OPTIMAL TIME-FREQUENCY KERNELS FOR SPECTRAL ESTIMATION OF LOCALLY
STATIONARY PROCESSES
Patrik Wahlberg and Maria Hansson
Signal Processing Group, Dept. Electroscience, Lund University,
Box 118, SE-221 00 Lund, Sweden, email pw@es.lth.se.
ABSTRACT
This paper investigates the mean square error optimal time-frequency kernel for estimation of the Wigner-Ville spec-trum of a certain class of nonstationary processes. The class of locally stationary processes have a simplified covariance structure which facilitates analysis. We give a formula for the optimal kernel in the ambiguity domain and conditions that are sufficient for the optimal time-frequency kernel to be a continuous function, decaying at infinity.
1. INTRODUCTION
Generalization of spectrum estimation from stationary to nonstationary stochastic processes is a delicate problem. First of all there is a problem of how to define the spectral den-sity of a nonstationary process. It turns out that not all good properties of the stationary spectral density, e.g. negativity and marginal properties, can be preserved for non-stationary processes. The Wigner-Ville spectrum (WVS) has however many nice properties and is a generalization of the stationary case [1].
The WVS of a process can be estimated from realiza-tions of the process using Cohen’s class of time-frequency (TF) representations, which are determined by a TF kernel function. The mean square error optimal solution to this problem has been obtained by Sayeed and Jones [2].
The optimal kernel is sometimes not an ordinary func-tion but a distribufunc-tion, containing Dirac measures. If the kernel is a Dirac measure at the origin (a case studied in [2]), no time-frequency averaging is performed, and the op-timal solution is to leave the time-frequency representation unaltered.
We study optimal kernels for a class of locally stationary processes (LSP). By this term we mean that the covariance function has a certain structure (as opposed to a large num-ber of researchers who use it to mean “almost stationary” in various senses). The covariance function of a LSP is deter-mined by two one-dimensional functions.
This project is supported by the Swedish Research Council.
We derive a formula, valid for LSPs, for the optimal ker-nel in the ambiguity domain. We give conditions on the pair of functions constituting the covariance of a LSP, that are sufficient for the optimal TF kernel to be a continuous func-tion, decaying at infinity. We restrict to circularly symmet-ric Gaussian processes.
2. OPTIMAL ESTIMATION OF WIGNER-VILLE SPECTRA
The autocovariance function for a zero mean stochastic pro-cessX(t)is defined by
r x
(t;s):=EfX(t)X(s)g; (1) and the process is called harmonizable [3] ifr
xhas a Fourier-Stieltjes representation r x (t;s)= 1 2 ZZ e i(t s) dR (;) (2) wheredRis a measure of bounded variation. The class of stationary processes is included in the class of harmoniz-able, since in this casedRis diagonal, and a positive mea-suredR
1then fulfills
dR (;)=dR 1
()Æ( ), accord-ing to Bochner’s theorem.
The WVS ofX(t)[4] is defined by W E (t;!):= Z r x (t+ =2;t =2)e i! d : (3) It exists due to the assumption of harmonizability. It follows immediately that the WVS reduces to the spectral density function when it is restricted to weakly stationary processes, sincer
x
(t;s)=r 1
(t s)is a function oft sonly then. The expected ambiguity function of the process is defined by A E (; ):= Z r x (t+ =2;t =2)e it dt; (4) and its Fourier transform is actuallyWE,
W E (t;!)= 1 ZZ A E (; )e i(t !) dd =
=F 1 1 F 2 fA E g; (5) whereF
1;2denotes Fourier transform in the first and second variable. It is denoted by capital letters and defined by
F(!):=(Ff)(!):= Z f(t)e i!t dt; (6) (F 1 F)(t):= 1 2 Z F(!)e i!t d!: (7)
The signal’s Wigner-Ville distribution (WVD) is defined by the stochastic integral
W(t;!):= Z X(t+ =2)X(t =2)e i! d ; (8) andEfW(t;!)g = W E
(t;!) holds under certain condi-tions on the process’ fourth order moments [2] which we assume fulfilled. The ambiguity process is defined by
A(; ):= Z
X(t+ =2)X(t =2)e it
dt: (9) The WVS can be estimated from process realizations using Cohen’s class of time-frequency representations,
d W E (t;!):=W(t;!)= = 1 2 ZZ W(t t 0 ;! ! 0 )(t 0 ;! 0 )dt 0 d! 0 ; (10) whereis a time-frequency estimation kernel. Sayeed and Jones [2] derived the optimal kernel in the mean square error sense, i.e. minimizing the integrated expected squared error
J()= ZZ Ej d W E (t;!) W E (t;!)j 2 dtd!: (11) In the ambiguity domain the Fourier transform converts the convolution (10) into a multiplication
c A
E
(; )=A(; ) (; ); (12) and the optimal kernel was deduced to be, in the ambiguity domain, opt (; )= jA E (; )j 2 EjA(; )j 2 : (13)
The time-frequency kernel is computed from the ambiguity domain kernel by a Fourier transformation
opt =F 1 F 2 f opt g: (14)
3. LOCALLY STATIONARY PROCESSES
A locally stationary process (LSP) [5] has, per definition, a covariance function determined by two functionsq,rand has the form
r x (t;s)=q t+s 2 r(t s): (15) It can be shown thatqcan be taken to be non-negative, and ris non-negative definite (i.e. the covariance of a station-ary process) [4, 5]. The normalizationr(0) = 1 is used without loss of generality (any other constant can be in-corporated into m). Such processes do exist [5]. For in-stance, ifris the covariance of any stationary process and q(
t+s 2
)is positive definite (a so called exponentially con-vex covariance [5, 6]). Then qis a real analytic function which is the Laplace-Stieltjes transform of a non-negative non-decreasing functionF, q(t)= Z e xt dF(x): (16)
It is exponentially growing either at+1or at 1, which means that the LSP is not harmonizable. But there also exists LSPs where the function q is bounded, and where q(
t+s 2
)is not a covariance (it is not non-negative definite).
Lemma 1. Ifr(t s)andq( t+s
2
)are non-negative def-inite functions, and if
q(t)e 2at 2 = Z e it dQ() (17)
wheredQis a bounded measure, then q 1 (t) = q(t)e 2at 2 andr 1 (t) = r(t)e (a=2)t 2 define a covariancer x (t;s) = q 1 ( t+s 2 )r 1 (t s)of a harmonizable LSP. Example 2. Letq(t)1,r(t)=e (b=2)t 2 )q 1 (t)= e 2at 2 ,r 1 (t)=e a+b 2 t 2 . Thenr 1and q 1define an LSP for anyb0.
LSPs can be seen approximately as stationary processes that have a time varying power described byq. The Cauchy-Schwartz inequality jr x (t;s)j 2 =jEfX(t)X(s)gj 2 EjX(t)j 2 EjX(s)j 2 (18) is necessary for a functionr
x to be the covariance of any stochastic process. For an LSP it is the inequality relation betweenqandr (q(t)) 2 jr( )j 2 q(t+ =2)q(t =2); 8t; : (19) The expected ambiguity function of a locally stationary process is separable (rank one),
A E
Restricting to circularly symmetric Gaussian distributed pro-cesses, the denominator of (13) reduces to
EjA(; )j 2 =jA E (; )j 2 +(F 1 1 F 2 fjA E j 2 g)( ;): (21) For the case of LSPs this givesEjA(; )j
2 = =jQ()j 2 jr( )j 2 +(Fjrj 2 )()(F 1 jQj 2 )( ): (22) The optimal kernel (13) is thus
opt (; )= = jQ()j 2 jr( )j 2 jQ()j 2 jr( )j 2 +(Fjrj 2 )()(F 1 jQj 2 )( ) = (23) = jQ()j 2 jr( )j 2 jQ()j 2 jr( )j 2 +(Fjrj 2 )()(qq)( ~ ) ; (24) whereq( )~ := q( ). It can be shown easily that the functionsFjrj 2 andF 1 jQj 2
are non-negative, which gives 0
opt
(; )1. A special case of LSPs is weakly sta-tionary processes,q( )c>0. Butq~qis not well defined then, so this case can not be treated with formula (24). From (14) and (24) we can directly make the following observa-tions concerning the kernels
opt
(; )and opt
(t;!). The variable support supp (
opt
) = supp(r). If supp(r) =K =compact then(t;!)is analytic in the!variable.
The variable support supp( opt
) = supp(Q). If supp(Q)=K =compact then(t;!)is analytic in thetvariable.
If supp(qq)~ supp(r), then opt
(; ) =1, 2 supp(r)nsupp(qq)~. In particular, if supp(qq)~ supp(r)=Rthen(t;!)will contain Dirac measure in the!variable. If supp(Fjrj 2 )supp(Q), then opt (; )=1, 2 supp(Q)nsupp(Fjrj 2 ). In particular, if supp(Fjrj 2 ) supp(Q) =Rthen(t;!)will contain Dirac mea-sure in thetvariable.
4. LSPS WITH CONTINUOUS OPTIMAL TF KERNELS
Here we give sufficient conditions for the time-frequency kernel
opt
(t;!) for a LSP to be a continuous function. Sayeed and Jones gave a characterization of when
opt
(; )cis a constant, i.e. when opt
(t;!)is a Dirac measure at the origin. This means that no averaging is per-formed in neither the time nor frequency domain by the esti-mator. The follwing (very restrictive) condition is necessary and sufficient for
opt
(; )cto hold.X(t)=X 0
u(t)
whereX
0is an arbitrary random variable with jX
0
j=const a.s., andu(t)is a deterministic function [2].
The pair of functionsq,rshould define a non-negative definite functionr
x
(t;s)in order to be the covariance. The two following conditions are sufficient for
opt 2 C
0 = continuous functions vanishing at infinity.
Theorem 3. If jQj 2 Fjrj 2 2L 1 (R); (25) then opt (t;!) 2 C 0
(t), i.e. it is a continuous function, vanishing at infinity, in thetvariable. If
jrj 2 qq~ 2L 1 (R): (26) then opt (t;!) 2 C 0
(!). i.e. it is a continuous function, vanishing at infinity, in the!variable.
Example 4. Letq(t) 1 )q 1 (t)=e 2at 2 ,r 1 (t)= e (a=2)t 2 r(t). Thenq 1 ~q 1 (t)=2 p =ae at 2 , and jr1j 2 q 1 ~q 1 = 1 2 p a=jr(t)j 2 . According to (26), opt (t;!) 2C 0 (!)if r2L 2 (R). Example 5. Letq(t) 1 )q 1 (t) =e 2at 2 ,r(t) = cos( 0 t) )r 1 (t)=e (a=2)t 2 cos( 0 t). Thenr2=L 2 (R), so opt (t;!)2C 0
(!)is not guaranteed by (26). The opti-mal kernel (24) is opt (; )= cos 2 ( 0 ) cos 2 ( 0 )+4e 2 0 =(4a) cosh( 0 2a ) : (27) For fixed,
optis oscillating as a function of
. Therefore
opt
(t;!) will contain Dirac measure as a function of!. From jQ 1 j 2 Fjr 1 j 2 = p =a 1+ 1 4 e 2 0 =a cosh( 0 =a) 1 cosh( 0 2a ) 2L 1 (R); (28)
we see by theorem 3 that opt
2C 0
(t).
5. LSPS WITH GAUSSIAN CONSTITUENT FUNCTIONS
Here we study the case when both q and r are Gaussian functions. Assume qis the fix Gaussian q( ) = e
2 =2 andr( ) = e c 4 2 =2 . r x
(t;s)is a covariance if and only if c 1. The two conditions (25), (26) of Theorem 1 are fulfilled if and only ifc>1. The limit casec=1results in
1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 c σ 2 Figure 1. Variances 2 t (solid), 2
!(dashed), and asymp-totes (dotted). opt (; ) 1 2 , opt (t;!)= 1 2 Æ(t;!). The kernel (24) is for arbitraryc opt (; )= 1 1+c 1=2 e (1 1 c ) 2 + c 1 4 2 : (29) We would like to study the influence of the parameterc, i.e. the quotient of the variances betweenq andr, on the TF kernelopt. The function
optis almost rank-one for small c 1, and of slowly increasing rank for largerc(ratio of largest eigenvalue to the sum of eigenvalues is0:9for1 c 100). Of this reason, it is reasonable to approximate
optby a rank-one function. We compute the variances of these functions, i.e. the variances off andgwhich fulfill
argmin f;g k opt (; ) f()g( )k 2 : (30) The variances are denoted by
2 and 2 , respectively. After Fourier transformation opt ! opt, ! t, ! !, the variance of
optwill approximately (exactly if
f, g were Gaussians) be 2 t =4 2
in thetdirection and 2 !
=4 2 in the!direction. Figure 1 shows
2 t and 2 ! as a function ofcin the interval1c9.
From the figure we conclude that 2 !
has a close to lin-ear increase. The increase is natural, since increasingc cor-responds tor approaching the covariance of a white pro-cess, implying that optimal estimation involves increased smoothing in the! direction. It can also be seen that
2 t grows slowly for largecand seems to have an upper bound. This can be expected since the functionq, which can be seen as representing the process power as a function of time, has a fixed variance=2.
6. CONCLUSIONS
The covariance function of a locally stationary processes is determined by two functionsq andr. Restricted to cir-cularly symmetric Gaussian distributed processes, we have obtained (i) a formula for LSPs for the optimal spectral es-timation kernel in the ambiguity domain, expressed inqand r, (ii) sufficient conditions onqandrto give a continuous TF kernel, vanishing at infinity, and (iii) various examples of combinations ofqandrleading to distinct types of op-timal kernels. Also, we have studied the case of Gaussian functionsqandrnumerically. It turned out that the result-ing kernel is close to rank-one.
7. REFERENCES
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[2] A. M. Sayeed and D. L. Jones, “Optimal kernels for nonstationary spectral estimation,” IEEE Trans Sig Proc, vol. 43, pp. 478–491, 1995.
[3] M Lo`eve, Probability theory, Van Nostrand, 1963. [4] P. Flandrin, Time-frequency/Time-scale analysis,
Aca-demic Press, 1999.
[5] R. A. Silverman, “Locally stationary random pro-cesses,” IRE Trans. Info. Theory, vol. 3, 1957.
[6] M. Lo`eve, “Fonctions al´eatoires a d´ecomposition or-thogonale exponentielle,” Rev. Sci., vol. 84, pp. 159– 162, 1946.
