Adaptive fractionally spaced blind CMA equalization: excess MSE

Abstract

The performance of the constant modulus algorithm (CMA)-a reference algorithm for adaptive blind equalization-is studied in terms of the excess mean square error (EMSE) due to the nonvanishing step size of the gradient descent algorithm. An analytical approximation of EMSE is provided, emphasizing the effect of the constellation size and resulting in design guidelines.

Dithered signed-error CMA: robust, computationally efficient blind adaptive equalization

“…Hence, we derive an expression for the excess MSE (EMSE) of DSE-CMA and discuss implications on step-size and equalizer-length selection. We note in advance that the EMSE expression for DSE-CMA bears close resemblance to an analogous expression derived for CMA in [12].…”

“…In the theorems below, the implications of (10) are formalized in terms of DSE-CMA behavior over specific ranges of Lemma 1: Define (11) The choice of dither amplitude ensures that for all equalizer outputs satisfying the output amplitude constraint Proof: By evaluating at the locations where , it can be seen that the "humps" of the cubic CMA error function (see Fig. 2) occur at heights Thus, is unique and well-defined for Since (10) implies that such values of prevent these humps from being clipped in forming the expected DSE-CMA error function, and are identical over the interval when For values is determined by the unique real-valued root of the cubic polynomial and can be expressed as (12) From (12), it can be shown that Writing the system output as for a (fixed) received vector and arbitrary equalizer allows the following equalizer-space interpretation of Lemma 1. Applying (10), we conclude that for any equalizer within the ball Note that the constant may be less than , in which case, there would exist isolated "CMA-like" neighborhoods around the ZF solutions-i.e., neighborhoods not contained in any "CMA-like" convex hull.…”

“…Equation (21) can be compared with an analogous expression for the EMSE of CMA [12]: (22) It is apparent that the EMSE of CMA and DSE-CMA differ by the multiplicative factor (23) via Note the dependence on both the dither amplitude and the source distribution. Table II presents values of for various -PAM sources and particular choices of (to be discussed in Section V-B).…”

Asymptotic properties of the algebraic constant modulus algorithm

“…It has been successfully applied to real data in a variety of scenarios for up to six sources simultaneously [29]. The potential performance of the CMA receiver, i.e., the minima of the modulus error cost function to which the adaptive CMA tries to converge, has been studied in detail recently in a series of papers by Tong, Johnson, and others [7], [9], [18], [32], [33]. These papers provide quantitative evidence for the observation already made by Godard that the minima of the constant modulus cost function are often very near the (nonblind) Wiener receivers or linear minimum mean square error (LMMSE) receivers.…”

Fractionally Spaced Constant Modulus Algorithm for Wireless Channel Equalization

“…Here we have used CMA algorithm [4] to estimate the channel variation and compare the performance of the Constant Modulus Algorithm with a more computationally efficient signed-error version of CMA (SE-CMA) [1,3,5,6]. For channel estimation we used to take a FIR channel with fractionally spaced equalizer (FSE) with sampling interval T /2 where T is taken as symbol period.…”