Questions and Answers of Digital Control System Analysis And Design

Demonstrate quantitatively that FIR filters based on the prefilter and the interpolation methods have lower sensitivity and reduced output roundoff noise than the minimax filters implemented using
Design a lowpass filter using the frequency-response masking method satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =2.0 \mathrm{~dB} \\A_{\mathrm{r}} & =40 \mathrm{~dB}
Design a highpass filter using the frequency-response masking method satisfying the specifications in Exercise 12.9. Compare the results obtained with and without an efficient ripple margin
Design a bandpass filter using the quadrature method satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =0.02 \mathrm{~dB} \\A_{\mathrm{r}} & =40 \mathrm{~dB}
Design a bandstop filter using the quadrature method satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =0.2 \mathrm{~dB} \\A_{\mathrm{r}} & =60 \mathrm{~dB}
Create a command in MATLAB that processes an input signal \(x\) with a frequencyresponse masking filter, taking advantage of the internal structure of this device. The command receives
Revisit Experiment 12.1 performing the following tasks:(a) Design a direct-form FIR filter using the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =0.1 \mathrm{~dB} \\A_{\mathrm{r}}
Consider the alternative parallel structure seen in Figure 13.56, whose transfer function is\[H(z)=h_{0}^{\prime}+\sum_{k=1}^{m} \frac{\gamma_{1 k}^{\prime} z+\gamma_{2 k}^{\prime}}{z^{2}+m_{1 k}
Derive the expressions in Equations (13.20) and (13.21). c2,j= 1112 1113 4-4 (13.20) -1213 4-4 (13.21)
Show that the \(L_{2}\) and \(L_{\infty}\) norms of a second-order \(H(z)\) given by\[H(z)=\frac{\gamma_{1} z+\gamma_{2}}{z^{2}+m_{1} z+m_{2}}\]are as given in Equations (13.28) and (13.29),
Derive the expressions for the output-noise PSD, as well as their variances, for odd-order cascade filters realized with the following second-order sections:(a) direct-form;(b) optimal state
Repeat Exercise 13.4 assuming floating-point arithmetic.Exercise 13.4Derive the expressions for the output-noise PSD, as well as their variances, for odd-order cascade filters realized with the
From Equations (13.55), (13.56), and the definitions of \(F_{i}^{\prime}(z)\) and \(G_{i}^{\prime}(z)\) in Figures 11.20 and 11.16 respectively, derive Equations (13.57) and (13.58). x(n) Fig.
Verify that Equations (13.72)-(13.73) correspond to a state-space structure with minimum output noise. m1 a11=- 2 a12=- m2 - m 4 a21=-a12 a22 = a11 Also, compute the parameters b, b, c1, and c, using
Show that when the poles of a second-order section with transfer function\[H(z)=d+\frac{\gamma_{1} z+\gamma_{2}}{z^{2}+m_{1} z+m_{2}}\]are complex conjugate, then the parameter \(\sigma\) defined in
Show that the state-space section free of limit cycles, Structure I, as determined by Equations (13.97) and (13.98), has minimum roundoff noise only
Design an elliptic filter satisfying the specifications below:\[\begin{aligned}A_{\mathrm{p}} & =0.4 \mathrm{~dB} \\A_{\mathrm{r}} & =50 \mathrm{~dB} \\\Omega_{\mathrm{r}_{1}} & =1000 \mathrm{rad} /
Repeat Exercise 13.10 with the specifications below:\[\begin{aligned}A_{\mathrm{p}} & =1.0 \mathrm{~dB} \\A_{\mathrm{r}} & =70 \mathrm{~dB} \\\omega_{\mathrm{p}} & =0.025 \pi \mathrm{rad} /
Design an elliptic filter satisfying the specifications of Exercise 13.10, using a cascade of direct-form structures with the ESS technique.Exercise 13.10Design an elliptic filter satisfying the
Design an elliptic filter satisfying the specifications of Exercise 13.10, using parallel connection of second-order direct-form structures, with \(L_{2}\) and \(L_{\infty}\) norms for scaling,
Derive Equation (13.132). Dj(z) Bi(z) [BS]-L-38-3] aj.j Dj- (13.132) Bi-1(z)
Derive the scaling factor utilized in the two-multiplier lattice of Figure 13.9, in order to generate the one-multiplier lattice of Figure 13.12. + Dj(z) -ajj (X) ajj + zB,(z) Fig. 13.9.
Derive the scaling factor to be used in the normalized lattice of Figure 13.13, in order to generate a three-multiplier lattice. D; (z) D(z) ajj (X) (1-a ( 1/2 zBj (z) D(z) + ( 2,1/2 (1-a) Fig.
Design a two-multiplier lattice structure to implement the filter described in Exercise 13.10.Exercise 13.10Design an elliptic filter satisfying the specifications
The coefficients of an allpass filter designed to generate doubly complementary filters of order \(N_{1}=5\) are given in Table 13.16. Describe what happens to the properties of these filters when we
The coefficients of an allpass filter designed to generate doubly complementary filters of order \(N_{1}=9\) are given in Table 13.17.(a) Plot the phase response of the allpass filter and verify the
Derive a realization for the circuit of Figure 13.45, using the standard wave digital filter method, and compare it with the wave lattice structure obtained in Figure 13.50, with respect to overall
Table 13.18 shows the multiplier coefficients of a scaled Chebyshev filter designed with the following specifications\[\begin{aligned}A_{\mathrm{p}} & =0.6 \mathrm{~dB} \\A_{\mathrm{r}} & =48
Revisit Experiment 13.1 and determine:(a) The peaking factor and the scaling factors for the 9-bit quantized cascade filter, as described in Table 13.5.(b) The relative output-noise variance for the
Write a table equivalent to Table 5.1 supposing that the ideal impulse responses have an added phase term of \(-(M / 2) \omega\), for \(M\) odd. Table 5.1. Ideal frequency characteristics and
Assume that a periodic signal has four sinusoidal components at frequencies \(\omega_{0}, 2 \omega_{0}, 4 \omega_{0}, 6 \omega_{0}\). Design a nonrecursive filter, as simple as possible, that
Given a lowpass FIR filter with transfer function \(H(z)\), describe what happens to the filter frequency response when:(a) \(z\) is replaced by \(-z\).(b) \(z\) is replaced by \(z^{-1}\).(c) \(z\)
Complementary filters are such that their frequency responses add to a delay. Given an \(M\) th-order linear-phase FIR filter with transfer function \(H(z)\), deduce the conditions on \(L\) and \(M\)
Determine the relationship between \(L, M\), and \(N\) which means that the overall filter in Figure 5.29 has linear phase. Fig. 5.29. x(n) N Linear-phase FIR filter of order M Overall filter block
Design a highpass filter satisfying the specification below using the frequency sampling method:\[\begin{aligned}M & =40 \\\Omega_{\mathrm{r}} & =1.0 \mathrm{rad} / \mathrm{s} \\\Omega_{\mathrm{p}} &
Plot and compare the characteristics of the Hamming window and the corresponding magnitude response for \(M=5,10,15,20\).
Plot and compare the rectangular, triangular, Bartlett, Hamming, Hann, and Blackman window functions and the corresponding magnitude responses for \(M=20\).
Determine the ideal impulse response associated with the magnitude response shown in Figure 5.30, and compute the corresponding practical filter of orders \(M=10,20,30\) using the Hamming window. 4 |
For the magnitude response shown in Figure 5.31, where \(\omega_{\mathrm{s}}=2 \pi\) denotes the sampling frequency:(a) Determine the ideal impulse response associated with it.(b) Design a
For the magnitude response shown in Figure 5.32:(a) Determine the ideal impulse response associated with it.(b) Design a fourth-order FIR filter using the Hann window. Fig. 5.32. 1- | H(ejo) | B| 4
Design a bandpass filter satisfying the specification below using the Hamming, Hann, and Blackman windows:\[\begin{aligned}M & =10 \\\Omega_{\mathrm{c}_{1}} & =1.125 \mathrm{rad} /
Plot and compare the characteristics of the Kaiser window function and corresponding magnitude response for \(M=20\) and different values of \(\beta\).
Design the following filters using the Kaiser window:(a) \(A_{\mathrm{p}}=1.0 \mathrm{~dB}\)\(A_{\mathrm{r}}=40 \mathrm{~dB}\)\(\Omega_{\mathrm{p}}=1000 \mathrm{rad} /
Determine a complete procedure for designing differentiators using the Kaiser window.
Repeat Exercise 5.14(a) using the Dolph-Chebyshev window. Compare the transition bandwidths and the stopband attenuation levels for the two resulting filters.Exercise 5.14(a)(a) \(A_{\mathrm{p}}=1.0
Design a maximally flat lowpass filter satisfying the specification below:\[\begin{aligned}& \omega_{\mathrm{c}}=0.4 \pi \mathrm{rad} / \mathrm{sample} \\& T_{\mathrm{r}}=0.2 \pi \mathrm{rad} /
Design three narrowband filters, centered on the frequencies \(770 \mathrm{~Hz}, 852 \mathrm{~Hz}\), and 941\(\mathrm{Hz}\), satisfying the specification below, using the minimax
Design Hilbert transformers of orders \(M=38,68\), and 98 using a Type IV structure and the Hamming window method.
Design a Hilbert transformer of order \(M=98\) using a Type IV structure and the triangular, Hann, and Blackman window methods.
Design a Hilbert transformer of order \(M=98\) using a Type IV structure and the Chebyshev method, and compare your results with those from Exercise 5.22.Exercise 5.22.Design a Hilbert transformer of
Determine the output of the Hilbert transformer designed in Exercise 5.23 to the input signal \(\mathrm{x}\) determined as Fs \(=1500 ; \mathrm{TS}=1 / \mathrm{Fs} ; \mathrm{t}=0: \mathrm{TS}:
The following relationship estimates the order of a lowpass filter designed with the minimax approach (Rabiner et al., 1975). Design a series of lowpass filters and verify the validity of this
Repeat Exercise 5.25 with the following order estimate (Kaiser, 1974):\[M \approx \frac{-20 \log _{10}\left(\sqrt{\delta_{\mathrm{p}}
Perform the algebraic design of a highpass FIR filter such that\[\begin{aligned}\omega_{\mathrm{p}} & =\frac{\omega_{\mathrm{s}}}{8}\\\delta_{\mathrm{p}} & =8 \delta_{\mathrm{r}}\end{aligned}\]using
Design a bandpass filter satisfying the specification below using the WLS and Chebyshev methods. Discuss the trade-off between the stopband minimum attenuation and total stopband energy when using
Repeat Experiment 5.1 with an input signal defined asand compare the results obtained with each differentiator system, by verifying what happens with each sinusoidal component in \(\mathrm{x}\). Fs
Change the values of Fs, total time length, and \(f_{C}\) in Experiment 5.1, one parameter at a time, and verify their individual influences on the output signal yielded by a differentiator system.
Change the filter specifications in Experiment 5.2, designing the filter accordingly, using the firpm command. Analyze the resulting magnitude response and the output signal to the input
Determine the normalized specifications for the analog, lowpass, Chebysev filter corresponding to the highpass filter:\[\begin{aligned}A_{\mathrm{p}} & =0.2 \mathrm{~dB} \\A_{\mathrm{r}} & =50
Determine the normalized specifications for the analog, lowpass, elliptic filter corresponding to the bandpass filter:\[\begin{aligned}A_{\mathrm{p}} & =2 \mathrm{~dB} \\A_{\mathrm{r}} & =40
Design an analog elliptic filter satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =1.0 \mathrm{~dB} \\A_{\mathrm{r}} & =40 \mathrm{~dB} \\\Omega_{\mathrm{p}} & =1000
Design a lowpass Butterworth filter satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =0.5 \mathrm{~dB} \\A_{\mathrm{r}} & =40 \mathrm{~dB} \\\Omega_{\mathrm{p}} & =100
Design a bandstop elliptic filter satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =0.5 \mathrm{~dB} \\A_{\mathrm{r}} & =60 \mathrm{~dB} \\\Omega_{\mathrm{p}_{1}} & =40
Design highpass Butterworth, Chebyshev, and elliptic filters that satisfy the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =1.0 \mathrm{~dB} \\A_{\mathrm{r}} & =40 \mathrm{~dB}
Design three bandpass digital filters, one with a central frequency of \(770 \mathrm{~Hz}\), a second of \(852 \mathrm{~Hz}\), and a third of \(941 \mathrm{~Hz}\). For the first filter, the stopband
Plot the zero-pole constellation for the three filters designed in Exercise 6.7 and visualize the resulting magnitude response in each case.Exercise 6.7Design three bandpass digital filters, one with
Create an input signal composed of three sinusoidal components of frequencies 770 \(\mathrm{Hz}, 852 \mathrm{~Hz}\), and \(941 \mathrm{~Hz}\), in MATLAB with \(\Omega_{\mathrm{s}}=8 \mathrm{kHz}\).
The transfer function\[H(s)=\frac{\kappa}{\left(s^{2}+1.4256 s+1.23313\right)(s+0.6265)}\]corresponds to a lowpass normalized Chebyshev filter with passband ripple \(A_{\mathrm{p}}=0.5\)
Transform the continuous-time highpass transfer function given by\[H(s)=\frac{s^{2}}{s^{2}+s+1}\]into a discrete-time transfer function using the impulse-invariance transformation method with
Repeat Exercise 6.11 using the bilinear transformation method and compare results from both exercises.Exercise 6.11Transform the continuous-time highpass transfer function given
Given the analog transfer function\[H(s)=\frac{1}{\left(s^{2}+0.76722 s+1.33863\right)(s+0.76722)},\]design transfer functions corresponding to discrete-time filters using both the impulseinvariance
Repeat Exercise 6.13 using \(\Omega_{\mathrm{s}}=24 \mathrm{rad} / \mathrm{s}\) and compare the results achieved in each case.Exercise 6.13Given the analog transfer
Repeat Exercise 6.13 using MATLAB commands impinvar and bilinear.Exercise 6.13Given the analog transfer function\[H(s)=\frac{1}{\left(s^{2}+0.76722 s+1.33863\right)(s+0.76722)},\]design transfer
Determine the original analog transfer function corresponding to\[H(z)=\frac{4 z}{z-\mathrm{e}^{-0.4}}-\frac{z}{z-\mathrm{e}^{-0.8}}\]assuming that the following method was employed in the analog to
Determine the original analog transfer function corresponding to\[H(z)=\frac{2 z^{2}-\left(\mathrm{e}^{-0.2}+\mathrm{e}^{-0.4}\right)
This exercise describes the inverse Chebyshev approximation. The attenuation function of a lowpass inverse Chebyshev filter is characterized as\[\begin{aligned}\left|A\left(\mathrm{j}
Show that the lowpass-to-bandpass and lowpass-to-bandstop transformations proposed in Section 6.4 are valid.Section 6.4 Usually, in the approximation of a continuous-time filter, we begin by
Apply the lowpass-to-highpass transformation to the filter designed in Exercise 6.4 and plot the resulting magnitude response.Exercise 6.4Design a lowpass Butterworth filter satisfying the following
Revisit Example 6.4, now forcing the highpass zero at \(\omega_{\mathrm{p}_{1}}=2 \pi / 3\). Plot the magnitude responses before and after the transformation. Example 6.4. Consider the lowpass notch
Given the transfer function\[H(z)=0.06 \frac{z^{2}+\sqrt{2} z+1}{z^{2}-1.18 z+0.94}\]describe a frequency transformation to a bandpass filter with zeros at \(\pi / 6\) and \(2 \pi / 3\).
Design a phase equalizer for the elliptic filter of Exercise 6.6 with the same order as the filter.Exercise 6.6Design highpass Butterworth, Chebyshev, and elliptic filters that satisfy the following
Design a lowpass filter satisfying the following specifications:\[\begin{aligned}M(\Omega T)=1.0, & \text { for } 0.0 \Omega_{\mathrm{s}}
The desired impulse response for a filter is given by \(g(n)=1 / 2^{n}\). Design a recursive filter such that its impulse response \(h(n)\) equals \(g(n)\) for \(n=0,1, \ldots, 5\).
Plot and compare the magnitude responses associated with the ideal and approximated impulse responses in Exercise 6.26.Exercise 6.26.The desired impulse response for a filter is given by \(g(n)=1 /
Design a filter with 10 coefficients such that its impulse response approximates the following sequence:\[g_{n}=\left(\frac{1}{6^{n}}+10^{-n}+\frac{0.05}{n+2}\right) u(n)\]Choose a few key values for
Compare the magnitude responses associated with the filters designed in Exercise 6.28 for several values of \(M\) and \(N\).Exercise 6.28Design a filter with 10 coefficients such that its impulse
Repeat Experiment 6.2 using \(F_{\mathrm{s}}=10 \mathrm{~Hz}\). Compare the magnitude response of the resulting discrete-time transfer function with the one obtained in the experiment. Experiment 6.2
Determine the Pascal matrix \(\mathbf{P}_{N+1}\) defined in Experiment 6.2 for \(N=4\) and \(N=5\). Experiment 6.2 Consider the analog transfer function of the normalized-lowpass Chebyshev filter in
Design an IIR digital filter to reduce the amount of noise in the two sinusoidal components in Experiment 1.3. Evaluate your specifications by processing x_noisy, as defined in that experiment, with
Use the periodogram algorithm to estimate the PSD of a length- \(L\) white-noise sequence generated by the MATLAB command randn. Average your results for \(N\) realizations of the white noise and
Use the periodogram algorithm to estimate the PSD of \(L=1024\) samples of the following signal:\[x(n)=\cos \frac{2 \pi}{20} n+x_{1}(n)\]where\[x_{1}(n)=-0.9 x_{1}(n-1)+x_{2}(n),\]where \(x_{2}(n)\)
Use the periodogram algorithm to estimate the PSD of the following signal:\[x(n)=\cos \frac{2 \pi}{4} n+x_{1}(n)\]where\[x_{1}(n)=-0.9 x_{1}(n-1)+x_{2}(n) \text {, }\]where \(x_{2}(n)\) is a white
An AR process is generated by applying white Gaussian noise, with variance \(\sigma_{X}^{2}\), to a first-order filter with transfer function\[H(z)=\frac{z}{z-a} .\]This process has the
In Exercise 7.4, consider \(a=0.8\) and plot the minimum-variance solution for window lengths equal to \(L=4, L=10\), and \(L=50\). Compare the estimated PSD in each case with the actual PSD and
Use the minimum-variance method to estimate the PSD of \(L=256\) samples of the following signal:\[y(n)=\sin \frac{\pi}{2} n+x_{1}(n)\]where\[x_{1}(n)=-0.8 x_{1}(n-1)+x_{2}(n) \text {, }\]with
(a) Use the MATLAB command roots to determine the pole locations of the AR approximation, using \(N=1,2,3,4\), for the ARMA system given in Example 7.3.(b) Find the AR impulse response for each value
Find an \(N\) th-order AR approximation for the ARMA system\[H(z)=\frac{1-1.5 z^{-1}}{1+0.5 z^{-1}}\]and compare the resulting magnitude response to the original ARMA system. Observe that in this
Given two first-order AR processes generated by the same zero-mean white noise with unit variance whose respective poles are located at \(a_{1}\) and \(-a_{1}\), with \(\left|a_{1}\right|
Find an \(N\) th-order AR approximation for the ARMA system\[H(z)=\frac{\left(1-0.8 z^{-1}\right)\left(1-0.9 z^{-1}\right)}{\left(1-0.5 z^{-1}\right)\left(1+0.5 z^{-1}\right)}\]and compare the
(a) Find an \(M\) th-order MA approximation for the ARMA system given in Example 7.3.(b) Use the roots command in MATLAB to determine the zero locations of the MA approximation for \(M=1,2,3,4\).(c)

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