Chapter 6, Problem 32
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Question
Answered step-by-step
Using the MATLAB functions ellipap, impulse and step:
(a) Determine the transfer function in polynomial form, and also factored to indicate the poles and zeros, of an elliptic filter with $\omega_p=1$, $A_p=1.2 \mathrm{~dB}, A_s=70 \mathrm{~dB}$, and $N=6$.
(b) Determine the impulse response and the step response for the filter of part (a).
(c) By multiplying the pole vector and the zero vector found in part (a) by $2 \pi 1000$ determine the transfer function of an elliptic filter with $f_p=1000 \mathrm{~Hz}, A_p=1.2 \mathrm{~dB}, A_s=70 \mathrm{~dB}$, and $N=6$.
(d) Determine and plot the magnitude frequency response of the filter of part (c) by using the MATLAB function freqs. Use a vertical scale in $d B$ and a linear horizontal scale from 0 to $5000 \mathrm{~Hz}$. Also determine and plot the phase response over this same frequency range. Use the MATLAB function unwrap rather than plotting the principle phase.
(e) By appropriately scaling the impulse response and the step response of part (b), determine and plot the impulse response and the step response of the filter of part (c). That is, the time axis for the step response needs to scaled by $1 /(2 \pi 1000)$, and the unit impulse response needs the same time-axis scaling and requires an amplitude scaling of $2 \pi 1000$
(f) Determine and plot the phase delay of the filter of part (c). Note that this is easily obtained from the phase response of part (d).
(g) Determine and plot the group delay of the filter of part (c). Note that this also is easily obtained from the phase response of part (d): $t_{g d}(n) \cong-[\phi(n)-\phi(n-1)] / S_s$, where $\phi(n)$ is the phase in radians at step $n$, and $S_s$ is the step size in $\mathrm{rad} / \mathrm{s}$.
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Using the MATLAB functions ellipap, impulse and step:(a) Determine the transfer function in polynomial form, and also factored to indicate the poles and zeros, of an elliptic filter with $\omega_p=1$, $A_p=1.2 \mathrm{~dB}, A_s=70 \mathrm{~dB}$, and $N=6$.(b) Determine the impulse response and the step response for the filter of part (a).(c) By multiplying the pole vector and the zero vector found in part (a) by $2 \pi 1000$ determine the transfer function of an elliptic filter with $f_p=1000 \mathrm{~Hz}, A_p=1.2 \mathrm{~dB}, A_s=70 \mathrm{~dB}$, and $N=6$.(d) Determine and plot the magnitude frequency response of the filter of part (c) by using the MATLAB function freqs. Use a vertical scale in $d B$ and a linear horizontal scale from 0 to $5000 \mathrm{~Hz}$. Also determine and plot the phase response over this same frequency range. Use the MATLAB function unwrap rather than plotting the principle phase.(e) By appropriately scaling the impulse response and the step response of part (b), determine and plot the impulse response and the step response of the filter of part (c). That is, the time axis for the step response needs to scaled by $1 /(2 \pi 1000)$, and the unit impulse response needs the same time-axis scaling and requires an amplitude scaling of $2 \pi 1000$(f) Determine and plot the phase delay of the filter of part (c). Note that this is easily obtained from the phase response of part (d).(g) Determine and plot the group delay of the filter of part (c). Note that this also is easily obtained from the phase response of part (d): $t_{g d}(n) \cong-[\phi(n)-\phi(n-1)] / S_s$, where $\phi(n)$ is the phase in radians at step $n$, and $S_s$ is the step size in $\mathrm{rad} / \mathrm{s}$.
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Design and Analysis of Analog Filters: A Signal Processing Perspective
Chapter 6
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
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Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Problem 15 Problem 16 Problem 17 Problem 18 Problem 19 Problem 20 Problem 21 Problem 22 Problem 23 Problem 24 Problem 25 Problem 26 Problem 27 Problem 28 Problem 29 Problem 30 Problem 31 Problem 32 Problem 33 Problem 34
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