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Let the equalizer coefficients obtained, say $w$ and the input to the equalizer is $y$ that is a noise corrupted received signal, $rx$ then the equalized signal, which is the output of the FIR based equalizer, should be equal to the transmitted signal / source signal. This is the working principle. But, we often perform the operation :

equalizer_output = filter(w,1,rx).

Question 1: Most equalizers are modeled as FIR filter. My Question is why? I do not understand how this operation gives the input or how is this performing deconvolution?

Question 2: For example considering a single input single output channel, $x$ is a BPSK modulated source signal and there is multipath fading with 3 taps (not Rayleigh), then the transmitted signal is $$y(t) = h(1).x(t-1) + h(2).x(t-2) + h(3).x(t-3) + x(t)+ w(t)$$ where $w$ is the Additive White Gaussian measurement / channel noise. The above model is FIR model and not any filter. This is not the equalizer. In order to devise an equalizer inorder to get the input $x$, do we again devise a FIR filter as an equalizer whose input would be $y$? $h$ is the channel impulse response but it is not any model as such.

Question 3: When do we model a channel as an IIR model and then the equalizer would be an IIR or FIR filter?

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The reason why almost all linear adaptive equalizers are implemented as FIR filters is that FIR filters are always stable and that there exist relatively simple and effective adaptation algorithms. Note that much work has been done on adaptive IIR filters (e.g., this book by Phillip Regalia), but in practice FIR filters are still the preferred option.

Note that no real-world channel can be ideally equalized by an FIR filter, but that's not as bad as it may seem because an FIR filter can approximate the ideal equalizer reasonably well if its filter length is chosen appropriately and if the adaptation algorithm is tuned to the application. Also, perfect equalization of the channel is not always desired if the channel is not only dispersive (i.e., acts a filter) but if it also introduces noise. In this case a combined criterion is used that takes into account the ISI due to the dispersive channel as well as the degradation due to the noise.

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Most of the books/articles on adaptive filters use the squared error as the optimization criterion. For FIR filters, the solution for the filter coefficients can be found using traditional least squares and can be made recursive via Recursive Least Squares (RLS). You can also get into order recursive schemes with lattice filter structure, but that is kind of out of scope for your question. Solution to the least squares is a global optimum and the filters are always stable.

In the case of recursive or IIR adaptive filters, using the squared error objective function leads to non-linear least squares, so the solutions/algorithms often involve approximations. There is also the risk that that the algorithm gets stuck in a local minimum. As Matt mentioned in his answer - with adaptive IIR filters there is often the risk that the filter may be unstable. The instability may be due to the explicit placement of a pole(s), or it may result from coefficient quantization.

As an example there is a notch filter developed by Nehorai ("A minimal parameter adaptive notch filter with constrained poles and zeros," Acoustics, Speech and Signal Processing, IEEE Transactions on (Volume:33 , Issue: 4 )) which explicitly places the poles inside the unit circle, but these types of filters tend to have a small number of poles and have a very specific objective.

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  • $\begingroup$ Thank you for your answer. Could you give some idea if a Kalman filter is used as an equalizer, then will the equalizer be modeled as FIR or IIR? This question stems from your statement in the answer that "in case of recursive or IIR adaptive filters..." so does it mean that all recursive adaptive filters are modeled as IIR? What is the decision criteria for modeling an equalizer as IIR or FIR? $\endgroup$ – Ria George Nov 8 '15 at 3:13
  • $\begingroup$ In the majority of cases IIR=recursive. There are some specially designed recursive filters that have an FIR - you can search this forum for details. For your Kalman question - I don't know. Whether you use IIR or not - depends on if you can live with the stability issues and possibly non-global optimum. $\endgroup$ – David Nov 10 '15 at 14:50

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