# Adaptive equalizer for IIR channel?

I am quite new to the idea of equalization. I have a few queries regarding the same.

• In my application I require to equalize a channel whose impulse response is an IIR response. Is it possible to design an adaptive equalizer based on LMS algorithm to equalize it? I tried some MATLAB simulations and it worked well (the estimation error reduced with each step and converged) for FIR channels but not for IIR responses. So is there any restrictions on the channel characteristics (like FIR/IIR response) for the adaptive filter to function properly?
• Also, from my simulations for FIR channels I observed that, in absence of noise, the adaptive filter after converging had a response that is inverse of the channel response which is the same as that of a Zero-Forcing equalizer. So is it always true that, LMS adaptive filter will tend to a Zero-forcing equalizing filter in the absence of noise? If yes, can someone refer me to the proof of this?

I have another doubt. Say I am measuring the channel output in presence of an additive zero mean noise,

$$Y(z)=X(z)H_c(z)+N(z),$$

Where $H_c(z)$ is the channel response, I still would like the equalizer to estimate the inverse response of the channel. I thought this should work, as if I try to derive the Weiner-Hopf filter, in presence of noise, I end up getting

$$R_{dx}(k) - R_xW -E\left[n(l)x^{*}(n-l]\right],$$

Where $R_{dx}$ is the cross correlation between the desired signal $d(n)$ and the filter output $x(n)$, and $R_x$ is the auto-correlation matrix, and $W$ is the filter weights. Since the noise is independent of $x$ and has zero mean, $E\left[n(l)x^{*}(n-l)\right]$ becomes 0 and I again get the same equation. So I thought that the same algorithm could potentially work.

Also in the LMS algorithm, instead of using $E\left[e(n)x^{*}(n)\right]$, directly as $e(n)x^{*}(n)$, I tried to take an $L$ point average of the product, with the intention that, since the error $e(n)$ will consist components both due to ISI as well as the additive noise, averaging it will remove the additive noise and retain the ISI, and so the filter coefficients will converge to give the inverse response of the channel.

But MATLAB simulations do not agree with my argument above. I want to know the mistake in my argument and also I would like to know if there is any means by which I could still estimate the inverse response of the channel to a good approximation even in presence of a zero mean noise.

• Can you plot the impulse response of your IIR channel? You can truncate it after it goes down to a certain value, consider it as the FIR response (N), count its taps and set the equalizer taps (M) accordingly (M >> N). And yes LMS yields zero forcing without noise. You can derive the Weiner-Hopf without noise to check. Commented Aug 18, 2014 at 12:41

Note that the inverse of an FIR system is IIR, and the same is true for the inverse of an IIR system, unless it is an all-pole system, the inverse of which would be FIR. So in most cases the ideal equalizer should have an infinitely long impulse response in order to perfectly invert the channel. In practice almost all adaptive equalizers are FIR filters which can only partly compensate for the channel response. This is why I think that there should be no reason for your equalizer to perform worse for IIR channels than for FIR channels. You could post some more information about the channel responses and the convergence of your equalizer to make it easier to see what's going on.

As for the zero forcing criterion, such criteria for designing an equalizer only make sense for noisy channels, because otherwise there is no trade-off between equalizing the channel response and noise enhancement. Zero-forcing and MSE criteria result in the same equalizer in the absence of noise. If you have no noise, the only problem is the (hopefully linear) distortion of the channel and naturally this is the only thing the equalizer would compensate for. However, with an FIR equalizer exact zero-forcing is usually impossible as discussed above.