Q:

Help in deriving the update rule for equalizer

Considering an FIR system excited by an input $s_n$. If the input is derived from a mapping, and a +1/-1 BPSK signal is derived. Considering the Constant Modulus Algorithm (CMA). The channel is modeled as an FIR. If the received signal is fed to the equalizer, the output is : $y_n = w^T(n) x_n$ where $w^T$ is the equalizer tap vector. The update equation for weights $w_{n+1} = w  - \mu \Delta J(w)$
Will there be a modification in the derivation of the CMA and the update rule? 

A:

The mapping from $z_n$ to $s_n$ is not bijective, so it can't be inverted. This means that even in the transmitter, $z_n$ cannot be recovered from $s_n$. This is why the constant modulus algorithm (CMA) need not (and cannot) be modified to estimate $z_n$. It can only be used to estimate $s_n$, and for this purpose no modification is necessary.
The thesis you referred to in a comment is mainly about blind signal separation. The update rules (3.16) and (3.17) apply to the iterative maximization of the log-likelihood function in order to compute the maximum likelihood solution of a source separation problem. The nonlinear vector-valued function $\mathbf{g}(\mathbf{u})$ depends on the probability density functions of the different source signals. The underlying model and the problem formulation are different from the model and the problem underlying the CMA, so there is no direct relation between the two, and the update rules mentioned above are not applicable to the CMA.

