Equalizer coefficients and channel coefficients

Question

Blind channel equalization methods equalize the channel without using the source data and without knowing the impulse response of the channel. Consider the channel to be a single input single output FIR system. The popular methods for blind channelequlization are Constant Modulus Algorithm and Least Mean Squares. Certainly, the equalizer coefficients are the not the estimated channel impulse response. It is not clear to me that once the equalizer weights are estimated, how does one get the estimates of the channel coefficients/ impulse response? I am missing a fundamental step somewhere, can somebody please help how we use the equalizer weights to estimate the channel?

Answer 1

With a blind equalization technique like the constant modulus algorithm (which is often implemented using a least mean squares (LMS) filter as you indicated), you aren't directly estimating the channel impulse response itself. Instead, the signal model is like this:

• The receiver observes the following signal: $$ x[k] = s[k] * c[k] + n[k] $$

  where:

  • $s[k]$ is the underlying desired signal,
  • $c[k]$ is the channel's impulse response, and
  • $n[k]$ is a noise process introduced by the channel (typically assumed to be additive white Gaussian noise)

• A blind equalizer operates by taking advantage of some constraint on the properties of the desired signal $s[k]$. For instance, if it is constant envelope, like phase-shift keying, then the constraint could be: $$ |s[k]| = 1 $$

• The CMA-LMS blind equalizer operates in two steps. First, it uses the current length-$L$ vector of LMS filter taps (where $L$ is a design parameter of the filter) $\mathbf{w}[k]$ to calculate its output: $$ y[k] = \mathbf{w}[k]^H\mathbf{x}[k] $$

  where $\mathbf{x}[k]$ is a vector of the last $L$ input samples to the filter. Next, it uses the calculated output to update its set of taps for the next iteration: $$ \mathbf{w}[k+1] = \mathbf{w}[k] - \mu \mathbf{x}[k](|y[k]|^2 - 1)y^*[k] $$

  When compared to the typical LMS filter structure, one can identify the error signal $e[k]$ as:

  $$ e[k] = (|y[k]|^2 - 1)y^*[k] $$

  You can see here how the error captures how far the filter output signal $y[k]$ deviates from the expected constant envelope of the desired signal $s[k]$.

Looking at this structure, you can see that at no point is the equalizer trying to directly estimate the channel's impulse response. Instead, it's more "given the knowledge that I have of the expected signal's structure, how do I need to adjust my filter taps to enhance that structure in the filter output?" So it's more of an inverse problem; the tap vector will tend to more closely resemble the inverse of the channel impulse response. Blind equalization techniques like this can do a surprisingly good job at doing so (even more surprisingly, techniques like CMA can even work for signal types that aren't constant envelope, like QAM).