Blind deconvolution (equalization): Some new results

SIGNAL PROCESSING

Signal Processing 53 (1996) 109-I 16

Blind deconvolution (equalization): Some new results Mithat C. Dogan, Jerry M. Mendel’

Department of Electrical Engineering Systems, Sigttal & Image Processing Institute, University of Southern California,

Los Angeles, CA 90089-2564, USA

Received 1 March 1995; revised 15 April 1996

Abstract

An interesting equation developed by Giannakis and Mendel (1989), and referred to by others (e.g., Friedlander and Porat (1989)) as the ‘GM equation’, links higher-order statistics to second-order statistics. In this paper, we show how this equation leads to a new universal relationship between a system and its inverse, and can be employed to obtain a new closed-form formula for a Shalvi-Weinstein-like super-exponential blind deconvolution algorithm.

Zusammenfassung

Eine interessante Gleichung, die von Giannakis und Mendel (1989) entwickelt wurde und von anderen (z.B. Friedlander und Porat (1989)) als ‘GM-Gleichung’ zitiert wird, stellt eine Verbindung zwischen Statistiken hijherer und zweiter Ordnung her. In diesem Beitrag zeigen wir, wie diese Gleichung auf eine neue, universelle Beziehung zwischen einem System und dem dazu inversen fLhrt und dazu verwendet werden kann, eine neue, geschlossene Fonnel fCr einen Superexponential-Ansatz zur blinden Entfaltung wie bei Shalvi-Weinstein zu erhalten.

R6sumC

Une Equation intkressante, dCveloppCe par Giannikis et Mendel (1989), et r&firen&e par d’autres (p.e. Friedlander et Porat (1989)) sous le nom d’6quation GM, met en relation les statistiques d’ordre sup&ieur et celles du deuxikme ordre. Dans cet article, nous montrons comment cette Equation conduit A une relation universelle entre un systeme et son inverse. Nous montrons aussi comment elle peut &e employ&e afin d’obtenir une nouvelle formule analytique pour un algorithme de d6convolution aveugle super-exponentiel semblable B celui de Shalvi-Weinstein.

1. Introduction

Blind deconvolution (blind equalization) is the

problem of determining a system’s input just from noisy measurements of its output, without knowledge of its impulse response, nor with the ability to apply a test input to the system so as to determine its impulse response. It is very important in a number of

* Corresponding author. Tel.: (213) 740-4445; fax: (213) 740-

465 1; e-mail: [email protected].

fields including communications, reflection seismo-

logy, and image restoration. If the system is minimum phase, then second-order statistics may lead to a satisfactory deconvolution filter, although, if the ad-

ditive measurement noise is colored with an unknown spectrum, then such a deconvolution filter will be severely degraded. In the applications just mentioned, the unknown input is known to be non-Gaussian and the system’s impulse response is often non-minimum phase; hence, deconvolution filters based only on second-order statistics do not perform well because

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PZI 0165-1684(96)00080-l

110 M. C. Dogan, J. M. Mendell Signal Processing 53 (1996) IO9-116

they do not exploit the available information. Con- sequently, during the past few years there has been intense interest in applying higher-order statistics to the design of deconvolution filters [8,2,4, 1 l-l 3, 15- 181 because higher-order statistics can be used to re- construct correct phase impulse responses and they are blind to any kind of Gaussian noise (although the variances of their estimates are indeed affected by such noises).

Recently, three seemingly unrelated results have been published which bear either directly or indirectly upon the blind deconvolution problem. Giannakis and Mendel [7] developed an interesting equation which relates the usual spectrum of a system’s output to a special polyspectrum. This equation, and especially its application to moving average (MA) systems, has come to be known as the ‘GM equation’ (Fried- lander and Porat [6] were the first to refer to it as the GM equation). To date, it has only been applied to the problem of identifying an unknown moving average system. We review the GM equation in Section 2, and later in the paper show how it can be used in two new ways.

Zheng et al. [16] showed that, for moving average systems, it is possible to determine the coefficients of a linear deconvolution filter by solving a system of equations that are linear in these coefficients. They were able to do this because they first obtained some very interesting and new theoretical formulas which relate the correlation between the deconvolution filter’s coefficients and either second-order, third-order or fourth-order statistics, to linear, quadratic or cu- bic powers of the moving average coefficients, respectively. They were not able to generalize their new results to higher than fourth-order statistics, and left the generalization as an open problem. We solve this problem in Section 3 using the GM equation, and, in fact, provide a generalization of Zheng et al’s results to arbitrary systems.

Shalvi and Weinstein [ 12,131 proposed a very interesting approach to blind deconvolution that leads to ‘super-exponential’ convergence. The underlying idea is that one wants the output of the deconvolution filter to be as impulsive as possible. Related results appear in Donoho [5]. Their algorithm is first for- mulated in terms of the composite system that is the convolution between the unknown system and the deconvolution filter. This is very clever, because it is

so clear what the objective of deconvolution should be in terms of the composite system. Because this system is not available (i.e., the system is unknown and the deconvolution filter has yet to be designed), they then transform their algorithm into one for designing the deconvolution filter. This new algorithm is in terms of signals that can be measured, and in the 1993 paper is in terms of higher-order statistics. In fact, their results are very general, and are given for arbitrary-order higher-order statistics; however, in their simulations they only use their general results for fourth-order statistics (third-order cumulants of communication signals are zero), no doubt because one prefers to work with the lowest-order higher- order statistic in order to keep the error variances of the cumulant estimates as small as possible. In Section 4 we show how to obtain a fourth-order statistics Shalvi-Weinstein-like super-exponential deconvolution algorithm using the GM equation. We believe that this is a more direct approach to the derivation of this algorithm.

2. GM equation

Our system of interest is depicted in Fig. 1. It is single-input and single-output, causal, linear and time- invariant, in which u(k) is stationary, kth-order white, and non-Gaussian with finite variance crz, and non- zero third-order (~3,~) or fourth-order cumulant (~4,~); n(k) is Gaussian noise (colored or white); u(k) and n(k) are statistically independent; and H(z) is an unknown asymptotically stable system. The deconvolution filter is given by O(z).

Starting with the following equation (the ‘Bartlett, Brillinger and Rosenblatt’ equation; Bartlett [l] and Brillinger and Rosenblatt [3]) for the kth-order cumulant of y(k) (for a derivation of most of the equations

channel decon filter

Fig. 1. Single-channel system and deconvolution filter.

M.C. Dogan, J. M. Mendell Signal Processing 53 (1996) 109-116 111

in this section, see Mendel [lo]): and

Ck,y(Z1,72,. . . , Tk-1)

‘XJ

= Yk,L’C h(n)0 + q)...h(n + Zk-1), (1) n=O

and setting ri = 72 = . . . = Zk-1 = z, Giannakis and Mendel [7] showed that the special polyspectrum Sky(c)) = DTFT{Ck,,(r, 7,. . . , z)} (‘DTFT’ is short for discrete-time Fourier transform) is related to the power spectrum of y(k), S,(w), by the following formula:

= (~~/Yk,v)H(O)Sk,y(W), k = 2,3,. . . , (2)

where

&_l(a) =H(o)@H(w)@“‘@ff(w)

(k - 2 complex convolutions) (3)

(e.g., when k =3, it is easy to show from (1) that

S3,y(w) = Y3,v m-o)W(w) @ H(w)1 = 73,” W-w)

H2(w) and S,(o) = ozH(w)H(-o), from which it follows that Hi& = (o~/~~,~)H(o)S~,~(O)). If signals are complex, then (3) must be changed to

ffk-l(a) = k even,

H(o)~H*(-o)~...~H*(-o)

k odd,

(k - 2 complex convolutions) (4)

in which ( )* denotes the complex conjugate of ( ). Although (4) was not stated by Giannakis and Mendel, its proof is so similar to the proof of (3) that we do not include it here. It is based on the following non-unique definition of the kth-order diagonal-slice cumulant for a complex process, y(k), which is stated next for even values of k: Ck,y(T) = cum(Y*(t>,Y(t + +Y*(t +

T), . . . > Y”@ + z), Y0 f z)).

If the channel is moving average, with moving average coefficients b(0) = 1, b(l), b(2), . . . , b(q), then (2) can be expressed in the time domain, for k = 3 and k = 4, as

5 b2(j)rv(m - j) = (d/~3,~\$ KW3Jm -A j=O

(sa)

5 b3(j)ry(m - j) = (~~/YJ,,)~ W)C4,y(~ -A j=O j=O

(5b)

where rY(m - j) denotes the autocorrelation function of y(k), Cs,,,(m - j) is short for C3,Jm - j,m - j), and &,(m - j) is short for C4Jm - j, m - j, m - j), i.e., Cs,,(m -j) and C4,,,(m - j) are diagonal slices of third- and fourth-order cumulants, respectively. These are the equations which Friedlander and Porat [6] refer to as the ‘GM equations’. Because they are special cases of (2), we prefer to call (2) the GM equation.

3. Linear equations for deconvolution filter coefficients

Zheng et al. [ 161 prove that, for real signals, if the system H(z) has no zeros on the unit circle and is a moving average system, and if the deconvolution system 13(z) is a (possibly) non-causal moving average system, where rl and r2 are the properly selected or- ders of the non-causal and causal parts of 0(z), respectively, then

. .

V J-'I 3

kjg, WC3,Ji + j) = { i2(i) (7)

and

Similar results can be obtained for complex signals. Their interesting proofs for these results are all similar, e.g., the proof of (7) computes E{u(k)y2(k+i)} in two different ways, one using the convolutional model for y(k -t- i), and the other using the deconvolution filter equation for u(k), after which the two results are equated. Because the cumulant of any Gaussian process is zero, it is trivial to extend their results from the noise-free measurement y(k) to the noisy measurement z(k); simply replace the subscript y in (7)-(8) by z.

112 M. C. Dogan, J. M. Mendel f Signal Processing 53 (1996) 109-116

Zheng et al. then conjecture that the generalization to (6)-(8) is

Although their computer simulations verify the truth of (9), they state ‘. . . a mathematical proof for the case k 2 5 still remains to be found . . .‘. Eq. (9) is indeed true and is a special case of the following theorem.

Theorem 1. If the system H(z) is linear, time- invariant and asymptotically stable, has no zeros on the unit circle, processes real signals, l/H(z) = e(z) exists, and the system’s input v(k) is zero-mean, non-Gaussian and i.i.d. (or higher-order white [7]), then the relation between H(z) and its inverseJilter g(z) can be expressed in terms of kth-order cumulants as

(l/~k,~)~ W)G,,ti +i> = hk-‘ti),

k = 2,3,... Wa)

When one has access only to noisy measurements, z(k) = y(k) + n(k), where n(k) is Gaussian and is independent of v(k), then

(l/rrl)C e( j)[r,(i + j) - r,(i + j) = h(i) (lob)

and

(l/yk,,)c g(j)&(i +j) = hk-‘(i),

k = 3,4,... (1Oe)

Before proving this theorem, note that (10a) reduces to (9) when H(z) is a moving average system (because in that case h(i) = b(i)) and we restrict j to the range [rl , Q]. In general, j can be associated with a finite or an infinite range of values. Note, also, that (lOa) is applicable to autoregressive and autoregressive moving average systems as well as to moving average systems; hence, (9) is a special case of our new result in (10a). When noisy measurements are available, we must be very careful about using (lob), because usually r,(i + j) is unknown. If (lob) is used

simply by ignoring the rn(i + j) term, errors will be incurred. Such errors do not occur in (10~) because the higher-order cumulants have (theoretically) sup- pressed the additive Gaussian noise.

We believe that (lOa), for example, represents a new universal type of relationship between a system and its inverse, a relationship that for the most part remains to be explored. If the system and its inverse are both unknown, it is presently unclear what one would exactly do with (10a). In the special case of a qth-order moving average system, the right-hand side of (10a) equals zero for all i > q. In this case, (10a) reduces to a system of equations in just the unknown inverse filter’s coefficients.

Proof of Theorem 1. Using the fact that S,(o) = ozH(o)H(-o), GM equation (2) can be expressed as

Hk-i(o)H(-w) = (l/Yk,v)&,y(w),

k = 2,3,...;

hence,

(11)

Hk--l(m) = (l/~k,v)s~,,(w)/H(-0)

= (l/Yk,“)sk,y(w)e(-w),

k = 2,3,. . . , (12)

where we have used the fact that, ideally, 0(o) = l/H(o). Eq. (10a) is obtained by taking the inverse DTFT of (12), using the formula for Hk-l(o) in (3). Eq. ( 1 Ob) is obtained by using the fact that rz( i + j) = rY(i + j) + rn(i + j). Eq. (10~) is obtained by using the fact that C&i + j) = 0, for all k > 3, because n(k) is Gaussian. 0

Observe that it is indeed very straightforward to obtain (IO) from the GM equation. Similar results apply to the case of complex signals, via (4).

A reviewer of this paper observed that (lOa) can also be obtained directly from (1) and the constraint H(z)g(z) = 1. This is correct, and represents a time- domain derivation of (lOa), whereas our derivation is a frequency-domain derivation. The time-domain derivation (for real signals) begins with (1) in which all of the lags are set equal to r = i+ j; then, multiply both sides of the resulting equation by e(j) and sum

hi. C. Dogan, J.M. Mendel f Signal Processing 53 (1996) 109-116 113

over all values of j; finally, use the fact that

c @.0h(t - j) = d(t) i

to obtain ( 10a).

4. Super-exponential blind deconvolution (equalization) algorithm

For the moment, let us ignore the additive noise which appears in Fig. 1, so that we focus our attention on the system depicted in Fig. 2. Shalvi and Wein- stein [ 12, 131 have developed the following iterative procedure for (to be consistent with the notation in the rest of this paper, we use s(k) and B(k) for Shalvi and Weinstein’s s,, and en, respectively) adjusting the unit sample response coefficients (taps) of the combined system (in this section, signals are complex):

s;(k) = s~_i(k)[sT-i(Qlq, I = 1,2,..., (13a)

Q(k) = &)lC l~‘r(~)12~ (13b)

where sl_ 1 (k) are the tap weights before the iteration, s!(k) are intermediate values, and sl(k) are the tap values after the iteration. Note that the summation in (13b) is over the number of tap weights, and that so(k) can be defined as so(k) = h(k).

Shalvi and Weinstein state further that “If we choose p and q to be nonnegative integers such that p + q 2 2, then the transformation in (13a) followed by the normalization in (13b) causes the taps with smaller magnitude to decrease more rapidly, forcing s/(k) to converge very quickly, as 1 increases, to the desired response in which one tap (the leading tap) approaches one in magnitude, while all other taps approach zero.” See Fig. 2 in [ 131 for an example that

illustrates this effect for so(k) = 0.81“l (see, also, our

(25)). Because the sl(k) taps are unavailable for direct

adjustment, Shalvi and Weinstein then “show how to implement the algorithm in the &domain using only the taps 0,, of the equalizer and some statistical cumulants of the observed data.” Here we show how to obtain their e-domain algorithm very directly by using the GM equation (2). We do this only for the case of p = 2 and q = 1, because, as we shall see, this is the case that involves fourth-order cumulants, which are known to be non-zero for communication signals. In practical applications, there presently is no need to use larger values of p and q.

We begin in the first iteration by designing 81(k). Choosing p = 2 and q = 1 in (13), we see that

Q(k) = Blad~>12&~) = BlW)12~(~), (14)

where /? is chosen so that C Jsi(k)12 = 1. Because we do not know h(k), we need to design 81(k) such that (14) is satisfied.

Theorem 2. To meet the requirement that s,(k) is given by (14), choose e,(w) such that

e,(w) = B[H(O) e 13*(-w) e ~(a OH

= P2s4,y, (d/f&, (u),

where BZ = lja&W

(15)

Before proving this theorem, note that (15) pro- vides a new closed-form formula for e,(w), and demonstrates that we can design &(o) by computing

s4,y, (0) = DTF’W4.y, ( z, 7, ~1) p DTFT{C4,&)}

and S,, (CD), in the absence of additive noise. The case of additive Gaussian noise is treated below.

---- v(k)

Fig. 2 Starting point for Shalvi-Weinstein design.

114 M.C. Dogan, J.M. Mendel/Signal Processing 53 (1996) 109-116

Proof of Theorem 2. We require St(o) = DTFT [j3]k(k)]2k(k)], where (Fig. 2) St(w) = H(w)&(w); hence,

01(w) = &(w)/H(o)

= ,f?DTFT{Jh(k)j2h(k)}/H(o). (16)

Applying the facts that DTFT{x(k)y(k)} = X(w) ~3 Y(w) (where the factor of 1/27r is absorbed into the definition of complex convolution) and DTFT{k*(k)} = H*(-o) to the numerator of (16), we obtain the first expression on the right-hand side of (15). Using the definition of Hs(o), given in (4) for k = 4, we can reexpress et(o) as

f%(o) = B~3(wYWo). (17)

Applying the GM equation (2), for Hs(w)/Z-Z(w) to ( 17), we obtain

h(w) = wdh,“N,y, (WYS,, (w), (18)

which is the second term on the right-hand side of (15). 0

Note that it is fairly straightforward to generalize these results to other values of p and q [see (13)], because of the generality of the GM equation (2).

Corollary. A time-domain equation for the design of the deconvolution Jilter’s cae@cients is

I$ &(k)ry,(z - k) = 82c4,y,(+ (19)

Proof. Express (15) as &s&Q,(w) = p&y, (0) and take the inverse DTFT of this equation to obtain (19). q

In (19) k ranges over the range of values assumed for the deconvolution filter’s coefficients, and r is chosen so that (19) constitutes a system of overdeter- mined linear equations, which can be expressed as

R& = d. (20)

Shalvi and Weinstein [ 131 also arrive at a linear system of equations which must be solved for Bi (our 81 is the same as their c; see their Eqs. (33a), (33b), (34), (35) and (36)); however the two systems are a bit different.

If only noisy measured values of yt(k) are available, i.e., we are working with the situation depicted in Fig. 1 instead of that depicted in Fig. 2, where n(k) is Gaussian, then (15) can be expressed as

et(o) = b2s4,zj <wf& t@; (21)

but S,,, (0) can be computed using fourth-order statistics (e.g. [lo]) as

S,,(w) = &%,Z,(~, 0, wY4,“w(o)12~ (22)

where S+, (o,O, 0) is a 1D slice of the full-blown trispectrum ofzi(k), i.e., &,,,(o,O,O) = S+,(wt =o, w2 = 0,03 = 0). Of course (22) requires H(0) # 0. The trispectnun is (theoretically) blind to Gaussian noise.

Our Shalvi- Weinstein-like super-exponential blind deconvolution algorithm (stated for the case of perfect measurements; see also Fig. 2) is: (1) Solve RI 81 = dI for 01, where RI and d, are computed using sample statistics of yl(k) 2 y(k). (2) Set j = 2. (3) Let yi(k)=gi_l(k)*yj_,(k), and compute Rj and dj US- ing the sample statistics of yj(k), and solve RjOj = dj forfIj.(4)Setj=j+landgoto(3)unlessj=J, in which case go to (5). (5) If you want an explicit form of the deconvolution filter, compute it as

B(k) = Bl(k) * eg(k)* . . * *eJ(k), (23)

otherwise, or after which. (6) Stop. By this procedure yJ(k) is the deconvolved sequence, i.e. 6(k) = y_t(k).

Each filter g,(k) takes yj(k), which has an efictive (overall) impulse response sj(k) (with so(k) = h(k)), and processes the data such that the effective impulse response, sj+l(k), of its output, yj+t (k), is

sj+l(k) = sj(k)lsj(k)12, j = 192,. . . ,J - 1, (24)

where sl(k) = h(k)/h(k)j2. This is possible, using Theorem 2, by considering each new stage constructed as the first stage of an equalizer, with the preceding combined system as the effective channel. Iterating (24), we find that

s;(k) = h(k)lh(k)jL3’-‘I 3 i=1,2 ,..., J, (25)

which demonstrates the super-exponential decay in the behavior of sj(k).

We do not include any simulations or performance analyses of our super-exponential blind deconvolution

M. C. Dogan, J. M. Mendell Signal Processing 53 (1996) 109-I 16 115

algorithm, because Shalvi and Weinstein [ 131 already provide these results.

5. Conclusions

The main point of this paper has been to demon- strate two new applications for the GM equation, an equation that links higher-order statistics to second- order statistics. Up until our work, the only application for the GM equation was to the identification of moving average coefficients. We have demonstrated that the GM equation (1) leads to a new universal relationship between a system and its inverse (Theorem l), and (2) can be employed to obtain a new closed-form formula for a Shalvi-Weinstein- like super-exponential blind deconvolution algorithm (Theorem 2).

The GM equation is based on the Bartlett- Brillinger-Rosenblatt equation ( 1). A generalization of the latter equation to multi-input and multi-output linear systems has been given by Swami and Mendel [14]. The extension of all the results in this paper to the multi-input and multi-output scenario, which can build upon the multi-input and multi-output Barlen-Brillinger-Rosenblatt equation, remains to be done.

Acknowledgements

This work was supported by an unrestricted gift from the Rockwell Science Corp., Thousand Oaks, CA. The authors would also like to thank the review- ers, whose comments helped to improve the final version of the paper.

An earlier conference version of this paper (“Ap- plication of the GM equation to blind deconvolution”, M.C. Dogan and J.M. Mendel, Proc. IEEE Signal Processing ATHOS Workshop on Higher-Order Statistics, pp. 459-463, Parador D’Aiguablavo, Begur, Girona, Spain, 12-14 June 1995) contained a corollary to Theorem 1 which maintained that substituting (10a) into the GM equation (2) leads to an identity; hence, doing this does not lead to a new system of equations, as claimed by Zheng et al. [16], which are linear in the coefficients of the deconvolution filter. In a private communication to the authors,

by Achilleas G. Stogioglou and Stephen McLaugh- lin, this corollary was shown to be incorrect because (10a) was obtained not only from the GM equation (2), but also from the spectrum equation for S,(w). Consequently, substituting (6) and (7) into (5a), or (6) and (8) into (Sb), does indeed lead to new linear systems of equations for the deconvolution filter coefficients. We wish to thank Stogioglou and McLaughlin for calling this to our attention.

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Blind deconvolution (equalization): Some new results