hw5slns

# hw5slns - Homework#5 Solutions(Total 60 points(5 5 5 = 15...

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Homework #5 Solutions (Total: 60 points)

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(5+5+5 = 15 points)
2. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 4 -50 -40 -30 -20 -10 0 -3 dB -30 dB p s Analog Elliptical Filters Frequency (rad/s) Magnitude (dB) No Prewarping Prewarping -5 -4 -3 -2 -1 0 1 2 3 4 5 x 10 4 -5 0 5 x 10 4 s-plane Re(s) Im(s) No Prewarping Prewarping

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0 0.5 1 1.5 2 2.5 3 -50 -40 -30 -20 -10 0 -3 dB -30 dB p s Digital Elliptical Filters Frequency (rad) Magnitude (dB) Bilinear, no prewarping Bilinear, prewarping Impulse Invariance -3 -2 -1 0 1 2 3 -1 -0.5 0 0.5 1 z-plane Re(z) Im(z) zero at 9.69 Bilinear, no prewarping Bilinear, prewarping Impulse Invariance N_Ellip = 4 Here, we see that without prewarping the frequencies, the bilinear transformation fails to meet the specs (it starts falling off too early). This is because the analog specs we used for the analog filter design do not map to the digital specs given to us without prewarping. If we do prewarp the frequencies, however, we do meet our spec. The poles in both cases get mapped to inside the unit circle, but when we use prewarping, the poles and zeros are slightly different since we are now using the correct frequency mapping. When we use impulse invariance, we see that the spec is not met in the stopband due to aliasing. We do, however, meet our passband spec. We have a zero from the imaginary axis on the analog filter mapped to far outside the unit circle in the digital filter using this method. (5+5+5+5 = 20 points)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 4 -50 -40 -30 -20 -10 0 -3 dB -30 dB p s Analog Butterworth Filters Frequency (rad/s) Magnitude (dB) No Prewarping Prewarping -5 -4 -3 -2 -1 0 1 2 3 4 5 x 10 4 -5 0 5 x 10 4 s-plane Re(s) Im(s) No Prewarping Prewarping

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0 0.5 1 1.5 2 2.5 3 -50 -40 -30 -20 -10 0 -3 dB -30 dB p s Digital Butterworth Filters Frequency (rad) Magnitude (dB) Bilinear, no prewarping Bilinear, prewarping Impulse Invariance -3 -2 -1 0 1 2 -1 -0.5 0 0.5 1 z-plane Re(z) Im(z) many 0's outside of plot range Bilinear, no prewarping Bilinear, prewarping
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hw5slns - Homework#5 Solutions(Total 60 points(5 5 5 = 15...

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