# Blind 1D equalization/deconvolution with some knowledge of filter kernel

Let $s_{\rm out}[n]$ be the 1D output signal of a system, $s[n]$ be the input, and $k[n,q]$ be the filter kernel for an element $n$ and for fixed value $q$. Then:

$s_{\rm out}[n] = s[n] \ast k[n,q]$

If I know $q$ as a real number, then I can compute the filter kernel $k[n,q]$ and determine $s[n]$.

The values for $q$ normally range between $q = 30$ and $q = 500$. The signal $s[n]$ is discrete data that is bounded between two values. Moreover, $s[n]$ is minimum-phase.

In the frequency domain, the filter kernel $k[n,q]$ modifies both the magnitude and phase of the input $s[n]$ to cause attenuation. This is a "lossy" transmission channel.

However, I do not know $q$ and $s[n]$, and so I am looking for a well-known, "classic" reference that details how to estimate $s[n]$ with some knowledge of $k[n,q]$. I know the mathematical form of $k[n,q]$, but I just don't know $q$.

I'm looking for an algorithm that works fairly well and is reasonably well-understood and applied. In this case, $s[n]$ could be contaminated by noise (as are all real signals), but the main goal is to remove $k[n,q]$ to estimate $s[n]$.

• Is there a manageably small discrete set of values of $q$ that you could search over? Do you know any characteristics of $s[n]$ that you can exploit? For example, the constant modulus algorithm can be used if you know that the input signal has constant envelope. You'll need to formulate some cost function that your equalizer aims to minimize. – Jason R Aug 2 '12 at 1:53
• JasonR: Thanks for this comment, Jason. I know that q ranges between 30 and 500. How do I know if the constant envelope assumption is good for my signal? The assumption can be made that the s[n] is minimum phase. How might I implement the constant modulus algorithm? – Nicholas Kinar Aug 2 '12 at 2:59
• You will get a constant envelope is modulations like BPSK, QPSK, and some kinds of M-QAM. The constant envelope just means that regardless of the phase of the bit that is being encoded, the envelope of the analytic signal is the same. (Which makes sense, since for example, in BPSK, the amplitude of the modulating sinusoid is always 1, but the phase is changing). CMA falls under blind equalization. – Spacey Aug 2 '12 at 18:16
• @Mohammad: Thanks for this comment. Is there a good, well-written and clear reference for CMA and these modulation signals (i.e. BPSK). – Nicholas Kinar Aug 2 '12 at 19:15
• @NicholasKinar BPSK modulations will fall under any comm book. Proakis and Salehi is one. CMA usually falls under adaptive filtering text-books. – Spacey Aug 2 '12 at 22:36