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?