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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-32, NO. 2, MARCH 1986 303 one can expect to identify pOI. For suppose dn,, = poi&, dt + Wiener filters. A general model of a suboptimal Wiener filter over a group dm,, 7 i = 1,2, and let n, = n,, + nzf. Then eventually all the is defined, which includes, as special cases, the known filters based on the observations of n, are almost entirely those of rz2,, which does discrete Fourier transform (DFT) in the case of a cyclic group and the not yield much information about pOl. Indeed C now becomes Walsh-Hadamard transform (WHT) in the case of a dyadic group. Statis- i 7 . Similarly, if lim, _ m+l,/+ZL = c E (0, cc), one can only tical and computational performances of various group filters are investi- [ 1 gated. The cyclic and the dyadic group filters are known to be computa- expect to identify cpOl + p02. tionally the best ones among all the group filters. However, they are not It might be difficult to check assumptions 2 and 3 of Theorem always the best ones statistically and other (not necessarily Abelian) group 2. Assumption 1 will in general be easy to verify. A sufficient filters are studied. Results are compared with those for the cyclic group condition for assumptions 1 and 2 to hold is, for example, filters (DFT), and the general problem of selecting the best group filter is +t - to (a > - l/2). A necessary condition for assumption 3 is posed. That problem is solved numerically for small-size signals (I 64) for that the eigenvalues of &#@~ ds are of the same order as t + cc. the first-order Markov process and random sine wave corrupted by white Assumption 3 is similar to the notion of persistence of excitation noise. For the first-order Markov process with the covariance matrix that appears in identification problems for ARMAX systems. II”,” = pi’-‘1 as p increases, the use of various non-Abelian groups Condition 3 of the theorem appears as a technical condition, results in improved statistical performance of the filter as compared to the necessary for the proof of Theorem 2. It seems,however, to be DFT. Similarly, for the random sine wave with covariance matrix B(“,‘) = related to cos X (s - I) as h decreases, non-Abelian groups result in a better statisti- 1 t WJZ- cal performance of the filter than the DFT does. However, that is lim ~ - dr > 0 almost surely (6.1) compensated for by the increased number of computations to perform the f ’ m P,T4r,J 0 Porps filtering. where Qr = /&& ds. Here (6.1) has an appealing interpretation. To see this, define a normalized version of (3.1) by I. INTRODUCTION In recent years interest has grown in utilizing orthogonal H,(P) = j&J,(P). (6.2) transforms in digital signal processing in order to improve statis- t tical or computational performance to permit a trade-off between Then minimization of H,( .) is equivalent with minimization of these two criteria by utilizing a certain chosen orthogonal trans- J,(.). One can easily check that for large t H/(p)l,=,, can be form PI, [31, 171,1141. approximated by (6.1). Hence (6.1) says that for t --f cc pO is A common quality shared by many fast transforms which indeed a minimum point of H,(e). enables their classification (see, e.g., [4], [5]) is that they can be We have not discussed the asymptotic distribution of the represented as Kronecker products of matrices which may or may estimates j, generated by (3.2) and (3.3). This issue will be not be sparse or structured. By virtue of this Kronecker product addressed in another publication. representation new transforms can be generated from old ones simply by using the Kronecker product. In a given problem, such REFERENCES as Wiener filtering with given statistical characteristics of a signal [l] P. Brt!maud, Point Processesund Queues. New York: Springer, 1981. and noise, one can select a computationally good approximating [2] C. Dellacherie and P.A. Meyer, Prohahilitk et potentiel. Paris, France: transform to a statistically optimal transform and the selection Hermann. 1980, chs. V-VIII. [3] Yu. Lin’kov, “Estimates of parameters of counting processes,” Prob. can be done out of the family of known fast transforms with a Inform. Trunsmission, vol. 18, pp. 63-16, 1982. Kronecker product representation. (See [lo], where a good refer- [4] Y. Ogata, “The asymptotic behavior of maximum likelihood, estimators ence list can be found, and [l].) for stationary point processes,” Ann. Inst. Statist. Moth., vol. 30, pt. A, Another approach to the same problem of Wiener filtering pp. 243-261, 1978. construct [5] H. Robbins and D. Siegmund, “A convergence theorem for nonnegative would be to a computationally good approximation to a almost supermartingales and some applications, ” in Optrmrzing Methods given statistically optimal transform. A possibility of solving that in Stutisttcs,J.S. Rustagi, Ed. New York; Academic, 1971, pp. 233-256. problem analytically for classesof signals defined by their covari- [6] J.H. van Schuppen, “Convergence results for continuous-time adaptive ante matrices (e.g., for signals whose covariance matrices are stochastic filtering algorithms,” J. M&h. Anul, A&., vol. 96, pp. 209-225. 1983. Toeplitz) has been pointed out in [12], [18], [19], [28] and this [7] V. Solo, “The convergence of AML,” IEEE Trans. Automut. Contr., vol. approach deserves further elaboration. Yet another approach is AC-24. pp. 958-962, 1979. to construct experimentally a computationally good approximat- ing transform to a transform which is known to be good statisti- cally. For example, the discrete cosine transform (DCT) has a nearly optimal statistical performance for highly correlated Markov signals (see [24]), and it has recently been approximated Statistical and Computational Performance of a Class by computationally convenient transforms [8]. Here even for of Generalized Wiener Filters small n (up to 32 vector-components of a signal) the problem is difficult, involves tedious trial and error procedures, and requires MARK G. KARPOVSKY, SENIOR MEMBER, IEEE, artistry rather than clear-cut methods. Another disadvantage is AND LAZAR A. TRACHTENBERG, MEMBER, IEEE that a success with approximating one transform (as DCT) for some n (say n = 16, 32) cannot be generalized to be used to Ahstrac~-A class of suboptimal Wiener filters is considered, and their approximate other transforms [8], [26]. computational and statistical performances (and the trade-off between the A number of researchers[15], [l], [3], [ll], [17] have selected a two) are studied and compared with those for known classes of suboptimal family of fast transforms which are group theoretic by their nature; i.e., they are based on group characters of corresponding Manuscript received October 31, 1983; revised August 8, 1985. Abelian groups: examples are the discrete Fourier transform This work was supported in part by the National Science Foundation under (DFT) in the case of a cyclic group and the Walsh-Hadamard Grants DCR-8317763 and ECS-8512748. This work was presented in part at the International Workshop on Fault Detection and Spectral Techniques, transform (WHT) (or simply the Walsh transform) in the case of Boston. MA, October 12-14, 1983. a dyadic group [l], [3], [ll], [15]-[17], [27]. The use of non-Abelian M. G. Karpovsky is with the College of Engineering, Boston University, groups was discussed in [13], [20]. Boston, MA 02215. These transforms exist for any number n, are computed ana- L. A. Trachtenberg is with the Department of Electrical and Computer Engineering, Drexel University, Philadelphia, PA 19104. lytically by formulas, and possess Kronecker product represen- IEEE Log Number 8406619. tations (which guarantee speed of computation for nonprime
0018-9448/86/0300-0303$01.00 01986 IEEE 304 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-32, NO. 2, MARCH 1986
n’s). Also, their statistical performances can be computed by In that case formulas, and as will be shown later, in many cases they compare favorably with other transforms. The cyclic and dyadic groups Do,,= l/n tr(4, - W,,BfJ. (3) are not always the best to use [3], [7], and attempts have been The case of uncorrelated signal and noise has been discussed in made to develop a general theory of approximation of systems P51, [31, v71. and signals by those over groups. Elements of this theory as The generalized Wiener filter utilizes a unitary transform which presented in [13], [14] embrace problems in digital signal is represented by the matrix T in Fig. 1 (see [15], [3]). Do f in (3) processing such as filtering, encoding, data compression, etc. is independent of T. The direct Wiener filtering (when f- is the (e.g., see [3]). identity matrix) is fastest as it requires at most n2 multiplications In the present work we apply methods of this theory to the and additions. The Karhunen-Loeve transform which uses the T , well studied problem of Wiener filtering. The applications of that diagonalizes W,,, is the slowest as it requires on the order of representations of noncommutative as well as Abelian groups are 2 n2 + n operations to perform the filtering W,,,f = the subject of our investigation. The results are compared with T-‘(TWo,,Tp’)Tf, where TW&,,T-’ is the diagonal matrix of those in the current literature, and it will be shown (see Sections eigenvalues of W,,, and multiplications by vectors are being II and III) that the DFT and the WHT are often not the best performed from the right. group transforms to use for suboptimal Wiener filtering. Using the examples of random sine and first-order Markov processes, we shall show that the use of a noncommutative group may be more advantageous than the use of a commutative group because they yield a better approximation for a given speed (see also [13, T-lW Tf sec. 51). OPf The correspondence comprises three sections. The suboptimal Wiener filtering problem is treated in Section II as that of Fig. 1. Generalized Wiener filter. constructing the best group Wiener filter approximation to a given classical Wiener filter over a given group. The general solution of this problem is presented and previously studied cases However, the mathematically optimal Dopt in (3) is never of cyclic and dyadic group filters are deduced. The results of achievable in reality (e.g., because of roundoff). We must there- computer experiments are given and analyzed in Section III for fore accept some degradation in performance. The idea in the stochastic sine wave and first-order Markov processes. suboptimal Wiener filter (see [15], [17], [3]) is to allow an accept- able performance degradation in filtering while increasing the II. GENERALIZED WIENER FILTERING OVER GROUPS computational performance of the filter. That is achieved by We consider the set 9 = (0, 1, . . . , n - l} of n integers without using a fast unitary transform for which a procedure exists to a group structure being imposed on it. Let u, e be random simplify the matrix TW,,,T-’ by, e.g., making it sparse or functions of signal and noise, respectively, defined on B and structured. The statistical performance D(c) must be kept within taking values in the field of complex numbers C; that is, u, e: 9 acceptable limits. For example (see [3], [15]-[17]), a family of -+ a3 with the mean values E( u( t)) = E( e(t)) = 0, t E 3. Let suboptimal Wiener filters has been obtained by setting all the off-diagonal entries in TWoptT-’ to zero, where T’s fast compu- w: 9 x 9 + C be a deterministic impulse response function of a digital device (a filter) which is processing the corrupted signal tational algorithm is based on it having a Kronecker product f = u + e. Then the classical Wiener filter problem of separating representation (such transforms are classified in [4], [5]). The WHT and DFT which have been used in [l], [3], [15]-[17] are the signal from noise is to find wept: 9 x 9 -+ C such that matrices T of characters of the dyadic group and of the cyclic group, respectively. We note that the choice of a dyadic or of a cyclic group G is, generally speaking, not optimal (see [3], [7]), where E,>(t) = C{G(BW(f, w(S)-0, t E 9; D(c,) = and moreover, replacement of a dyadic or of a cyclic group by c ,,9D(c,(t)); D(c,(t)) = E(c,(t)c,(t))/n is the dispersion some (not necessarily Abelian) group may result in the reduction squared which depends upon the choice of impulse response of D(c) (see the next section and [13, 261). Such group trans- function w( . , .) of the filter; and E represents the expectation forms exist for any integer n, can be computed analytically by operator. Using the notation, u = (u(O), u(1); . . , u( n - 1))r and formulas, possess Kronecker product representations [20], [21], e = (e(O), e(1);. ., e( n - 1))’ where T stands for the transpose [23], and as will be shown later, in many cases compare favorably of a row vector, we reformulate the same problem (1) of the with other transforms. Therefore, attempts have been made to optimal filtering of the zero-mean n-vector of noise e from the develop a general approximation theory of systems and signals corrupted, zero-mean signal f = u + e by means of the (n x n)- over groups. Elements of this theory are presented in [13], [14] matrix of the optimal filter IV&, = (Wd,s;‘)), Wd,“;‘) = and embrace approximation problems in digital signal processing woPt(s, r), s, I = 0,l; . ., n - 1 such that such as filtering, encoding, data compression, etc. (see also [l], 131). m${VnE(IIW- 4)) = l/nE(IIK,ptf- uII>, (1’) In what follows, we apply ideas and methods of that theory to construct optimal filters over an arbitrary group. Let us introduce where a group structure in 9= {O,l; . ., n - 1). That is, G stands for the group of cardinal&y n defined on the underlying index set 9 l/nE( ]]r,,J2) B l/nE(tr(e&)) = c o(e,,,
considered here are the DCT which is asymptotically equivalent has covariance matrix to the KLT for the first-order Markov process [24] and the DFT which is represented by the matrix of characters of the cyclic B,, = (cosX(s - I)), s,I=O,l;..;n-1. (20) group G = C,, of integers 0, 1, . . . , n - 1. We shall consider two The noise e is assumed to be white with identity covariance nonAbelian groups, namely, S, (the symmetric group of the third ’ matrix and the signal and noise are assumed to be uncorrelated. order) and Q2 ,(the quatemion group of the eighth order, The KLT is computed for the first-order Markov process in and C,, (the cyclic group). Their duals is, Q, , ?n are given in [22], [15]. In the case of the random sine wave, the matrix of eigenvec- [23]. We shall use the direct products S, X C,, C, X S,, Q2 x tors of the corresponding Wept in (2) (the KL’I) was computed C,,, C,, x Q2, S3 X Q2, Q2 x S,, S3X &, Q2 X Q2. The corre- for each n. The DCT is asymptotically equivalent to the KLT of sponding duals are computed by (6kusing the Kronecker product the Markov process (see [24]) and is defined in [6], [25]. property of group representations. The computation of dispersion The computer experiments for the first-order Markov process squared Dz in (18) is done using the fast Fourier transform are summarized in Tables II and III. Their purpose is to compare algorithm over the group G (see [20], [21], [23]). The number n statistical performance of suboptimal group filters (see Table II) Cicln, of operations needed to compute the Fourier transform as well as to compare the known results for the DFT with other (5) (see Section II) is the upper bound on the computational (not necessarily Abelian) groups. It follows (see Tables II, III) complexity. The actual number of operation: depends upon the that as p increases, the use of various nonAbelian groups results number of zeros in the elements of the dual G for a given group in improved statistical performance as compared with the DFT. G. For example, for G = S, four zeros are among the elements of That is, however, compensated for by the increased number of w = 2. Therefore, to compute (5) for S, X C, we need not computations. 12(6 + 2) = 96 computer operations but only 12 . 8 - 2 . 4 = 88 It follows also that D$ increases as p decreases, e.g., for operation?. Analogously, 16 zeros are among the elements of G = S, x C, see Table III. That happens because B,, ap- o = 4 E Q2. Hence, e.g., to compute (5) for Q2 x Q, we need proaches the identity matrix as p decreases. It follows that as p 64 . 16 - 8 * 16 - 8 . 16 = 3 . 2’ = 768 computer operations. increases, various noncommutative groups compare favorably It was pointed out in the discussion of Fig. 1 in Section II (see with C,,. We note that the order of the groups G, and G, in their also [15], [l], [3]) that when using a transform i”, the filter direct product affects the dispersion without affecting the compu- T-‘(TWT-‘)Tf is performed in the following three steps, which tational speed; i.e., the number of operations required to com- determine the overall amount of computer operations: pute the dispersion is the same. In the general case it is a difficult problem to select the optimal group among all the groups of a 1) Tf, given order [13]. However, this problem can be solved numeri- 2) (TWT-‘)(Tf ), cally for moderate values of n. 3) T-‘((TWT-‘)(Tf)). The computer experiments for the random sine wave process The numbers of computer operations required to perform gener- are summarized in Table IV. It follows (see Tables I, IV) that as alized filtering in the cases of identity, Karhunen-Loeve, and X decreases, the use of various nonAbelian groups results in DFT’s are n2 (step 2 only), 2n2 f n (n2 operations at steps 1, 3, improved statistical performance as compared with the DFT. For and n operations to multiply a diagonal matrix TWT-’ by a example, for n = 64 and X = 0.01 (or X = 0.05) the statistical vector Tf ), nz + 2n log, n, n = 2’( n log, n at steps 1 and 3 and gain in 17.59 percent (or 20.95 percent) for Q2 X Q2 as com- n2 operations at step 2), respectively. pared with C,, That is, however, compensated for by an increase The suboptimal filtering results in reducing the number of in speed of nearly 100’percent in the DFT as compared with the operations at step 2 to the order of n at the expense of setting all group Q2 x Q2 (see Table I). It follows also that the dispersion the off-diagonal entries in TWT-’ to zero (as in the case of D$ increases as h increases. This happens because B,,, ap- DFT, WHT, or any Abelian group based transform T) or by proaches the matrix all of whose entries are ones as X decreases, structuring TWT-’ to a canonical block diagonal form, uniquely i.e., correlation between u-components increases and the disper- determined by the group G (as in the case of a noncommutative sion decreases. Alternatively, as X increases, B,,, approaches the group, see [13]). identity matrix and the dispersion increases. It follows that for Based on these considerations, the results of comparing differ- small X various noncommutative groups compare favorably stat- ent transforms (see [3], [9] for DFT and DCT) in suboptimal istically with the cyclic group C,. We note that as X increases, the filtering from the point of view of the number of operations group transforms compare favorably with the DCT; that is, the required are given in Table I. Various transforms are compared cyclic group C,, provides a better approximation to the KLT than by their statistical performance in what follows. does the DCT. It can be expected that the more a stochastic process differs from the first-order Markov (in the behavior of its TABLE I covariance matrix), the worse the results of employing the DCT NUMBEROF OPERATIONSREQUIRED FORVARIOUS will be because DCT is equivalent to the KLT of the first-order SUBOPTIMALFILTERS Markov process [24]. At the same time, a great variety of fast group transforms can always be used to choose the best from Transform using (17) and (18). Therefore, group filters might find their use II KLT DCT et VT) Cx, Qz x Q2 in practical situations in which we do not know computationally 8 136 92 56 104 good approximating transforms for the KLT for a given process. 16 528 252 144 212 32 2080 652 352 736 Iv. SUMMARY 64 X256 1612 832 2048 1600 Different group transforms have been compared among them- selves and with the DFT in the problem of constructing a suboptimal Wiener filter. Computational and statistical perfor- We consider the following signals: mance of various filters was considered as well as the problem of trade-off between the two. 1) the first-order Markov process with the covariance matrix The best group model is constructed for a given group which B,,, = (p”-‘I), s,I=O,l;..,n- 1, performs the filtering using a fast group transform algorithm. (19) The problem of choosing an optimal group among all the groups where 0
statistics of the signal and noise. IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-32, NO. 2, MARCH 1986 307
TABLE II GROUP WITH THE OPTIMAL STATISTICAL PERFORMANCE FOR THE FIRST-ORDER MARROV PROCESS
n P 6 8 12 16 18 24 30 32 36 40 42 48 54 56 60 64
‘.9 S, Q2 G X s3 CM Cl, C C C '36 GO C c c54 c56 1.99 s, Q, C, x S, C, x Q2 C, x S, C, TQ, C, ?:es, C, TQ, s3 X S, Q, X c5 s3 k s, x""Q, S, x C, Q2 x C, S, ?C,, Q22Q;
TABLE III DISPERSION FOR THE GROUP FILTER WITH G = S, X C, P 0.5 0.6 0.7 0.8 0.9 0.92 0.94 0.96 0.99 Ds3xc2 0.4545 0.4294 0.3944 0.3437 0.2628 0.2398 0.2128 0.1800 0.111
TABLE IV GROUP WITH THE OPTIMAL STATISTICAL PERFORMANCE FOR THE RANDOM SINE WAVE
X 6 8 12 16 18 24 30 32 3: 40 42 48 54 56 60 64
1.01 c, c, c,, cl6 S, x ~3 S3 x C, S3 x C, Q2 x C, S3 X G Q2 X G s, x c, S, x C, S, x C, Q2 x C, s3 x Cl0 ep2 ; g3
1.05 2 ;: 22 “, 2 z;x”,‘: s, x c, s, x c, s, x c, Q2 x C, S, x S, Q2 x C, S3 x C, S3 X Q2 S, x C, Q2 x G S3 x Go Q: x Q: s3 1.1 S3 Q2 C2 x S, G x Q2 S, X C, S, X c4 s3 X c5 c, X Q2 c, X s3 c, x Q2 c, x s3 c,, G4 c56 C 60 C 64
ACKNOWLEDGMENT PO1 M. G. Karpovsky, “Fast Fourier transforms on finite non-Abelian groups,” IEEE Truns. Comput., vol. C-26, pp. 1028-1030, 1977. The authors wish to thank Mr. 0. Zimmerman for conducting WI P. J. Nicholson, “Algebraic theory of finite Fourier transforms,” J. the computer experiments of Section III. Comput. Syst. Sci., pp. 524-547, 1971. WI L. Domhoff, Group Representation Theory. New York: Marcel Dekker, 1971. REFERENCES 1231 M. G. Karpovsky and E. A. Trachtenberg, “Fourier transform over VI D. F. Elliott and K. R. Rae, Fast Transforms: Algorithm, Analyses und finite groups for error detection and error correction in computation Applications. New York: Academic, 1983. channels,” Inform. Contr., vol. 40, pp. 335-358, 1979. PI H. C. Andrews, Picture Processing and Digital Filtering. New York: [241 M. D. Flickner and N. Ahmed, “A derivation for the discrete cosine Springer-Verlag. 1975. transform,” Proc. IEEE, vol. 70, pp. 1132-1134, 1982. I31 N. Ahmed and K. R. Rae, Orthogonal Transform for Digitul Signal 1251 M. Hamidi and J. Pearl, “Comparison of cosine and Fourier transforms Processing. New York: Springer-Verlag. 1975. of Markov-1 signals,” IEEE Trans. Acourt., Speech, Signcrl Processing, [41 B. J. Fino and V. R. Algazi, “A unified treatment of discrete fast unitary vol. ASSP-24, pp. 428-429,1976. transforms,” SIAM J. Computing, vol. 6, pp. 700-717, Dec. 1977. WI E. A. Trachtenberg, “Systems over finite groups as suboptimal Wiener [51 H. C. Andrew and K. L. Caspari, “Degrees of freedom and modular filters: A comparative study,” presented at 1983 Int. Symp. on Muthe- structure in matrix multiplication,” IEEE Trans. Comput., vol. C-19, pp. nwtical Theory of System and Networks, Beer-Sheva, Israel, 1983. 16-25, 1970. v71 M. G. Karpovsky, Finite Orthogonal Series in the Design of Digital [61 N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” Deuices. New York: Wiley, 1976. IEEE Trans. Comput., vol. C-23, pp. 90-93, 1974. WI E. A. Trachtenberg, “Construction of Group Transforms Subject to [71 J. Pearl, “Basis restricted transformations and performance measures for Several Performance Criteria,” IEEE Tmns. Acourt. Speech, Signal spectral representations,” IEEE Trms. Inform. Theory, vol. IT-17, pp. Processing, vol. ASSP-33, pp. 1521-1531, 1985. 751-752, 1971. WI H. S. Kwak, R. Srinivasan, and K. R. Rao, “C-matrix transforms,” submitted to IEEE Truns. Acoust. Speech, Signul Processing. [91 W. M. Chen, C. M. Smith, and S. C. Fralick, “A fast computational algorithm for the discrete cosine transform,” IEEE Truns. Commun., vol. COM-25, pp. 1004-1009.1977. PO1 K. R. Rao, J. G. K. Kuo, and M. A. Narasimhan, “Slant-Haar trans- form,” Int. J. Computer M&h., sec. B, 7, pp. 73-83, 1979. Ull E. A. Trachtenberg, “Fast Wiener filtering computation technique,” in Proc. 1979 Int. Symp. on the Muthemuticul Theory of Networks and Maximum Likelihood Estimation for Multivariate Swtems. vol. 3, Delft, The Netherlands pp. 174-177. Mixture Observations of Markov Chains WI E. A. Trachtenberg, “Construction of fast unitary transforms which are equivalent to Karhunen-Loeve spectral representations, in Proc. 1980 B. H. JUANG, STEPHEN E. LEVINSON, SENIOR MEMBER, IEEE, IEEE Int. Symp. on EMC, Baltimore, MD, USA, pp. 376-379. AND M. M. SONDHI v31 M. G. Karpovsky and E. A. Trachtenberg, “Some optimization prob- lems for convolution systems over finite groups,” In/arm. Corm., vol. 34, pp. 227-247.1977. Abstract-To use probabilistic functions of a Markov chain to model v41 J. Pearl, “Optimal dyadic models of time-invariant systems,” IEEE certain parameterizations of the speech signal, we extend an estimation Trum. Comp;t., vol. C-24, 1975, pp. 598-602. technique of Liporace to the cases of multivariate mixtures, such as [I51 W. K. Pratt, “Generalized Wiener filtering computation technique,” IEEE Truns. Comput., vol. C-21, pp. 636-641, 1972. Gaussian sums, and products of mixtures. We also show how these V61 J. Pearl, “Walsh processing of random signals,” IEEE Truns. Electro- problems relate to Liporace’s original framework. mugn. Cornput., vol. EMC-13, pp. 137-141, 1971. (171 -, “On coding and filtering stationery signals by discrete Fourier transform,” IEEE Truns. Inform. Theory, vol. IT-19, pp. 229-232, 1973. Manuscript received September 10, 1984; revised July 12, 1985. This work IlRl -. “Asymptotic equivalence of spectral representations,” IEEE Truns. was presented at the 1985 IEEE International Symposium on Information Acoust.. Speech, Signul Processing, vol. ASSP-23, pp. 547-551. 1975. Theory, Brighton, England, June 24-28. I1 91 Y. Yemini and J. Pearl, “Asymptotic properties of discrete unitary The authors are with Acoustics Research Department, Bell Laboratories, transforms,” IEEE Trans. Pattern Anal. Muchine Intell., vol. PAMI-1, Murray Hill, NJ 07974. pp. 366-371. 1979. IEEE Log Number 8406633.
001%9448/86/0300-0307$01.00 01986 IEEE