in Proc. IEEE-SP Int. Sympos. Time-Frequency Time-Scale Analysis (TFTS-94), Philadelphia (PA), Oct. 1994, pp. 417–420
Copyright IEEE 1994
CORRELATIVE TIME-FREQUENCY ANALYSIS AND
CLASSIFICATION OF NONSTATIONARY RANDOM PROCESSES*
Werner Kozekl
Franz Hlawatsch2
Heinrich Kirchauer2
Uwe Trautwein3
‘NUHAG, Dept. of Mathematics, University of Vienna, A-1090 Vienna, Austria ([email protected])
21NTHFT, Vienna University of Technology, A-1040 Vienna, Austria ([email protected])
3FG EMT, Fak. EI, Ilmenau University of Technology, D-98684 Ilmenau ([email protected])
2 THE E X P E C T E D A M B I G U I T Y F U N C T I O N
Abstract-The
ezpected ambiguity function (EAF) is
shown to provide a generalization of stationary correlation analysis to nonstationary random processes. Important properties of the EAF are discussed, and the EAFs
of special processes are considered. Based on the EAF,
a fundamental classification (underspread/overspread) of
nonstationary processes is introduced and shown to be relevant to timevarying spectral analysis.
The (generalized) ambiguity function (AF) [SI of a signal
x ( t ) is
1 INTRODUCTION
The correlative analysis of stationar processes using the
autocorrelation function (ACF) p , ( ~ f = E { z ( t )x*(t - 7))
is of fundamental importance [l].In particular, the power
spectrum S,(f) is the Fourier transform of the ACF
( Wiener-Khintchine relation)’,
(1)
where a is a realvalued parameter. We define the ezpected
(generalized) ambiguity function (EAF) of a nonstationary
random process s ( t )as the expectation of the AF,
EA?)(^,
e‘
E { A ? ) ( ~ ,VI} .
It follows that the EAF is the Fourier transform of the
a-parameterized ACF
with respect to t ,
For a nonstationary process x ( t ) , the ACF
Jt
r z ( t i , t z ) = E { x ( t i )x * ( t 2 ) }
is a 2-D function [l],and the power spectrum S,(f)is replaced by a time-varying power spectrum T,(t, f ) such as
the physical spectrum, the (generalized) Wagner-Ville spect m m , or the evolutionary spectrum [ 2 , 3, 41. Nonstationary processes exhibit spectral correlation [5] as measured
by the spectral ACF R Z ( f 1 , f 2 =
) E{X(f’)X*(fi)} (assuming existence of the Fourier transform X ( f ) of z ( t ) ) .
We now ask if there exists a joint time-frequency (TF)
correlation function which combines the temporal ACF
r , ( t i , t 2 ) and the spectral ACF R , ( f l , f i in a meaningful way, and which is related to a meanin ul time-varying
spectrum by a Fourier transform (generalization of the
Wiener-Khintchine relation (1)). In this paper, we show
that a satisfactory answer to this question is provided by
the expected ambiguity function (EAF) recently proposed
in [ S , 7 . We demonstrate that the effective support region
of the A F provides useful indications about the type of
nonstationarity inherent in a rocess. We then introduce
a fundamental classification &nderspread/overspread) of
nonstationary processes. For underspread processes, various timevarying spectra (such as the generdized WignerVille spectra and the evolutionary spectra) are shown to be
effectively equivalent. Furthermore, the underspread property is relevant to timevarying spectrum estimation, and
finally, the physical spectrum of an underspread process is
a complete second-order description of the process.
d
k
‘Funding by grant 4913 of the Jubilaumsfondf der Osterreichischen Nationalbank and FWF grant P10012-OPH.
‘Integrals go from -cu to
00
unless specified otherwise.
0-7803-2127-8/94 $4.00 01994 E E E
417
Since the ACF can be recovered from the EAF by inversion of the Fourier transform (2) followed by a simple substitution to obtain r , ( t l , t z ) from r p ) ( t , T ) ,the EAF is a
complete second-order description of the process.
Interpretation as TF Correlation. An intuitively reasonable measure for the statistical correlation between two
T F points ( t l ,f i ) and ( t 2 , f2) 1s
where g 1 ( t ) and g2(t) are two normalized “test signals”
TF-localized about ( t l ,f1) and ( t 2 , fz), respectively (see
Fig. l ( a ) ) . The inner product (2, g i ) = z ( t )g: ( t )dt measures the “content of x ( t ) about the T F point ( t i ,f;).” It is
easily shown that the T F correlation C z ( t l ,f l ; t 2 , f2) can
be derived from the EAF EA?)(.r, v) as
c z ( t ’ , f l ; t 2 , f 2 )= (EA:)( , A ( a )
=
lbp)(~,
v) A k ! i 2 ( 7v)
, d7dv (3)
Ag!g2(~,
where
v) is the cross-AF of the test signals g i ( t )
and g2(t). For signals g1(t) and g 2 ( t ) TF-localized about
( t l ,fi) and ( t 2 , fi), respectively,
U) is known to be
v)concentrated about the points ( f 7 1 2 , f v 1 2 ) in the (7,
plane, where 7 1 2 = tl - t 2 and v 1 2 = f l - f2 are the time la
and frequency lag, respectively. If these points ( f 7 1 2 , *vi27
are well outside the effective support of the EAF (as shown
Ag!g2(~,
and it represents the maximum EAF magnitude,
vt
ft
IEALa)(r,v)l 5 E A y ( 0 , O ) .
0
/ / I E A p ) ( r , v ) l2 d r d v = ~ ~ ~ 2 1 r . ( t i , t 2 ) 1 2 d ~ l d t 2 .
z
T
0
Fig. 1. T F correlation interpretation of the EAF:
(a) TF plane, (b) TF lag plane.
L
r
EA%)(O,v)=
r2(tr
t - T ) dt
R,(f,f-v)df.
Jf
Karhunen-L&ve Representation. The ACF can be
expanded as
r t ( t 1 ,t2)
=
xk
Uk(t1)Ul(t2)
(5)
k
with the nonnegative, realvalued Karhunen-Lodve (KL)
eigenvalues x k and the orthonormal K L eagenfinctaons
Uk(t) 111. It follows from ( 5 ) that the EAF is a weighted
superposition of the AFs of the KL eigenfunctions,
b
(EA?),~ e ) )
lTlvIEAg)(r,v)I2 d r d v = C kA:.
(4)
Ck
v
3 SPECIAL PROCESSES
This is a generalization of the Wiener-Khintchine relation
(1) to nonstationary processes.
It is instructive to study the EAFs of important special
types of nonstationary processes (see Fig. 2).
Properties. We next summarize some basic properties
of the EAF.
The EAFs obtained for different choices of a are equal
up to a phase factor,
Stationary Processes. For a (wide-sense) stationary
process with ACF r 2 ( t l , t 2 ) = p2(tl - t z ) , the EAF effectively reduces to the 1-D ACF p z ( r ) since
~ ~ p z ) (=~e J,2 a~( a)i - a z ) ~ vE A ? l ) ( r , v ) .
This shows that the EAF’s magnitude (which is usually of primary interest) is independent of a.
The EAF’s magnitude satisfies the symmetry property
IEA%)(-.r,-v)I
EAP)(r,v) = p 2 ( 7 ) S(v) .
Note that the EAF of a stationary process is zero for v # 0,
which indicates the absence of spectral correlation.
Nonstationary White Noise. In the dual case of
nonstationary white noise [l] with ACF r 2 ( t i , t 2 ) =
q 2 ( t l )S ( t 1 - t2) ( q z ( t ) 1 0 ) , the EAF is
= IEAP’(7,v)I
.
For a realvalued process z ( t ) ,the EAF’s magnitude is
symmetric with respect to U ,
IEA?)(T,
4
EAP)(r,0) =
k
r ? ) ( t , r ) e - j z n f Td r .
J J EA?)(^, v) e-jZ*(fT-tv)d7 d u .
r,v).
2
Thus, when all KL eigensignals U k ( t ) are well concentrated
in the T F plane (in which case the AFs of the Uk(t) will
be well concentrated about the origin of the r,v)-plane),
then also the EAF will be well concentrated a out the origin of the (7, v)-plane. Conversely, if the KL eigensignals
U k ( t ) are poorly TF-concentrated, this does not necessarily imply poor concentration of the EAF. We also note the
relations
= X k , EA?)(o, 0) =
x k , and
It can be shown that the generalized Wigner-Ville spectrum is essentially the 2-D Fourier transform of the EAF,
EWia)(t,f) =
TFshifting a process by ti,me 70 and frequency vo
such that 5(t) = z ( t - 70)eJ2n”otleaves the EAF invariant up to a phase factor,
EA<^)(^, = eJ2n(v0r--r0v) EA(”)(
2
Relation with Wigner-Ville Spectrum (Generalized Wiener-Khintchine Relation). The generalized
Wigner-Ville spectrum E W p ) ( t ,f ) [3].is a timevarying
power spectrum defined as the expectation of the generalized Wigner distribution Wi”’(t,f ) [9,lo],
erE{ W i ” ) ( t ,f ) } =
U
On the r and v axis, respectively, the EAF is
in Fig. l ( b ) ) , then (EA?),AE!g,) = 0 and, due to (3), also
C 2 ( t l , f 1 ; t 2 , f 2 )= 0 . We have thus shown the followin
important result: If the EAF is zero about a given “TB
lag point” (712,v12), then any two TF points ( t 1 , f i ) and
(t2,f 2 ) with t l -t2 = 7 1 2 and f 1 - f 2 = v 1 2 are unwrrelated.
Conversely, if the TF lag points ( f r 1 2 , f v 1 2 ) are inside the EAF’s effective support so that EAp)(r,v)and
AZ!g2( r ,v) overlap, this does not necessarily imply that
the TF points ( t l , f i ) and (t2, f 2 ) are correlated: since both
EA?)(r, v) and AZ!g2(r,v) are typically oscillatory func) still be zero. Thus, the EAF
tions, ( E A P ) , A g ! g 2may
EA?)(r, v) indicates the potential correlation between T F
points separated by time lag r and frequency lag U.
EWi”)(t,f )
The integral of the EAF’s squared magnitude is
=
IEA?)(~,V)~
EA?)(r,v) = S(T) Q 2 ( v ) ,
where Q 2 ( v )is the Fourier transform of q 2 ( t ) . Note that
the spread of Q 2 ( v )grows with increasing temporal vari-
for z ( t ) E R.
The EAF’s value at the origin of the (r,v)-plane
equals the expected energy of the process,
ations of the average intensity function q 2 ( t ) (i.e., increasing nonstationarity of the process), and characterizes the
amount of spectral correlation. The EAF is zero for r # 0,
which indicates the absence of temporal correlation.
E A ? ) ( O , 0) = E{ l l z ( t ) 1 2 d t } = l r , ( t , t ) dt ,
418
0 for (v > vmax” is a limitation of the spectral correlation wi th or, equivalently, of the degree of nonstationarity. The underspread property combines these two properties but allows to exchange one property for the other. For
example, a quasistationary process (with small vm,) and
a “nearly white” process (with small 7”) may both be
underspread. Some important consequences of the underspread property are discussed in the following.
d
‘5
‘5
Equivalence of Time-Varying Spectra. For underspread processes, many different time-varying spectra are
essentially equivalent. First, the deviation between two different generalized Wagner-Ville spectra EWJ”)(t,f ) can be
bounded (for U, < 2 and la1 -021 < 1) as [ll]
‘5
f)l
I E W p ’ ) ( t , f )- E W p 2 ) ( t ,
<
Iai-az)
with C = $ Ckx k where x k are the KL eigenvalues of z ( t ) .
Thus, if U, << 1, then EWL”’)(t,f) x E W e z ) ( t ,f) for all
T F points (t,f). A similar bound can be derived [ll]for
the deviation between a generalized Wigner-Ville spectrum
EWi”)(t,f) and the evolutionary spectrum ES,(t,f) [3, 41
(defined by self-adjoint factorization of the ACF [ll]):For
U, < 4 and la1 < 1/2, there is
Fig. 2. Effective EAF support for special rocesses:
(a) stationary, (b) nonstationary white, (cfstationory
white, (d) quasistationary, (e) finite correlation width,
(f) underspread.
Stationary White Noise. In the case of stationary
white noise with ACF r , ( t l , t z ) = q, b(tl - t z ) (the intersection of the two last-mentioned cases), the EAF indicates
the absence of temporal or spectral correlation:
f)l
(ES,(t,f ) - EW,(”)(t,
+
< D
&r
where D = (U, a/32 ILYJ
b) with a = (Ek
and
b = Ckx k . Thus, ifu, << 1,then a z ( t ,f) E! E W ; ” ) ( t , f ) .
Fig. 3 shows the similarity of the Wigner-Ville spectrum
E W i o ) ( t ,f), the Rihaczek spectrum EW;’’’)(t, f), and the
evolutionary spectrum E&(t,f) in the case of a specific
underspread process.
EA?)(r, v ) = q x 6(r)6(v)
Cyclostationary Processes. The ACF of a cyclostationary process with period T satisfies r,(tl +T, tZ+T) =
rx(tl,tz). It follows that the EAF is ideally concentrated
at equally spaced frequency lags,
Estimation of Time-Varying Spectra. The EAF
of an underspread process is restricted to a rectangle
[--7inax,~m=] x [-vmax,~max]
with area U, < 1. Due
to (4), the generalized Wigner-Ville spectrum EW,(“)(t,f)
is a 2-D lowpass function that is uniquely characterized
by its samples taken on a rectangular grid with time period T = 1/(2vmax) and frequency period F = 1/(27m,x)
[12]. The underspread property then implies T F =
1/(4 Tmax vm=) = l/u, > 1. This property is of relevance
to the problem of estimating the generalized Wigner-Ville
spectrum E W e ) ( t ,f) from a single realization of the process z ( t ) . Since EW,(Q)(t,
f) is completely specified by its
samples wkl = E W , ( “ ) ( ~ TI ,F ) , the spectrum estimation
problem reduces to the problem of estimating the samples
wkl. Let us assume that the process z ( t ) is bandlimited
with bandwidth B,, and can thus be represented by its
samples z, = z(nT,) with T, = 1/(2 B,). If a realization of
x ( t ) is observed over a time interval of length Tabs = N T,,
then our data set comprises N samples x,,. On the other
hand, the number M of parameters to be estimated equals
the number of samples W k l falling into the rectangular T F
region with time length Tabs and frequency length 2B,,
Processes with Finite Correlation Width. If, for
all t , the ACF r$l/’)(t,T) = r,(t,t-r) is zero for 171 >
Tmm, then also E A p / 2 ) ( r , v )= F t - , v r ~ l ” ) ( t , rwill
) be
zero for 171 > rmax. Since the EAF with arbitrary Q equals
EA:”’)(r,v) up to a phase factor, the same will be true
also for EAFs with LY # 1/2. Thus, the EAF’s spread in
the r-direction indicates the correlation width of a process.
Quasistationary Processes. The ACF r$’/’)(t,r ) =
r,(t,t - r ) of a quasistationary process is slowly vary-
ing with respect to t [l]. This entails small spread of
EAP”)(r, v) = F+,v r$’”)(t,T ) in the v-direction. Since
any arbitrary EAF equals EA:/’)(r, v) up to a phase, the
EAF’s spread in the v-direction is a “spectral correlation
width” indicating the “degree of nonstationarity.”
4 UNDERSPREAD PROCESSES
Definition. Let [-Tmax, ~ m a x ]x [-vmax, ~ m a x ]be the
smallest rectangle (centered at the origin of the ( 7 , ~ ) plane) which contains the effective support of the EAF, i.e.,
E A F ) ( r ,v) x 0 for 17) > Tmax or JvJ> vmax. Furthermore,
let U= = 4 rmax Vmax denote the area of this rectangle. We
call a process underspread if U= < 1 (see Fig. 2(f)) and
overspread if U, > 1 [6]. This definition is independent of
a since the support of EAP)(r,v) is independent of a.
The property “EA?)(T,v) x 0 for IT^ > Tmax” is a limitaT ,M
tion of the temporal correlation width, and “ E ~ ? ) (v)
419
Fig. 3. Time-varying spectra of an underspread process:
(a) Wigner-Valle spectrum, (b) Rihaczek spectrum, (c)
evolutionary spectrum. The time length is 128 samples.
For robust estimation, the number of parameters to be estimated must be smaller than the size of our data set, i.e.
M < N which implies U, < 1. Thus, the underspread p r o p
erty uz < 1 is a necessary condition for robust estimation
of the generalized Wigner-Ville spectrum.
Sufflciency of the Physical Spectrum. The physical
spectrum [3]
I/
&‘“)(t,f) = E{
z(t‘)w*(t‘-t)e-i2*ft’dt’
t’
is defined as the expectation of the spectrogram of z ( t )
with analysis window w(t). It can be shown that
PSLw)(t,
f)
=//
EWp)(t‘,f’) Wia)*(t‘-t, f’- f) dt‘df’.
t’
f’
$3
In general, the smoothing described by (6) makes it impossible to recover E W , ( = ) ( t , f )(and, in turn, the ACF
r , ( t l , t 2 ) ) from PSiw)(t,f), which means that the physical spectrum is not a complete second-order description
f) requires a
of the process. Indeed, recovering EW,(Q)(t,
deconvolution to invert (6). Taking the 2-D Fourier trans-
-
(w)
form of (6) yields PS, (T,u)= EA?)(T,u) A?)*(T,u)
-(w)
where PS, ( 7 , ~ =
) Ft+vF~~,{PS~w)(t,f)}.
The deconvolution then corresponds to performing the divi--(U)
sion E ~ ? ) ( T , v )= PS, ( ~ , u ) / A p ) * ( ~which
, u ) is illconditioned if A?)(T,u) s~ 0 for T F lag points ( 7 , ~E) S,
where S denotes the effectivesupport of EA?)(T, U).
In the case of an underspread process, the EAF’s support
S is contained in a rectangle with area U, < 1. Here,
one can always find windows w(t) whose AF is sufficiently
bounded away from zero for (7,U ) E S (see Fig. 4),
-(w)
The EAF can then be recovered from PS, ( 7 , ~by
) per(U)
forming the division EA?)(T, U) = PS, ( 7 ,U ) / A?)*(T,U )
for ( T , U ) where IA?)(T, U) 2 e, and setting EA?)(T, U ) =
0 elsewhere. From the EA , the ACF can finally be derived
as explained in Section 2 . Thus, the physical spectrum of
an underspread process is a complete second-order descri
tion of the process, provided that the analysis window w&
used in PSLw’(t,f) is matched to the process in the sense
that the E A F s effective support is contained in the effective support of the window’s AF. For example, a long
-
A
Fig.
4.
The effective support of E ~ ? ) ( T
U ), is contained in
the eflectiwe support of A ? ) ( T , U).
420
window will be suited to a quasistationary process whereas
a short window will be suited to a “near1y white” process. Note that the underspread property guarantees the
existence of “matched” windows. Techniques for optimum
window matching can be found in [6,7,111.
5 CONCLUSIONS
The expected ambiguit function (EAF) is a useful timefrequency correlation Knction of nonstationary processes
which indicates both the temporal and spectral correlation
widths. The EAF allows the definition of the class of underspread processes for which various timevarying spectra
(the generalized Wigner-Ville spectra and the evolutionary spectrum) are essentially equivalent. The underspread
property is a necessary condition for robust estimation of
the generalized Wigner-Ville spectrum. For underspread
processes, the physical spectrum (with suitable analysis
window) is a complete second-order description.
The underspread property is important in man other respects as well, such as nonstationary Wiener &ers [13],
the Gabor expansion [7],and the short-time Fourier transform [SI. A class of time-varying spectrum estimators for
underspread processes is studied in [14].
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