Effect of Source Distributions on Multimodulus Blind Equalization Algorithm

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

“…Lemmas 2.a and 2.b of [1] and Lemma 2.c indicate explicitly that the source s must be sub-Gaussian with kurtosis k, < 2 such that both (2k s -4)1 < 0 (or (2k s -4)H <0) and (2-k s)I>0 (or (2-k s)H>0) hold for MMA to converge and such that both (2k s -4)G<0 and (2-k s)G>0 hold for CMA to converge. Computer simulations performed in this study demonstrate that once blind equalization started with a sub-Gaussian source, a non-zero component of h with maximum magnitude square, r 2 ( k) , rose, and the sum of the magnitude squared of the remaining M -1 nonzero components, Lr 2 (I) , fell, eventually diminishing to zero.…”

“…The curvatures computed from part (b) of Lemmas 2.a and 2.b of [1] and Lemma 2.c, which are always negative for a sub-Gaussian source s, correspond to the rates of convergence of the (M -1) non-zero components of h with small modulus to the origin (i.e., the removal speed of residual lSI at the receiver). This is because part (b) of Lemmas 2.b of [1] and Lemma 2.c implies that the origin corresponds to a local minimum in both the CMA and MMA cost functions in terms of the (M -1) non-zero components of h during the transient mode operation. For example, from Lemma 2.b of [1], the residual lSI of CMA is mainly determined by (2k s -4)G, and it can be readily shown that 1(2k s -4)GI~1(2k s -4)11> 1(2k s -4)HI .…”

“…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.…”

“…II. CURVATURES OF MMAAND CMA The general formulas for both the MMA and the CMA error surface curvatures at all of the stationary points via the Hessian have been summarized in Lemmas 2.a and 2.b of [1]. Lemma 2.c described below is a special case of Lemma 2.a of [1] when SR and Sf are inde endent of each other.…”