Chapter 7, Problem 20

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### Question

Answered step-by-step

Using the MATLAB functions BESSELDE, impulse and step:

(a) Determine the transfer function in polynomial form, and also factored to indicate the poles, of a Bessel filter with $\omega_p=1, A_p=2 d B$, and $N=6$.

(b) Determine the impulse response and the step response for the filter of part (a).

(c) By multiplying the pole vector found in part (a) by $2 \pi 1000$ determine the transfer function of a Bessel filter with $f_p=1000 \mathrm{~Hz}, A_p=2 \mathrm{~dB}$, 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 to display the smooth phase response rather than 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 BESSELDE, impulse and step:(a) Determine the transfer function in polynomial form, and also factored to indicate the poles, of a Bessel filter with $\omega_p=1, A_p=2 d B$, and $N=6$.(b) Determine the impulse response and the step response for the filter of part (a).(c) By multiplying the pole vector found in part (a) by $2 \pi 1000$ determine the transfer function of a Bessel filter with $f_p=1000 \mathrm{~Hz}, A_p=2 \mathrm{~dB}$, 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 to display the smooth phase response rather than 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 7

** Chapter 1

** Chapter 2

** Chapter 3

** Chapter 4

** Chapter 5

** Chapter 6

** Chapter 7

** Chapter 8

** Chapter 9

** Chapter 10

** Chapter 11

Sections

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

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