fil-
tering and waveform multiplication, which in turn in-
a cos (2aat + 0) with a # 0 (1)
clude the special cases of time delay and multipath if the Fourier coefficient
propagation, bandlimiting, frequency conversion, and
time sampling. These results are used to derive the
spectral-correlation density function for the random-
amplitude modulated sine wave, the random-ampli- is not zero, in which case (1)gives
tude modulated pulse train, and the binary phase-shift
keyed sine wave. The spectral-correlation density M : = Aae"
functions for some other types of phase-shift keyed
signals are also described graphically. Finally in this In (2), the operation .
( ) is the time-averaging opera-
section, the measurement of (estimation of the ideal) tion
spectral-correlation density function is briefly dis-
cussed and illustrated with a simulation of a phase-
shift keyed signal.
(.) A lim
T-- T
5 TI2

-r12
(.) dt

The fourth section of this article contains the payoff

for working through the preceding two sections. It pro- In this case, the power spectral density (PSD) of At)
vides explanations of how the spectral redundancy includes a spectral line a t frequencyf= a and its image
that is inherent in signals that exhibit cyclostationar- f = -a.(The PSD is defined in the next section.) That is,
ity can tie exploited in a variety of statistical signal the PSD contains the additive term'
processing tasks. The spectral redundancy can gen- IM1I2[S(f- a) + S(f + all (3)
erally be exploited to enhance the accuracy and reli-
ability of information gleaned from measurements of where 6(.) is the Dirac delta, or impulse, function. For
corrupted signals. Such information includes the fol- convenience in the sequel, it is said that such a signal
lowing: exhibits first-order periodicity, with frequency a.
Let x(t) be decomposed into the sum of its finite-
A decision as to the presence or absence of a ran- strength sine-wave component, with frequency a , and
dom signal with a particular modulation type in a its residual, say n(t),
background of noise and other modulated signals,
A classification of multiple received signals in x(t) = a cos (2aat + 0) + n(t) (4)
noise according to their modulation types,
An estimate of a signal parameter, such as carrier where n(t) is defined to be that which is left after sub-
phase, pulse timing, or direction of arrival, or of traction of (1)from x(t).It is assumed that n(t) is ran-
the number of signals being received simulta- dom. Here, the term random is used to denote nothing
neously, based on noise-and-interference-cor-
'The strength of the spectral line is lM;12 as indicated in (3) if and
rupted measurements, only if the limit (2) exists in the temporal mean square sense with
An estimate of a message being communicated by respect to the time parameter U obtained by replacing t with t + U in
a signal over a channel corrupted by noise, inter- (2) [ I . Chapter 15, exc. 61.

IEEE SP MAGAZINE APRIL 1991

more than the vague notion of erratic or unpredictable The discussion begins with a couple of examples.
behavior. If the sine wave is weak relative to the ran- Example 1 :AM. Let a(t)be a random lowpass signal
dom residual, it might not be evident from visual in- (say lowpass filtered thermal noise) with the PSD Sa(f )
spection of the signal that x ( t )contains a periodic com- shown in Fig. la, which contains no spectral lines. If
ponent. Hence, it is said to contain hiddenperiodicity. a ( t )is used to modulate the amplitude of a sine wave,
However, because of the associated spectral lines, hid- we obtain the amplitude modulated (AM) signal
den periodicity can be detected and in some applica-
tions exploited through techniques of spectral analy-
sis.
This article is concerned with signals that contain whose PSD S,(f) is given by [ I , Chapter 3 , Sec. D]
more subtle types of hidden periodicity that, unlike
first-order periodicity, do not give rise to spectral lines
in the PSD. but that can be converted into first-order
periodicity by a nonlinear time-invariant transforma-
tion of the signal. In particular, we shall focus on the as shown in Fig. l b . Although the PSD is centered
type of hidden periodicity that can be converted by a about f = fo andf = -fo. there is no spectral line a t fo
quadratic transformation to yield spectral lines in the or -fo. The reason for this is that, as shown in Fig. la,
PSD. there is no spectral line in Sa(f ) a t f = 0. This means
that the dc component

M: (a@)) (7)

is zero, since the strength of any spectral line atf = 0

is IM",I".
Let us now square x ( t )to obtain

y(r) = x2(t) = u2(r)cos2 (27rf0t)

f = $ [b(f)+ b(t) COS (47rfot)l (8)
B
where

b ( f )= U2(f) (9)

Since b(t)is nonnegative, its dc value must be positive:

ME > 0. Consequently, the PSD of b(t)contains a spec-
tral line a t f = 0, as shown in Fig. IC.The PSD for y ( t )
f
is given by
f0

and, as shown in Fig. Id. it contains spectral lines at f

= -+2fo as well a s atf = 0. Thus, by putting x(t)through
a quadratic transformation (a squarer in this case) we
have converted the hidden periodicity resulting from
the sine-wave factor cos ( 2 r f o t ) in (5) into first-order
f periodicity with associated spectral lines. This is par-
2B ticularly easy to see if a ( t )is the asynchronous random
telegraph signal, which switches back and forth a t
random times between + 1 and - 1, because then b(t)
= 1 and y ( t )in (8)is therefore a periodic signal

y(t) = ;+ f cos (4Tfot)

Example 2: PAM. A s another example, we consider
the pulse-amplitude modulated (PAM)signal

0 2f0
Fig. 1 . a) Power spectral d e n s i t y (PSD) of a l o w p a s s
signal. b) PSD of an a m p l i t u d e - m o d u l a t e d ( A M ) sig-
nal. e) PSD of a s q u a r e d l o w p a s s signal. d / PSD of a
squared A M signal. where the pulse p ( t ) is confined within the interval
APRIL 1991 IEEE SP MAGAZINE 17
 The PSD for y(t) is given by

where Q(f ) is the Fourier transform of q(t). Because of

the spectral line a t f = 0 in S,(f), which is shown in
Fig. 2. A pulse-amplitude-modulated (PAM) signal Figure IC, we have spectral lines in S,(f) at the har-
w i t h p u l s e w i d t h less t h a n interpulse t i m e . monics &To (for some integer values of m)of the pulse
rate l/To, as shown in Fig. 3b. Thus, again, we have
converted the hidden periodicity in x(t) into first-order
( -T0/2, T0/2)so that the pulse translates do not over- periodicity with associated spectral lines by using a
lap, as :jhown in Fig. 2. The PSD of x ( t )is given by [ 1, quadratic transformation. This is particularly easy to
Chapter 3, Sec. D] see if a(nTo)is a random binary sequence with values
+ I , because then b(nTo)= 1 and y(t) in (13) is there-
fore a periodic signal

where Sa(f 1 is shown in Fig. la, which contains no

spectral lines. and where P( f ) is the Fourier transform
of p ( t ) . Since there are no spectral lines in S,(f) (or
P ( f ) since p ( t ) has finite duration), there are none in The cyclic autocorrelation function
S,(f), ELS shown in Fig. 3a, regardless of the periodic
repetition of pulses in x ( t ) .But, let us look at the square Although the squaring transformation works in these
of x(t ) : examples, a different quadratic transformation involv-
ing delays can be required in some cases. For example,
if a ( n T o )is again binary, but p ( t ) is flat with height 1
and width To,a s shown in Fig. 4, then y(t) = x"(t) = 1,
which is a constant for all t. Thus, we have a spectral
where line a t f = 0 but none a t the harmonics of the pulse
rate. Nevertheless, if we use the quadratic transfor-
mation

y(t) = x(t)x(t - 7) (17)

for any of a number of nonzero delays 7,we will indeed

obtain spectral lines atf = &To. That is,

My = (y(t)e-'*""')

= (X(t)X(t - 7)e -'>=a') # 0 (18)

4
for Q = &To for some integers m.

Fig. 3. a ) Power spectral d e n s i t y (PSD) o f a p u l s e - a n i -

plitude--modulated(PAM) signal w i t h 67% duty-cycle Fig. 4. A binary pulse-amplitude-modulated (PAM)
pulses. b) PSD of t h e squared PAM signal. signal w i t hfull duty-cycle p u l s e s .

18 IEEE SP MAGAZINE APRIL 1 9 9 1

 The most general time-invariant quadratic transfor- where
mation is simply a linear combination of delay prod-
ucts of the form (17), viz.,

y(t) = s h(7)X(t)X(t - 7) d7

Recall that multiplying a signal by e iiaol' shifts the

for some weighting function h(7)that is analogous to
spectral content of the signal by +01/2. For example,
the impulse-response function for a linear transfor-
the Fourier transforms of u ( t )and u(t)(if they exist) are
mation. This motivates u s to define the property of
second-order periodicity as follows: The signal x(t )
contains second-order periodicity if and only if the PSD U ( f ) = X ( f + a/2) (254
of the delay-product signal (17) for some delays 7 con- and
tains spectral lines a t some nonzero frequencies 01 #
V(f)= X(f - cd2) (25b)
0, that is, if and only if (18)is satisfied.
In developing the theory of second-order periodicity Similarly, their PSDs are
it is more convenient to work with the symmetric de-
lay product

(The complex conjugate * is introduced here for gen-

erality 1 o accommodate complex-valued signals, but it
It follows from (23)-(24) that x ( t )exhibits second-order
is mentioned that for some complex-valued signals, the
periodicity ((20)is not identically zero as a function of
quadratic transformation without the conjugate can
7 for some a # 0) if and only if frequency translates
also be useful [ l ,Chapter 10, Sec. C].) Thus, the fun-
(frequency-shifted versions) of x ( t ) (namely u(t) and
damenl a1 parameter ( 18) of second-order periodicity
u ( t ) ) are correlated with each other ((23)is not identi-
becom E'S
cally zero as a function of 7 for some (Y # 0 in (24)).
This third interpretation of R 3 7 ) suggests a n appro-
priate way to normalize R 3 7 ) as explained next.
A s long as the mean values of the frequency trans-
which is the Fourier coefficient May, of the additive sine- lates u(t)and u ( t ) are zero (which means that x ( t )does
wave component with frequency (Y contained in the de- not contain finite-strength3 additive sine-wave com-
lay-product signal yT(t). ponents at frequencies +a/2 and, therefore, that S,(f)
The notation R 3 7 ) is introduced for this Fourier coef- has no spectral lines at f = +a/2), the crosscorrelation
ficient because, for Q = 0, (20) reduces to the conven- Ruu(7)= R 3 7 ) is actually a temporal crosscovariance
tional a utocorrelation function KU,(7).That is,

for which the notation Rx(7)is commonly used. Fur-

thermore, since R 3 7 ) is a generalization of the auto-
correlation function, in which a cyclic (sinusoidal)
weighting factor e-i2an' is included before the time av- An appropriate normalization for the temporal cross-
eraging; is carried out, R:(7) is called the cyclic auto- covariance is the geometric mean of the two corre-
correlation function.' Thus, we have two distinct sponding temporal variances. This yields a temporal
interpr'etations of R 3 7 ) = M;,.In fact, we have yet a correlation coefficient, the magnitude of which is up-
third distinct interpretation, which can be obtained by per bounded by unity. It follows from (24) that the two
simply factoring in order to reexpress (20) as variances are given by
R37) = ([X(t + ~ / 2 ) e - ' " ~ ]([ ~X ( t+ ~ '7/2)e
- ~) +'Tu(' - 7'2)
I*)
(22)
That is, R 3 7 ) is actually a conventional crosscorrela-
tion function

Therefore, the temporal correlation coefficient for fre-

quency translates is given by
'Although some readers will recognize the similarity between the
cyclic autocorrelation function and the radar ambiguity function, the
relationship between these two functions is only superficial. The con-
cepts and theory underlying the cyclic autocorrelation function. as
summarized in this article, have little in common with the concepts
and theory of radar ambiguity (cf. [ 1. Chapter 10, Sec. C]). 31t does contain infinitesimal sine-wave components.

APRIL 1991 IEEESPMAGAZINE 19

Hence, the appropriate normalization factor for R ~ ( Tis)
simply 1 /R,(O).
This is a good point at which to introduce some more
terminology. A signal x(t)for which the autocorrela-
tion RJs-)exists (e.g., remains finite as the averaging does contain finite-strength additive sine-wave com-
time T goes to infinity) and is not identically zero (as ponents with frequencies a = f 2 f 0 , since (30b) ren-
it is for lransient signals) is commonly said to be sta- ders one or the other ofthe last two terms in the quan-
tionary (in the wide sense or of second order). But we tity
need to I-efine the terminology to distinguish between
those stationary signals that exhibit second-order pe-
riodicity (R:(T)f 0 for some a # 0) and those station-
ary signals that do not (R:(T)= 0 for all a # 0). Con- + + 7/2)a*(t - ~ / 2 ) e - ' ~) " ~ '
sequently, we shall call the latter for which R$(T)= 0
purely stationary (of second order) and the former for
which R;[T) f 0 cyclostationary (of second order). We
shall also call any nonzero value of the frequency pa-
rameter 01 for which R:(T) f 0 a cyclefrequency. The
case a = 0 can be considered to be a degenerate cycle
frequency, since earn'= d ' = 1 is a degenerate sinu-
soid. Thus, some stationary signals are also cyclosta- nonzero for a = f2fo. That these are the only two cycle
tionary or exhibit cyclostationarity; only stationary frequencies a # 0 follows from the fact that (30c) ren-
signals t;hat do not exhibit cyclostationarity are purely ders (33)equal to zero for all a except (Y = 0 and a =
stationary. The discrete set of cycle frequencies is f 2fo. Thus, the cycle spectrum consists of only the two
called the cycle spectrum. For example, if a signal ex- cycle frequencies a = f 2 f 0 and the degenerate cycle
hibits only one fundamental period of second-order pe- frequency a = 0.
riodicity the cycle spectrum contains only harmonics Hence, the versions u(t)and u ( t ) of x ( t ) obtained by
(integer multiples) of the fundamental cycle fre- frequency shifting x ( t ) up and down by a12 = fo are
quency, which is the reciprocal of the fundamental pe- correlated. This is not surprising since (31)reveals that
riod. But if there are multiple incommensurate pe- x ( t )is obtained from a(t)by frequency shifting up and
riods, then the cycle spectrum contains harmonics of down by& and then adding. In conclusion, from (33)we
each of the incommensurate fundamental cycle fre- have the cyclic autocorrelation function
quencies.
We conclude this section by reconsidering the AM ex-
ample and determining the cyclic autocorrelation
function for the AM signal.
Example 1 continued: AM. Let a(t)be a real random lo otherwise
purely stationary signal with zero mean:
from which it follows that the temporal correlation
coefficient is given by

( a ( / + d2)a*(t - d 2 ) ) f 0

Equation (30c)guarantees that

(30b)

Thus, the
Y37) =
LI e-tr28

strength of
0
~ ~ ( 7 )for a
otherwise

correlation
= f2f,
(354

between
x(t + 7/2)e- i d t + 7/21 and x ( t - d2)etn"('-"'), which is
given by

We consider the amplitude-modulated sinewave

+ 19)
x ( t ) = a(t) cos (2.lrA)r
can be substantial for this amplitude-modulated sig-
nal, e.g., IY:(o) =
A s a n especially simple specific example of a ( t ) ,we
consider as before a stationary random telegraph wave,
Because of (30d),a(t)contains no finite-strength addi-
which switches back and forth between + 1 and - 1 at
tive sine-wave components and, therefore (together
random [Poisson distributed) switching times ( 2 ,
with (30a)). x(t)contains no finite-strength additive
Chapter 6, exc. 121. If we consider 7 = 0 in (32). we
sine-wave components. This means that its power
obtain
spectral density contains no spectral lines. However,
the quadratic transformation yo(t) = / x ( t ) / ' = /a(/)12cos' (27rhr + I9)

= f + f cos ( 4 T h t + 20)
20 IEEE SP MAGAZINE APRIL 1991
which clearly contains finite-strcngth additive sine- are used for the PSD and then measure the temporal
wave components with frequencies 01 = f2f0. In fact, correlation of the filtered signals to obtain
in this very special case, there is no random compo-
nent iri yo(t). On the other hand. for 7 # 0. y,(t) roii-
tains both a sine-wave component and a random
component.
Other examples of cyclostationary signals can be which is called the spectral-correlatiori density (SCD)
similarly viewed a s mixtures of stationarity and peri- function. Fig. 6 . This yields the spectral density of cor-
odicity. Examples are cited in the introductory Sec- relation in u( t )and u ( t ) a t frequencyf, which1 is iden-
tion. T,ypical cycle spectra include harmonics of pulse tical to the spectral density of correlation in x(t)a t fre-
rates, keying rates, spreading-code chipping rates, fre-
quency hopping rates, code repetition rates, doubled exp( - i xa t )
carrier frequencies, and s u m s and differences of these
[ I . Chapter 121.

THE SPECTRAL-CORRELATION
DEN:SITY FUNCT I ON
Definition of the SCD

In the same way that it is beneficial for some pur- exp(+i x a t )

poses to localize in the frequency domain the average Fig. 6. One channel-pair of a spectral-correlation
power ( (x(t)I2> = R J O ) in a stationary random signal. analyzer (or a cyclic-spectrurn ana1yzer)for measur-
it can be very helpful to localize in frequency the cor- ing the spectral-correlation density (or cyclic spectral
relation ( u ( t ) u * ( t>) = ( /x(t)I2e - 1 2 a ' y =
' ) K ~ ( o of
) fre- density).
quency-shifted signals u ( t )and u ( t )for a cyclostation-
quenciesf + 01/2 a n d f - d 2 . Fig. 7. That is, SZ(7) is
ary random signal x(t).I n the former case of localizing
the bandwidth-normalized (i.e.?divided by B) correla-
the power. we simply pass the signal of interest x ( t )
tion of the amplitude and phase fluctuations of the
through a narrowband bandpass filter and then mea-
sure the average power a t the output of thc filtcr. By
doing this with many filters whose center frequencies
x (v)
are separated by the bandwidth of the filters, we can
I
partition any spectral band of interest into a set of con-
tiguous narrow disjoint bands. In the limit a s the
bandwidths approach zero, the corresponding set of
measurements of average power, normalized by the
bandwidth. constitute the power spectral density (PSD)
function. That is, a t any particular frequency f , the
f - 7 jU
- f f+q
PSD for x ( t )is given by

where 0 denotes convolution and h k ( t )is the impulse

response of a one-sided bandpass filter with center fre-
quency f , bandwidth B , and unity gain a t the band
center, Fig. 5. U
- f
- 2
Center Frequency = f

Bandwidth = 0
Fig. 5. One channel of a spectrum analyzer for mea-
suring the power spectral density (PSD). (The symbol
z ind;cates that the output only approximates the
idealfunction S,(f)forfinite 7' arid B . ) +zf U

Fig. 7. Illustration of spectral bands used in the

In the latter case of localizing the correlation. we sim- measurement of the spectral-correlation density
ply PESS both of the two frequency translates u(t)and S:(f). ( v is a dummyfrequency uariable: the light and
u ( t ) of x(t)through the same set of bandpass filters a s dark bands are the bands selected by the BPFs.)
APRIL 199 I IEEE SP MAGAZINE 21
narrowband spectral components in x ( t ) centered a t the covariance becomes a correlation coefficient:
frequencies f + a12 and f - a12, in the limit as the
bandwidth B of these narrowband components ap-
proaches zero.
It is well known (see, for example, [ 1, Chapter 3, Sec.
C] for a proof) that the PSD obtained from (36)is equal
to the Fourier transform of the autocorrelation func- it is a con-
Since I p : ( f ) l is bounded to the interval [0,1],
tion, venient measure of the degree of local spectral redun-
dancy that results from spectral correlation. For ex-
ample, for I p E ( f ) l = 1, we have complete spectral
redundancy a t f + a12 and f - 0112.
Let us now return to the AM example considered in
Similarly, it can be shown [ 1, Chapter 11. Sec. C] that the preceding Section.
the SCD obtained from (37)is the Fourier transform of Example 1 continued: AM. By Fourier transforming
the cyclic autocorrelation function, (34) and invoking the cyclic Wiener relation (39).we
obtain the following SCD function for the amplitude-
modulated signal (31):

e””S,(f) for a = +2J,

Relation (38)is known as the Wiener relation (see, for
example 11, Chapter 3, Sec. C]), and (39)is therefore t S,(f + A,) + a S,(f - A,) for a = 0
called the cyclic Wiener relation [ 1, Chapter 1 1 , Sec. 0 otherwise
C]. The cyclic Wiener relation includes the Wiener re-
lation as the special case of a = 0. (In the probabilistic (43)
framewcrk of stochastic processes, which is based on
expected values [ensemble averages] instead of time
averages, the probabilistic counterpart of (38)is known
a s the Wiener-Khinchin relation, and therefore the
probabilistic counterpart of (39) is called the cyclic
Wiener-Khinchin relation [2,Chapter 12,Sec. 12.21.)
Because of the relation (39).the SCD is also called the
cyclic spectral density (CSD) function [ 1, Chapter 10, / - /
Sec. B]. Unlike the PSD which is real valued, the SCD / f /
is in general complex valued. -f. fo

It follows from (39) and the interpretation (23) of Fig. 8. Magnitude of the spectral-correlation density
R :(7) as RU,(7)that the SCD is the Fourier transform function f o r a n AM signal graphed as a height above
of the crosscorrelation function R,,(T) and is therefore the biji-equencyplane with coordinates f and a.
identical to the cross-spectral density function for the
frequency translates u(t ) and u ( t ) :
The magnitude of this SCD is graphed in Fig. 8 as the
height of a surface above the bifrequency plane with
coordinatesfand a. For purposes of illustration, a ( t )is
where S , , , ( f ) is defined by the right hand side of (37) assumed to have a n arbitrary low-pass PSD for this
for arbitrary u ( t )and u ( t ) . This is to be expected since graph. Observe that, although the argument f of the
the cross-spectral density S , , ( f ) is known (cf. 11, SCD is continuous, as it always will be for a random
signal, the argument a is discrete, as it always will be
Chapter 7, Sec. A]) to be the spectral-correlation den-
since it represents the harmonic frequencies of peri-
sity for spectral components in u(t) and u ( t ) at fre-
quency.f, and u(t)and u ( t ) are frequency-shifted ver- odicities underlying the random process (the sine-wave
sions of x ( t ) .The identity (40)suggests a n appropriate carrier in this example).
I t follows from (43)that the spectral correlation coef-
normalization for S:( f ) :as long as the PSDs of u(t)and
u ( t ) contain no spectral lines a t frequency f , which
ficient is given by
means that the PSD of x ( t ) contains no spectral lines
S , ( f )e+r28
at either of the frequenciesf f 0112, then the correla- P,a(f) =
tion of the spectral components (40) is actually a co- {[S,(f +2 0 + S , ( f ) l IS,(f) + S,(f - 2A,)l}’!2
variance since the means of the spectral components
for a = k2J, (444
are zero [I. Chapter 11, Sec. C]. When normalized by
the geometric mean of the corresponding variances,
Thus, the strength of correlation between spectral
which are given by
component in x ( t )a t frequenciesf + a12 andf - a12 is
S,(f, = S,(f + (4 1 4 unity:

and

22 IEEE SFIMAGAZINE APRIL 1991

provided that a(t)is bandlimited to If[ Ifo, Example 3: Time Delay. A s our first example of (49).
we consider a filter that simply delays the input by to;
then h(t) = 6 ( t - to)and H ( f ) = e-''*fto. Therefore, for
z ( t ) = x(t - to),we obtain from the input-output SCD
This is not surprising since the two spectral compo- relation (49)
nents in x(t)a t frequenciesf *
a / 2 = f +fo are obtained
from the single spectral component in a(t) at fre-
quency f simply by shifting and scaling. Thus, they
are perlectly correlated. That is, the upper (lower)side- which indicates that, unlike the PSD, the SCD of a cy-
band fcrf > 0 carries exactly the same information as clostationary signal is sensitive to the timing or phase
the lower (upper) sideband forf < 0. Techniques for of the signal.
exploiting this spectral redundancy are described in a
later section. rn
Before considering other examples of the SCD, let us
Example 4: Multipath Propagation. A s a second ex-
first gain a n understanding of the effects of some basic
ample of (49),if x(t) undergoes multipath propagation
signal processing operations on the SCD. This greatly
during transmission to yield a received signal
facilital es the determination of the SCD for commonly
encountered manmade signals.
z(t) = Cn a,x(t - t,,)

Fil ter:ing where a, and t, are the attenuation factor and delay
of the nth propagation path, we have
When a signal x(t)undergoes a linear time-invariant
(LTI) transformation, (i.e., a convolution or a filtering H(f) = (52)
" a,e-'2d''
operation),
and therefore (49)yields

S , * ( f ) = S.:(f) c
r7.m
an4

the spectral components in x ( t ) are simply scaled by

the complex-valued transfer function H( f ) , which is
the Fourier transform

H(f) = im
-m
h(t)e-'2"Pdt (47) Example 5 : Bandpass Signals. A s a third example
of the utility of the relation (49), let us determine the
support region in the ( f, a ) plane for a bandpass signal
of the impulse-response function h(t ) of the transfor- with lowest frequency b and highest frequency B. To
mation A s a result, the PSD gets scaled by the squared enforce such a spectrum, we can simply put any signal
magnitude of H(f) (see, for example, [ 1,Chapter 3, Sec. x(t ) through an ideal bandpass filter wikh transfer
C] or [ 2 , Chapter 10, Sec. 10.11) function

S,Cf) = IH(f)I2Uf) (48) 1 forb < If1 < B

H ( f ) = [O otherwise
Equation (48) can be derived from the definition (36)
of the PSD. Similarly, because the spectral compo-
nents of x(t) a t frequencies f +_ 01/2 are scaled by It then follows directly from the input-output SCD re-
H(f f a / 2 ) , the SCD gets scaled by the product lation (49)that the SCD for the output of this filter can
H(f + cr/2)H*(f - d 2 ) : be nonzero only for I l f 1 - 1011/21 > b and I f I Ia ) / 2 +
< B:
Sl(f) = H(f + d 2 ) H * ( f - &/2)S.?(f) (49)

This result, called the input-output SCD relation for

filtering. which can be derived from the definition (37)
of the SCD, includes (48) a s the special case of a = 0.
Observe that it follows from (49)and the definition (42)
that This shows that the support region in the ( f , a ) plane
for a bandpass signal is the four diamonds located a t
the vertices of a larger diamond, depicted in Fig. 9a.
By letting b + 0, we obtain the support region for a
lowpass signal, and by letting B + 03, we obtain the
That is, the magnitude of the spectral correlation coef- support region for a highpass signal. This is shown in
ficient IS unaffected by filtering (if H(f f a / 2 ) # 0). Figs. 9b and 9c.

APRIL 1991 IEEE SP MAGAZINE 23

 s(r) = cos (2Tf;,t + 0)
the product modulator becomes a frequency converter
when followed by a filter to select either the up-con-
verted version or the down-converted version of r(t).
By applying first the input-output SCD relation (57)for
the product modulator (which applies since a sinusoid
is statistically independent of all time-series [ 1, Chap-
ter 15, Sec. A]), and then (49)for the filter, we can de-
termine the up-converted or down-converted SCD. To
illustrate, we first determine the SCD for the sinusoid
s(t).By substituting the sinusoid s(t) into the definition
of the cyclic autocorrelation, we obtain

+ cos ( 2 ~ f ; , 7 ) for a = 0
RP(7) = [ : e k r 2 ’ for a = +2J, (58)
f otherwise

Fourier transforming then yields the SCD

4 6(f - A,) + 4 6(f + A,) for a = 0

Fig. 9. t i ) S u p p o r t region i n t h e bifrequency p l a n e f o r for a = f2f, (59)
t h e spectral-correlation d e n s i t y f u n c t i o n of a b a n d -
p a s s signal. b) S u p p o r t region f o r a l o w p a s s signal. c) otherwise
Suppori region f o r a h i g h p u s s signal.
which is illustrated in Fig. loa. Using (57). we con-
Waveform multiplication volve this SCD with that of a stationary signal r ( t ) ,for
which
When finite segments of two signals are multiplied
together, we know from the convolution theorem that
their Fourier transforms get convolved. From this, we
expect some sort of convolution relation to hold for the
SCDs of signals passing through a product modulator.
In fact, it can be shown [ l , Chapter 11, Sec. C], (2,
Chapter 12, exc. 411 that if x(t)is obtained by multi- / - /
plying together two statistically independent4 time-se- / f /
ries r ( t );and s(t), -f. +fo

x(t) = r(t)s(t) (55)

then the cyclic autocorrelation of x(t) is given by the

discrete convolution in cycle frequency of the cyclic
autocorrelations of r ( t )and s(t): ___)
41J3
-4f0/3
f

where, for each a , (3 ranges over all values for which

R f ( 7 )f 0. By Fourier transforming (56).we obtain the
i n p u t - o u t p u t SCD relation f o r w a v e f o r m multiplica- A A A
tion:

s:(f) = jIm$ s f ( ~ ) S : ! - ’ ?-
f V) dv (57)
--ao
which is a double convolution that is continuous in the
variable f and discrete in the variable 01. /
-fo fo
Example 6:Frequency Conversion. A s an example
of (57),if s(t) is simply a sinusoid, Fig. 10. a) Magnitude of t h e spectral-correlation d e n -
s i t y (SCD)f o r a s i n e wave of f r e q u e n c y f o . b) SCD f o r
4Time-seriesare statistically independent if their joint fraction-of-time a l o w p a s s stationary signal. c) SCD m a g n i t u d ef o r t h e
probability densities factor into products of individual fraction-of-timr product of s i g n a l s corresponding to a) a n d b), ob-
probability densities. a s explained in 11, Chapter 15. Sec. Al. tained b y convolving t h e SCDs i n a) a n d b).

24 IEEE SP MAGAZINE APRIL 1991

 the SCD S z ( f ) for the sequence of samples { x ( n T s ) }is
depicted in terms of the single diamond support region
for a lowpass waveform x ( t ) ,which is shown in Fig. 9b.

which is illustrated in Fig. lob. The result is that the cy

1
SCD of the stationary signal simply gets replicated and
scaled at the four locations of the impulses in the SCD
of the sinusoid, as illustrated in Fig. 1Oc.
Exarn,ple 7: Time Sampling. Another important sig- 2B+2/T
nal processing operation is periodic time sampling. It
is known that for a purely stationary signal x ( t ) ,the
PSD S,( f ) of the sequence of samples { x ( n T s )n: = 0 ,
+1, k 2 , . . . } is related to the PSD S,( f ) of the
waveform by the aliasing formula [ 1, Chapter 3 , Sec.
E], [2, Chapter 11, Sec. 11.11

It is shown in 11, Chapter 11, Sec. C], [2, Chapter 12,

Sec. 12.$1 that this aliasing formula generalizes for the
SCD to

We also can obtain essentially the same result (except

for a n additional factor of U T ) by applying the input-
output !3CD relation for waveform multiplication (57)
to a n impulse sampler, which is a product modulator
with one input equal to a periodic train of impulses,
Fig. Illustration of support regions in the bifre-
m
quency plane for the spectral-correlation densities
s(f)= c
n= -02
6(t - nT,) (63) that are aliased by periodic time-sampling.

This alternative approach can be carried out (cf. [ l ,

Chapter 11, Sec. C]) by formally expanding (63) into Discrete time
the Fourier series
Before leaving the topic of time sampling, it is ex-
plained that the discrete-time counterpart of the sym-
metric definition of the cyclic autocorrelation function
(201, which uses delays of +7/2 and -712, is not appro-
priate since time samples midway between the given
samples { x ( n T s ) }are not available. Therefore, the
Observe that, when x ( t )is not purely stationary (i.e., asymmetric definition
when S c (f ) f 0 for (Y = m / T , for some nonzero inte-
gers rn), the conventional PSD aliasing formula (61) l?:(kT,) 2 (x(nT, + kT,)x*(nT,)e-'2"""T')e-'aahT3 (66)
must be corrected according to (62) evaluated at (Y =
0: (where ( . ) denotes discrete-time averaging over n ) ,
which uses delays of 7 and 0 and which includes the
correction factor e-iaarfor T = kTs that makes the
asymmetric definition agree with the symmetric defi-
nition, has been adopted [ l ,Chapter 11, Sec. C], [2,
This rellects the fact that, when aliased overlapping Chapter 12, Sec. 12.41.
spectral components add together, their PSD values The discrete-time counterpart of the SCD can be de-
add only if they are uncorrelated. When they are cor- fined just as is done at the beginning of this Section
related, as in a cyclostationary signal, the PSD value (but with a discrete-time bandpass filter instead of the
of the sum of overlapping aliased components depends continuous-time filter used there). The SCD can then
on the particular magnitudes and phases of their cor- be shown to be the discrete-time Fourier transform of
relations. The SCD aliasing formula (62) is illustrated the cyclic autocorrelation [ l ,Chapter 11, Sec. C], 12,
graphically in Fig. 11, where the support regions for Chapter 12, Sec. 12.41:

APRIL 1991 IEEE SP MAGAZINE 25

 Example 2 continued: PAM. Let {a,} be a purely sta-
tionary random sequence, and let u s interpret these
random variables as the time-samples of a random
waveform, a, = a(nT,), with PSD Sa(f ) . We consider
Periodlically time-variant filtering the PAM signal

Many signal processing devices such as pulse and

carrier modulators, multiplexers, samplers. and scan-
ners, can be modeled as periodically time-variant fil-
ters, especially if multiple incommensurate periodici- where p ( t )is a deterministic finite-energy pulse and E
ties (i.e.,periodicities that are not harmonically related) is a fixed pulse-timing phase parameter. To determine
are included in the model. By expanding the periodi- the SCD of x(t ) ,we can recognize that x(t ) is the output
cally time-variant impulse-response function in a Fou- of a periodically time-variant linear system with input
rier series a s explained shortly, any such system can a(t ) , and impulse response
be represented by a parallel bank of sinusoidal product
m
modulators followed by time-invariant filters. Conse-
quently, the effect of any such system on the SCD of h(t, U ) = c
,,= --m p ( t - nT, + E)@U - nT,,)
its inpul. can be determined by using the SCD relations
for filters and product modulators. In particular, it can We can then use the input-output SCD relation (68).
be shqwn [ I . Chapter 11, Sec. D] that the SCD of the Or we can recognize that this particular periodically
output z ( t )of a multiply-periodic system with input x ( t ) time-variant system is composed of a product modu-
is given by lar, that implements an impulse sampler, followed by
a linear time-invariant pulse-shaping filter with im-
Sq(f) = C
U.?EA
G,(f + d2)G: (f - a/2) pulse-response function h(t) = p ( t ) ,as shown in Fig.
12. We can then use the input-output SCD relation
)"' 2 (57). as it applies to impulse sampling, together with
the relation (49) for filtering. The result is

where Go(f ) are the transfer functions of the filters and

A is the set of sinusoid frequencies associated with the
product modulators in the system representation.
More specifically, for the input-output equation

the mu1tiply-periodic impulse-response function h(t, U )

can be (expanded in the Fourier series
2 Filter

Fig. 12. Interpretation ofPAM signal generator as the

where the Fourier coefficients (for each 7)are given by cascade of an impulse sampler and a pulse-shaping
filter.

1
It follows from (69)-(70) that the filter output can be S';(f) = -P(f + 01/2)P*(f - 0112)
expressed as Tt

z(t) =
OEA
C o
[ ~ ( t ) e " ~ ' ~g~B ]( t ) (72)

where g u ( t )are the impulse-response functions of the Using the SCD aliasing formula (62)for a ( t )we can re-
filters with corresponding transfer functions Go( f ). express (74) as
Thus, periodically time-variant filters perform time-in-
variant filtering on frequency-shifted versions 1
s:(f) = -~ ( + fa / 2 ) ~ * (-f a/2)S,"(f)e'2Tc" (75)
x(t)ei2"'"of the input. This results in summing scaled, T,
frequency-shifted, cycle-frequency-shifted versions of
the SC13 for the input x ( t ) to obtain the SCD for the where S z ( f ) is the SCD for the pulse-amplitude se-
output z(t),as indicated in (68). quence { a,}. Having assumed that {a,} is purely sta-
Let us now consider a couple of additional examples tionary, and using the periodicity property (exhibited
of modulation types, making use of the results ob- by all SCDs for discrete time-series [ 1, Chapter 11, Sec.
tained in the preceding paragraphs to determine SCDs. Cl
26 IEEE SP MAGAZINE APRIL 1991
 + a/2) for a = k/T,, For a white-noise amplitude-sequence as in (79). (81)
(76) reduces to
otherwise
1
f o r k = 0,k l , +2, . . , we can express (75)as R : ( T ) = - r;(T)e'2aaf for a = k/T,, (83)
T"

X(f) = and, for a rectangular pulse as in (78).this yields the

I 1 temporal correlation coefficient

-P(j- + a / 2 ) P * (f - a/2)$,(f + a/2)e'2""' for a = k/T,
T"
yy(T) = sin [aa(T1- IT1)] ef2ffaf for IT[ 5 To
(84a)
T,
0 Otherwise which peaks for a = UTo at T = T0/2, where it takes
(77) on the value
A graph of the magnitude of this SCD for the full-duty-
cycle relztangular pulse ly:(7',,/2)1 = l/a f o r a = 1/T, (84b)

1 for It( 5 Tc,/2 That is, the strongest possible spectral line that can be
p(t) =
[ 0 otherwise
(78) regenerated in a delay-product signal (cf. remark made
following (20)) for this particular PAM signal occurs
when the delay equals half the pulse period. In con-
and a white-noise amplitude sequence with PSD trast to this, when the more bandwidth-efficient pulse
whose transform is a raised cosine is used, the optimal
SAf, = 1 (79) delay for sine-wave regeneration is zero.
An especially simplekxample of a sequence of pulse
is shown in Fig. 13. amplitudes {a,} is a binary sequence with values f 1.
It from (77) that for all a = k/To for which If we consider 7 = 0 in the delay-product signal, then
S,(f f #r/2)z o and P( f + a / 2 ) ~ * (-f a / 2 ) + 0, the we obtain
spectral correlation coefficient &( f ) is unity in mag-
nitude: m

Thus, all spectral components outside the band 1 f I < If the pulses do not Overlap if p ( t ) = for It(
112 To are completely redundant with respect to those T0/2),this reduces to
inside this band. Techniques for exploiting this spec- m
tral redundancy are described in the next section. yo(t) =
n=-m
c &*(t - nT,, + E)

A =
n = -m
p2(t - nT, + E)

6/T0
which is periodic with period To and therefore contains
finite-strength additive sine-wave components with
frequencies k/T, (except when p ( t )is flat as in (78)).In
- - this very - special
- case where {a,) is binary and the
-3/Ta ___) pulses do not overlap, there is no random component
f in y o ( t ) ;but, for T # 0, y , ( t ) contains both sine-wave
Fig. 13. M a g n i t u d e of t h e spectral-correlation d e n - components and random components (even when p ( t )
s i t y for a PAM signal w i t h full duty-cycle rectangular is flat).
pulses. Example 8: ASK a n d PSK. By combining the ampli-
tude-modulated sine wave and the digital amplitude-
By inverse Fourier transforming the SCD (771, we ob- modulated pulse train, we obtain the amplitude-shift-
tain the cyclic autocorrelation function keyed (ASK) signal
r - 0 :

R:(T) = I'
10
- c
To n = - m
R,(nT,)r,"(T - nTJe'2""f for a

otherwise
= kIT,
where
x(t) = U(?) cos (2af,t + e)

where
(81)
a(r) = c
"= -m a,p(t - nT, + E)

r;(~) A im+-m
p(t 7/2)p*(t - ~ / 2 ) e - ' * ~d" ' (82)
and {a,} are digital amplitudes. By using the SCD re-
lation (57) for waveform multiplication and the result
APRIL1991 IEEESPMAGAZINE 27
(77) for the SCD of a ( t ) ,we can obtain the SCD for the of the SCD a t cycle frequencies associated with the
signal (€15)by simply convolving the SCD functions carrier frequency (viz., a = +2f0 + m/Tofor all integers
shown in Figs. 10a and 13. The result is shown in Fig. m) in QPSK. Similarly, the pulse staggering by T,/2
14a, where the cycle frequencies shown are a = +2f0 (between the in-phase and quadrature components)
+ m/To ilnd 01 = m/T0 for integers m, and f o = 3.3/T0. present in SQPSK but absent in QPSK results in the
SCDs being cancelled a t a = +2f0 + m/T, only for even
integers m, and at 01 = m/T, only for odd integers m in
SQPSK. This again illustrates the fact that the SCD
contains phase and timing information not available
in the PSD. In fact, as formulas (43) and (77) reveal,
the carrier phase 0 in (31)and the pulse timing E in (73)
are contained explicitly in the SCDs for these carrier-
and pulse-modulated signals.

The ideal SCD function (37) is derived by idealizing

the practical spectral correlation measurement de-
picted in Fig. 6, by letting the averaging time T in the
correlation measurement approach infinity and then
letting the spectral resolving bandwidth B approach
zero. Consequently, the practical measurement with
finite parameters T and B can be interpreted a s a n es-
timate of the ideal SCD. This estimate will be statisti-
cally reliable only if T B >> 1. Numerous alternative
implementations of this practical measurement are
described in [ I , Chapter 131, and computationally ef-
ficient digital architectures for some of these, which
are developed in [ 3 ]and [4],are presented in this issue
in [SI. The statistical behavior (bias and variance) of

-5/To - f
5/To

Fig. 14. M a g n i t u d e of spectral-correlation densities.

a) BPSE:, b) QPSK, a n d c) SQPSK. [ E a c h signal has a
rectangular k e y i n g envelope.)

For a binary sequence with a, = + 1, this amplitude-

shift kelyed signal, with the pulse (78). is identical to
the binary phase-shift keyed (BPSK) signal ___)
f

2 ~ f , r+ 0 +2 5
,!=-e=
a,,p(t - nT<,)] (86)

since shifting the phase of a sine wave by f d 2 is the

same as shifting it by 7d2 and multiplying its ampli- n
tude by f 1. Other commonly used types of phase-shift-
keyed signals include quaternary phase-shift keying
(QPSK) and staggered QPSK (SQPSK). The details of
these signal types are available in the literature (see,
for example 11. Chapter 12, Sec. E], [2,Chapter 12, Sec.
12.51).Only their SCD-magnitude surfaces are shown
here in Figs. 14b and 14c, where again f o = 3.31T0.
It is ernphasized that the three signals BPSK, QPSK, Fig. 15. Magnitude of m e a s u r e d spectral-correlation
and SQI?SK differ only in their carrier phase shifts and d e n s i t y [SCD) e s t i m a t e d f r o m a f i n i t e d a t a record f o r
t h e QPSK signal w h o s e ideal S C D is s h o w n i n Fig.
pulse tiining and, as a result, they have identical PSDs, 14b. a) Record l e n g t h i s 128 t i m e s a m p l e s , a n d f o u r
as shown in Fig. 14 (consider a = 0).However, as also adjacent f r e q u e n c y [f ) b i n s a r e averaged together. b)
shown in Fig. 14, these differences in phase and tim- Record l e n g t h i s 32,768 a n d 1,024 adjacent f r e -
ing result in substantially different SCDs (consider a q u e n c y [ f ) b i n s a r e averaged together. ( T h e sam-
# 0). That is, the phase-quadrature component pres- pling r a t e i n b o t h a) a n d b) is 1 O/To, w h e r e l/To i s t h e
ent in QPSK but absent in BPSK results in cancellation keying rate of t h e QPSK signal.)
ia IEEE SP MAGAZINE APRIL 1991
such estimates is analyzed in detail in [ l ,Chapter 15, ables it to be readily exploited is its distinctive char-
Sec. B]. For the purpose of making the applications de- acter. That is, most manmade signals exhibit spectral
scribed in the next section more concrete, it suffices redundancy, but most noise (all noise that is not cy-
here to simply point out that because the SCD S:( f ) clostationary) does not. Furthermore, in many practi-
is equivalent to a particular case of the conventional cal situations where multiple signals of interest, as well
cross spectral density S,,,( f ) (cf. (40)). one can envi- as signals not of interest (interference),overlap in both
sion any of the conventional methods of cross spectral time and frequency, their spectral redundancy func-
analysis as being used in the applications. tions are nonoverlapping because their cycle frequen-
Example 9:QPSK. A s an example, the result of using cies CY are distinct. This is a result of signals having
the Wiener-Daniel1 method [ 11, based on frequency distinct carrier frequencies and/or pulse rates or key-
smoothing of the cross-periodogram of U(t )and U ( t )(the ing rates, even when occupying the same spectral
conjugate product of their FFTs), is illustrated in Fig. band.
15 for EL QPSK signal with carrier frequencyf, = 1/4T, The distinctive character of spectral redundancy
and keying rate 1/T, = l/8Ts, where UT, is the sam- makes signal selectivity possible. Specifically, for the
pling rate. An FFT of length 128 (T = 128Ts) was used received signal
in Fig. 15a, and only four frequency bins were aver-
aged together (B = 4 / T ) , whereas in Fig. 15b, the FFT L

length used was 32,768 ( T = 32,768Ts)and 1,024 bins x(r) = c ”((t)+ n(t)
(’=
(87)
were averaged together (B= 1,024/T).It is easily seen
by comparing with the ideal SCD in Fig. 14b that un- where the {s?(t)};include both signals of interest and
less TB >> 1, the variability of the SCD estimate can interference-all of which are statistically indepen-
be very large. dent of each other-and where n(t) is background
noise, we have the SCD

But if the only signal with the particular cycle fre-

quency (Yk is S k ( t ) . then (for measurement time T + m)
we have
The existence of correlation between widely sepa-
rated spectral components (separation equal to CY)can
be interpreted as spectral r e d u n d a n c y . The meaning
of the term r e d u n d a n c y that is intended here is essen-
regardless of the temporal or spectral overlap among
tially the same as that used in the field of information ; also n(t).
{ s l ( t ) }and
theory and coding. Specifically, multiple randomly
fluctuating quantities (random variables) are redun- Example 1 0 : BPSK Signal i n Multiple AM lnterfer-
dant if they are statistically dependent, for example, e n c e a n d Noise. To illustrate the concept of signal se-
correlai.ed. In coding, undesired redundancy is re- lectivity, let us consider the situation in which a
moved from data to increase the efficiency with which broadband BPSK signal of interest is received in the
it represents information, and redundancy is intro- presence of white noise and five interfering AM signals
duced in a controlled manner to increase the reliability with narrower bandwidths that together cover the en-
of storage and transmission of information in the pres- tire band of the BPSK signal. The noise and each of the
ence of noise by enabling error detection and correc- five interfering signals have equal average power.
tion. Therefore, the total signal-to-interference-and-noise
Here, redundancy is to be exploited to enhance the rate (SINR) is approximately -8 dB. The BPSK signal
accuracy and reliability of information gleaned from has carrier frequency fo = 0.25/Ts and keying rate CY,
the measurements of corrupted signals, but the term = 0.O625/Ts. It has full-duty-cycle half-cosine enve-
informution is interpreted in a broad sense. For in- lope, which results in an approximate bandwidth of Bo
stance, it includes the six examples outlined in the in- = 0. 1875/Ts.The five AM signals have carrier frequen-
troductory section. In all these examples, the perfor- cies f i = 0.156/Ts,f2 = 0.203/Ts,f3 = 0.266/Ts,f4 =
mance of the signal processors that make the decisions 0.313/Ts,f5 = 0.375/Ts, and bandwidths B, = 0.04/Ts,
and/or produce the estimates can be substantially im- 8 2 = 0.05/Ts, B3 0.045/Ts, B4 = 0.04/Ts, B5
proved by suitably exploiting spectral redundancy. 0.O8/Ts. With the use of the same measurement pa-
The degree of improvement relative to the perfor- rameters (FFT length = 32,768) a s in the preceding
mance of more commonly used signal processors that Example 9 for the measurement of the SCD of the QPSK,
ignore spectral redundancy depends on both the se- the SCD for these six signals in noise was measured.
verity of the signal corruption (noise, interference, dis- The resultant SCD magnitude is shown in Fig. 16a.
tortion] and the degree of redundancy in the signal x(t), Also shown in Figs. 16b and 16c are the SCD magni-
as measured by the magnitude of the spectral corre- tudes for the BPSK signal alone and for the five AM
lation coefficient I&( f )I defined in the preceding sec- interferences plus noise alone. Although all six signals
tion. exhibit strong spectral redundancy (Io,”,(f ) l = 1). the
The primary feature of spectral redundancy that en- cycle frequencies CY at which this redundancy exists

APRIL 1991 IEEE SP MAGAZINE 29

 We can see from Fig. 16 that knowing the particular
pattern of the SCDs for BPSK and AM signals (see Figs.
8 and 14) enables us to detect the presence of six sig-
nals and to classify them according to modulation
type. This would be impossible if only PSD (SCD at CY
= 0) measurements were used. One approach to ex-
ploiting the spectral redundancy of a signal to detect
its presence is to generate a spectral line at one of its
cycle frequencies and then detect the presence of the
spectral line (cf. the earlier section on SCD).It has been
shown that the maximum-SNR spectral-line generator
for a signal s ( t )is additive Gaussian noise and interfer-
ence with PSD S,,(f ) produces the detection statistic
[ 1 , Chapter 14, Sec. E]

for comparison to a threshold. In (90).S $ ( f )is a crude

estimate of S g ( f ) obtained by deleting the time-aver-
aging operation ( . ) and the limiting operation from
(37) and choosing B equal to the reciprocal of the rec-
ord length of x ( t ) .It can be shown that (90)is equiva-
___)
lent to whitening the noise and interference using a
f
filter with transfer function l / [ S , ( f ) ] 1 ’and
2 , then cor-
relating the measured SCD for the noise-and-interfer-
ence-whitened data with the ideal SCD of the signal
(transformed by the whitener) to be detected [ 1 , Chap-
ter 14, Sec. E].
A detailed study of both optimum (e.g., maximum-
likelihood and maximum-SNR)and more practical sub-
0
optimum detection on the basis of SCD measurement
is ’reported in [ 6 ] and
, receiver operating characteris-
tics for these detectors obtained by simulation are pre-
sented in 171.

Once the six signals have been detected and classi-

Fig. 16. Magnitudes of estimated spectral-correla- fied, their carrier frequencies and phases and the key-
tion deiisities (SCDs). a) SCD magnitude for a BPSK ing rate and phase of the BPSK signal can-with suf-
signal corrupted by white noise and five AM interfer- ficiently long signal duration-be accurately estimated
ences. 12) SCD magnitude for the BPSK signal alone. from the magnitude and phase of the SCD (cf.,fa. 0 in
c) SCD !magnitudefor the white noise andfive AM in-
(43) and To, t in (77)).It is clear from the theory dis-
terferences. (The power levels, center frequencies.
and bandwidths for the signals and noise are speci- cussed in preceding sections that SCD measurement
fied in the text; the record length used is 32.768 time- is intimately related to the measurement of the ampli-
samples and 1,024 adjacent frequency ( f ) bins are tudes and phases of sine waves generated by quadratic
averaged together.) transformations of the data. Thus, the fact that a n SCD
feature occurs a t CY = 2f0 for each carrier frequency fo
is a direct result of the fact that a sine wave (spectral
line) with frequency CY = 2f0 and phase 2 0 can be gen-
are distinct because the carrier frequencies are all dis- erated by putting the data through a quadratic trans-
tinct. Thus, a n accurate estimate of the SCD for the formation. Similarly, for the SCD feature at CY = l / T o ,
BPSK signal is easily extracted from the SCD for the where 1/T, is the keying rate, a spectral line with fre-
corrupted measurements. Similarly, accurate esti- quency CY = l / T oand phase E can be quadratically gen-
mates of the SCDs for each of the five AM signals can erated. Consequently, SCD measurement is useful
be extracted. Consequently, any information con- either directly or indirectly for estimation of synchro-
tained in these SCDs can be reliably extracted. nization parameters (frequencies and phases) required
In connection with this example, let us briefly con- for the operation of synchronized receivers. The link
sider some of the signal processing tasks outlined in between synchronization problems and spectral re-
the introductory section. dundancy is pursued in [ 8 ] .

30 IEEESPMAGAZINE APRIL 1991

 The cross SCD S & ( f )for two signals x ( t )and w(t) is Thus, nulling signals other than sy( t )in the output x ( t )
defined in a way that is analogous to the definition (37) of the linear combiner reduces the denominators in
and (24-1of the auto SCD S:(f). That is, x ( t )in (24a) is (93) and (94) but not the numerators. Hence, I&(f)l
simply replaced with w(t). If we were to compute the and /y:(7)1 can be increased by nulling any of the sig-
cross SCD for two sets of corrupted measurements ob- nals other than sp(t).Moreover, the linear combiner
tained from two reception platforms, then the cross needs no knowledge of the reception characteristics of
SCD m,agnitude would look very similar to that in Fig. the array (no calibration) to accomplish this nulling. A
16 (except that the low flat feature at LY = 0, which rep- thorough study of spectral-coherence-restoral algo-
resents the PSD of the receiver noise, would be ab- rithms that perform this blind adaptive spatial filter-
sent), but the phase of the cross SCD would contain a ing is reported in [ 111.
term linear i n f a t each value of CY where the auto SCD
of one of the six signals is nonzero. The slope of this
linear phase equals the time-difference-of-arrival
(TDOAIl of the wavefront at the two platforms for the
particular signal with that feature. That is, for x ( t )from
one platform given by (87) and w(t) from the other We can take this approach one step further if we do
platform given by indeed have calibration data for the reception charac-
teristics of an antenna array because we can then also
L
exploit signal selectivity in LY to perform high-resolu-
w(t) = C
F= I
sp(t - tu) + m(r) (91) tion direction finding (DF) without some of the draw-
backs (described below) of conventional methods for
where { t y }are the TDOAs, we have high-resolution DF, such as subspace fitting methods
[12], that do not exploit spectral redundancy. In par-
ticular, let us consider the narrowband model

provided that s a ( t )is the only signal with cycle fre-

quency C Y . Consequently, accurate estimates of the
TDOAs of each of these signals can be obtained from
the cross SCD measurement, regardless of temporal for the analytic signal (or complex envelope) x ( t )of the
and spectral overlap or of the closeness of the individ- received data vector of dimension r, where a(0,)is the
ual TDOAs. In other words, the signal selectivity in the direction vector associated with the &th received sig-
CY domain eliminates the problem of resolving TDOAs nal s , ( t ) ,and the function a(* ) is specified by the cali-
of overlapping signals. Detailed studies of signal-selec- bration data for the array. Then, by working with the
tive TDOA estimation are reported in [9] and [lo], magnitude and phase information contained in the r
where various algorithms are introduced and their x r cyclic correlation matrix
mean-squared-error performance is evaluated.

for some fixed 7 (where t denotes conjugate trans-

pose), instead of working with the information con-
Continuing in the same vein, we consider receiving tained in the conventional correlation matrix
these same six signals in noise with a n antenna array.
Then we can use the signal selectivity in CY to blindly
(without any training information other than knowl-
edge of the cycle frequencies CY of the signals) adapt a
linear combiner of the outputs from the elements in
the array to perform spatial filtering. Specifically, by we can avoid the need for advance knowledge of the
directing the linear combiner to enhance or restore correlation properties of the noise RJO) and interfer-
spectral redundancy in its output at a particular cycle ence RJO) for P # k, and we can avoid the constraint
frequency C Y , the combiner will adapt to null out all imposed by conventional methods that the number of
other signals (if there are enough elements in the array elements in the array exceed the total number L of sig-
to make this nulling possible). This behavior of the nals impinging on the array. Also, by resolving signals
combiner can be seen from the fact that the spectral in CY,we need not resolve them in direction of arrival.
correlation coefficient for x ( t )in (87) is (from (89)) Consequently, superior effective spatial resolution is
another advantage available through the exploitation
of spectral redundancy. A s an example of a cyclic DF
method, we can exploit the fact that the r x r matrix
in (96)has a rank of unity and the ( r - 1)-dimensional
and, sirnilarly, the temporal correlation coefficient for null space of this matrix is orthogonal to a(&).There-
the frequency-shifted versions of x ( t )is fore, we can choose as our estimate of & that value 0,
APRIL 1991 IEEE SP MAGAZINE 31
which renders a(&)most nearly orthogonal to the null
space of an estimate of the matrix R:(T) obtained from
finite-time averaging. A thorough study of this ap-
proach to signal-selective DF is reported in [ 131,where
various algorithms are introduced and their perfor-
mances ;are evaluated.
In the preceding paragraphs of this section, the sig-
nal processing tasks (with the exception of spatial fil-
tering) involve decisions or parameter estimation, but
do not involve estimating (or extracting) a n entire sig-
nal or a n information-bearing message carried by the
signal. Nevertheless. for the signal-extraction prob-
lem, the utility of spectral redundancy is just a s ap-
parent, its explained in the following paragraphs.

Spectrally redundant signals that are corrupted by

other interfering signals can be more effectively ex-
tracted In some applications by exploiting spectral f
correlation through the use of periodic or multiply-pe-
riodic linear time-variant filters, instead of the more
conventional time-invariant filters. These time-variant
filters enable spectral redundancy to be exploited for Fig. 17. Illustration of power spectral densities
signal extraction, because such filters perform fre- (PSDs)for cochannel-interference removal with min-
quency-shifting operations (cf. (72))as well as the fre- imal signal distortion. a) P S D for AM signal plus in-
quency-dependent magnitude-weighting and phase- terference. bl P S D after interference removal by time-
shifting operations performed by time-invariant fil- invariant filtering. c) PSD after distortion removal by
ters. The utility of this is easily seen for the simple ex- freq uency-sh ifti ng.
ample in which interference in some portions of the
spectral band of the signal is so strong that it over- PAM, ASK, PSK, or digital QAM (quadrature AM). It
powers the signal in those partial bands. In this case, can be shown that, regardless of the degree of spectral
a time-invariant filter can only reject both the signal and temporal overlap, each of the two interfering sig-
and the interference in those highly corrupted bands, nals can be perfectly extracted by using frequency
whereas a time-variant filter can replace the rejected shifting and complex weighting, provided only that
spectral components of the signal of interest with they have either different carrier frequencies or phases
spectral components from other uncorrupted (or less (AM, ASK, BPSK) or different keying rates or phases
corrupted) bands that are highly correlated with the (PAM, ASK, PSK, digital QAM) and at least 100%ex-
rejected components from the signal. cess bandwidth (bandwidth in excess of the minimum
AM is a n obvious example of this because of the com- Nyquist bandwidth for zero intersymbol interference).
plete redundancy that exists between its upper side- In addition, when the excess bandwidth is ( L - 1)
band (above the carrier frequency) and its lower side- loo%, L spectrally overlapping signals can be sepa-
band (below the carrier frequency). Although this rated if they have the same keying rate but different
redundancy is exploited in the conventional double keying phases or carrier frequencies. Also, when
sideband demodulator to obtain a 3 dB gain in S N R broadband noise is present, extraction of each of the
perform,mce, it is seldom exploited properly when par- signals can in many cases be accomplished without
tial-band interference is present. The proper exploita- substantial noise amplification.
tion in this case is illustrated in Fig. 17. Figure 17a To illustrate how spectrally overlapping signals can
shows the spectral content (Fourier transform magni- be separated, we consider the case of two QPSK sig-
tude of a finite segment of data) for a n AM signal with nals with unequal carrier frequencies and unequal
partial-band interference in the upper sideband. Fig- keying rates and 100%excess bandwidth. The graphs
ure 17b shows the spectral content after the interfer- in Fig. 18 show the overlapping spectra for these two
ence has been rejected by time-invariant filtering. The signals. Starting from the top of this figure, each pair
signal distortion caused by rejection of the signal com- of graphs illustrates the result of one filtering and fre-
ponents along with the interference can be completely quency-shifting stage. The sub-band shaded with a
removed by simply shifting replicas of perfectly cor- single set of parallel lines represents spectral compo-
related components from the lower sideband into the nents from one signal that are not corrupted by the
upper sideband, and then properly adjusting their other signal. These components are selected and com-
magnitudes and phases, as suggested in Fig. 17c. A plex-weighted by a filter and then frequency-shifted to
less easily explained example involves two spectrally cancel the components in another sub-band, which is
overlapping linearly modulated signals such as AM, identified by crosshatched shading. The result of this

32 IEEESPMAGAZINE APRIL 1991

cancellation is shown in the second graph (which con- multipath propagation. Straightforward amplification
tains no shading) of each pair. After five such stages, in faded portions of the spectrum using a time-invari-
a full sideband of each of the two QPSK signals has ant filter suffers from the resultant amplification of
been completely separated. In each stage the complex noise. In contrast to this, a periodically time-variant
spectral redundancy between components separated filter can replace the faded spectral components with
by the keying rate is being exploited, and this same stronger highly correlated components from other
spectral redundancy can be used to reconstruct the bands. If these correlated spectral components are
entire [QPSK from either one of its sidebands. weaker than the original components before fading
there will be some noise enhancement when they are
amplified. But the amount of noise enhancement can
be much less than that which would result from the
time-invariant filter, which can only amplify the very
weak faded components.
Detailed studies of the principles of operation and the
mean-squared-error performance of both optimum and
adaptive frequency-shift filters are reported in [ 1 ,
Chapter 14, Secs. A, B ] , 12,Chapter 12, Sec. 12.81, 131,
[14]-[17].

- - If a signal is correlated with time-shifted versions of

itself (i.e., if it is not a white-noise signal), then its past
can be used to predict its future. The higher the degree
of temporal coherence 1 y;(~)l, the better the prediction.
A signal that exhibits cyclostationarity is also corre-
lated with frequency-shifted versions of itself. Conse-
quently, its future can be better predicted if frequency-
shifted versions of its past are also used, so that its
spectral coherence a s well as its temporal coherence
can be exploited. For example, if x(nT,) has cycle fre-
quencies { a 1 . . . . , then we can estimate the
future value x [ ( n+ k)T,] for some k > 0 using a linear
combination of the pasts of the N signals

x , ( n ~ , )= x ( n ~ ,er2raqr'75
) f o r q = 0. ... ,N -- I (98)

That is, the predicted value is given by

M-I Y-l

i[(n + k ) T,] = Z: oC= o h,(rn) x,[(n - rn) T,]

n,=0
(99)

where M is the memory-length of the predictor. The set

of MN prediction coefficients that minimize the time-
averaged (over n ) squared magnitude of the prediction
f --1 f 2 f3
f + -1 error x [ ( n+ k )T,] - x [ ( n+ k)T,] can be shown to be
T2 T2 fully specified by the cyclic correlation functions for
Fig. 18. Illustration of power spectral densities for the N cycle frequencies. Specifically, the set of M N
cochannel-QPSK-signalseparation. The keying rates coefficients { h , ( m ) }is the solution to the set of M N
of the two signals are different and the carrier fre- simultaneous linear equations
quencies also are different. Each QPSK signal has a
positiiie-frequency bandwidth equal to twice its key-
ing rate.

The five cascaded stages of filtering, frequency-shift-

ing, and adding operations can be converted into one f o r n = k , . . . , M + k - 1 a n d p = 0 , . . . , N - 1.
parallel connection of frequency-shifters, each fol- Also, the percent accuracy of prediction is determined
lowed by a filter, simply by using standard system- solely by the temporal coherence functions for the fre-
transformations to move all frequency shifters to the quency translates, which are the discrete-time ana-
input. logues of (29).It can be shown that for each cycle fre-
A final example involves the reduction of the signal quency ayqexploited, there is a corresponding increase
distortion due to frequency-selective fading caused by in the percent accuracy of the prediction.
APRIL 1 9 9 1 IEEESP MAGAZINE 33
 In the same way that time-invariant autoregressive (which includes S:(f) in the earlier section on SCD, by
model-fitting of stationary time-series data is mathe- choosing xl(t)= u(t)and x2(t)= u ( t ) , a s a special case)
matically equivalent to time-invariant linear predic- is zero for all a = ( - ) f l + ( - ) f 2 # 0. (In (105)(*) de-
tion [ l , Chapter 9, Sec. B], it can be shown that notes independent optional conjugation and in (107)
frequency-shift (or multiply-periodic time-variant) au- ( - ) denotes corresponding independent optional mi-
toregressive model-fitting is mathematically equiva- nus sign. That is, depending on the particular spectral
lent to fresquency-shiftlinear prediction. Studies of this moment of interest, one can either include or exclude
problem are reported in [ 181-[24]. any of the optional conjugations and corresponding
minus signs.)
The same is true for cyclostationarity and spectra1
HIG W E R - 0 R DER CYC LOSTATIONAR ITY moments associated with pulse and keying rates of
some severely bandlimited digital signals such as par-
Some types of modulated signals like QPSK and dig- tial-response-coded digital PAM with positive-fre-
ital QAM (quadrature AM) exhibit second-order cyclo- quency pulse-bandwidth less than half the pulse rate.
stationarity associated with the carrier only after the For such signals, exploitation of the higher-order cy-
signal has gone through a nonlinear transformation, clostationarity or spectral redundancy is, in principle,
like a signal squarer or signal quadrupler. In other possible, but little progress h a s yet been made. A not-
words, fourth-order (or higher-order) time-delay prod- able exception is the long established work on syn-
ucts, suc:h as (for real x(t)) chronization of communication receivers, where
higher-order nonlinearities are commonly used to gen-
erate sine waves at harmonics of carrier frequencies to
be used to synchronize local oscillators for demod-
exhibit a spectral line at the fourth (or higher) har- ulation of received signals. Recent progress on this
monic CY of the carrier frequency, problem is reported in [25].
A much more recent application of higher-order cy-
( y T l. . . ,(t) e -i2na‘ >go clostationarity is reported in [26],where new methods
(102)
for identifying the input-output relations of nonlinear
even though the second order time-delay product systems with memory are developed. These methods
make use of second-order, third-order, fourth-order,
and so on, cyclostationary inputs to identify the sec-
ond-order (quadratic),third-order (cubic),fourth-order
(quartic), etc., nonlinear components of the overall
(which iricludes y,(t) x(t + d 2 ) x(t - d 2 ) from the nonlinear system.
earlier section headed “CYCLOSTATIONARITY” as a spe- The foundation for developing the generalization of
cial case) does not exhibit a spectral line at the second the spectral redundancy theory of cylcostationarity
harmonic, from the second order [ 1, Chapter 10-141, 12, Chapter
121 to higher orders is presented in (271and [28].Also,
the foundation for the strict sense theory of cyclosta-
tionarity, based on fraction-of-timeprobability (or tem-
As a result, such signals exhibit nonzero nth order mo- poral probability), in contrast to the wide-sense theory
ments of spectral components only for n 2 4; that is, described in this article, which is based on temporal
the fourth-order (or higher-order) spectral moment moments, is developed in ( 1, Chapter 151 and [29].

CONCLUSION AND FURTHER

READING
Spectral correlation and more general spectral re-
dundancy associated with higher-order spectral mo-
ments are common in manmade signals. Almost all
types of modulated signals encountered in communi-
cations and telemetry systems and also in some con-
trol, radar, and sonar systems exhibit spectral redun-
dancy as a direct result of underlying periodicity
associated with the modulation. With the substantial
increase in the sophistication of signal processors that
can be built by using modern digital technology, newly
developing techniques for exploiting spectral redun-
can be nonzero for some a # 0, even though the sec- dancy promise significant improvements in the capa-
ond order spectral moment function bility of signal processors for extracting information
from corrupted signals for such purposes as detection
and estimation.
This article provides a concise introduction to the rel-
34 IEEE SP MAGAZINE APRIL 1991
atively new spectral-correlation theory of modulated REFERENCES
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Springer for Signal Processing

D.L. Snyder and M.I. Miller, both, Washington University, St. Louis, MO S. Ranka, Syracuse University, Syracuse, M: and S. Sahni, University of
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