Questions and Answers of Digital Control System Analysis And Design

In the conversion of a discrete-time signal to a continuous-time signal, the practical D/A converters, instead of generating impulses at the output, generate a series of pulses \(g(t)\) described
Suppose a discrete-time signal \(x(n)\) is obtained by sampling a bandlimited, continuoustime signal \(x_{\mathrm{a}}(t)\) so that there is no aliasing. Prove that the energy of \(x_{\mathrm{a}}(t)\)
Given a sinusoid \(y(t)=A \cos \left(\Omega_{\mathrm{c}} t\right)\), using Equation (1.170), show that if \(y(t)\) is sampled with a sampling frequency slightly above the Nyquist frequency (that is,
As seen in Example 1.15, cinema is a three-dimensional spatio-temporal signal that is sampled in time. In modern cinema, the sampling frequency in order to avoid aliasing is \(\Omega_{\mathrm{s}}=24
Verify that the cross-correlation \(r_{X, Y}\) and the cross-covariance \(c_{X, Y}\) of two random variables \(X\) and \(Y\), as defined in Equations (1.222) and (1.223), satisfy the relation\[c_{X,
Show that the autocorrelation function of a WSS process \(\{X\}\) presents the following properties:(a) \(R_{X}(0)=E\left\{X^{2}(n)\right\}\)(b) it is an even function; that is:
Show that the autocorrelation function of the output of a discrete linear system with impulse response \(h(n)\) is given by\[R_{Y}(n)=\sum_{k=-\infty}^{\infty} R_{X}(n-k)
The entropy \(H(X)\) of a discrete random variable \(X\) measures the uncertainty about predicting the value of \(X\) (Cover \(\&\) Thomas, 2006). If \(X\) has the probability distribution
For a continuous random variable \(X\) with distribution \(p_{X}(x)\), the uncertainty is measured by the so-called differential entropy \(h(X)\) determined as (Cover \& Thomas,
Compute the \(z\) transform of the following sequences, indicating their regions of convergence:(a) \(x(n)=\sin (\omega n+\theta) u(n)\)(b) \(x(n)=\cos (\omega n) u(n)\)(c) \(x(n)= \begin{cases}n, &
Prove Equation (2.35). z"-1 dz [0, 2j Jc 1, for n #0 for n=0 (2.35)
Suppose that the transfer function of a digital filter is given by\[\frac{(z-1)^{2}}{z^{2}+\left(m_{1}-m_{2}\right) z+\left(1-m_{1}-m_{2}\right)}\]Plot a graph specifying the region of the \(m_{2}
Compute the impulse response of the system with transfer function\[H(z)=\frac{z^{2}}{4 z^{2}-2 \sqrt{2} z+1}\]supposing that the system is stable.
Compute the time response of the causal system described by the transfer function\[H(z)=\frac{(z-1)^{2}}{z^{2}-0.32 z+0.8}\]when the input signal is the unit step.
Determine the inverse \(z\) transform of the following functions of the complex variable \(z\), supposing that the systems are stable:(a) \(\frac{z}{z-0.8}\)(b) \(\frac{z^{2}}{z^{2}-z+0.5}\)(c)
Compute the inverse \(z\) transform of the functions below. Suppose that the sequences are right handed and one sided.(a) \(X(z)=\sin \left(\frac{1}{z}\right)\)(b) \(X(z)=\sqrt{\frac{z}{1+z}}\).
An alternative condition for the convergence of a series of the complex variable \(z\), namely\[S(z)=\sum_{i=0}^{\infty} f_{i}(z)\]is based on the function of \(z\)\[\begin{equation*}\alpha(z)=\lim
Given a sequence \(x(n)\), form a new sequence consisting of only the even samples of \(x(n)\); that is, \(y(n)=x(2 n)\). Determine the \(z\) transform of \(y(n)\) as a function of the \(z\)
Determine whether the polynomials below can be the denominator of a causal stable filter:(a) \(z^{5}+2 z^{4}+z^{3}+2 z^{2}+z+0.5\)(b) \(z^{6}-z^{5}+z^{4}+2 z^{3}+z^{2}+z+0.25\)(c) \(z^{4}+0.5 z^{3}-2
Given the polynomial \(D(z)=z^{2}-(2+a-b) z+a+1\) that represents the denominator of a discrete-time system:(a) Determine the range of values of \(a\) and \(b\) such that the system is stable, using
For the pole-zero constellation shown in Figure 2.20, determine the stable impulse response and discuss the properties of the solution obtained.Figure 2.20, -11 Im(z) triple zero 12 11 2 1171 1 Re(z)
Compute the frequency response of a system with the following impulse response: (-1)", |n]
Compute and plot the magnitude and phase of the frequency response of the systems described by the following difference equations:(a) \(y(n)=x(n)+2 x(n-1)+3 x(n-2)+2 x(n-3)+x(n-4)\)(b)
Plot the magnitude and phase of the frequency response of the digital filters characterized by the following transfer functions:(a) \(H(z)=z^{-4}+2 z^{-3}+2 z^{-1}+1\)(b)
If a digital filter has transfer function \(H(z)\), compute the steady-state response of this system for an input of the type \(x(n)=\sin (\omega n) u(n)\).
A given piece of hardware can generate filters with the following generic transfer function:\[H(z)=\frac{\delta_{0}+\delta_{1} z^{-1}-\delta_{2} z^{-1} f(z)}{1-\left(1+m_{1}\right) z^{-1}-m_{2}
Given a linear time-invariant system, prove the properties below:(a) A constant group delay is a necessary but not sufficient condition for the delay introduced by the system to a sinusoid to be
Suppose we have a system with transfer function having two zeros at \(z=0\), double poles at \(z=a\), and a single pole at \(z=-b\), where \(a>1\) and \(0
If the signal \(x(n)=4 \cos [(\pi / 4) n-(\pi / 6)] u(n)\) is input to the linear system of Exercise 2.6e, determine its steady-state response.Exercise 2.6e,(e) \(\frac{1-z^{2}}{\left(2
Compute the Fourier transform of each of the sequences in Exercise 2.1.Exercise 2.1.Compute the \(z\) transform of the following sequences, indicating their regions of convergence:(a) \(x(n)=\sin
Compute \(h(n)\), the inverse Fourier transform of\[H\left(\mathrm{e}^{\mathrm{j} \omega}\right)=\left\{\begin{aligned}-\mathrm{j}, & \text { for } 0 \leq \omega.\]and show that:(a) Equation
Prove that the Fourier transform of \(x(n)=\mathrm{e}^{\mathrm{j} \omega_{0} n}\) is given by Equation (2.216) by computing\[X\left(\mathrm{e}^{\mathrm{j} \omega}\right)=\lim _{N \rightarrow \infty}
Prove the properties of the Fourier transform of real sequences given by Equations (2.229) to (2.232). Re{X(e)) = Re{X (e-)). (2.229)
State and prove the properties of the Fourier transform of imaginary sequences that correspond to those given by Equations (2.229)-(2.232). Re{X(e)) Re{X (e-j)). (2.229)
Show that the direct-inverse Fourier transform pair of the correlation of two sequences is\[\sum_{n=-\infty}^{\infty} x_{1}(n) x_{2}(n+l) \longleftrightarrow X_{1}\left(\mathrm{e}^{-\mathrm{j}
Show that the Fourier transform of an imaginary and odd sequence is real and odd.
Show that the Fourier transform of a conjugate antisymmetric sequence is imaginary.
We define the even and odd parts of a complex sequence \(x(n)\) as\[\mathcal{E}\{x(n)\}=\frac{x(n)+x^{*}(-n)}{2} \quad \text { and } \quad \mathcal{O}\{x(n)\}=\frac{x(n)-x^{*}(-n)}{2}\]respectively.
Solve Exercise 1.22 using the concept of the transfer function.Exercise 1.22Compute the inverse Fourier transform of\[X\left(\mathrm{e}^{\mathrm{j} \omega}\right)=\frac{1}{1-\mathrm{e}^{-\mathrm{j}
Prove that\[\begin{equation*}\mathcal{F}^{-1}\left\{\sum_{k=-\infty}^{\infty} \delta\left(\omega-\frac{2 \pi}{N} k\right)\right\}=\frac{N}{2 \pi} \sum_{p=-\infty}^{\infty} \delta(n-N p) \tag{2.262}
Show that the PSD function of a WSS random process \(\{X\}\) satisfies the following properties:(a) \(\Gamma_{X}(0)=\sum_{v=-\infty}^{\infty} R_{X}(v)\).(b) It is an even function; that is:
Show that\[\sum_{n=0}^{N-1} W_{N}^{n k}= \begin{cases}N, & \text { for } k=0, \pm N, \pm 2 N, \ldots \\ 0, & \text { otherwise }\end{cases}\]
Given the DFT coefficients represented by the vector\[\mathbf{X}=\left[\begin{array}{llllllll}9 & 1 & 1 & 9 & 1 & 1 & 1 & 1\end{array}\right]^{\mathrm{T}}\](a) determine its length-8 IDFT;(b)
Suppose the DFT \(X(k)\) of a sequence as represented by the vector\[\mathbf{X}=\left[\begin{array}{llllllll}4 & 2 & 2 & 2 & 2 & 2 & 2 & 4\end{array}\right]^{\mathrm{T}} .\](a) Compute the
Consider the sequence\[x(n)=\delta(n)+2 \delta(n-1)-\delta(n-2)+\delta(n-3)\](a) Compute its length-4 DFT.(b) Compute the finite-length sequence \(y(n)\) of length- 6 whose DFT is equal to the
Show how, using a single DFT of length \(N\), one can compute the DFT of four sequences: two even and real sequences and two odd and real sequences, all with length \(N\).
Show how to compute the DFT of two even complex length- \(N\) sequences performing only one length \(N\) transform calculation. Follow the steps below:(i) Build the auxiliary sequence
Repeat Exercise 3.9 for the case of two complex antisymmetric sequences.Exercise 3.9Show how to compute the DFT of two even complex length- \(N\) sequences performing only one length \(N\) transform
Show how to compute the DFT of four real, even, length- \(N\) sequences using only one length- \(N\) transform, using the results of Exercise 3.9.Exercise 3.9Show how to compute the DFT of two even
Compute the coefficients of the Fourier series of the periodic sequences below using the DFT.(a) \(x^{\prime}(n)=\sin \left(2 \pi \frac{n}{N}\right)\), for \(N=20\).(b)
Compute and plot the magnitude and phase of the DFT of the following finite-length sequences:(a) \(x(n)=2 \cos \left(\pi \frac{n}{N}\right)+\sin ^{2}\left(\pi \frac{n}{N}\right)\), for \(0 \leq n
Compute the linear convolution of the sequences in Figure 3.28 using the DFT. Fig. 3.28. 10 1.0 0 1 2 -0.5 -1.0 Sequences from Exercise 3.14. 1.0 0 n -0.5 n
Compute the linear convolution of the sequences in Figure 3.29 using DFTs with the minimum possible lengths. Justify the solution, indicating the DFT properties employed. x1(n) 1 0 1 2 3 4 Fig. 3.29.
Given the sequences\[\left.\begin{array}{l}\mathbf{x}=\left[\begin{array}{lll}1 & a & \frac{a^{2}}{2}\end{array}\right]^{\mathrm{T}} \\\mathbf{h}=\left[\begin{array}{lll}1 & -a &
We want to compute the linear convolution of a long sequence \(x(n)\), of length \(L\), with a short sequence \(h(n)\), of length \(K\). If we use the overlap-and-save method to compute the
Repeat Exercise 3.17 for the case when the overlap-and-add method is used.Exercise 3.17We want to compute the linear convolution of a long sequence \(x(n)\), of length \(L\), with a short sequence
Express the algorithm described in the graph for the decimation-in-time FFT in Figure 3.11 in matrix form.Figure 3.11 x(0) x(4) W x(2) x(6) x(1) x(5) x(3) N4 W N/2 X(0) X(1) X(2) X(3) W2 WN2 -1 WN
Express the algorithm described in the graph of the decimation-in-frequency FFT in Figure 3.13 in matrix form. Fig. 3.13. x(0) x(1) x(2) -1 WN2 x(3) W x(4) WN x(5) W W4 x(6) -1 W x(7) -1 -1 -1 X(0)
Determine the graph of a decimation-in-time length-6 DFT, and express its algorithm in matrix form.
Determine the basic cell of a radix-5 algorithm. Analyze the possible simplifications in the graph of the cell.
Repeat Exercise 3.22 for the radix- 8 case, determining the complexity of a generic radix-8 algorithm.Exercise 3.22Determine the basic cell of a radix-5 algorithm. Analyze the possible
Show that the number of complex multiplications of the FFT algorithm for generic \(N\) is given by Equation (3.176). M(N) = N(N1 + 2 + + N-1+ N-1). (3.176)
Compute, using an FFT algorithm, the linear convolution of the sequences (a) and (b) and then (b) and (c) in Exercise 3.13.Exercise 3.13.Compute and plot the magnitude and phase of the DFT of the
Show that:(a) The DCT of a length- \(N\) sequence \(x(n)\) corresponds to the Fourier transform of the length- \(2 N\) sequence \(\tilde{x}(n)\) consisting of \(x(n)\) extended symmetrically; that
Show that the discrete cosine transform of a length- \(N\) sequence \(x(n)\) can be computed from the length \(N\) DFT of a sequence \(\hat{x}(n)\) consisting of the following reordering of the even
Prove the relations between the DHT and the DFT given in Equations (3.224) and (3.225). H(k) Re{X(k)) - Im{X (k)) X(k) E(H(k))- jO(H(k)), (3.224) (3.225)
Prove the convolution-multiplication property of the DHT in Equation (3.228) and derive Equation (3.229) for the case \(x_{2}(n)\) is even. Y(k) = H(k)E(H(k)) + H(k)O(H(k)}, (3.228)
Show that the Hadamard transform matrix is unitary, using Equation (3.230). Hn 1 H-1 2H-1 HR-1 (3.230) -HR-1
The spectrum of a signal is given by its Fourier transform. In order to compute it, we need all the samples of the signal. Therefore, the spectrum is a characteristic of the whole signal. However, in
Linear convolution using \(\mathrm{f} f t\) in MATLAB.(a) Use the \(\mathrm{fft}\) command to determine the linear convolution between two given signals \(x(n)\) and \(h(n)\).(b) Compare the function
Given the signal\[x(n)=\sin \left(\frac{\omega_{\mathrm{s}}}{10} n\right)+\sin \left[\left(\frac{\omega_{\mathrm{s}}}{10}+\frac{\omega_{\mathrm{s}}}{l}\right) n\right]\]then:(a) For \(l=100\),
Design a lag-lead compensator for the system of Problem 8.4-1. Note that \(G_{p}(\mathrm{~s})=1 /\left(s^{2}+sight), T=1 \mathrm{~s}\), and \(H=1\).(a) Use the lag compensator of Problem 8.4-1 and
Design a lag-lead compensator for the system of Problem 8.2-2. Note that \(G_{p}(\mathrm{~s})=1 /\left(s^{2}+sight)\), \(T=0.2 \mathrm{~s}\), and \(H=1\).(a) Use the lag compensator of Problem 8.4-2
Use the MATLAB pidtool to design a PID controller for the system of Problem 8.6-1. Start with these parameters, then try to improve the design \(\left(G_{p}=1 /\left(s^{2}+sight), T=1, H=1, P_{m}=45,
Use the MATLAB pidtool to design a PID controller for the system of Problem 8.6-2. Start with these parameters, then try to improve the design \(\left(G_{p}=1 /\left(s^{2}+sight), T=0.2, H=1,
Use the MATLAB pidtool to design a PID controller for the system of Problem 8.6-5. Start with these parameters, then try to improve the design \(\left(G_{p}=20 /\left(s^{2}+6 sight), H=1, T=0.05,
Use the MATLAB pidtool to design a PID controller for the system of Problem 8.6-6. Start with these parameters, then try to improve the design \(\left(G_{p}=10 / s^{2}, H=0.02, T=0.1, P_{m}=45,
For the system of Problem 7.2-5 and Fig. P7.2-5:(a) Plot the \(z\)-plane root locus.(b) Plot the \(w\)-plane root locus.(c) Determine the range of \(K\) for stability using the results of part
For the antenna control system of Problem 7.5-5 and Fig. P7.5-5, let \(K=1\).(a) The frequency response for \(G(z)\) was calculated by computer and is given in Table P7.7-2. Sketch the Nyquist
Given the pulse transfer function \(G(z)\) of a plant. For \(w=2 \mathrm{rad} / \mathrm{s}, G\left(\varepsilon^{j \omega T}ight)\) is equal to the complex number \(1.3 \angle\left(-25^{\circ}ight)\).
For each case below, plot the Nichols chart and closed-loop frequency response. If the system is stable find the gain margin, phase margin, peak closed-loop frequency response, and system
For the antenna control system of Problem 7.5-5, use MATLAB to:(a) Find the range of stability for KK (by rlocus).(b) Find the gain KK for critical damping (by rlocus).(c) Find the gain and phase
Consider the system of Fig. P6.2-1, with \(D(z)=1\). Use the results of Problem 6.2-1 if available.(a) Find the system time constant \(\tau\) for \(T=4 \mathrm{~s}\).(b) With the input a step
Consider the temperature control system of Problem 6.2-4 and Fig. P6.2-4.(a) Let \(T=5 \mathrm{~s}\), and solve for the response to the input \(R(s)=0.2 \mathrm{~s}^{-1}\). Plot this response on the
Consider the system of Fig. P6.2-4, with \(D(z)=1\). Use the results of Problems 6.2-4 and 6.2-5 if available.(a) Find the system time constant \(\tau\) for \(T=0.8 \mathrm{~s}\).(b) With the input a
The block diagram of an attitude control system of a satellite is shown in Fig. P6.4-2. Let \(T=1 \mathrm{~s}, K=200\), \(J=0.2, H_{k}=0.04\), and \(D(z)=1\).(a) Find the damping ratio \(\zeta\), the
For the system of Fig. 6-13, \(G_{p}(s)=20(s+5) /(s(s+4)(s+6))\), with \(T=0.05\) and \(D(z)\) implementing the following difference equation\[m(k)=10 e(k)-16 e(k-1)-6.3 e(k-2)+m(k-1)\](a) Find the
Repeat Problem 6.5-5 with \(T=0.02\).(a) Find the system type.(b) Plot the unit-step response.(c) Find the rise time, overshoot, and settling time.(d) Compare your results with those of Problem 6.5-5
Repeat Problem 6.6-1, but with the process transfer function given by\[G(s)=\frac{20}{s(3 s+1)}\]Solve for the unit-step responses for \(0 \leq t \leq 1.0 \mathrm{~s}\).Problem 6.6-1Consider the
Repeat Problem 4.2-3 with \(T=0.01 \mathrm{~s}\).Problem 4.2-3. Find the \(z\)-transform of the following functions, using \(z\)-transform tables. Compare the pole-zero locations of \(E(z)\) in the
Find the \(z\)-transforms of the following functions:(a) \(E(s)=\frac{\varepsilon^{3}-1}{\varepsilon^{3} s(s+1)}, \quad T=0.2 \mathrm{~s}\)(b) \(E(s)=\frac{(0.8 s+1)\left(1-\varepsilon^{0.5}
Repeat Problem 4.3-2 for the case that \(T=0.2 \mathrm{~s}\) and the plant transfer function is given by:(a) \(G_{p}(s)=\frac{8}{s^{2}+3 s+2}\)(b) \(G_{p}(s)=\frac{5}{s^{2}+4 s+4}\)Verify each
Repeat Problem 4.4-3 for the case that the filter solves the difference equation\[m(k+1)=0.5 e(k+1)-(0.5)(0.8) e(k)+0.485 m(k)\]the sampling rate is \(10 \mathrm{~Hz}\), and the plant transfer
Consider the system of Fig. P4.4-5. The filter transfer function is \(D(z)\).(a) Express \(C(z)\) as a function of \(E\).(b) A discrete state model of this system does not exist. Why?(c) What
Consider again the system of Fig. P4.4-5. Add a sampler for \(E(s)\) at the input. Given\[G_{1}(s)=\frac{1}{s+10} \quad D(z)=\frac{z-0.5}{z-1} \quad G_{2}(s)=\frac{s}{s^{2}+9 s+23}\]find \(c(k T)\)
Find the modified \(z\)-transform of the following functions:(a) \(E(s)=\frac{6}{(s+1)(s+2)(s+3)}\)(b) \(E(s)=\frac{4}{s(s+2)^{2}}\)(c) \(E(s)=\frac{s^{2}+2 s+2}{s(s+2)^{2}}\)(d)
Find the \(z\)-transform of the following functions. The results of Problem 4.5-3 may be useful.(a) \(E(s)=\frac{6 \varepsilon^{-0.3 T s}}{(s+1)(s+2)(s+3)}\)(b) \(E(s)=\frac{4 \varepsilon^{-0.6 T
Repeat Problem 4.8-1 after replacing the digital filter with this transfer function:\[D(z)=\frac{z-0.5}{(z-0.8)(z-1)}\]
The model of a continuous-time system with algebraic loops is given as\[\begin{aligned}\dot{x}_{1}(t) & =-x_{1}(t)+2 \dot{x}_{2}(t)+u_{1}(t) \\\dot{x}_{2}(t) &
Consider the system of Fig. P4.10-1. The plant is described by the first-order differential equation\[\frac{d y(t)}{d t}+0.04 y(t)=0.2 m(t)\]Let \(T=2 \mathrm{~s}\).(a) Find the system transfer

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