Hi I was presented with this problem I would like some guidance to solve:

I am given the following DT channel model: $$C(z) = 1-0.3z^{-1}$$

I am first asked to design an IIR linear zero forcing equalizer.

I am then asked to approximate the IIR filter obtained above with an FIR filter, where 80% of the energy of the IIR filter is captured in the FIR filter.

My initial thought was to transform the channel from the z domain to: $$ y[n] = c[n] + 0.3c[n-1] + z[n] $$ - where z is gaussian distributed noise

Apply a zero forcing 3 tap filter to this and obtain the 3 coefficients.

Once I obtain these coefficients however I am unsure to proceed. Do I then simply take the convolution to find the output of this filter, is that considered the IIR zero forcing equalizer?

Also once this is done how do I ensure 80% of the energy of the IIR filter is then captured in the FIR filter?

Any help would be greatly appreciated


1 Answer 1


This could be homework, so I'll only give you a few hints to help you solve the problem yourself.

  1. Remember that a ZF equalizer just inverts the channel, so if $D(z)$ is the equalizer's transfer function, what must be the result of the product $C(z)D(z)=?$. From this equation you obtain $D(z)$, which is IIR.

  2. Let the coefficients of $D(z)$ be $d[n]$, i.e., $$D(z)=\sum_{n=0}^{\infty}d[n]z^{-n}$$ The energy of the filter is $$E_d=\sum_{n=0}^{\infty}\big|d[n]\big|^2$$ The FIR filter you're looking for just uses the first $N$ coefficients $d[n]$. If $N$ is large enough, the approximation will be sufficiently good. According to the requirement, you need to choose a number $N$ such that $$\sum_{n=0}^{N-1}\big|d[n]\big|^2=0.8\sum_{n=0}^{\infty}\big|d[n]\big|^2$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.