# Approximate IIR filter with FIR filter with energy restrictions

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. 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 $$\sum_{n=0}^{N-1}\big|d[n]\big|^2=0.8\sum_{n=0}^{\infty}\big|d[n]\big|^2$$