4.2 Lab 2: prelab (part 2)

Filter-design exercise

Using the zero-placement method, design the FIR filters for the multirate system in Multirate Filtering: Introduction . Recall that the $z$ -transform of a length- $N$ FIR filter is a polynomial in $z^{-1}$ , and that this polynomial can be factored into $N-1$ roots.

Use this relation to design a low-pass filter (for the anti-aliasing and anti-imaging filters of the multiratesystem) by placing twelve complex zeros on the unit circle at $(\frac{3\pi }{8})$ , $(\frac{\pi }{2})$ , $(\frac{5\pi }{8})$ , $(\frac{3\pi }{4})$ , $(\frac{7\pi }{8})$ , and $(\pi )$ . This filter that you have just designed will serve for both FIR 1 and FIR 3. For filter FIR 2 (operatingat the decimated rate), use four equally-spaced zeros on the unit circle located at $(\frac{\pi }{4})$ and $(\frac{3\pi }{4})$ . Be sure to adjust the resulting filter coefficients to ensure that the gain does not exceed oneat any frequency.

Design your filters by writing a MATLAB script to compute the filter coefficients from the given zero locations. The MATLABfunction poly is very useful for this; type help poly in MATLAB for details.

Once you have determined the coefficients of the filters, use MATLAB function freqz to plot the frequency responses. You will find that the frequency response of thesefilters has a large gain. Adjust the resulting filter coefficients to ensure that the largest frequency gain is lessthan or equal to one by dividing the coefficients by an appropriate value. Do the frequency responses match yourexpectations based on the locations of the zeros in the z-plane?