Fractionally spaced equalization using CMA: robustness to channel noise and lack of disparity

In the noise-free case, the fractionally spaced equalization using constant modulus (FSE-CM) criterion has been studied previously. Its minima were shown to achieve perfect equalization when zero-forcing (ZF) conditions are satisfied and to be able to still achieve fair equalization when there is lack of disparity. However, to our best knowledge, the effect of additive channel noise on the FSE-CM cost-function minima has not been studied. In this paper, we show that the noisy FSE-CM cost function is subject to a smoothing effect with respect to the noise-free cost function, the result of which is a tradeoff between achieving zero forcing and noise enhancement. Furthermore, we give an analytical closed-form expression for the loss of performance due to the noise in terms of input-output mean square error (MSE). Under the ZF conditions, the FSE-CM MSE is shown to be mostly due to output noise enhancement and not to residual intersymbol interference (ISI). When there is lack of disparity, an irreducible amount of ISI appears independently of the algorithm. It is the lower equalizability bound for given channel conditions and equalizer length-the so-called minimum MSE (MMSE). The MMSE lower bound is the sum of the MMSE and of additional MSE mostly due to noise enhancement. Finally, we compare the FSE-CM MSE to this lower bound.