r,v
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 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 ,
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]. References [l] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd edition. McGraw-Hill, 1991. [2] W. Martin and P. Flandrin,“Wi er-Ville s ectral analysis of nonstationary processes,” IE% %ns. jmust., Speech, Signal Processing, vol. 33,no. 6,Dec. 1985,pp. 1461-1470. [3] P. Flandrin, “Timedependent spectra for nonstationary stochastic processes,” in Time and Frequency Representation of Signals and Systems, eds. G. Longo and B. Picinbono. Wien: Springer, 1989,pp. 69-124. [4] M. Priestley, Spectral Analysis and Time Series, Vol. 11. London: Academic Press, 1981. [5] W. A. Gardner, Statistical Spectral Analysis: A Nonpmbabilistic Theory. New Jersey: Prentice Hall, 1988. [6] W. Kozek, “Optimally Karhunen-Loevelike STFT expansion of nonstationary processes,” Pmc. IEEE ICASSP-93, Minneapolis, MN, April 1993,vol. 4, pp. 428-431. [7] W. Kozek, “Matched generalized Gabor expansion of nonstationary processes,” Pm. 27th Asilomar Conf., Pacific Grove, CA, Nov. 1993, pp. 499-503. [8] F. Hlawatsch, “Duality and classification of bilinear timefrequency signal representations,” IEEE %ns. Signal Pmcessing, vol. 39, no. 7,July 1991,pp. 1564-1574. [9] F. Hlawatsch and R. Urbanke, “Bilinear time-frequencx representation of si nals: The shift-scale invariant class, IEEE %ns. Signa? Processing, vol. 42, no. 2, Feb. 1994, pp. 357-366. [lo] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, “The Wigner distribution-A tool for time-frequency signal analysis,” Part 111, Philips J. Res., vol. 35, pp. 372-389,1980. [ll] W. Kozek, “Matched We 1 Heisenberg expansions of nonstationary environments,”Ph.D. Thesis, Vienna University of Technology, in preparation. [12] W. Kozek, “On the generalized Weyl correspondence and its application to time-frequency analysis of linear time-varying s stems,” Proc. IEEE Int. Symp. on TimeFrequency a n i Time-Scale Analysis, Victoria, Canada, Oct. 1992,pp. 167-170. [13] H. Kirchauer, F. Hlawatsch, and W. Kozek, “Time-frequency formulation and desi n of nonstationary Wiener filters,” submitted to IEEE I8ASSP-95. [14]W. Kozek and K. Ftiedel, “Quadratic timevarying spectral estimation for underspread processes,” Pmc. IEEE Int. S mp. on Time-Frequency and Time-Scale Analysis, Phildelphia, PA, Oct. 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 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 ,
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]. References [l] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd edition. McGraw-Hill, 1991. [2] W. Martin and P. Flandrin,“Wi er-Ville s ectral analysis of nonstationary processes,” IE% %ns. jmust., Speech, Signal Processing, vol. 33,no. 6,Dec. 1985,pp. 1461-1470. [3] P. Flandrin, “Timedependent spectra for nonstationary stochastic processes,” in Time and Frequency Representation of Signals and Systems, eds. G. Longo and B. Picinbono. Wien: Springer, 1989,pp. 69-124. [4] M. Priestley, Spectral Analysis and Time Series, Vol. 11. London: Academic Press, 1981. [5] W. A. Gardner, Statistical Spectral Analysis: A Nonpmbabilistic Theory. New Jersey: Prentice Hall, 1988. [6] W. Kozek, “Optimally Karhunen-Loevelike STFT expansion of nonstationary processes,” Pmc. IEEE ICASSP-93, Minneapolis, MN, April 1993,vol. 4, pp. 428-431. [7] W. Kozek, “Matched generalized Gabor expansion of nonstationary processes,” Pm. 27th Asilomar Conf., Pacific Grove, CA, Nov. 1993, pp. 499-503. [8] F. Hlawatsch, “Duality and classification of bilinear timefrequency signal representations,” IEEE %ns. Signal Pmcessing, vol. 39, no. 7,July 1991,pp. 1564-1574. [9] F. Hlawatsch and R. Urbanke, “Bilinear time-frequencx representation of si nals: The shift-scale invariant class, IEEE %ns. Signa? Processing, vol. 42, no. 2, Feb. 1994, pp. 357-366. [lo] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, “The Wigner distribution-A tool for time-frequency signal analysis,” Part 111, Philips J. Res., vol. 35, pp. 372-389,1980. [ll] W. Kozek, “Matched We 1 Heisenberg expansions of nonstationary environments,”Ph.D. Thesis, Vienna University of Technology, in preparation. [12] W. Kozek, “On the generalized Weyl correspondence and its application to time-frequency analysis of linear time-varying s stems,” Proc. IEEE Int. Symp. on TimeFrequency a n i Time-Scale Analysis, Victoria, Canada, Oct. 1992,pp. 167-170. [13] H. Kirchauer, F. Hlawatsch, and W. Kozek, “Time-frequency formulation and desi n of nonstationary Wiener filters,” submitted to IEEE I8ASSP-95. [14]W. Kozek and K. Ftiedel, “Quadratic timevarying spectral estimation for underspread processes,” Pmc. IEEE Int. S mp. on Time-Frequency and Time-Scale Analysis, Phildelphia, PA, Oct. 1994.
