Homework 8 5 = 1, (e) = 0, Of course, you obtained the unit sample response for this filter in the previous problem. We will be designing a filter for use with speech that is sampled at 16 kHz, a standard sampling rate for speech. As we will discuss in a week when we introduce the sampling theorem, the maximum unique discrete-time frequency. , corresponds to half the sampling frequency, or 8000 Hz. We would like the cutoff frequency for our lowpass filter to be 1000 Hz, which corresponds to Q/8 for our sampling rate. Use the MATLAB command audioread to read in the familiar utterance welcome16k.wav While you have used this command in previous homework sets, feel free to consult with the help files MATLAB commands as needed. Use the MATLAB command soundsc to listen to the waveform. (b) In order to make the practical discrete-time lowpass filter finite in duration, we simply multiply the infinite-duration desired unit sample response hp[n] by a finite- duration window function w[n]: hp [n] = hp [n]w[n] LP Two popular window functions are the rectangular window window w,, [n] which are defined as follows: Rectangular window W, [n]= 1, 0 n M 0, otherwise Hamming window W, [n] = w, [n] and the Hamming 0.54 0.46 cos(2m/M), 0 n M 0, otherwise where in this case the parameter MN - 1 where N is the length of the window function. For this problem we will choose the specific value of N = 127, which has deliberately been chosen to be an odd number for N, (and hence an even value for M). Note that the rectangular window is just what you would get if you simply snipped a short segment of signal away from a longer segment. The design of the Hamming window is rather clever, but it is beyond the scope of this discussion. In addition to having the practical filter be finite in duration, we normally want it to exhibit linear phase if possible. This ensures that signals that lie entirely in the passband will not be distorted, although they may be delayed. It can be shown (based