Sign algorithms for blind equalization and their convergence analysis

CIRCUITS SYSTEMS SIGNAL PROCESS VOL. 10, NO. 4, 1991

SIGN ALGORITHMS FOR BLIND EQUALIZATION AND THEIR CONVERGENCE ANALYSIS *

Vijitha Weerackody, 1'2 Saleem A. Kassam, 1 and Kenneth R. Laker 1

Abstract. In blind equalization a communication channel is adaptively equalized wi~out resorting to the usual training sequence. In this paper we have introduced two new\algorithms for blind equalization, which hard limit the equalizer input or the error at the output of the equalizer. These new algorithms are simple to implement and reduce the number of multiplications by approximately one-half. We show by way of simulations that the performance of the algorithm resulting from hardlimiting the error is comparable with the performance of the corresponding algorithm in which the error is not hardlimited. We formulate the new sign-error algorithm as a stochastic minimization of an error functional and demonstrate that the case of zero intersymbol interference corresponds to local minima of this error functional. We also present convergence analysis to predict the output mean square error in both these sign algorithms. Since the algorithms are highly nonlinear we incorporate several simplifying approximations and provide heuristic justifications for the validity of these approximations when the algorithms are operated in a typical practical environment. Computer simulations demonstrate the accuracy of the predicted convergence behavior.

1. Introduction

Adaptive equalization of communica t ion channels is usually accomplished using a known training sequence in proper synchronizat ion with the transmitted data sequence. In blind equalization, which has received more at tention recently, channel equalization is achieved without recourse to such a training sequence. The importance of blind equalizers with special reference to mult ipoint communica t ion networks has been underscored by G o d a r d [1] in one of the early contr ibutions in this area. Several techniques for blind

* Received May 18, 1990. This research was supported in part by the Office of Naval Research under Contract N00014-89-J-1538.

1 Department of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsyl- vania 19104, USA.

Now at AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974, USA.

394 WEERACKODY, KASSAM, AND LAKER

equalization have appeared in the literature [,-1]-[,6] since Sato [-7] introduced the concept of blind equalization for PAM data transmissions. A particularly useful blind equalization algorithm that is referenced widely is the generalization of Sato's algorithm for complex or QAM data transmissions which we refer to as the Generalized Sato Algorithm (GSA). This generalization was introduced independently by Benveniste and Goursat [-3] and by Godard and Thirion [-2] as one of many "reduced constellation type" algorithms.

In conventional adaptive equalizers, the equalizer coefficients are usually updated using the stochastic gradient algorithm to minimize the mean square value of the difference between the equalizer output and the desired data symbol, which is obtained from a training sequence. In blind equalization the desired data symbol is inaccessible; therefore, a simple teehnique to update the equalizer taps is sought by employing a stochastic gradient approach to minimize an error functional based upon the equalizer output and the transmitted data constellation statistics. The error functional that is minimized in the GSA is given by

E{I e(n)12} = E{I z(n) - 7 csgn(z(n))12}, (1.1)

where z(n) is the equalizer output and csgn(z(n)) is the complex sign function defined by

csgn(z(n)) --- sgn(zr(n)) + j sgn(zi(n))

with subscripts r and i denoting the real and imaginary components, respectively. The quantity 7 is a gain constant, the value of which will be considered shortly.

It is clear from (1.1) that 7 esgn(z(n)) acts as an estimate for the desired transmitted data symbol which is unavailable at the receiver. In decision- directed equalization schemes an estimate for the desired data symbol is formed by quantizing the output of the equalizer to points of the data constellation using a nearest-neighbor rule. However, it is well known that this technique will work satisfactorily only when the output of the quantizer is correct with a high probability, i.e., the distortion introduced by the transmission channel is low. Since blind equalizers are employed in situations where channel distortion is usually high, it may be anticipated that a coarser quantization scheme for the equalizer output will lead to satisfactory results. Figure 1 depicts the quantization of the equalizer output, z(n), to 7 csgn(z(n)) for 16-QAM and 64-QAM rectangular data constellations. In essence, the quantization is carried out in such a manner that the output of the equalizer is with reasonable probability correct upto a quadrant.

The stationary points of the error functional (1.1) as a function of the coefficients of the composite system resulting from cascading the channel with the equalizer have been investigated in great detail by Benveniste et al. [8]. They show that for an equalizer having an infinite number of taps and for

SIGN ALGORITHMS FOR BLIND EQUALIZATION 395

? -

csgn (z(n)) , \ ,

+

I ?

X z(n)

?

I

(a) (b)

Figure 1. Illustration of coarse quantization of equalizer output z(n) to V csgn(z(n)). (a) 16-QAM data constellation, (b) 64-QAM data constellation.

a certain class of non-Gaussian input signals the error functional (1.1) has minima corresponding to the case of zero intersymbol interference. It should be noted that these minima have the same magnitude and they correspond to different shifts of the taps of the infinite length equalizer.

Using the stochastic gradient of (1.1) to update the equalizer taps, we get

C(n + 1) = C(n) - ~(z(n) - Y csgn(z(n)))X*(n), (1.2)

where C(n) is the tap vector of the equalizer at time n, X(n) is the vector of input values stored in the equalizer at time n, and ~ is a constant known as the step-size parameter. In (1.2) the * denotes the complex conjugation. The above algorithm is depicted in Figure 2.

In the steady state we may approximate the equalizer output, z(n), by the data symbol, a(n). It is seen from (1.2) that in the steady state the expected value of the tap update term is zero. Therefore, using z(n) = a(n) in the steady state of (1.2) and assuming that the real and imaginary components of a(n) are IID and independent of each other, we get

E{a~(n)} E{a~(n)} ? - E{I a,(n)1} - E{] ai(n)1}" (1.3)

/~ ~y__csgn(z(n))

a(n) ] Channel ] x(n) I Equilizer -[ h(.) ] C(n)

/ Figure 2. Simplified discrete representation of the channel and the equalizer. The equalizer coefficients are updated so that E{e2(n)} is minimized.


In the above we have assumed that the components of X(n) are given by the convolution of the data symbol sequence {a(n)} with the channel response {h(n)}.

The need for increased speed of computation and simplicity of operation necessitates that various quantities associated with an adaptive filtering algorithm be quantized appropriately. To this end we introduce two modifi- cations of the algorithm of (1.2) by employing a one-bit quantization for either the error signal or the components of the input vector of the equalizer. The resulting algorithms are referred to as the Generalized Sato Sign Algorithms (GSSA). The absence of multiplications in the adaptation process of these algorithms make them attractive especially in high-speed applications such as microwave digital radio. Considering the fact that many equalizers are implemented as fractionally tap-spaced equalizers [9], the reduction in multiplications brought about by these modified sign algorithms is quite significant. Specifically, the sign variants of the GSA which we consider are

GSSA(e): C(n + 1) = C(n) - ~ csgn(z(n) - Ye csgn(z(n)))X*(n), (1.4)

GSSA(x): C(n + 1) = C(n) - c~(z(n) - Yx csgn(z(n))) csgn(X*(n)). (1.5)

In the above, as in the GSA, 7e and 7x are real constants that determine the gain of the equalizer and csgn(X) is a complex vector whose ith element is obtained by taking the complex sign of the ith component of X. Since it is desired to make the output of the equalizer converge toward a(n), in Section 2

we evaluate 7e by setting z(n) = a(n). The value of 7e thus obtained is K/x/~ , where K is the maximum value of the real or the imaginary component of the transmitted complex data symbol a(n), the components of which are assumed to be continuous and uniform IID random variables. The continuous distribution approximation for the data symbols can be justified easily for higher-order QAM distributions. We have observed that, even for a 16-QAM data constellation, ye computed in this manner gives satisfactory results. We do not have an approximate value for 7~ in (1.5) which is independent of the unknown channel samples and which drives the output of the equalizer toward a(n). In Section 4 we have incorporated a simple AGC loop after the equalizer to circumvent this problem.

The resemblance of the GSA and its sign variants to the Widrow-Hoff LMS algorithm and its corresponding sign variants is apparent. As noted previously, in the LMS forms, y csgn(z(n)), the estimate for the transmitted data symbol, is replaced by the data symbol itself. It is well known that for the same step-size parameter e, the steady-state excess mean square error (MSE) in the sign-error and sign-data LMS algorithms are significantly larger than that of the unmodified LMS algorithm [-10]. Additionally in the sign-data algorithm, which is the LMS analogue of (1.5), stability is an issue that has to be given careful consideration [11], [12]. The larger steady-state MSE in the


LMS sign-error algorithm can be explained as follows. The tap update term in this algorithm is sgn[z(n) - a(n)]X(n) (assuming real data). Because of the finite step-size parameter, in the steady state z(n) will only be a close approximation to a(n). Since the error is hardlimited the update term will be the same in magnitude irrespective of the size of the error, leading to a larger variance in the taps and thereby a larger steady-state output MSE. In contrast, the update term in the LMS algorithm is [z(n) - a(n)]X(n), which is proportional to the error and leads to a smaller tap variance and smaller output MSE for the same value of 0r. It should be noted that the tap update in the sign-error algorithm results from taking the stochastic gradient of the convex cost function E{[ z(n) - a(n)[}, for which the stationary points are well defined [13].

The performance of the GSA and its sign-error version, the GSSA(e), can be examined in the above context. It is clear that the error functional (1.1) does not vanish (except for a four-level QPSK system) even in the case of zero intersymbol interference, when it attains its minimum value. Therefore, the tap update term of(1.2), [z(n) - ~ csgn(z(n))]X*(n) , will be quite significant in the steady state. If it is necessary to reduce the output MSE smaller values of

will have to be used, which in turn results in slower convergence. Hence, from the above it may be anticipated that an algorithm resulting from hardlimiting this error functional may not affect the steady-state performance as drastically as in the LMS algorithm. We have in fact observed that the performance of GSSA(e) is comparable with that of the GSA, and comes about with a reduction in number of multiplications.

The paper is organized as follows. In Section 2 we examine the stationary points of the error functional which is minimized by the GSSA(e). We present convergence analysis to predict the output MSE trajectory of the real versions of the GSSA(e) and GSSA(x) in Section 3 and provide simulation results in Section 4. Section 5 contains some conclusions.

2. Stationary points of the error functional of the GSSA(e)

The GSSA(e) given in (1.4) results from taking the stochastic gradient of the following error functional:

E{~(n)} = E{lzr(n) - ~)e sgn(zr(n))]} + E{lz~(n) - V~ sgn(z~(n))[}, (2.1)

where the subscripts denote the real (r) and imaginary (i) components. This can be shown by expressing zr(n ) and zi(n ) in terms of z(n) and z*(n) and then taking the gradient of (C(n) with respect to the tap weight vector, C(n). Before examining the stationary points of the complex system (2.1) we consider the corresponding real system. We then extend these results to the complex case characterized by (2.1).


(a) Stationary points of the real GSSA(e)

It follows from (2.1) that the corresponding error functional for the real system is given by

E{((n)} = E{] z(n) - 7e sgn(z(n))1}, (2.2)

where z(n) is the equalizer output which is a real quantity in this case and 7e is a positive constant, which together with sgn(z(n)) forms an estimate for the desired data symbol, a(n).

In this subsection we assume that the equalizer has an infinite number of taps and that the data symbols a(n) are zero mean, continuous, IID symbols uniform on [ - K , K]. A similar assumption has been used by Benveniste et al. [8] in examining the stationary points of the GSA error functional given in (1.1). However, in that situation they consider a more general continuous distribution for the data symbols a(n).

Denote by s(k) the samples of the composite impulse response, i.e., the impulse response of the channel and the equalizer cascaded. Then we may write, for the equalizer output,

z(n) = ~ s(k)a(n - k). (2.3) k

For ease of analysis we rewrite the error functional in (2.2) as

E{((n)} = E{I ]z(n)[ - Yel}- (2.4)

Differentiating (2.4) with respect to s(k) and setting the resulting expression to zero, we have

E{sgn(lz(n)[- 7~) sgn(z(n))a(n - k)} = 0, k = 0, + 1 , . . . . (2.5)

In seeking a solution to (2.5), let us assume that the solution for the sequence {s(k)} has only N (N > 0) nonzero components. Since sgn(z(n)) is undefined for z(n)= 0, the case for N = 0 is considered separately in Appendix A. Using (2.3) to substitute for z(n) in (2.5) and observing that the random variables a(n) are independent of each other, we get

E{sgn( l~s(l)a(n - l) - 7~) sgn(t~s(l)a(n -1))a(n - k)} = O,

k = 0, + 1 , . . . , (2.6)

where the set ~: consists of the indices of the nonzero components of {s(k)}. For k ~ K we get N nontrivial equations from (2.6). Since a(n) are IID and symmetrically distributed about the origin it follows that a solution for (2.6) exists when all the N nonzero components of {s(k)} have the same magnitude. It should be stated that SN, the common magnitude of these nonzero components of {s(k)}, is different for different values of N. Suppose m is the median of the random variable 2 whose probability density function (pdf) f~


is given by f~(x)= xfz(x)/~g x s x > 0 and f~(x)= 0 for x < 0. Here f~(x) is the pdf ofz(n) given by (2.3) with the N nonzero components of {s(k)} set to unity. Then we show in Appendix A that SN = ye/m. It should be noted that we have not addressed the issue of the existence of stationary points other than those considered above. In [8] Benveniste et al. examine the nature of the stationary points of the error functional used in the GSA when the input data symbols are IID and have a non-Gaussian distribution. In that paper they demonstrate that at a stationary point of this error functional all the nonzero components of the composite impulse response have the same magnitude. Also, the stationary points of one of the error functionals introduced by Godard in [1] (the case for p = 2 in [1]) exhibit a similar behavior for data symbols which come from a discrete uniform distribution. Therefore, because of the similarity of the error functionals used in these blind equalization algorithms we may conjecture that when the data symbols are uniformly distributed the stationary points of (2.4) are represented by N equal magnitude samples of the composite impulse response.

From (2.3) it can be seen that the N = 1 cases correspond to the situation of zero intersymbol interference. We show in Appendix A that these cases correspond to local minima of the error functional (2.4). In order to determine ye we set the magnitude of the nonzero component of {s(k)} equal to unity. As seen from (2.3), this condition ensures that the output of the equalizer converges to points of the data constellation instead of a scaled version of it. Using this in (2.6) we have

E{sgn([ a(n -- j)[ -- 7e) sgn(a(n -- j ) )a(n -- j)} = 0, (2.7)

where s(j) is the only nonzero component of the composite impulse response. Assuming that a(n) is uniform in [ - K , K] it can be shown using (2.7) that

7~ = K/x//~. (2.8)

We show in Appendix A that the N = 0 case corresponds to a local maximum of the error functional (2.4) and cases corresponding to 1 < N < 20 are "saddle points" or points of unstable equilibrium of (2.4). Further- more, by extending the results obtained for 1 < N _< 20 to N > 20, we may conjecture that all cases for N > 1 correspond to points of unstable equilibrium. Therefore, this error functional may be used as a legitimate error functional for blind equalization. We point out that the case for N = 1 corresponds to multiple local minima of (2.4). These multiple local minima have the same magnitude and correspond to different shifts of the nonzero sample of the composite impulse response sequence. In a practical situation this problem can be easily overcome by suitably initializing the equalizer so that the shifting of the tap weights can be predetermined. Next, we extend the above results obtained for the real system to the complex system whose error functional is given by (2.1).


(b) Stationary Points of the (Complex) GSSA(e)

For ease of analysis we rewrite (2.1) as follows:

E{~C(n)} = E{llzr(n)l- 7el} + E{[]zi(n)[- 7el}. (2.9)

We assume that the real (ar(n)) and the imaginary (ai(n)) components of the complex data symbol, a(n), are zero-mean uniform IID random variables independent of each other. Then it follows that the components of the equalizer output, zr(n) and zi(n), are identically distributed. Using this in (2.9) we get

E{(C(n)} = 2E{[ ]z,(n)[- 7~1}. (2.10)

Now, the composite impulse response samples, s(k), of the channel and the equalizer are complex. Let us denote the real and imaginary components of s(k) as sr(k) and si(k), respectively. Introduce the new sequence t(k) as follows:

t(2k) = s~(k), (2.1 i)

t(2k + 1) = - s i ( k ).

Also, introduce the new IID sequence b(n) as

b(n - 2k) = a~(n - k), (2.12)

b(n - 2k - 1) = ai(n - k),

where k takes on any integer value. Using (2.11) and (2.12) we have

zr(n) = ~ t(k)b(n - k). (2.13) k

From (2.10) and (2.13) it follows that the complex error functional given by (2.9) is very similar to the real error functional of (2.4) considered in the previous subsection with t(k) and b(n) being, respectively, the corresponding variables for s(k) and a(n) in (2.4). Therefore, we conclude that in the complex case the error functional has local minima only when a single component of t(k) is nonvanishing, i.e., for the ease of zero intersymbol interference. Also, it

follows from (2.8) that 7r = K/~//~, where we have assumed that ar(n) and ai(n) are distributed uniformly in [ - K , K]. From (2.11), (2.12), and (2.13) we see that if the index 1 of the nonzero sample of {t(k)} is odd, then

z~(n) = t(l)ai(n-l~+2~l ) . (2.14)

That is, this situation corresponds to a rotation of the data symbol constellation by 90 ~ This is a common phenomenon in blind equalization algorithms when a symmetric density function is assumed for the data symbols. Since the transmitted data symbols are usually differentially encoded, this "modulo - n/2" uncertainty of the data symbols at the receiver can be ignored. In


passing we state that Benveniste and Goursat [3] have used a similar argument as that presented in this subsection to extend the results obtained for the real GSA to the complex GSA.

3. Convergence analysis to predict the MSE trajectory

In this section we present convergence analysis to predict the evolution of the MSE trajectory of the real versions of the GSSA(e) and GSSA(x). The highly nonlinear nature of the algorithms makes the task of analyzing these algorithms difficult. We introduce several simplifying approximations to alleviate this problem. We present justifications for the validity of these approximations when the algorithms are employed in typical practical environments. Since the complex system (i.e., for QAM data symbols) gives rise to added complexity without much additional insight, in this section we concentrate only on the real system (i.e., for PAM data symbols). The extension to the complex system is straightforward as seen from a similar analysis of the GSA [14].

The similarity of these blind equalization algorithms to the corresponding LMS versions has already been noted. The approach that we adopt to analyze these algorithms is in general similar to the analysis of some LMS- related algorithms presented in [15] and [16]. Specifically, we derive an expression for the MSE of the equalizer at each time instant in terms of the first- and second-order moments of the equalizer taps. Next, recursive relations are derived for the first- and second-order moments of the taps with the statistics of the data sequence, the channel impulse response, and the step- size parameter, ~, as the variables. The nonlinearities introduced in these algorithms would give rise to higher-order moments of the equalizer taps in an exact analysis. However, we have used reasonable approximations to avoid the appearance of these higher-order terms in the analysis. We assume that the equalizer is initialized in such a manner that its output asymptoti- cally approaches a(n), rather than a time-shifted version of a(n). Since in blind equalization intersymbol interference dominates the degradation of the channel output, in this analysis we ignore the effects of additive noise.

For ease of analysis we make use of a transformation that orthogonalizes the equalizer input vector X(n) = [x(n + L) . . . x(n - L)] T, with (2L + 1) being the number of taps of the equalizer and x(n) given by

x(n) = {h(n)} �9 {a(n)}. (3.1)

Here �9 denotes the convolution operation and h(n) is the channel impulse response. Denote by R the (2L + 1) x (2L + 1) channel covariance matrix with elements given by

Rij = E{Xi(n)Xj(n)}, 1 < i,j < 2L + 1, (3.2)


where X~(n) is the ith component of X(n). We assume that the input data are stationary so that R is independent of the time index n. Using (3.1) and (3.2) above, we can show that

R,; = E{aZ(n)} ~, h(k)h(k + i - j ) . (3.3) k

Since R is a positive definite matrix, there exists a unitary matrix U (i.e., U r = U-1), such that

p = U R U T, (3.4)

where p is the diagonal matrix formed by the eigenvalues, 2i of the channel autocorrelation matrix R, given by

p = diag{21, ,~2 . . . . . )]'2L+ 1}. (3.5)

The output of the equalizer, z(n), is given by

z(n) = CT(n)X(n)

= WT(n)Y(n), (3.6)

where the vectors W(n) and Y(n) are respectively the tap weight vector and the equalizer input vector in the transformed domain given by UC(n) and UX(n). It can be seen from (3.2) and (3.4) that E{Y(n)yT(n)} = p.

Next we list the primary assumptions that are employed in the analysis and their justifications. These assumptions are necessary to make the analysis tractable while giving rise to satisfactory results. They are similar to those assumptions we have made in [14] in analyzing the convergence of the GSA.

(a) Primary assumptions

(i) The data symbols, a(n), are zero-mean IID symbols derived from a PAM data constellation whose amplitude levels are uniformly distributed.

(ii) The equalizer input vector X(n) conditioned on the particular data symbol, a(n), is a Gaussian random vector.

(iii) The equalizer taps in the transformed domain, W(n), are uncorrelated. That is

E{W~(n)W~(n)} = ~E{W~(n)}E{W~(n)}, i ~ j, (3.7) (F,(n), i ----- j.

Here F~(n) is defined as E{WZ(n)}. (iv) The tap weight vector, C(n), is independent of the equalizer input

vector, X(n). (v) The equalizer output, z(n), conditioned on a(n) and C(n) is a Gaussian

random variable with mean and variance of the distribution approxi-


mated as follows:

E{z(n)]a(n), C(n)} = E{z(n)[a(n)}, (3.8)

z (3.9) var{z(n) la(n), C(n)} = a~,

2 is the MSE at the output of the equalizer given by E{e2(n)}. where r

Comments on the assumptions. (i) Many PAM systems meet approximately the conditions of Assumption (i); this is a reasonable assumption.

(ii) Since {X(n)} is the output of a linear channel with IID non-Gaussian inputs, {a(n)}, the sequence X(n), given a(n), cannot be strictly Gaussian. However, to make the analysis tractable we make use of this assumption. For typical communication channels this key assumption may be justified by the central limit theorem. This assumption leads to good agreement between the theoretical and simulation results presented in Section 4.

(iii) To justify the validity of this assumption we present a simulation result in Figure 3. The figure depicts the correlation coefficient between the first and the rest of the taps of the 11-tap equalizer in the transformed domain for the real GSSA(e). The channel impulse response used is shown in Figure 4(b) and the two curves in Figure 3 correspond to the ensemble averaged (over 300

1.0

r

r

0

8

0.8

0.6

0.4

0.2

1000 iterations

5000 iterations

0.0 1 ~ ~

,0 2 4 6 8 10 separation in tap spacings

Figure 3. Correlation coefficient for the components of the tap weight vector used to justify Assumption (iii). The plot depicts the correlation coefficient between W~(n) and Wj(n) for j = 1 . . . . . 2L.


1.0 0.98

0.27

(a)

0.27

0.06 0.03 I

0.01 I

0.05

0.58

0.?3

0.36

(b)

0.,02 0'107 0.01 0.01

0.854

0.009 0.049 . , i I

0.005 0.()24

0.218 Real Component of h(n)

0.52

0.273

0.03 0.02

0.016 0.004 [ J 0.104 0.074

Imaginary Component of h(n) (c)

Figure 4. Impulse responses of the channels. (a) Channel (1), (b) Channel (2), (c) real and imaginary parts of a typical voice-band communication channel.

independent runs) correlation coefficient after 1000 and 5000 iterations. By observing the learning curves in Figure 5(c) it can be seen that the result at the end of 1000 iterations corresponds to the transient state and the result at the end of 5000 iterations corresponds to the steady state. Therefore, we may conclude that the correlation between the taps in the transformed domain is small both in the transient and the steady states. We note that this assumption has been shown to be valid under some general conditions for the complex LMS algorithm [17].

(iv) For small values of the step-size parameter, c~, we may assume that C(n) is independent of the more recent data samples and therefore X(n). This assumption is frequently encountered in the adaptive filtering literature [18], [19].

(v) From Assumption (ii) the equalizer input, X(n), conditioned on a(n) is Gaussian. Therefore it follows that the equalizer output, z(n), conditioned on a(n) and C(n) is also Gaussian. In order to substantiate the approximation E{z(n)la(n), C(n)} = E{z(n)la(n)}, note that for small values of ~ the variations of C(n) about its mean value are very small. Therefore, it may be assumed that the trajectory of C(n) is approximately deterministic and is given by the mean value trajectory of C(n). In passing we note that an


assumption similar in style to the above has been employed by Chou and Mathews 1-20] in the analysis of the LMS sign-error algorithm. Also, Honig and Messerschmitt [21,] use a similar assumption for the PARCOR coefficients in the analysis of an adaptive lattice structure.

Next to justify (3.9), it follows from the preceeding discussion that var{z(n) la(n), C(n)} = var{z(n) la(n)}. Next, if s(n) is the composite impulse response of the channel and the equalizer cascaded it can be shown that

var {z (n ) [a (n ) } = E{a2(n ) } Z s2(m) mr

and 2 E{eZ(n)} = at = E{a2(n)}[(s(O) - 1) 2 + ~ s2(m)],

m ~ O

where we have used the fact that e(n) = z(n) - a(n), In the steady state we know that s(0) is approximately equal to unity. Furthermore, the dominant component of the channel response is usually positioned at the center. Also, the equalizer is initialized so that only the center tap has a nonvanishing value which is given by unity. Considering the above we may approximate s(0) by unity even during the transient stages. Therefore, we may write var{z(n) la(n)} = E{e2(n)} = rr~ z, and hence (3.9).

It should be noted that blind restoration of a general communication channel cannot be achieved if the input data are Gaussian [8]. In the above, although we approximate the equalizer input conditioned on the data symbol to be Gaussian we do not assume that either the data sequence or the output of the equalizer is Gaussian.

(b) Equalizer output M S E expression

The output MSE, E{eZ(n)} = a 2, may be written as

2 rr~ = E{[z(n) - a(n)'] 2}

= E{zZ(n)} + E{aZ(n)} - 2E{z(n)a(n)} . (3.10)

Since in Assumption (iv) we have assumed that C(n) and X(n) are independent, it follows that in the transformed domain W(n) and Y(n) are also independent. Using this and considering the fact that the components of u are uncorrelated, we get from (3.6)

E{zZ(n)} = p~r(n), (3.11)

where Pl = [21, 22 . . . . . 22L+ 1]T is defined as the vector of eigenvalues and F(n) is the vector of second-order moments of the equalizer taps in the transformed domain with the ith component given by Fi(n ) = E{WZ(n)} .

Next we consider E{z(n)a(n)} in (3.10). From (3.6), the independence Assumption (iv), and (3.l) it follows that

E{z(n)a(n)} = E{aZ(n)}Mr(n)~l , (3.12)


where M(n) is the mean tap weight vector in the transformed domain with the ith component given by Mi(n)= E{W/(n)} and r I is the channel impulse response vector in the transformed domain given by

q = Uh, (3.13)

with h = [h(L)... h ( -L)] r being the channel impulse response vector. Substituting (3.11) and (3.12) in (3.10), we get

2 9[F(n) + E{a2(n)} 2E{a2(n)}Mr(n)q. (3.14) 0" 6 ~_

Equation (3.14) expresses the output MSE as a function of the mean vector M(n) and the mean-square vector F(n) of the transformed tap weights. Next we consider the real versions of GSSA(e) and GSSA(x) and derive recursive relations for M(n) and F(n).

(c) Convergence analysis of the real. GSSA(e)

From (1.4) we have, for the real GSSA(e),

C(n + 1) = C(n) - ~ sgn(z(n) - 7e sgn(z(n)))X(n). (3.15)

Premultiplying (3.15) by U, we get

W(n + 1) = W(n) = ct sgn[z(n) - 7e sgn(z(n))]Y(n). (3.16)

Analytical results for the mean tap vector. To obtain a recursive relation for the mean tap weight vector, we take expectations on both sides of (3.16):

M(n + 1) = M(n) - ~E{sgn[z(n) -- Ye sgn(z(n))]Y(n)}. (3.17)

Using the assumptions stated in Section 3(a) the expectation on the right- hand side of (3.17) is evaluated in Appendix B. From (B1.4) in Appendix B and (3.17) we get

M(n + 1) = M(n) -- ct{211[--f~ - f ~ ) + fo(?~)]

E{a(n)2}(p -- llqr)M(n) +

0"~

Here the functions f~ and f l ( x ) are respectively defined as E{a(n)Fo((x + N/a,)} and E{FI(I x + # I/~h)} with the expectation taken over the data symbol constellation. In the above the functions Fi(x) and the

variable # are defined as Fi(x)= ( 1 / ~ ) ~ z i exp(-z2/2)dz, i = 0, 1, and # = E{z(n)la(n), W(n)}. From (B1.2) in Appendix B we have # = a(n)Mr(n)q.


In (3.18), p and q are respectively the diagonal matrix of channel eigenvalues and the channel impulse response vector in the transformed domain, ct and 7e are constants, and fo and f~ can be computed for a given data constellation and channel impulse response, assuming that ae and M(n) are available. Next, we present a recursive relation for the mean square tap weights.

Analytical results for the mean square tap weights. Squaring each component of (3.16) and taking expectations on both sides, we have

Fk(n + 1) = Fk(n) + ~2E{ y2(n)} - 2~E{sgn[z(n) - 7e sgn(z(n))] Yk(n)Wk(n)},

k = 1 , . . . , (2L + 1). (3.19)

Next we derive expressions for the components on the right-hand side of (3.19). Since Y(n)= UX(n), we see from (3.4) that E{Y](n)} = Pkk. The second expectation on the right-hand side of (3.19) is given by (B1.5) in Appendix B. Combining these in (3.19) we get

Fk(n + 1)

E{a(n)2} = Fk(n) + ~2pkk -- 2~ 2~lkMk(n) { - fO(0) -- fO(_ 7e) + f0(7~)} + - -

O" e

X { ~kk~k(ri)-~ ir ~')kiMk(n)Mi(n)}

k = l , 2 . . . . . ( 2 L + l ) , (3.20)

where the matrix f~ = [p - i!I!r]. The mean square tap vector recursions of (3.20) together with the mean tap vector recursions of (3.18) and the MSE expressions (3.14) describe the evolution of the MSE trajectory of the real GSSA(e). The parameters for these recursions are p and r/which depend on the channel impulse response, constants ~ and 7~, and the functions fo and f l . As noted previously, the functions f o and f1 can be computed for a given data constellation and channel impulse response when the mean tap vector, M(n), and the MSE, aft, are known.

(d) Convergence analysis of the real GSSA(x)

The Generalized Sato Sign-Data Algorithm (GSSA(x)) for the real system follows from (1.5):

C(n + 1) = C(n) - ~(z(n) - 7~ sgn(z(n))) sgn(X(n)). (3.21)

Here sgn(X(n)) is the vector whose ith component is given by hardlimiting the ith component of X(n). As in the preceeding subsection premultiplying (3.21)


by U we have

W(n + 1) = W(n) - e(z(n) - 7~ sgn(z(n)))O(n), (3.22)

where 0(n) = U sgn[X(n)], and W(n) is the transformed tap weight vector given by UC(n). The MSE at the output of the equalizer in terms of the mean (transformed) tap vector M(n) and the mean square (transformed) tap vector F(n) is given by (3.14). Next, as in the previous subsection, we derive recursive relations for M(n) and F(n).

Analytical results for the mean tap vector. Taking expectations of both sides of (3.22) and recalling that the mean tap weight vector in the transformed domain is M(n) = E{W(n)}, we have

M(n + 1) = M(n) - c~E{(z(n) - 7x sgn(z(n)))O(n)}. (3.23)

In Appendix B we evaluate the quantities E{z(n)O(n)} and E{sgn(z(n))O(n)} using the assumptions listed in Section 3(a). Denote by Y~ = E{a2(n)} (R - hhr)urw(n) the covariance vector of z(n) and X(n) conditioned on a(n) and W(n) and the matrix A as A = E{a2(n)}(R -- hhr). Define the functions

P~m as f t /a(n)hi'Q ptm(i) = E~a ( n ) F , d - - / t - , l, m = 0, 1, (3.24) t

where the expectation is taken over the random variable a(n). Then using (B2.2) and (B2.5) in Appendix B to substitute in (3.23) we have

y~ [AU~q,MSn) Mk(n +.1) = Mk(n ) -- ~ _;" Uki 2plo(i)Mr(n)R + J

In the above we have defined 4o(0 as

{f? 2 E 1 q~(i) = ~ x/1 _ x 2

x e x p ( - a 2 ( n ) //[MT(n)q]2 2(1 - x 2) \ ~r 2

EY 4Fo f a(n)h''~F o(a(n)Mr (n)"'~ ~, + t t )J

k = 1 , 2 , . . . , ( 2 L + 1). (3.25)

2Mr(n)qhl h~ dx}

(3.26)

with the expectation taken over the data symbol a(n) and ~5 k =

Ei[AUT]kiMi(n)/ae~kk being the correlation coefficient between z(n) and Xk(n ) conditioned on a(n) and W(n).


Equation (3.26) gives a recursive relation for the mean tap vector in the transform domain in terms of the channel impulse response, the data constellation statistics, and the MSE error at the output of the equalizer. The matrices U and A and the vector !1 can be computed when the channel impulse response is given. Since it is assumed that the data symbols, a(n), come from a discrete distribution with a finite number of levels, the quantities Pol, Plo, and ~0(i) can be evaluated in a simple manner when the channel impulse response, the data constellation, the mean tap vector, M(n), and the MSE are given. From (3.14) it is seen that the output MSE is a function of M(n) and the F(n). In the following subsection we derive a recursive relation for the mean square (transformed) tap weights, F(n).

Analytical results for mean square tap weights. To obtain a recursive relation for the mean square of the transformed tap weights, squaring each component of (3.21) and taking expectations of the resulting set of equations, we get

Fk(n + 1) ---- Fk(n) -- 2~tE{[z(n) -- ~'x sgn(z(n))]Ok(n)Wk(n)} 2 2 + ctZE{[zZ(n) - 2Vx]z(n)[ + 7x]Ok(n)},

k = 1 , 2 . . . . , ( 2 L + l ) . (3.27)

Next, in order to compute the last term on the right-hand side of (3.27), we introduce the following additional assumptions.

(vi) The equalizer output is given by z (n )= a(n)+ e(n) with e(n), the 2 Also, it error, distributed normally with mean zero and variance o, .

is assumed that e(n) is independent of the IID data sequence {a(n)}. (vii) The equalizer input data vector X(n) conditioned on the data symbol

a(n) and the equalizer output z(n) conditioned on a(n) are independently distributed.

In order to justify Assumption (vi) consider the composite impulse response of the channel-equalizer cascade, s(n). We see from the justifications of Assumption (v) in Section 3(a), that e(n) = a(n)(s(O)- 1) + ~ , , , o s(m)a(n - m). If none of the quantities, s(m) for m # 0, and (s(0) - 1) is dominant, then it follows from the central limit theorem that e(n) is approximately a zero-mean Gaussian random variable. Next, consider the following: E{e(n)a(n)} = E{aZ(n)}(s(O) - 1). In the steady state we know that s(0) ~ 1. Also, as discussed previously, because of the equalizer initialization we may assume that s(0) is close to unity even during the transient stages. Therefore, e(n) and a(n) are uncorrelated. We may further assume that they are independent.

Since in the steady state s(0) ~ 1 independent of the equalizer initialization scheme, we may argue that the above justification will be more reasonable in the steady state rather than in the transient state. In fact, these assumptions (Assumptions (vi) and (vii)) will be used to compute quantities which have a significant contribution only in the steady state. In


order to substantiate this, it can be seen from the Appendix B that Assump- tions (vi) and (vii) are used only to compute the second term (i.e., the term containing ~z) in (3.27). Since ~ is small the contribution from this term to the total MSE is small when compared with that of the first term in (3.27) which is multiplied by ~. However, in the steady state, from (3.23) we see that E{[z(n)- Vx sgn(z(n))]Ok(n)} = 0. In the steady state, for small values of c~, the random variable Wk(n ) will be slowly varying when compared with that of [ z ( n ) - ~ sgn(z(n))] Ok(n). Therefore, from the averaging principle [22] it follows that the random variables Wk(n) and [ z ( n ) - 7x sgn(z(n))] Ok(n) are approximately uncorrelated. Since in the steady state E{[z(n)--v~sgn(z(n))]Ok(n)}=O , it follows that E{[z(n)- ~ sgn(z(n))]Ok(n)Wk(n)} will be very small in the steady state. Therefore, the contribution to the MSE from the second term in (3.27) will be significant in the steady state. Since, as noted before, the contribution to the MSE from the second term in (3.27) is insignificant during the transient stages, it follows that Assumptions (vi) and (vii) are used only to compute quantities which have a significant effect on the MSE only in the steady state.

Next, in order to justify Assumption (vii), we consider the correlation between z(n) and X(n) when conditioned on a(n). From Assumption (vi) and (B3.2) in Appendix B it follows that E{z(n)X(n)la(n)}= E{(a(n) + e(n))X(n)[a(n)} = a2(n)h. Also, it can be seen that E{z(n)l a(n)} --- a(n) and E{X(n)la(n)} = a(n)h. Therefore, it follows from the above that z(n) and X(n) conditioned on a(n) are uncorrelated. Since z(n)= Cr(n)X(n), we have already noted that z(n) and X(n) conditioned on a(n) and C(n) are jointly Gaussian. The desired result follows if we assume that the tap variations are small and therefore z(n) and X(n) conditioned on a(n) are jointly Gaussian.

In Appendix B we derive expressions for the individual components on the right-hand side of (3.27). Suppose O(ij)= E{sgn(Xi(n))sgn(Xj(n))l a(n)}; using Price's theorem [23] to evaluate this quantity we have, from (B2.12) in Appendix B,

f ( ( h2 a2(n) h~ 2h~hj 'J "~'~dx 2 ~'J 1 exp - - x + 0(/.7)=~oo x/I: x 2 2(]-:x 2) ~ ~ A##//]

,~ fa(n)hi'~_ fa(n)hj'~ § (3.28) ' q ' / " o - - / ' o - -

Here the quantity ~;j is the correlation coefficient between Xi(n ) and Xj(n) conditioned on a(n). Define the quantities ql(ij) and q2(ij) as follows:

ql(ij) = E{a(n)O(ij)Fo(~) } (3.29)


and

q2(ij)= E{O(ij)F~(~)}, (3.30)

where the expectation is over the data symbol a(n). Using (B2.7), (B2.10), (B2.14), and (B2.15) in Appendix B to substitute for the expectations in (3.27), we have

Fk(n+l)=Fk(n)--2o~Uki{2Plo(i)Mr(n)Mk(n)q+E~!--2Pol(i)]

x ~ E A,jUtiM,(n)Mk(n) x/ A . l~k

-b ~j AijUkjI"k(n)]-- )~x~O(i)Mk(n)}

{ 22 sin-I(R/J~ + E{a2(n)O(ij)} + a2E E Uk~Ukj a~ -- , j 7t \RiJ

{ ~: 2 . _fl'Rij'~ - 27x 2qt(ij) + a , - sm / - - / \ R J

2 . _ ll/Rij'~) - - - sin / - - / Y ~ 2a,q2(ij) + 7~ r~ \Ruf j

k = l , 2 . . . . . ( 2 L + l ) . (3.31)

The above equation together with (3.25) and (3.14) give a set of recursive relations to predict the evolution of the MSE at the output of the equalizer for the real version of the GSSA(x). When the channel impulse response and the data constellation statistics are given the quantities Plo, Pol, cO, ~, ql, and q2 can be computed at a given instant of the recursive process.

4 . S i m u l a t i o n r e s u l t s

We have carried out simulations to determine the accuracy of the analytical results obtained in Section 3 and also to examine the performance of the sign versions of the GSA when employed in a typical practical environment. Figure 4(a) and (b) depicts the impulse responses of the two channels which were used in conjunction with the real versions of the GSSA(e) and GSSA(x) and Figure 4(c) shows a typical voice-band communication channel which has both amplitude and phase distortion [6]. A key assumption in our analysis is the Gaussianity of the equalizer input vector conditioned on the data symbol. Clearly, Channel (1) in Figure 4(a) will be a poor choice for such


a model because it has a dominant component together with only two other components in its impulse response.

We have employed a four-level PAM data sequence together with an 11- tap equalizer to check the validity of the analytical results derived in Section 3. The equalizer was initialized so that the center tap was set to unity and all the other taps were set equal to zero. Also, the average power level of the input data sequence was normalized to unity. For the real GSSA(e) 7e = 1.265 and for the real GSSA(x) 7x was set to unity. The value of 7e was evaluated by assuming that the pdf of the data symbols was uniformly distributed in [ - 4 x 0.447, 4 x 0.447]. Since in blind equalization intersymbol interference is the dominant cause of distortion the effect of noise has been ignored both in our analysis as well as in the simulations which verify the accuracy of the analytical results. The analytical and ensemble-averaged (over 300 independent runs) simulation results for the evolution of the MSE trajectory are shown in Figure 5(a)-(f). The analytical results for the real GSSA(e) were obtained from the MSE expression (3.14), the mean tap vector recursions in (3.18), and the mean square tap weights in (3.20). For the real GSSA(x) we used (3.14), (3.25), and (3.31). The mean (transformed) tap weights were initialized so that the center tap of E{C(n)} was set equal to unity and all the other taps set equal to zero. The mean square tap weights were initialized so that Fk(n)= M~(n). The initial MSE for Channel (1) (Figure 4(a)) was 0.145 and that for Channel (2) (Figure 4(b)) was 0.479.

In Figure 5(a) and (b) we show the results for the real GSSA(e) for Channel (1) with c~ = 0.001 and e = 0.0005, respectively. The corresponding results for Channel (2) are shown in Figure 5(c) and (d). From these results it can be seen that the analytical model closely predicts the evolution of the MSE trajectory for the real GSSA(e). The learning curves for the real GSSA(x) are shown in Figure 5(e) and (t) for Channel (2) with e = 0.001 and c~ = 0.0005. As can be seen the analytical model for the real GSSA(x) predicts the evolution of the MSE trajectory reasonably well for Channel (2). However, we have observed that for Channel (1) the agreement between the analytical and simulation results is poor. This could be attributed to the fact that, for Channel (1), the key assumption of Gaussianity of the equalizer input vector conditioned on the data symbol is poor. Figure 6(a) depicts E{Cs(n)} for Channel (1) with the real GSSA(e) and Figure 6(b)-(d) depicts the mean value of the center tap, E{Ca(n)}, for the real GSSA(e) and GSSA(x) with Channel (2). It is clear from these simulations that the analytical model for the real GSSA(e) predicts very closely the MSE and the mean tap-vector trajectories whereas the analytical model for the real GSSA(x) gives a reasonably close result for Channel (2).

The performance of the GSSA(e) and GSSA(x) when employed with the channel response shown in Figure 4(c) is examined next. The 16-QAM and 64-QAM data constellations that are used here are shown in Figure 1. In our simulations we have normalized the average power level of the input data


"sAina~1321;d

- 1 5

-20 1500 3000 4500 Data points

(a)

--I0 AnaIysis

I ~Simulated

0 2000 4000 6000 8000 10000

Data points (b)

0

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1500 3000 4500 Data points

(c)

~ A . n a l y s i s

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Data points Data points

(e) (0

Figure 5. M S E from ensemble averaging and the trajectory predicted from analysis for real G S S A ( e ) ( (a)- (d)) and real G S S A ( x ) ((e) and (f)). (a) Channel (1), ~ = 0.001, y, = 1.265; (b) Channe l (1), e = 0.0005, 7e = 1.265; (c) Channe l (2), c~ = 0.001, ?e = 1.265; (d) Channe l (2), e = 0.0005, 7e = 1.265; (e) Channe l (2), ~ = 0.001, 7x = 1; (f) Channe l (2), c~ = 0.0005, 7x = 1.


0.00 ~ x ~ .

~176176 1 \ { -0.20 j ~ o N~gT____ Analysis = ~ l a t e d

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2000 4000 6000 8000 Data points

(a)

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0.9 aaaaaa a

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Data points Data points (c) (d)

1.5E+04 2.0E +04

Figure 6. Mean tap coefficient trajectory from ensemble averaging and analysis for real GSSA(e) ((a) and (b)) and real GSSA(x) ((c) and (d)). (a) Trajectory of Cs(n ), Channel (1), ~ = 0.0005; (b) Trajectory of C6(rl), Channel (2), ~ = 0.0005; (c) Trajectory of C6(rt), Channel (2), ~ = 0.001; (d) Trajectory of C6(n), Channel (2), ~ = 0.0005.

sequence to unity. A 21-tap complex equalizer is used in the simulations with a signal-to-noise ratio of 30 dB. In Section 1 we noted that 7~ in GSSA(x) determines the gain of the equalizer and that we do not have an approximate value for 7~ which is independent of the channel impulse response. Therefore, in this section we set 7~ = 1 and use an AGC loop to normalize the power level of the equalized Output sequence to unity. Suppose z(n) is the output of the equalizer, then the output of the AGC device is ~(n) = z(n)/a(n), where or(n) is given by

~r2(n) = (1 - fl)o'2(n - 1) + fllzZ(n)[.

In the case of the GSA the parameter 7 evaluated using (1.3) is given respectively by 0.79 and 0.81 for the 16-QAM and 64-QAM data constellations. For the 16-QAM data format 7e for the GSSA(e) is 0.89. This is computed by assuming that the real and the imaginary components of the data symbol are distributed uniformly in [ - 4 x 0.316, 4 x 0.316]. For the


-5-

-10"

-15-

--20 0.0E + 00 5.0E +03 1.0E+04 1.5E+04 2.0E + 04

Data points (a)

-10

-15

-20 O.0E + 00 1.5E+04

-s4 0 -15

-20 0.E+00

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1.E+04 2 .E+04 3,E+04 4.E+04 Data points

(b)

~ f

3.0E + 04 4.5E+IN 6.0E+04 Data points

(c)

Figure 7. (a) Learning curves for GSA, GSSA(e), and GSSA(x) for channel in Figure 4(c) with 16-QAM data and signal-to-noise ratio = 30 dB. (1) GSA, e = 0.0015,

= 0.79; (2) GSSA(e), e = 0.0008, 7e = 0.89; (3) GSSA(x), e = 0.0008, 7x = 1, fl = 0.001. (b) Learning curves for GSA and GSSA(e) for channel in Figure 4(c) with 64- QAM data and signal-to-noise ratio = 30dB. (1) GSA, ct = 0.0011, 7 = 0.81; (2) GSSA(e), c~ = 0.00045, 7~ = 0.818. (c) Learning curves for GSSA(x) for channel in Figure 4(c) with 64-QAM data and signal-to-noise ratio = 30 dB. e = 0.0007, Vx = 1,

= 0.001.

64 -QAM format we have observed that the best results for the performance of the algori thm can be obtained by comput ing 7e assuming that the real and the imaginary components of the data symbol are distributed uniformly in [ - 7 . 5 x 0.154, 7.5 x 0.154]. The value of ~)e thus computed is 0.818.

The learning curves for the GSA and its sign versions are shown in Figure 7(a) for a 16-QAM data constellation. The step-size parameter ct, for the GSA, GSSA(e), and GSSA(x) are given by 0.0015, 0.0008, and 0.0008, respectively. For the GSSA(x), /? was set to 0.001. Figure 7(b) and (c) shows the corresponding results for a 64 -QAM data format. F r o m these observa- tions it may be concluded that the rate of convergence of the GSSA(e) is comparable with that of the GSA. In the case of the GSSA(x) we have


observed convergence but the speed of convergence is much slower than that of the GSA.

5. Conclusions

We have introduced two new sign algorithms for blind equalization. These algorithms are simple to implement and are especially useful at high-speed applications. By way of computer simulations we have demonstrated that the convergence rate of one of these new sign algorithms is comparable with the convergence rate of the corresponding unsigned algorithm. The problem of cartier tracking has not been addressed here. The techniques employed in [24] may be extended to accommodate cartier tracking with these new algorithms. We have examined the stationary points of the error functional minimized by the sign-error algorithm and show that the case of zero intersymbol interference corresponds to the local minima of the error functional. We have also presented convergence analysis to predict the MSE trajectory at the output of the equalizer. In the analysis we make use of several simplifying assumptions which can be justified when the algorithms are employed in typical practical situations. We have observed that for the real GSSA(e), with Channel (1), which does not meet satisfactorily the key Gaussianity assumption, the predicted results are in close agreement with those obtained from simulations. This demonstrates the robustness of the convergence analysis presented for the real GSSA(e). Therefore we may conclude that for the analysis of the GSSA(e) the conditions imposed by the assumptions need not be met strictly. The analysis for the GSSA(x) gives satisfactory results when used with Channel (2). However, for low-distortion channels (which do not satisfy Assumption (ii) the analytical results fail to predict the simulation results. It should be noted that this is not a limitation in the analysis because Channel (1) is a low-distortion channel response when compared with many channel responses encountered in practice. The analytical results presented here gives some insight into the convergence behavior of these blind equalization algorithms. These analytical results may be used to determine the time constants of the convergence process and impose bounds on the step-size parameter.

Appendix A

In this appendix we classify the stationary points of the error functional used in the real version of the GSSA(e).

(1) Classification of the stationary points of E { l l z ( n ) l - ~'el}

To classify the stationary points of E{[ [z(n)[- ~e[} we consider its second derivatives with respect to the samples of the composite impulse response,


s(k). It was demonstrated in Section 2(a) that stationary points of the above error functional occur when the sequence {s(k)} has N (N > 0) nonzero components all having the same magnitude. Introduce the sequence {g(k)} such that its components have a one-to-one correspondence to the components of {s(k)}, Index the N nonvanishing components of {s(k)} as g(0), g(1) . . . . . g(N - 1). Since at a stationary point of the error functional these N nonzero components have the same magnitude, we have

Ig(0)l = Ig(1)l . . . . . Ig(N - 1)[ = 8~r (ml.1)

where SN is defined as the magnitude of the N nonzero samples of {s(k)} at a stationary point of (2.4).

We consider the infinite-dimensional, finite energy space spanned by the components of the sequence {s(k)}, that is, ~-~k S2(k) < o0. In this appendix we show the existence of a local maximum of the error functional for N = 0, local minima for N = 1, and points of unstable equilibrium for 1 < N < 20.

Case (a): N = 0. This corresponds to the trivial case of vanishing output. From (2.4) we see that when N = 0, E{((n)} = 7e. To demonstrate that this case corresponds to a local maximum, we examine the behavior of the error functional E{((n)} when it is subjected to a small perturbation of the sequence {s(k)} from its all-zero value corresponding to the N = 0 case. Noting that z(n) is zero-mean and f~, the pdf of z(n) in (2.3), is symmetric about the origin we may write (2.4) as

f) f; E{~(n)} = - 2 xf~(x) dx + 4 x s dx =0 =?e

+ 2y e dx - 27e fz(x) dx. (A1.2) =~r e

Since 7e > 0 and 27r f~e Crx) dx < (27e)�89 we have from (A1.2) , I x = O J z ~,

E{~(n)} < - 2 Xfz(X) clx + 4 xf~(x) dx + ~ = 0 = y e

Next, let m i be defined as

f; mi = xif~(x) dx, i = 1, 2, 3, 4. (A1.4) = 0

It can be seen from (2.3) and (A1.4) that m2 = �89 ~k s2(k) �9 Since we have assumed a finite energy space, the components of {s(k)} can be perturbed in any direction in this space such that m2 is arbitrarily small. Furthermore, for arbitrarily small values of m 2 it can be readily seen from


(2.3) that m 4 < m 2. Next, denote by 1?1 and 172 the first- and second-order moments of the random variable x' > 0 whose pdf is given by x' fz(x ' ) /m 1. Then we have

171 = m2/ml, (A1.5)

172 = ma/mp (AI.6)

Since the variance ofx ' is positive, we have 172 ~" /']I" Using this and (A1.5) and (A1.6), we get

m 2 < mlrn 3, (A1.7)

Similarly considering the first- and second-order moments of the random variable whose density function is given by X2fz(x)/m2, x > 0, we get

m E < mEre4. (A1.8)

Using (A1.7), (A1.8), and the fact that m 2 > m 4 we get

m 2 < m 1. (A1.9)

Now, for any x > 0 and cr > 0 using the fact that P{x > ~} < E{x}/o~ [25], we get

~x ~ m2 xf~(x) dx <_ - - . (AI.10) =~o ml mlTe

Next, considering the components on the right-hand side of (A1.3), we get from the above

4 x fz (x) dx < 4 m2. (A1.11)

Using (A1.9) in (AI . l l ) we get for ~e > 2

4 x f z (x ) dx < 2 fz(x) dx. (A1.12) = Y e

Finally, from (A1.3) and (A1.12) it follows that

e(~(n)} < ~e- (A1.13)

Therefore, we see that sufficiently small perturbations of {s(k)} from its all-zero position always decreases the value of the error functional. Hence, we conclude that the N = 0 case corresponds to a local maximum. It should be noted that the condition ye > 2 does not restrict the generality of the proof. This is so because as seen from (2.4) scaling ?~ results in all the coordinate axes being scaled by the same amount without altering the shape of the error surface. Case (b): 1 < N < 20. Here we assume that all the components of {s(k)} are positive. Since a(n) has a symmetric distribution it follows that the results so


derived are equally applicable to the case when the components of {s(k)} take negative values. Expanding the error functional (2.4) and assuming that s(i) > 0, we get

I ~ - (~,~ + a(n - i)s(i))

e{~(n)} = E - j _ ~ (z ' (n) + s( i )a(n - i) + ~'0

~ - a(n - i)s(i)

+ (z'(n) + s(i)a(n -- 1) + 7e) d - (Ye + a(n - i)s(i))

[" ( re - a(n - i)s(i))

- - | (z '(n) + s(i)a(n - i) -- 7~) �9 1 - a(n - i)s(i)

+ I 1 (z '(n) + s(i)a(n -- i) - ~e) f~,(z'(n)) dz'(n),

(Al.14)

where we have made use of (2.3). In the above the variable z'(n) is given by

z'(n) = ~ s(k)a(n - k), (Al.15)

the function f~, is the probability density function of z'(n), and the expectation is over the random variable a(n - i).

Next, in order to get the/fh component of the Hessian matrix of the error functional we differentiate (Al.14) first with respect to s(i) and then with respect to s(j). It should be noted that s(j) is contained in z'(n) given in (Al.15). Substituting the components of the sequence {g(k)} for the components of {s(k)} we have at a stationary point, for i r j,

~2E{~(n)} - 0, i , j q~ [0, g -- 1] or g -- 1, (Al.16) ~(0 ~(J)

c32E{~(n)}sg(i) Og(j) -- 2E{a2(n)}sN - 2 a ( n - - j ) a(n - j ) + w(n) + S~ N

- j ) + w(n)]f~(a(n - j ) + w(n))~ + a ( n j)[a(n )

• f~(a(n - j))fw(W(n)) da(n - j) dw(n),

i , j ~ [ O , N - 1 ] , N > I , (Al.18)


f . ( x ) =

where i satisfies

a2E{((n)}~g2(i) 2E{aE(n)}sN 2a2(n - i ) fN-1 - a ( n - i) +

- a2(n - i)fN_ 1 ( - a ( n - i)) t

x f,,(a(n - i))da(n - i), i e [0, N - 1]. (Al.19)

In the above g(k) = SN for k e [0, N - 1]. Denote by x the set consisting of the indices of the nonzero components of {s(k)}. Then the variable w(n) in (Al.18) is given by ~ k ~ , k , i , j a(n -- k), and the value of the quantity 7e/SN is evaluated using (2.5) and (2.3), in Section (2) of this appendix and is depicted in Table 2. The function fN is the pdf of the random variable ~k~,, a(n -- k) given by

E a(m,J) xm, i = - , . . . . . (A1.20) j=0 m=0

2iK < x < 2(i + 1)K (N even),

(2i - I)K < x < (2i + 1)K (N odd).

In the above [x] denotes the integer component of x, and a(m,j) is given by

a(m, j ) = ( N ---1)i-(2K) ~ j m

At a local minimum of the error functional the Hessian matrix is positive definite and indefinite at points of unstable equilibrium. The special structure of the Hessian matrix given by (A1.16)-(Al.19) enables us to evaluate its eigenvalues in a simple manner. Suppose H~j, the ij th (i # j ) component of the Hessian matrix, is such that

H i j ~- O,

H ii = ,~1,

H~1 = 22,

Hit = 23,

i,j ~ [0, N - 1],

~ [0, N - 1],

i, j e [0, N - - 1],

i e [0, N - 1].

Then it can be shown that the eigenvalues of H are as follows:

(N - 1)22 + 23 (multiplicity one), (A1.21)

- 2 2 + 23 (multiplicity N - 1), (A1.22)

21 (infinite multiplicity). (A1.23)

The eigenvalues of the Hessian matrix given by (A1.21)-(A1.23) and evaluated using (AI.16)-(AI.19) are depicted in Table 1, for N = 1, 2 . . . . . 20. Since for N = 1 all the eigenvalues of H are positive we conclude that this case corresponds to a local minimum. For N > 1 (N < 20) we see that

Tab

le 1

. E

igen

valu

es o

f th

e H

essi

an m

atri

x. T

he n

umbe

r of

non

zero

com

pon

ents

of

{s(

k)}

are

give

n by

N.

N 1

2 3

4 5

6 7

8 9

10

(21)

Su/K

0.

33

0.0

0.03

49

0.02

42

0.01

71

0.01

30

0.01

03

0.00

85

0.00

71

0.00

60

(--2

1 "-k

).3)S

N/K

1.0

-0.5

--

0.08

84

--0.

0475

-0

.031

4 --

0.02

24

--0.

0170

--

0.01

36

-0.0

111

-0.0

093

((N

-

1),~

2 +

J.3)

SN/K

1.

0 0.

5 0.

3864

0.

3364

0.

2966

0.

2692

0.

2480

0.

2312

0.

2174

0.

2058

N

11

12

13

14

15

16

17

18

19

20

(;~I)S

N/K

0.

0052

0.

0046

0.

0041

0.

0036

0.

0033

0.

0030

0.

0027

0.

0025

0.

0023

0.

0021

(-- 2

2 q-

23)

SN/K

-

0.00

79

- 0.

0069

-

0.00

60

- 0.

0054

-0

.004

8 -0

.004

3 -

0.00

39

-0.0

036

- 0.

0033

-

0.00

30

((N

-

1)22

+

23)S

s/K

0.

1959

0.

1873

0.

1797

0.

1730

0.

1669

0.

1615

0.

1566

0.

1521

0.

1479

0.

1441

03

;>

O

,.q

O

Z s ba

,..]

:Z

to


(--)~2 ~- /~3) ~ 0 and [(N -- 1)• 2 -~ )~3] > 0. Therefore, we c o n c l u d e tha t

the stationary points for 1 < N _< 20 correspond to points of unstable equilibrium.

(2) Evaluating 7e/SN for 1 < N <_ 20

Using (2.3) and (AI.1) we get, for the equalizer output at a stationary point,

z(n) = SN ~ I(i)a(n -- i), (A2.t)

where I takes on either + 1 or - t , the data symbols a(n) are uniform in [ - K , K], SN is a constant introduced in (AI.1), and x is the set whose elements constitute the indices of the N nonzero components of {sk}. The value of 7e/SN is given by the solution of (2.5). Using (2.3) in (2.5) it can be shown that

E{sgn(lz(n) l - 7e) lz(n)[} = 0. (A2.2)

Let us introduce a new variable, zl(n) = [z(n)I/SN. Using this in (A2.2) we have

From (A2.1) it follows that the pdf of za(n), fz~, is given by

f=,(x)= ~ ~ a(m,j)x m, i = 0 , 1 . . . . . . j=0 m=O

(A2.3)

x > 0 ,

(A2.4)

where i satisfies

2iK < x <_ 2(i + 1)K (N even),

(2i - 1)K < x <_ (2i + 1)K (N odd).

In the above [ ] denotes the integer component and a(m,j) is given by

a(m, j) = (N 2 [(N - 2j)K] n - l - m

From (A2.3) it follows that

f ~/SN f i-r xf~(x) dx = xfx,(x ) dx. (A2.5) e/SN Now, if m is the median of the random variable whose pdf is given by xf~(x)/S~ xf~(x) dx, x > O, then from (A2.5) it follows that SN = ve/m. For a


Table 2. The value of y~/S~r

N

1 2 3 4 5 6 7 8 9 10

(ye/S:c)K 0.707 1.000 1.189 1.372 1.531 1.674 1.807 1.931 2.046 2.156

N

11 12 13 14 15 16 17 18 19 20

(7e/SN)K 2.261 2.361 2.457 2.549 2.638 2.724 2.808 2.889 2.968 3.044

given value of N, 7e/Su can be computed using (A2.4) and (A2.5). These values are tabulated in Table 2.

Appendix B

In this appendix we evaluate some expectations that appear in Section 3 in the analysis of the real versions of the GSSA(e) and GSSA(x).

(1) Evaluating expectations in (3.17) and (3.19) for the real GSSA(e)

E{sgn[z (n) - Ye sgn(z(n))]Y(n)}. We see from Assumption (v) that z(n) conditioned on a(n) and C(n) is a Gaussian random variable (RV), Therefore, it follows that z(n) la(n), W(n) is also a Gaussian RV with mean,/~, given by

# = E{z(n)[a(n), W(n)}

= Mr(n)E{Y(n)la(n)}, (BI.1)

where we have made use of Assumption (iv) and (3.6). Using (B3.6), for E{Y(n) la(n)}, we get

# = a(n)Mr(n)q. (B1.2)

Since z(n)= Wr(n)Y(n), z(n) and Y(n) conditioned on a(n) and W(n) are jointly Gaussian. Therefore, using the result (B4.7) for jointly Gaussian RVs,


we get

E{sgn[z(n) -- 7~ sgn(z(n))]Y(n)Ja(n), W(n)}

12

E{a(n)2}(p - llqr)W(n) -+ fie

where we have used (B3.6) and (B3.10). The functions Fi(x ) are defined as

Lfo - - z i dz, i = O , 1 . Fi(x ) = exp

Finally, taking expectations of (B 1.3) over the independent variables a(n) and W(n), we get

E{sgn[z(n) - 7e sgn(z(n))]u

e { a(n)Z }(p -- lplT)M(n) = 2q{-f~ - f ~ ) + fO(Te)} +

0"~

x + 2f1(0) - 2fl(--Te) -- 2ft(Te) , (B1.4)

where the functions f~ and f~(x) are respectively defined as E{a(n)Fo((X +/O/a~)} and E{F,(i x + I.tl/o'~)} with the expectation taken over the data symbol constellation.

E{sgn[z(n) - 7e sgn(z(n))] Yk(n)Wk(n)}. To compute this expectation we mul- tiply the kth component of (B1.3) by Wk(n) and take expectations over the independent variables a(n) and W(n) while observing that the components of the transformed taps are uncorrelated (Assumption (iii)). We have

E{sgn[z(n) - 7~ sgn(z(n))] Yk(n)Wk(n)}

= 2tlkMk(n){fo(o) + f o ( _ 7~) - f~ q- - -

x { f~kkFk(n) + ~" f~kIMk(n)Mi(n)

X { ~ q / ~ + 2 f l ( O ) - - 2 f l ( - - V r 1 6 2

where the matrix f2 = [p - llllrl.

E(a(.) } O" e

(B1.5)


(2) Evaluatin9 expectations in (3.23) and (3.27) for the real GSSA(x)

E{z(n)Ok(n)}. We note that since z(n) = cr(n)X(n) = Wr(n)UX(n) it follows from Assumption (ii) that z(n) and X(n) conditioned on a(n) and W(n) are jointly Gaussian RVs. The parameters of the joint distribution follow from (3.9) and (BI.1), (B3.2), (B3.4), and (B3.9). Using these in the expression for E{z sgn(y)} in (B4.13), we get

a(n)hl T U k i ( 2 F o ( ~ a ( n ) M (n)q E{z(n)Ok(n)la(n), W(n)} = t \ x/ A,,.]

Z, [ ~_ 2vl(a(n)h,~l], (B2.1)

where s = E{a2(n)}(R - hhr)urw(n) is the covariance vector of z(n) and X(n) conditioned on a(n) and W(n) and matrix A is the conditional covariance of X(n) defined as A = E{aZ(n)}(R - hhT). Taking expectations over the conditional variables in (B2.1), we have

E{z(n)Ok(n)} = ~ Uk'{ 2pl~ + [AUTM(n)]I

�9

(B2.2)

Here the functions Ptm are defined as

plm(i) = E #a'(n)Fm(a(n)hi~ ~, t \,f 7)J

where the expectation is taken over the random variable a(n).

(B2.3)

E{sgn[z(n)]Ok(n)}. In order to simplify the computation of this term we introduce another approximation. We assume that the conditional covariance cov{z(n), X(n)[a(n), W(n)} is approximated by cov{z(n), X(n)[a(n)}. This can be justified by an argument similar to that used in the justification of Assumption (v). Denote by #k the correlation coefficient of z(n) and Xk(n) conditioned on a(n) and W(n). Using the above approximation and (B3.4) and (B3.9) it follows that

Y, #k = i (B2.4)

O. %/~kk

where We have made use (3.9) for the conditional variance of z(n). Next we use Price's theorem [23] to evaluate e{sgn[z(n)]Ok(n)la(n), W(n)} and then take expectations over the conditional variables a(n) and W(n) while observing


their independence. We have

E{sgn[z(n)]Ok(n)} = ~ Uk,~O(i), (B2.5) i

where we have defined q~(i) as

~p(i) = _2 E 1 N~- ] - - X 2

x exp 2(~-5x2) \ a2 a ~ u x + A,iJJ J

+ E~.4Fofa(n)h'~Fo (a(n)Mr-(n)~l~]., (B2.6)

The expectation in (B2.6) is taken over the data symbol a(n).

E{[z(n) - 7x sgn(z(n))]Ok(n)Wk(n)}. From (B2.1) and (B2.5) it follows that

E{[z(n) - 7x sgn(z(n))]Ok(n)Wk(n)}

1

(B2.7)

In the above we have assumed that the components of the transformed tap weight vector are uncorrelated (Assumption (iii)).

E(zZ(n)O~(n)}. Since z(n) = e(n) + a(n) using Assumptions (vi) and (vii) we have

E{zZ(n)O2(n)} = a2E{O2(n)} + E{a2(n)OZ(n)}. (B2.8)

We note that by definition e(n) = U sgn[X(n)]. Hence

E{aZ(n)O2(n)} = ~_, Z Uki Uk~E{a2(n) sgn(Xi(n)) sgn(X~(n))}. (B2.9) i j

To compute (B2.9) we assume that Xi(n ) and Xj(n) are jointly Gaussian RVs. The parameters of this joint distribution are given by (B3.1) and (B3.3), later in this appendix. Therefore, using Price's theorem it can easily be shown that

2s in- ( ) (B2.10) _ 1 R i j , E{sgn(Xi(n)) sgn(Xj(n))} = n

where R~j = E{X~(n)Xj(n)} is the /jth element of the channel covariance matrix. Note that using only Assumption (ii) we can compute the above


expectation without assuming that X,(n) and X~(n) are jointly Gaussian RVs. However, in view of (3.1) and the central limit theorem we can justify the use of this simplification. It follows from Assumption (ii) that X~(n) and Xj(n) conditioned on the data symbol, a(n), are jointly Gaussian. Define @(/j) as

~(ij) = E{sgn(Xi(n)) sgn(X j(n))la(n)}. (B2.11)

Using Price's theorem to evaluate E{sgn(X~(n))sgn(X~(n))la(n)}, it can be shown that

2 fP'J ~ e x p ( a2(n) h~ 2h,hj h 2 X1-x t + jW}

. ~ /a(n)hi'x ~ /a(n)hs'~ + q ~ o / - - / ~ o / ~ - , (B2.12)

where we have used (B3.2) and (B3.4) for the parameters of the distribution of X(n)[a(n), the quantity Pu is the correlation coefficient between Xi(n) and Xj(n) conditioned on a(n), and the matrix A = E{a(n)2}(R- hh r) is the covariance matrix of X(n)la(n). Using (B2.9)-(B2.11) in (B2.8) we get

(B2.13)

E{lz(n)tO~(n)}. Substituting z(n) = e(n) + a(n) and noting that e(n) is independent of a(n) and X(n) (Assumption (vi)), it can be shown that

E{ Iz(n) l sgn(X,(n)) sgn(Xj(n))}

where the expectation is over the data symbol, a(n). In the above we have made use of the fact that X,(n) and Xj(n) conditioned on the data symbol are jointly Gaussian and that e(n) is a zero mean Gaussian RV. The functions F~,

i = 0, 1, are given by Fi(x)= ( 1 / x / ~ ) ~ z~exp(-zZ/2)dz. Finally, since 0(n) = U sgn[X(n)], we have

E{[z(n)lO~(n)} 2

where the functions ql(/J) and qz(ij) are defined by

ql(0) = E{a(n)O(ij)Fo(~)} (B2.16)


and

q2(ij) = E { ~ b ( i j ) F t ( ~ ) } ,

with the expectation being taken over the data symbol a(n).

(B2.17)

(3) Joint statistics of equalizer input and output

Here we derive the means and covariances of the joint distribution of the equalizer input and the output conditioned on the data symbol a(n), and the tap weight vector C(n), or the transformed tap weight vector W(n). Since a(n) is assumed to be zero mean and {x(n)} = (h(n)}*{a(n)}, we have

E{X(n)} = 0, (B3.1)

E{X(n)la(n)} = a(n)h, (B3.2)

cov{X(n), X(n)} = R, (B3.3)

cov{X(n), X(n) la(n)} = E{a(n)2}(R -- hhr). (B3.4)

Here, h = [h(L) . . . . , h ( - L ) ] r is defined as the channel impulse response vector and X(n) as the vector of equalizer input samples. Next, we consider the transformed input vector, Y(n) --- UX(n). Using the above we have

E{Y(n)} = 0, (B3.5)

E{Y(n) la(n)} = a(n)~l, (B3.6)

cov{Y(n), Y(n)} = p, (B3.7)

cov{Y(n), Y(n) la(n)} = E{a(n)2}(p - nqT), (B3.8)

where q = Uh is the channel impulse response vector in the transformed domain and p is the diagonal matrix of channel eigenvalues. Since the equalizer output z(n) = CT(n)X(n), the conditional covarianee of z(n) and X(n) will be given by

cov{X(n), z(n)[a(n), C(n)} = E{a(n)2}(R - hhT)C(n), (B3.9)

where we have observed the independence of C(n) and X(n), Finally, writing the covarianee in the transform domain, we get

cov{Y(n); z(n) la(n), W(n)} = E{a(n)2}(p -- qqT)W(n). (B3.10)

(4) Evaluat& 9 E{sgn(z - 7 sgn(z))y}, 7 > 0, and E{z sgn(y)}

In this section we compute the above quantities when z and y are jointly Gaussian. Let z and y be jointly Gaussian RVs with the parameters of the

2 and E. distribution given by #, r/, az z, o'r,


E { s g n ( z - ? sgn(z))y}. Expressing E { s g n ( z - 7 sgn(z))y} in terms of the conditional expectations we get

E{sgn(z - 7 sgn(z))y} --- E{sgn(z - 7 sgn(z))E{y[z}}. (B4.1)

The random variable y lz is a Gaussian random variable with mean given by

[z E{ylz} = r /+ ~r~ - #]" (B4.2)

Substituting (B4.2) in (B4.1),

E{sgn(z - V sgn(z))y} = F r / - E~-~lE{sgn(z - ? sgn(z))} L o'z A

Z, E{z sgn(z sgn(z))}. (B4.3) O" z

It can be shown in a straightforward manner that

E{sgn(z -- V sgn(z))} = --2F o o ~ - j + 2Fo (B4.4)

and

E{z sgn(z - ? sgn(z))}

= 2 r l - F ~ \ az } \ az }J

\az} \ az }J

where F o and F 1 are defined as

(0 - - z i dz, Fi(x ) = exp ~ - -

Finally, using (B4.4) and (B4.5) in (B4.3) we get

E{sgn(z - ? sgn(z))y}

d o(0 ,+,

(B4.5)

E{z sgn(y)}. We may write E{z sgn(y)} as follows:

E{z sgn(y)} --- E{sgn(y)E{z[y}}. (B4.8)

i = 0, 1. (B4.6)


For jointly Gaussian RVs z and y we have

E E { z l y } = # + L~ [Y -- r/]. (B4.9)

O-y

Substituting (B4.9) in (B4.8)

E{z sgn(y)} = E{sgn(y)} ~ % ay - ~ + ~ E { l y l } . (B4.10)

It can be shown that for Gaussian RVs

E{sgn(y)} = 2 F o ( a ~ ) (B4.11)

a n d

2 a y F l ( ~ 2~Fo(~l ~. y kay] kay/ ,qn

Using (B4.11) and (B4.12) in (B4.10) we get

(B4.12)

(B4.13)

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Documents

Sign algorithms for blind equalization and their convergence analysis