Recursive filtering and smoothing for reciprocal

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where U L + ~is the right singular vector of G(kl, k2 ) associated with the ( L 1)th singular value. The choice of kl and k2 may affect the performance of the algorithm. However, it is not difficult to combine the cyclic statistics for some or all (distinct) k l ’ s and k?’s. In doing so, the criterion in (5 1 ) can be modified by replacing G(ICl, k2 ) by

+

G = [ G t ( 0 . 1 ) , . . . , G = 1 ( 0 , T - l ) , G = 1 (,1 . 2 ) . . .. - 1), . . . , ( T - 2.

-

l)lt.

(53)

Such modification may improve the performance.

V. CONCLUDING REMARKS We established several different necessary and sufficient conditions for the identifiability of a possibly nonminimum phase channel from its output cyclic autocorrelation functions. In comparison to the timedomain approach presented earlier in [ 121, the frequency-domain approach to the channel identification problem gives more insight into the issue of channel identifiability. It also provides the basis for new channel identification algorithms.

Recursive Filtering and Smoothing for Reciprocal Gaussian Processes-Pinned Boundary Case E. Baccarelli, R. Cusani, Member, IEEE, and G. Di Blasio, Associate Member, IEEE Abstruct- The least square estimation problem for pinned-to-zero discrete-index reciprocal Gaussian processes in additive white noise is solved, thus completing and extending some previous results available in the literature. In particular, following the innovations approach a (finite) set of recursive equations is obtained for the filter and for the three standard classes of smoothers (fixed-point, fixed-interval, fixed-lag). Recursive expressions for the mean square performance of the proposed estimators are also given. Index Terms-Reciprocal timators*

processes, innovations method, recursive es-

I. INTRODUCTION A discrete-index multivariate real reciprocal Gaussian random process (RGP) { . r ( k ) E R”, A4 5 k 5 N } defined on an assigned probability space (0, A, P) is described by the well-known selfadjoint second-order noncausal difference model [ 1 ]

i!o(k)r(k) M + ( k ) s ( k + 1)- rVr,’(k - l ) s ( k - 1) = e ( k ) ,

ACKNOWLEDGMENT

M + 1 5 k 5 JV - 1 (1) The authors wish to thank H. Liu at the University of Texas at Austin for his comments.

where { M O( k ) } ,{ M+ ( k ) } are deterministic sequences of matrices. The conjugate process

71

x

11

REFERENCES M. Goursat, A. Benveniste, and G. Ruget, “Robust identification of a nonminimum phase system: Blind adjustment of a linear equalizer in data communication,” IEEE Trans. Automat. Contr., vol. AC-25, no. 6, pp. 385-399, June 1980. R. R. Bitmead, S. Y. Kung, B. D. 0. Anderson, and T.Kailath, “Greatest common divisors via generalized Sylvester and Bezout matrices,” IEEE Trans. Automat Contr., vol. AC-23, no. 6, pp. 1043-1047, Dec. 1978. G. D. Fomey, “Minimal bases of rational vector spaces, with applications to multivariable linear systems,” SIAM J. Contr., vol. 13, no. 3, pp. 493-520, May 1975. W. Gardner, “A new method of channel identification,” IEEE Trans. Commun., vol. 39, no. 6, pp. 813-817, June 1991. W. A. Gardner and L. E. Franks, “Characterization of cyclostationary random signal processes,” IEEE Trans. Inform. Theory, vol. IT-2I , no. 1, pp. 4-14, Jan. 1975. B. Hassibi, personal communication, Aug. 1992. T.Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. S. Y. Kung, T. Kailath, and M. Morf, “A generalized resultant matrix for polynomial matrices,” in Proc. IEEE Con$ on Decision and Control, 1976, pp. 892-895. Y. Li and 2. Ding, “Blind channel identification based on secondorder cyclostationary statistics,” in Proc. IEEE Int. Con$ on Acoustics, Speech, and Signal Processing, vol. IV (Minneapolis, MN, Apr. 1993), pp. 81-84. L. Tong, “Blind channel identification using cyclic spectra,” in Proc. 26th Con$ on Information Sciences and Systems (Princeton, NJ, Mar. 1992), pp. 711-716. L. Tong, G. Xu, and T. Kailath, “A new approach to blind identification and equalization of multipath channels,” in 25th Asilomar Con$ (Pacific Grove, CA, Nov. 1991). -, “Blind identification and equalization based on second-order statistics: A time domain approach,” IEEE Trans. Inform. Theoiy, vol. 40, no. 2, Mar. 1994. J. K. Tugnait, “On blind identifiability of multipath channels using fractional sampling and second-order cyclostationary statistics,” in Proc. 1993 Global Commun. (Houston, TX, Dec. 1993).

is bi-orthogonal to { ~ ( k ) }that , is,

E ( s ( k ) J ( s ) }= I h ( k ,s ) . and its covariance matrix is given by 11, eqs. (3.4a), (3.4b)l. Boundary random values with singular probability measures (pinned-to-zero) are considered in this paper, i.e., s ( M ) = . E ( - % ~ ) = 0, P-as. It is assumed that the sequence {z(k ) } is amplitude-modulated and then sequentially transmitted through a noisy communication channel for increasing values of the index k . As a consequence, the observed sequence { y ( k ) E R‘} is modeled as

y(k;)= r ( k ) s ( l ; )

+ u l ( k ) , nr + 1 5 I; 5 N - 1

(2)

where { w ( k ) E R‘} is an additive white Gaussian noise (AWGN) process, independent of { x ( k ) } , with covariance matrices { R(k ) } : { r(k ) } are known and uniformly limited T x 11 matrices. It is also assumed that the matrices { M O( k ) } and { R( k ) } are nonsingular. In this correspondence, the problem of estimating a pinned-to-zero RGP in AWGN is addressed. Pinned-to-zero boundary conditions represent an important subclass of the more general Dirichlet boundary conditions [l]. In this case, an RGP does not always admit a wellbehaved first-order causal white innovations representation over the whole parameter space (e.g., including both ending points) giving a Markov version (in the stochastic sense) of the assigned process. As a consequence, the pertaining estimation problems cannot be directly Manuscript received June 29, 1993; revised July 4, 1994.The material in this paper was presented at the IEEE Intemational Symposium on Information Theory, Trondheim, Norway, June 27-July I , 1994. The authors are with the INFOCOM Department, University of Rome ‘La Sapienza,’ 00184 Rome, Italy. IEEE Log Number 9407260.

0018-9448/95$04.00 0 1995 IEEE

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k

Fig. 1. MSE of the filter (continuous line) and of the fixed-interval smoother (dashed line) for an univariate homogeneous RGP with A4 = 1, N = 21, MO = 3 . 3 3 , M+ = 1 in ( 1 ) and r ( k ) = 1 , r = 1 in (2). The variance of the observation noise is R ( k ) = 0.2 ( a ) , 0.5 (b), 3 (c). Simulation results (+ for the filter, x for the smoother) based on 1000 independent trials are also added.

solved by standard recursive algorithms which are reported in the literature for Gauss-Markov processes (GMP’s) [ 2 ] . Furthermore, the solutions of the estimation problems derived for pinned-to-zero boundary conditions could be extended to solve the corresponding problems for general Dirichlet boundary conditions. Some preliminary results about these more general problems are also reported in

Starting from the integral form of the difference model in (1) (reported, for example, in [ l , eq. (3.13)]) and on the basis of ( 2 ) , after some algebra the one-step predictions sequence in (3) can be recursively calculated as

.?(k I k - 1) = M t l ( k ) [AtfT(k- 1) + M + ( k ) H ( k - 1,AT)]

PI. The solution for the fixed-interval smoothing (often considered as the natural problem for RGP’s) has been given in [l], [3]. However, to the authors’ knowledge no explicit formulas are available for its performance neither for the filter nor for the fixed-point and fixed-lag smoothers. Regarding this point, we observe that in some real-time applications the delay associated to the fixed-interval smoother cannot be tolerated so that the filter may constitute an effective alternative if the resulting accuracy loss is small enough. The aim of this correspondence is then to give recursive expressions for the filter, the smoothers, and their mean square performance in a unified approach based on the innovations theory.

11. THEFILTER In order to derive the minimum mean square error (MMSE) filtering formulas for the pinned RGP’s, let us denote by n E-’={y(M l),. . . , y(k)} the observed sequence until step a k ; by Fk=u{Yk}), k 2 A4 1, the u-algebra generated by Y k (with

+

+

I

A

I F k P m } , the filtered sequence (for m = 0) and the one-step predictions sequence (for m = 1); by S ( k 1 k - m ) , with m = 0 , 1 , their corresponding error covariance matrices. On the basis of the observation model in ( 2 ) and using the well-known Martingale Difference representation theorem, the filter equation is given by F M ~ { @ , R } )by ; .?(k

k - m)=E{X(k)

iya I k ) = iqlc I ,t - 1 ) + ~

.i(k- 1 I k - l ) ,

H ( k - 1,N ) 2 G(lc + 1,k ; k - l,X)M,’(k

M

S(k I k - 1) =Mi1(k)

-

+ M;l(k){C(k)s(k- 1 1 k - l)CT(k)

+ M + ( k ) G ( k + 1, k + 1; k - 1,N ) M , T ( k ) . 11 - S ( k , N

-

l)]

+ 2Vs(k,k - 1; N ) } M ; l ( k ) ,

1. (4)

(7)

with

C ( k ) ) M F ( k - 1) + M + ( k ) H ( k - l , l v ) , M+l

+ R ( ~ ) I - ~ , and

+ 1 5 k < iv

(6)

and { G ( T ,s; k - 1,N ) , ( k - 1) 5 T 5 N , k 5 s 5 N - l} is the Green’s function associated to (1) over the subinterval [ k - 1,N] of the whole assigned index-space [ M ,N] (it is defined here according to [ l , eq. (3.10)]). It is worthwile observing that the recursive computation of the one-step prediction sequence in ( 5 ) requires the use of the integral form of the model in (1) (and then, the use of the Green’s functions of (6)), so that the effect of the assigned boundary values is taken into account in the solution of the problem. This is a direct consequence of the noncausal structure of the model in (1). In order to express the gains sequence of (4) in a recursive form, first of all the second-order statistics of the estimator must be calculated. Regarding the covariance matrix of the one-step prediction error, from (1)-(5) and from the bi-orthogonality property mentioned in Section I we obtain

From the assumptions of Section I it follows that the gains sequence { G ( k ) }in (3) is necessarily Fk-predictable, so that it can be expressed as

G(x.)= [s(kI k - i ) r T ( k ) ] [ r ( k ) s ( k I L - i ) r T ( k )

- I),

M + l < k < N - l

M + l < k I N - l

(3)

(5)

where

( k ) [ ~( kr () k ) i ( k I IC - 1)1, M+l