CMA fractionally spaced equalizers: stationary points and stability under i.i.d. and temporally correlated sources

Abstract

A common assumption in blind equalization schemes using the Constant Modulus Algorithm (CMA) is that the source sequence is an independent identically distributed (i.i.d.) sequence with equiprobable symbols. Much of the analysis demonstrating the global convergence of CMA in a noiseless channel to an open-eye setting uses this assumption. This work investigates the effect of source statistics (distributions and correlations) on the location of CMA stationary points in the fractionally-sampled equalizer case under the conditions of equalizability. The work identifies the stationary points as the solution set of a system of multivariate polynomial equations with monomial coefficients given by the source moments.

Effect of Channel Noise on Blind Equalization and Carrier Phase Recovery of CMA and MMA

“…Moreover, (22) along with Z Z α = k s /2 reveals that when the CMA is used with various sources, a larger source kurtosis, k s , is associated with greater resistance of the CMA to channel noise as long as the sources remain sub-Gaussian. However, a large k s is widely known to inhibit CMA (or MMA) equalization [3], [13], [24], [25], [29].…”

“…A large (small) curvature of the error surface at a stationary point may indicate that the rates of approach and escape around the saddle point are fast (slow) before converging to the desired minimum [24], [13]. Unlike the investigation in [13] that examined the effect of source distributions on the equalizer convergence and the phase tracking for a noiseless channel, this section investigates the effect of additive channel noise on the equalizer convergence and phase tracking by employing various source distributions.…”

Decision directed adaptive blind equalization based on the constant modulus algorithm

“…In a multichannel system, it is also possible, under certain conditions, to choose the length of the equalizer in such a way that the nullspace of H is trivial [21]. Other related works about the convergence of adaptive blind algorithms can be found in [22][23][24][25][26][27][28].…”

Effects of source distributions on CMA and MMA using symmetric two-dimensional signal constellations

“…This paper continues the study of effects of source distributions on the constant modulus algorithm (CMA) and the multimodulus algorithm (MMA) in [1], which was motivated by the investigation in [2]. Although the CMA is known to work efficiently in the presence of carrier phase offsets [3], [4], it needs to be followed by a blind carrier phase recovery algorithm (CPRA) [i.e.…”

“…The equalizer input, un = sn *c n is then sent to a tap-delay-line blind equalizer, fn ' intended to equalize the distortion caused by lSI without a training signal. The output of the blind equalizer, Yn =j,,* *un =sn *h; , can be used to recover the transmitted data symbols, sn' where * denotes complex conjugation and h n =in* *c n denotes the impulse response of the combined channel-equalizer system whose parameter vector can be written as the time-varying N x 1 vector h = [hn(l),h; (2) there is no possibility of confusion, the notation is simplified by suppressing the time index n, e.g., sn=s= sll+jsf and hn(k)=h(k), where SR and Sf denote the real and imaginary parts of sn' Due to space limitations, this paper has been written as an extension of the results developed in [1]. Therefore, all the notations used in [1] directly applies to this work and, furthermore, both the CMA and the MMA cost functions as well as Lemmas 1.a, 1.b, 2.a, 2.b, and 3 described in [I] will not be re-written herein and will be directly used in this work.…”