Digital Filter Design

A low$-$pass digital filter is to be designed according to the specifications of passband frequency p $=0\mathrm{.}3$$,$ ripple $1,$ stopband frequency s $=0\mathrm{.}5$ rad$/$s$,$ and ripple $2.$

a$)$ Filter Characteristics

Illustrate the filter characteristics based on the design specifications.

b$)$ FIR Filter Design Using Window Function

Explain step by step the FIR filter design approach using the window function. Design a linear$-$phase FIR filter using both rectangular and Hamming window approaches. Verify that the designed filter coefficients meet the specifications by obtaining the frequency response of the filter.

c$)$ IIR Filter Design Through Analog$-$to$-$Digital Conversion

Explain step by step the IIR filter design approach through digitization from an analog filter. Use the Butterworth approach to obtain a digital filter from an analog filter that meets the same specifications. Choose a sampling frequency of $4$ kHz and digitize using the Tustin approach.