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

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

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

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

Determine the poles for the transfer function of Problem 9.1.

Determine the poles for the transfer function of Problem 9.2.

Determine the poles for the transfer function of Problem 9.3.

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 $f_s=26 \mathrm{kHz}$.

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 $f_s=13 \mathrm{kHz}$.

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

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

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

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

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.

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.

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.

Indicate how Figure 3.11 could be used to obtain the plot of the unit step response for each of the two filters in Problem 9.7.

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

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

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

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

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

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

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

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

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

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

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 $f_{p_{H P}}=26 \mathrm{kHz}$.

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 $f_{p_{H P}}=13 \mathrm{kHz}$.

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.

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.

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.

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.

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.

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.

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).

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).

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 $B_p=200 \mathrm{rad} / \mathrm{s}$.

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 $B_p=200 \mathrm{rad} / \mathrm{s}$.

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

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

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 $B_s=26 \mathrm{kHz}$.

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 $B_s=13 \mathrm{kHz}$.

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.

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

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.

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

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.

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

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.

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

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 $B_p=500 \mathrm{rad} / \mathrm{s}$.

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).

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 $B_p=200 \mathrm{rad} / \mathrm{s}$.

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 $B_p=200 \mathrm{rad} / \mathrm{s}$.

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

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

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 $B_p=26 \mathrm{kHz}$

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 $B_p=13 \mathrm{kHz}$.

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.

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

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.

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

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.

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

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.

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

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 $B_p=500 \mathrm{rad} / \mathrm{s}$.

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).