Questions and Answers of Digital Control System Analysis And Design

Show that, for a causal AR system with a zero-mean input $x(n)$ and output $y(n)$ :\[E\{x(n) y(n-v)\}= \begin{cases}\sigma_{X}^{2}, & \text { for } v=0 \\ 0, & \text { for } v>0\end{cases}\]
Solve the Yule-Walker equations for a second-order AR system\[H(z)=\frac{1}{1+a_{1} z^{-1}+a_{2} z^{-2}}\]determining $R_{Y}(v)$ for a white-noise input of variance $\sigma_{X}^{2}$.
Solve the Yule-Walker equations provided by Equation (7.58) for the ARMA system given in Example 7.3, assuming a unit-variance white-noise input. Example 7.3. Approximate the ARMA system by an
Determine the second-order optimal predictor in the minimum MSE sense for the output process of an MA system\[H(z)=1+2 z^{-1}+3 z^{-2}\]to a unit-variance white-noise input.
Show that the autocorrelation method yields an AR system whose poles are not outside the unit circle in the $z$ plane.
Show, by a simple numerical example, that the covariance method may yield an AR system with poles outside the unit circle in the $z$ plane.
Given two first-order AR processes, generated by the same zero-mean white noise with unit variance, whose respective poles are located at $z=0.8$ and $z=-0.8$, generate a new process by adding
Given an MA process generated by applying a zero-mean white noise with unit variance to a system described by\[H(z)=0.921-1.6252 z^{-1}+z^{-2} \text {, }\]estimate the third-order AR model for the
Use the covariance method to estimate the PSD of $L=1024$ samples for the following signal:\[x(n)=\cos \frac{2 \pi}{L} n+x_{1}(n)\]where \[x_{1}(n)=a x_{1}(n-1)+x_{2}(n),\]with $x_{2}(n)$ being a
Estimate the PSD of the signal\[x(n)=\mathrm{e}^{\mathrm{j}(2 \pi / 8) n}+x_{1}(n)\]with $x_{1}(n)$ being a white Gaussian noise with unit variance.
Solve Exercise 7.21 by estimating the AR parameters using the autocorrelation method.Exercise 7.21Use the covariance method to estimate the PSD of $L=1024$ samples for the following
Verify that the Burg reflection coefficients, as given in Equation (7.108), determine the minimum value of $\xi_{\mathrm{B},[i]}$ defined in Equation (7.102). SB,[i] = f,[i] + $b[i] 2 (7.102)
Solve Exercise 7.21 by estimating the AR parameters using the Levinson-Durbin recursions with the Burg reflection coefficients.Exercise 7.21Use the covariance method to estimate the PSD of $L=1024$
Show that the Burg reflection coefficients provided in Equation (7.108) are such that \(\left|k_{i}\right| ki = L-1 2x [i-1] (n)xb[i-1](n - 1) L-1 n=i (x-11(n-1)+x-11(n)) n=i (7.108)
Determine closed-form expressions for the Levinson-Durbin reflection coefficients $k_{1}, k_{2}$, and $k_{3}$ as functions of $R_{Y}(v)$.
Estimate the PSD of the ARMA system output in Exercise 7.11 to a unit-variance white-noise input. Use the standard periodogram method and the autocorrelation method with $N=1,2,3,4$ and compare the
Show that the minimum MSE value achieved by the Wiener solution is given by Equation (7.118). Emin = E{y (n)} - px RX Pxx (7.118)
A random process $x(n)$ is generated by applying a white noise $w(n)$ with unit variance as input to a system described by the following transfer function:\[H(z)=\frac{1}{z^{2}-0.36}\]Compute the
Deduce the two noble identities (Equations (8.30) and (8.31)) using an argument in the time domain. DMX (2))H(z) = DM{X (2)H(z)}, (8.30)
Prove Equations (8.28) and (8.29). x (mM/L), {x m-N, k Z y(m) = 10. m=kL, k Z otherwise (8.28)
The sequence\[x=0.125,0.25,0.5,1,2,4\]is filtered by a filter with transfer function\[H(z)=\frac{1}{3}\left(1+z^{-1}+z^{-2}\right)\]and the result is decimated by 2 . The output is then upsampled by
Show that decimation is a time-varying, but periodically time-invariant, operation and that interpolation is a time-invariant operation.
Design two interpolation filters, one lowpass and the other multiband (Equation (8.22)), with specifications\[\begin{aligned}& \delta_{\mathrm{p}}=0.0002 \\& \delta_{\mathrm{r}}=0.0001
Show the polyphase decomposition structure, as given in Equation (8.35), for an FIR filter with impulse response\[h(n)=0.25,0.5,0.5,1,1,0.5,0.5,0.25\]for $n=0,1, \ldots, 7$. Try to minimize the
Show the polyphase decomposition structure, as given in Equation (8.37), for an FIR filter with impulse response\[h(n)=-0.375,0.25,-0.5,1,-1,0.5,-0.25,0.375\]for $n=0,1, \ldots, 7$. Try to minimize
Show the polyphase decomposition structures, as given in Equations (8.35) and (8.37), for an FIR filter with impulse response\[h(n)=a,b, c,d, e,-d,-c,-b,-a\]for $n=0,1, \ldots, 8$, minimizing the
Show that the zeroth polyphase component of an $L$ th band filter is constant in the frequency domain.
Prove that for a linear-phase filter whose impulse response has length $M L$, its $M$ polyphase components (Equation (8.35)) should satisfy\[E_{j}(z)= \pm z^{-(L-1)} E_{M-1-j}\left(z^{-1}\right)
Design a lowpass filter using one decimation/interpolation stage satisfying the following specifications:\[\begin{aligned}& \delta_{\mathrm{p}}=0.001 \\& \delta_{\mathrm{r}} \leq 0.0001 \\&
In the serial-to-parallel converter of Figure 8.25, see also Figure 8.19, assume $M=2$ and $L=3$ and that the input signal is a sequence given by\[x(n)=0,0,a, b,c, d,e, f, g, h, i, 0,0\]for
In the parallel-to-serial converter of Figure 8.25, see also Figure 8.18, assume $M=2$ and $N=3$ and that the input signals are given by\[\begin{aligned}& x_{1}(n)=0,0, a \\&
In Exercise 8.7, a nonminimum delay overlapped solution is possible by choosing\[\mathbf{C}_{l}(z)=\left[\begin{array}{cccc}D_{0}(z) & R_{1}(z) & 0 & 0 \\0 & R_{0}(z) & R_{1}(z)
Given the matrix\[\mathbf{C}(z)=\left[\begin{array}{ccc}R_{0}(z) & R_{1}(z) & R_{2}(z) \\z^{-1} R_{2}(z) & R_{0}(z) & R_{1}(z) \\z^{-1} R_{1}(z) & z^{-1} R_{2}(z) & R_{0}(z)\end{array}\right]\]verify
Given the matrix\[\mathbf{C}(z)=\left[\begin{array}{cc}R_{0}(z) & R_{1}(z) \\z^{-1} R_{1}(z) & R_{0}(z)\end{array}\right]\]show that its inverse is pseudo-circulant.
Assume you want to implement an FIR filter of length 16 using the fast convolution structure of Figure 8.27 described by Equation (8.82). Compute the delay and the number of multiplications per
Implement the transfer function below using the fast convolution structure of Figure 8.27 with the subfilters of length one:\[H(z)=1+z^{-1}+2 z^{-2}+4 z^{-3} .\] x(n) 12 C(z) 100 Co(2) + C(2) 10 121
Implement the transfer function below using the fast convolution structure of Figure 8.27 with the subfilters of length one:\[H(z)=0.25+0.5 z^{-1}-0.5 z^{-2}-0.25 z^{-3} .\] x(n) 12 C(z) 100 Co(2) +
Design the filter satisfying the following specifications using the minimax approach and show its submatrices of overlapped blocking filtering, for $M=L=4$ and $N=2$
Design the filter of Exercise 8.26 with the minimax approach and show its submatrices of overlapped blocking filtering for $M=N=4$ and for $L=2$.Exercise 8.26Design the filter satisfying the
Design the filter of Exercise 8.26 with the WLS approach and derive its overlapped structure I of Figure 8.26.Exercise 8.26Design the filter satisfying the following specifications using the minimax
Design the filter of Exercise 8.26 with the WLS approach and derive its overlapped structure II of Figure 8.27.Exercise 8.26Design the filter satisfying the following specifications using the minimax
Design a filter satisfying the following specifications with the minimax and WLS approaches and show their implementations employing the overlapped structure I of Figure
Design a filter satisfying the following specifications with the minimax approach and show its submatrices of overlapped blocking filtering for $M=L=2$ and for $N=1$
Solve Exercise 8.31 using the WLS design approach.Exercise 8.31Design a filter satisfying the following specifications with the minimax approach and show its submatrices of overlapped blocking
Prove Equation (8.52). Y(z) = -(N-1) 21] C(z) M-I 1 (zWM)- X (zW). 1-0 (WM)-(-1) (8.52)
Show that, for a realization of a WSS process $\{X\}$ applied as input process to a serial-to-parallel converter, the PSD matrix $\Gamma_{X}(z)$ is pseudo-circulant.
Let us consider the case where a vector $\{\mathbf{X}(n)\}$ represents an $M \times 1$ WSS process, and that\[Y_{i}(n)=W_{i}(n) X_{i}(n)\]for $i=0,1, \ldots, M-1$. Show that
Show that if you apply an input process WSCS with period $N$ to a linear periodically time-varying system with period $N$, the output process will also be WSCS with period $N$.
Give two distinct realizations for the transfer functions below:(a) $H(z)=0.0034+0.0106 z^{-2}+0.0025 z^{-4}+0.0149 z^{-6}$.(b) \(H(z)=\left(\frac{z^{2}-1.349 z+1}{z^{2}-1.919
Write the equations that describe the networks in Figure 4.34, by numbering the nodes appropriately. Fig. 4.34. x(n) x(n) 13 % y(n) -m -m x(n) Signal flowgraphs of three digital filters.. y(n) 1 y(n)
Show that:(a) If $H(z)$ is a Type I filter, then $H(-z)$ is Type I.(b) If $H(z)$ is a Type II filter, then $H(-z)$ is Type IV.(c) If $H(z)$ is a Type III filter, then $H(-z)$ is Type
Determine the transfer functions of the digital filters in Figure 4.34. Fig. 4.34. x(n) IN 03 32 1 -m 1 x(n) K x(n) -1 % %0 Signal flowgraphs of three digital filters. 1 Z y(n) -m 1 y(n) TN 1 1 y(n) 1
Show that the transfer function of a given digital filter is invariant with respect to a linear transformation of the state vector\[\mathbf{x}(n)=\mathbf{T} \mathbf{x}^{\prime}(n)\]where
Describe the networks in Figure 4.34 using state variables. Fig. 4.34. x(n) x(n) 13 % y(n) -m -m x(n) Signal flowgraphs of three digital filters.. y(n) 1 y(n)
Implement the transfer function below using a parallel realization with the minimum number of multipliers:\[H(z)=\frac{z^{3}+3 z^{2}+\frac{11}{4}+\frac{5}{4}}{\left(z^{2}+\frac{1}{2}
Given the realization depicted in Figure 4.35:(a) Show its state-space description.(b) Determine its transfer function.(c) Derive the expression for its magnitude response and interpret the result.
Consider the digital filter of Figure 4.36:(a) Determine its state-space description.(b) Compute its transfer function and plot the magnitude response. y(n) -m -m2 x(n) Fig. 4.36. Digital filter
Determine the transfer function of the digital filter shown in Figure 4.37 using the state-space formulation. x (n) 12 + 12 + 13 + y (n) Fig. 4.37. Second-order lattice structure for Exercise 4.10.
Given the digital filter structure of Figure 4.38:(a) Determine its state-space description.(b) Compute its transfer function.(c) Show its transposed circuit.(d) Use this structure with
Given the digital filter structure of Figure 4.39:(a) Determine its transfer function.(b) Generate its transposed realization. x(n) Fig. 4.39. 1/L -21/L Digital filter structure for Exercise 4.12. T
Given the structure shown in Figure 4.40:(a) Determine the corresponding transfer function employing the state-space formulation.(b) Generate its transposed realization.(c) If
Find the transpose for each network in Figure 4.34. Fig. 4.34. x(n) x(n) 13 % y(n) -m -m x(n) Signal flowgraphs of three digital filters.. y(n) 1 y(n)
Determine the transfer function of the filter in Figure 4.41, considering the two possible positions for the switches. x(n) -z-2 -z-2 -z-2 z-4 z-4 Fig. 4.41. Signal flowgraph of a digital filter. -Z
Determine and plot the frequency response of the filter shown in Figure 4.41, considering the two possible positions for the switches. x(n) -z-2 -z-2 -z-2 z-4 z-4 Fig. 4.41. Signal flowgraph of a
Determine the impulse response of the filter shown in Figure 4.42. x(n) 0.5 z-1 -0.75 0.5 y(n) Z-1 Fig. 4.42. Signal flowgraph of a digital filter.
Show that if two given networks are described by $Y_{i}=\sum_{j=1}^{M} T_{i j} X_{j}$ and $Y_{i}^{\prime}=$ $\sum_{j=1}^{M} T_{i j}^{\prime} X_{j}^{\prime}$, then these networks are
Some FIR filters present a rational transfer function:(a) Show that the transfer function\[H(z)=\frac{\left(r^{-1} z\right)^{-(M+1)}-1}{r e^{\mathrm{j} 2 \pi /(M+1)} z^{-1}-1}\]corresponds to an FIR
Plot the pole-zero constellation as well as the magnitude response of the transfer function of Exercise 4.20 for $M=6,7,8$ and comment on the results.Exercise 4.20Some FIR filters present a
Design second-order lowpass and highpass blocks, and combine them in cascade, to form a bandpass filter with passband $0.3 \leq \omega \leq 0.4$, where $\omega_{\mathrm{s}}=1$. Plot the resulting
Design second-order lowpass and highpass blocks, and combine them in parallel, to form a bandstop filter with stopband $0.25 \leq \omega \leq 0.35$, where $\omega_{\mathrm{s}}=1$. Plot the
Design a second-order notch filter capable of eliminating a $10 \mathrm{~Hz}$ sinusoidal component when $\omega_{\mathrm{s}}=200 \mathrm{rad} / \mathrm{sample}$ and show the resulting magnitude
For the comb filter of Figure 4.31, choose $L=10$ and compute the magnitude and phase responses for the cases where $a=0, a=0.6$, and $a=0.8$. Comment on the result. x(n) + Fig. 4.31.
For the comb filter of Equation (4.110), compute the value of the normalization factor that should be multiplied by the transfer function such that the maximum value of the magnitude response becomes
An FIR filter with the transfer function\[H(z)=\left(z^{-L}-1\right)^{N}\]with $N$ and $L$ integers is also a comb filter. Discuss the properties of this filter regarding its zero positions and
Given the transfer function\[H(z)=\frac{z\left[z-\cos \left(\omega_{0}\right)\right]}{z^{2}-2 \cos \left(\omega_{0}\right) z+1}\](a) Where are its poles located exactly on the unit circle?(b) Using
With a state-space structure, prove that, by choosing $a_{11}=a_{22}=\cos \left(\omega_{0}\right)$ and $a_{21}=$ $-a_{12}=-\sin \left(\omega_{0}\right)$, the resulting oscillations in states
Create Matlab commands to fill in the blanks of Table 4.3. Table 4.3. List of MATLAB commands for transforming digital filter representations. Direct Zero-pole Cascade Direct tf2zp roots Zero-pole
Characterize the systems below as linear/nonlinear, causal/noncausal and time invariant/time varying:(a) $y(n)=(n+a)^{2} x(n+4)$(b) $y(n)=a x(n+1)$(c) $y(n)=x(n+1)+x^{3}(n-1)$(d) \(y(n)=x(n)
For each of the discrete signals below, determine whether they are periodic or not. Calculate the periods of those that are periodic.(a) $x(n)=\cos ^{2}\left(\frac{2 \pi}{15} n\right)$(b)
Consider the system whose output $y(m)$ is described as a function of the input $x(m)$ by the following difference equations:(a) $y(m)=\sum_{n=-\infty}^{\infty} x(n) \delta(m-n N)$(b)
Compute the convolution sum of the following pairs of sequences:(a) $x(n)=\left\{\begin{array}{ll}1, & 0 \leq n \leq 4 \\ 0, & \text { otherwise }\end{array} \quad\right.$ and \(\quad h(n)=
For the sequence\[x(n)= \begin{cases}1, & 0 \leq n \leq 1 \\ 0, & \text { otherwise }\end{cases}\]compute $y(n)=x(n) * x(n) * x(n) * x(n)$. Check your results using the MATLAB function conv.
Show that $x(n)=a^{n}$ is an eigenfunction of a linear time-invariant system by computing the convolution summation of $x(n)$ and the impulse response of the system $h(n)$. Determine the
Supposing that all systems in Figure 1.27 are linear and time invariant, compute $y(n)$ as a function of the input and the impulse responses of each system. x(n) h(n) h(n) h(n) Fig. 1.27. Linear
We define the even and odd parts of a sequence $x(n), \mathcal{E}\{x(n)\}$ and $\mathcal{O}\{x(n)\}$ respectively, as\[\begin{aligned}\mathcal{E}\{x(n)\} & =\frac{x(n)+x(-n)}{2}
Find one solution for each of the difference equations below:(a) $y(n)+2 y(n-1)+y(n-2)=0, y(0)=1$ and $y(1)=0$(b) $y(n)+y(n-1)+2 y(n-2)=0, y(-1)=1$ and $y(0)=1$.
Find the general solution for the difference equation in Example 1.9 when $a=b$. Example 1.9. Solve the difference equation y(n) + ay(n 2) = b" sin(n)u(n) assuming that ab and y(n) = 0, for n < 0.
Determine the solutions of the difference equations below, supposing that the systems they represent are initially relaxed:(a) \(y(n)-\frac{1}{\sqrt{2}} y(n-1)+y(n-2)=2^{-n} \sin \left(\frac{\pi}{4}
Write a Matlab program to plot the samples of the solutions of the difference equations in Exercise 1.9 from $n=0$ to $n=20$.Exercise 1.9Find one solution for each of the difference equations
Show that a system described by Equation (1.63) is linear if and only if the auxiliary conditions are zero. Show also that the system is time invariant if the zero auxiliary conditions are defined
Compute the impulse responses of the systems below:(a) $y(n)=5 x(n)+3 x(n-1)+8 x(n-2)+3 x(n-4)$(b) $y(n)+\frac{1}{3} y(n-1)=x(n)+\frac{1}{2} x(n-1)$(c) $y(n)-3 y(n-1)=x(n)$(d) \(y(n)+2
Write a MatLaB program to compute the impulse responses of the systems described by following difference equations:(a) $y(n)+y(n-1)+y(n-2)=x(n)$(b) $4 y(n)+y(n-1)+3 y(n-2)=x(n)+x(n-4)$.
Determine the impulse response of the following recursive system:\[y(n)-y(n-1)=x(n)-x(n-5) .\]
Determine the steady-state response of the system governed by the following difference equation:\[12 y(n)-7 y(n-1)+y(n-2)=\sin \left(\frac{\pi}{3} n\right) u(n) \text {. }\]
Determine the steady-state response for the input $x(n)=\sin (\omega n) u(n)$ of the filters described by(a) $y(n)=x(n-2)+x(n-1)+x(n)$(b) $y(n)-\frac{1}{2} y(n-1)=x(n)$(c) \(y(n)=x(n-2)+2
Write a MATLAB program to plot the solution to the three difference equations in Exercise 1.18 for $\omega=\pi / 3$ and $\omega=\pi$.Exercise 1.18Determine the steady-state response for the input
Discuss the stability of the systems described by the impulse responses below:(a) $h(n)=2^{-n} u(n)$(b) $h(n)=1.5^{n} u(n)$(c) $h(n)=0.1^{n}$(d) $h(n)=2^{-n} u(-n)$(e) \(h(n)=10^{n}
Show that $X_{\mathrm{i}}(\mathrm{j} \Omega)$ in Equation (1.170) is a periodic function of $\Omega$ with period $2 \pi / T$.
Suppose we want to process the continuous-time signal\[x_{\mathrm{a}}(t)=3 \cos (2 \pi 1000 t)+7 \sin (2 \pi 1100 t)\]using a discrete-time system. The sampling frequency used is 4000 samples per
Write a MATLAB program to perform a simulation of the solution of Exercise 1.22.(i) Simulate the continuous-time signals $x_{\mathrm{a}}(t)$ in MATLAB using sequences obtained by sampling them at

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