Chapter 9, FREQUENCY TRANSFORMATIONS Video Solutions, Design and Analysis of Analog Filters: A Signal Processing Perspective | Numerade (2024)

Paarmann L.D.

Chapter 9

FREQUENCY TRANSFORMATIONS - all with Video Answers

Educators

Chapter Questions

Problem 1

Similar to Example 9.1, determine the transfer function of 3rd-order Butterworth lowpass filter with $\mathrm{an}_c$ of $5000 \mathrm{rad} / \mathrm{s}$.

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Problem 2

Determine the transfer function of a 3rd-order Chebyshev Type I lowpass filter with $A_p=1.5 d B$ and $\omega_p=1000 \mathrm{rad} / \mathrm{s}$.

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Problem 3

Repeat Problem 9.2 for $\omega_c=1000 \mathrm{rad} / \mathrm{s}$.

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Problem 4

Determine the poles for the transfer function of Problem 9.1.

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Problem 5

Determine the poles for the transfer function of Problem 9.2.

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Problem 6

Determine the poles for the transfer function of Problem 9.3.

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Problem 7

Given that the desired specifications of a Butterworth lowpass filter are as follows: $\quad A_p=3 \mathrm{~dB}, \quad A_s=70 \mathrm{~dB}, \quad \omega_p=2,500 \mathrm{rad} / \mathrm{s}, \quad$ and $\omega_s=10,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specifications. Repeat the above for $A_p=3 d B, A_s=70 d B$, $f_p=6,500 \mathrm{~Hz}$, and $f_s=26 \mathrm{kHz}$.

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Problem 8

Given that the desired specifications of a Chebyshev Type I lowpass filter are as follows: $A_p=1.2 \mathrm{~dB}, \quad A_s=75 \mathrm{~dB}, \quad \omega_p=3,500 \mathrm{rad} / \mathrm{s}, \quad$ and $\omega_s=7,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specifications. Repeat the above for $A_p=1.2 d B, A_s=75 d B$, $f_p=6,500 \mathrm{~Hz}$, and $f_s=13 \mathrm{kHz}$.

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04:55
Problem 9

Determine the Filter Selectivity, $F_S$, for each of the two filters in Problem 9.7.

Chapter 9, FREQUENCY TRANSFORMATIONS Video Solutions, Design and Analysis of Analog Filters: A Signal Processing Perspective | Numerade (11)

Amit Srivastava

Numerade Educator

04:55
Problem 10

Determine the Filter Selectivity, $F_S$, for each of the two filters in Problem 9.8.

Chapter 9, FREQUENCY TRANSFORMATIONS Video Solutions, Design and Analysis of Analog Filters: A Signal Processing Perspective | Numerade (14)

Amit Srivastava

Numerade Educator

Problem 11

Determine the Shaping Factor, $S_a^b$, for each of the two filters in Problem 9.7, where $a=3 d B$ and $b=70 d B$.

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Problem 12

Determine the Shaping Factor, $S_a^b$, for each of the two filters in Problem 9.8, where $a=1.2 d B$ and $b=75 d B$.

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Problem 13

Indicate how Figure 3.8 could be used to obtain the plot of phase delay for each of the two filters in Problem 9.7.

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Problem 14

Indicate how Figure 3.9 could be used to obtain the plot of group delay for each of the two filters in Problem 9.7.

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Problem 15

Indicate how Figure 3.10 could be used to obtain the plot of the unit impulse response for each of the two filters in Problem 9.7.

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Problem 17

Similar to Example 9.2, determine the transfer function of 3rd-order Butterworth highpass filter with an $\omega_c$ of $5000 \mathrm{rad} / \mathrm{s}$.

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Problem 18

Determine the transfer function of a 3rd-order Chebyshev Type I highpass filter with $A_p=1.5 d B$ and $\omega_p=1000 \mathrm{rad} / \mathrm{s}$.

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Problem 19

Repeat Problem 9.18 for $\omega_c=1000 \mathrm{rad} / \mathrm{s}$.

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Problem 20

Determine the poles and zeros for the transfer function of Problem 9.17.

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Problem 21

Determine the poles and zeros for the transfer function of Problem 9.18.

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Problem 22

Determine the poles and zeros for the transfer function of Problem 9.19.

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Problem 23

Using MATLAB, plot the magnitude frequency response and the phase response for the filter of Problem 9.17.

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Problem 24

Using MATLAB, plot the magnitude frequency response and the phase response for the filter of Problem 9.18.

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Problem 25

Using MATLAB, plot the magnitude frequency response and the phase response for the filter of Problem 9.19.

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Problem 26

Using MATLAB, plot the magnitude frequency response and the phase response for the highpass filter of Example 9.3.

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Problem 27

Given that the desired specifications of a Butterworth highpass filter are as follows: $A_p=3 d B, \quad A_s=70 d B, \quad \omega_{s_{H P}}=2,500 \mathrm{rad} / \mathrm{s}, \quad$ and $\omega_{p_{H P}}=10,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specifications. Repeat the above for $A_p=3 d B, A_s=70$ $d B, f_{s_{H P}}=6,500 \mathrm{~Hz}$, and $f_{p_{H P}}=26 \mathrm{kHz}$.

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Problem 28

Given that the desired specifications of a Chebyshev Type I highpass filter are as follows: $A_p=1.2 \mathrm{~dB}, \quad A_s=75 \mathrm{~dB}, \quad \omega_{s_{d I P}}=3,500 \mathrm{rad} / \mathrm{s}$, and $\omega_{p_{H P}}=7,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specífications. Repeat the above for $A_p=1.2 \mathrm{~dB}, A_s=75 \mathrm{~dB}$, $f_{s_{H P}}=6,500 \mathrm{~Hz}$, and $f_{p_{H P}}=13 \mathrm{kHz}$.

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Problem 29

Determine the Filter Selectivity of the highpass filter of Problem 9.17 in two ways: (a) by use of (9.14) and (3.7), and (b) computationally, using MATLAB.

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Problem 30

Determine the Filter Selectivity of the highpass filter of Problem 9.18 in two ways: (a) by use of (9.14) and (4.9), and (b) computationally, using MATLAB.

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Problem 31

Determine the Filter Selectivity of the highpass filter of Problem 9.19 in two ways: (a) by use of (9.14) and (4.9), and (b) computationally, using MATLAB.

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Problem 32

Determine the Shaping Factor of the highpass filter of Problem 9.17 in two ways: (a) by use of (9.15) and (3.10), and (b) computationally, using MATLAB.

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Problem 33

Determine the Shaping Factor of the highpass filter of Problem 9.18 in two ways: (a) by use of (9.15) and (4.12), and (b) computationally, using MATLAB.

Problem 34

Determine the Shaping Factor of the highpass filter of Problem 9.19 in two ways: (a) by use of $(\mathbf{9 . 1 5})$ and (4.12), and (b) computationally, using MATLAB.

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Problem 35

For the Butterworth highpass filter of Problem 9.17, determine a closed-form expression for the group delay, similar to Example 9.4. Using MATLAB, plot the response of your expression. For comparison, determine and plot the group delay response as obtained by computational manipulation of the phase response (the traditional computational approach).

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Problem 36

Using MATLAB, plot the magnitude frequency response, phase response, phase delay response, group delay response, unit impulse response, and unit step response for a 10th-order Chebyshev Type I highpass filter with $A_p=1 d B$ and $\omega_{p_{H P}}=1000 \mathrm{rad} / \mathrm{s}$. That is, confirm the results plotted in Figure 9.1 through ${ }^{\text {HPp }}$ Figure 9.7 (Example 9.6).

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Problem 37

Similar to Example 9.11, determine the poles and zeros of an 8th-order Butterworth bandpass filter, with $\omega_o=1000 \mathrm{rad} / \mathrm{s}$, and $B_p=200 \mathrm{rad} / \mathrm{s}$.

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Problem 38

Determine the poles and zeros of a 6th-order Chebyshev Type I bandpass filter, with $1 \mathrm{~d} B$ of ripple, $\omega_o=1000 \mathrm{rad} / \mathrm{s}$, and $B_p=200 \mathrm{rad} / \mathrm{s}$.

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Problem 39

Using MATLAB, plot the magnitude frequency response and the phase response for the filter of Problem 9.37.

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Problem 40

Using MATLAB, plot the magnitude frequency response and the phase response for the filter of Problem 9.38.

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Problem 41

Given that the desired specifications of a Butterworth bandpass filter are as follows: $\quad A_p=3 d B, \quad A_s=70 d B, \quad B_p=2,500 \mathrm{rad} / \mathrm{s}, \quad$ and $B_s=10,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specifications. Repeat the above for $A_p=3 d B, A_s=70$ $d B, B_p=6,500 \mathrm{~Hz}$, and $B_s=26 \mathrm{kHz}$.

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Problem 42

Given that the desired specifications of a Chebyshev Type I bandpass filter are as follows: $A_p=1.2 \mathrm{~dB}, A_s=75 \mathrm{~dB}, \quad B_p=3,500 \mathrm{rad} / \mathrm{s}$, and $B_s=7,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specifications. Repeat the above for $A_p=1.2 \mathrm{~dB}, A_s=75 \mathrm{~dB}$, $B_p=6,500 \mathrm{~Hz}$, and $B_s=13 \mathrm{kHz}$.

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Problem 43

Determine the Filter Selectivity of the bandpass filter in Problem 9.41 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.31) and (3.7), and (b) computationally, using MATLAB.

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Problem 44

Repeat Problem 9.43 for $\omega_o=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 45

Determine the Filter Selectivity of the bandpass filter in Problem 9.42 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.31) and (4.9), and (b) computationally, using MATLAB.

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Problem 46

Repeat Problem 9.45 for $\omega_o=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 47

Determine the Shaping Factor of the bandpass filter in Problem 9.41 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.32) and (3.10), and (b) computationally, using MATLAB.

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Problem 48

Repeat Problem 9.47 for $\omega_0=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 49

Determine the Shaping Factor of the bandpass filter in Problem 9.42 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.32) and (4.12), and (b) computationally, using MATLAB.

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Problem 50

Repeat Problem 9.49 for $\omega_o=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 51

Using the closed-form procedure of Example 9.14 , compute the group delay of a 6th-order Butterworth bandpass filter at $\omega_{p_1}, \omega_{p_2}$, and $\omega_o$ where $\omega_o=5000 \mathrm{rad} / \mathrm{s}$ and $B_p=500 \mathrm{rad} / \mathrm{s}$.

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Problem 52

Using MATLAB, plot the magnitude frequency response, phase response, phase delay response, group delay response, unit impulse response, and unit step response for a 10th-order Chebyshev Type II bandpass filter with $A_p=3 \mathrm{~dB}, A_s=80 \mathrm{~dB}, \omega_o=1000 \mathrm{rad} / \mathrm{s}$, and $B_p=200 \mathrm{rad} / \mathrm{s}$. That is, confirm the results plotted in Figure 9.8 through Figure 9.14 (Example 9. 15).

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Problem 53

Similar to Example 9.20 , determine the poles and zeros of an 8th-order Butterworth bandstop filter, with $\omega_o=1000 \mathrm{rad} / \mathrm{s}$, and $B_p=200 \mathrm{rad} / \mathrm{s}$.

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Problem 54

Determine the poles and zeros of a 6th-order Chebyshev Type I bandstop filter, with $1 \mathrm{~dB}$ of ripple, $\omega_o=1000 \mathrm{rad} / \mathrm{s}$, and $B_p=200 \mathrm{rad} / \mathrm{s}$.

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Problem 55

Using MATLAB, plot the magnitude frequency response and the phase response for the filter of Problem 9.53.

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Problem 56

Using MATLAB, plot the magnitude frequency response and the phase response for the filter of Problem 9.54.

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Problem 57

Given that the desired specifications of a Butterworth bandstop filter are as follows: $\quad A_p=3 \mathrm{~dB}, \quad A_{\mathrm{s}}=70 \mathrm{~dB}, \quad B_{\mathrm{s}}=2,500 \mathrm{rad} / \mathrm{s}, \quad$ and $B_p=10,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specifications. Repeat the above for $A_p=3 d B, A_s=70$ $d B, \quad B_s=6,500 \mathrm{~Hz}$, and $B_p=26 \mathrm{kHz}$

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Problem 58

Given that the desired specifications of a Chebyshev Type I bandstop filter are as follows: $A_p=1.2 \mathrm{~dB}, \quad A_s=75 \mathrm{~dB}, \quad B_s=3,500 \mathrm{rad} / \mathrm{s}$, and $B_p=7,000 \mathrm{rad} / \mathrm{s}$, determine the minimum required filter order to meet or exceed these specifications. Repeat the above for $A_p=1.2 d B, A_s=75 d B$, $B_s=6,500 \mathrm{~Hz}$, and $B_p=13 \mathrm{kHz}$.

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Problem 59

Determine the Filter Selectivity of the bandstop filter in Problem 9.57 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.45) and (3.7), and (b) computationally, using MATLAB.

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Problem 60

Repeat Problem 9.59 for $\omega_o=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 61

Determine the Filter Selectivity of the bandstop filter in Problem 9.58 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.45) and (4.9), and (b) computationally, using MATLAB.

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Problem 62

Repeat Problem 9.61 for $\omega_o=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 63

Determine the Shaping Factor of the bandpass filter in Problem 9.57 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.46) and (3.10), and (b) computationally, using MATLAB.

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Problem 64

Repeat Problem 9.63 for $\omega_o=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 65

Determine the Shaping Factor of the bandpass filter in Problem 9.58 with $\omega_o=15,000 \mathrm{rad} / \mathrm{s}$ in two ways: (a) by use of (9.46) and (4.12), and (b) computationally, using MATLAB.

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Problem 66

Repeat Problem 9.65 for $\omega_o=10,000 \mathrm{rad} / \mathrm{s}$.

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Problem 67

Using (9.48), and the procedure in Example 9.22, compute the group delay of a 6th-order Butterworth bandstop filter at $\omega_{p_1}, \omega_{p_2}$, and $\boldsymbol{D C}$, where $\omega_o=5000 \mathrm{rad} / \mathrm{s}$ and $B_p=500 \mathrm{rad} / \mathrm{s}$.

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Problem 68

Using MATLAB, plot the magnitude frequency response, phase response, phase delay response, group delay response, unit impulse response, and unit step response for a 10th-order elliptic bandstop filter with $A_p=1 d B$, $A_s=80 \mathrm{~dB}, \omega_o=1000 \mathrm{rad} / \mathrm{s}$, and $B_p=200 \mathrm{rad} / \mathrm{s}$. That is, confirm the results plotted in Figure 9.15 through Figure 9.21 (Example 9.23).

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Chapter 9, FREQUENCY TRANSFORMATIONS Video Solutions, Design and Analysis of Analog Filters: A Signal Processing Perspective | Numerade (2024)

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