3.1 Design Two Lowpass Filters Design two lowpass FIR filters with M = 30 and = 0.5x = 2x(fe/fs) = 2x(2500/10000), one using a Hamming window, the other a Rectangular window. For the measurement of passband and stopband edges, there are two approaches: use the filterdesign GUI and read numbers from the plot, zooming when necessary, or export the filter coefficients from the GUI and use MATLAB to make plots of the magnitude of the frequency response using freekz (or freqz) and plot. In MATLAB, zooming would be more precise and reliable because the frequency sampling can be specified in the call to freekz. (a) For the filter obtained with the rectangular window, determine an accurate measurement of the pass-band edge (@p) assuming the passband ripple specification is 8, = 0.1, i.e., 1 +0.1. (b) For the filter obtained with the rectangular window, determine an accurate measurement of the stop- band edge (@s) assuming the stopband ripple specification is 8, = 0.1. (C) For the filter obtained with the Hamming window, determine an accurate measurement of the edge (o) assuming the passband ripple specification is 8, = 0.01, i.e., 10.01. (d) For the filter obtained with the Hamming window, determine an accurate measurement of the stopband edge (s) assuming the stopband ripple specification is 8=0.01. (e) Question: is the cutoff frequency half way between (p) and (s) for both filters? pass-band Verification of Completion 3.2 Transition Zone of the LPF The difference between the stopband edge and the passband edge is called the transition width of the filter: A = ng - , . The smaller the transition width, the better the filter because it is closer to the ideal filter which has a transition width of zero. (a) For the two lowpass filters from Section 3.1, determine the transition width. Comment: "when comparing two Meh order filters, the one with a smaller transition width will have larger ripples." (b) Design a new Hamming-window LPF that has the same cutoff frequency, c = 0.52, but twice the order, i.e., M=60. Repeat the measurement of op, og and A for this LPF. (c) Compare the values of A from parts (a) and (b); when the order doubles, describe what happens to the transition width. Use this observation to explain that the following Hamming window design formula should = be true 6 = L and find the value of the constant C. 3.1 Design Two Lowpass Filters Design two lowpass FIR filters with M = 30 and = 0.5x = 2x(fe/fs) = 2x(2500/10000), one using a Hamming window, the other a Rectangular window. For the measurement of passband and stopband edges, there are two approaches: use the filterdesign GUI and read numbers from the plot, zooming when necessary, or export the filter coefficients from the GUI and use MATLAB to make plots of the magnitude of the frequency response using freekz (or freqz) and plot. In MATLAB, zooming would be more precise and reliable because the frequency sampling can be specified in the call to freekz. (a) For the filter obtained with the rectangular window, determine an accurate measurement of the pass-band edge (@p) assuming the passband ripple specification is 8, = 0.1, i.e., 1 +0.1. (b) For the filter obtained with the rectangular window, determine an accurate measurement of the stop- band edge (@s) assuming the stopband ripple specification is 8, = 0.1. (C) For the filter obtained with the Hamming window, determine an accurate measurement of the edge (o) assuming the passband ripple specification is 8, = 0.01, i.e., 10.01. (d) For the filter obtained with the Hamming window, determine an accurate measurement of the stopband edge (s) assuming the stopband ripple specification is 8=0.01. (e) Question: is the cutoff frequency half way between (p) and (s) for both filters? pass-band Verification of Completion 3.2 Transition Zone of the LPF The difference between the stopband edge and the passband edge is called the transition width of the filter: A = ng - , . The smaller the transition width, the better the filter because it is closer to the ideal filter which has a transition width of zero. (a) For the two lowpass filters from Section 3.1, determine the transition width. Comment: "when comparing two Meh order filters, the one with a smaller transition width will have larger ripples." (b) Design a new Hamming-window LPF that has the same cutoff frequency, c = 0.52, but twice the order, i.e., M=60. Repeat the measurement of op, og and A for this LPF. (c) Compare the values of A from parts (a) and (b); when the order doubles, describe what happens to the transition width. Use this observation to explain that the following Hamming window design formula should = be true 6 = L and find the value of the constant C
