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?

• Your signal is $s_n$ and the CMA will estimate this signal. The signal $z_n$ cannot be recovered because it is already lost in the transmitter due to the mapping which can't be inverted. I'm not sure I understand your question, but if your goal is to recover $z_n$ then this is impossible. – Matt L. May 28 '15 at 8:49
• @MattL.: I do not want to recover $z_n$. I want to know how to represent this transformation function in the update equation; if there is any difference in the update law on using this mapping or will the update law remain unchanged? If it is unchanged then what is the difference or use of using any nonlinear function? The nonlinear function acts as the driving source. So, it can be a neural net signum function or a Volterra system that is used to derive the BPSK or QPSK signal source. Want to examine the performance of CMA for different input signals. – SKM May 28 '15 at 16:21
• Your input signal is $s_n$. It doesn't matter how it is generated. CMA will estimate $s_n$, and no modification is necessary. The generation of $s_n$ via the hidden variable $z_n$ doesn't enter the picture. Just like CMA doesn't care whether your data come from an audio signal, a video signal, or anything else. – Matt L. May 28 '15 at 16:26
• I see, thank you. Then this Question is no longer valid. – SKM May 28 '15 at 16:57
• OK, do you want me to add that information as an answer so you can accept it and the question gets marked as answered / problem solved? – Matt L. May 28 '15 at 17:01

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.
• I found an article univ-brest.fr/lest/tst/publications/pdf/… which says that for the Bernoulli Map, the mapping between symbol space and the number space is bijective. In that case, the mapping from $z_n$ to $s_n$ is bijective nad so $z_n$ can be recovered. I will have to study this more carefully from other sources as well. In case, it is bijective, then it may be possible to recover $z_n$ by CMA. Any thoughts on this? I really appreciate all your insights and helpful guidelines. – SKM May 29 '15 at 16:28
• Although I am not quite sure if the mapping from $z_n$ to $s_n$ is not bijective. In the book : The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks Under Section 1.1.15, there is a hint that we can inverse the mapping to obtain the real number (initial condition) from which the dynamical system was iterated. For example: as given in book notation, $x_0 = \sigma_0\sigma_1...$ and each $\sigma_i$ forms the BPSK input. Then if CMA can equalize to obtain the $\sigma_i$ then by doing the inverse of the symbolism we may get the number $x_0$. – SKM May 29 '15 at 20:34
• @SKM: If the mapping from $z_n$ to $s_n$ is bijective, you can run the (unmodified) CMA to recover $s_n$ and then apply the inverse mapping to obtain $z_n$. There would still be no need to modify the CMA. – Matt L. May 30 '15 at 17:19