Questions and Answers of Digital Control System Analysis And Design

Consider the third-order continuous-time LTI system\[\begin{aligned}\dot{\mathbf{x}} & =\mathbf{A x}+\mathbf{B} u \\y & =\mathbf{C x}\end{aligned}\]\[\begin{array}{r}\text { with }
Show that, for an \(M\)-band filter bank whose analysis and synthesis filters are ideal filters, perfect reconstruction demands that their frequency responses are M, M Ho (ejo) = M, H2k-1 (ej) = |w =
Show that a perfect reconstruction linear-phase filter bank with causal filters must be such that \(H_{0}(z) H_{1}(-z)\) has an odd number of coefficients and that all but one of its odd powers of
Prove, using an argument in the frequency domain, that the system depicted in Figure 9.10 has a transfer function equal to \(z^{-M+1}\). x(n) +14 xo(m) M x(m) XM-1(m) y(n) M M Fig. 9.10. M-band
Find the matrices \(\mathbf{E}(z)\) and \(\mathbf{R}(z)\) for the filter bank described by Equations (9.131)(9.134), and verify that their product is equal to a pure delay. 1 Ho(2)=(-1+221 +62-2
Modify the causal analysis and synthesis filters of the filter bank in equations (9.131)(9.134) so that they constitute a zero-delay perfect reconstruction filter bank. Do the same for the filter
Design a Johnston two-band filter bank with 64 coefficients.
Design a Johnston two-band filter bank with at least \(60 \mathrm{~dB}\) of stopband attenuation.
Consider a perfect reconstruction filter bank which satisfies the QMF condition, \(H_{1}(z)=H_{0}(-z)\), and has the following lowpass analysis filter:\[H_{0}(z)=z^{-5}+5 z^{-3}+z^{-2}+6 z^{-1}+4
Repeat Exercise 9.9 for the lowpass analysis filter equal to\[H_{0}(z)=\frac{1}{4} z^{-5}+z^{-3}+z^{-2}+z^{-1}+2\]Exercise 9.9Consider a perfect reconstruction filter bank which satisfies the QMF
Repeat Exercise 9.9 for the lowpass analysis filter equal to\[H_{0}(z)=z^{-3}-z^{-2}+z^{-1}+1 .\]Exercise 9.9Consider a perfect reconstruction filter bank which satisfies the QMF condition,
Design a CQF bank with at least \(55 \mathrm{~dB}\) of stopband attenuation.
Given the lowpass analysis filter of a two-band FIR perfect reconstruction filter bank\[H_{0}(z)=z^{-3}+a z^{-2}+b z^{-1}+2\]determine the analysis and synthesis filters and discuss the class of
Design a two-band QMF filter bank satisfying the following specifications:\[\begin{aligned}\delta_{\mathrm{p}} & =0.1 \\\delta_{\mathrm{r}} & =0.05 \\\Omega_{\mathrm{p}} & =0.4
Show that a halfband filter can be designed from a Hilbert transformer as follows:\[h(n)=0.5\left[\delta(n)+(-1)^{(n-1) / 2} h_{\mathrm{h}}(n)\right]\]where \(h(n)\) is the halfband impulse response
Design a two-band perfect reconstruction filter bank using the CQF design satisfying the following specifications:\[\begin{aligned}\delta_{\mathrm{p}} & =0.05 \\\delta_{\mathrm{r}} & =0.05
Design a perfect reconstruction filter bank using the cosine-modulated filter bank with six sub-bands with at least \(40 \mathrm{~dB}\) of stopband attenuation and \(0.5 \mathrm{~dB}\) of passband
Design a perfect reconstruction filter bank using the cosine-modulated filter bank with 10 sub-bands with at least \(40 \mathrm{~dB}\) of stopband attenuation and \(0.5 \mathrm{~dB}\) of passband
Design a perfect reconstruction filter bank using the cosine-modulated filter bank with 10 sub-bands with at least \(35 \mathrm{~dB}\) of stopband attenuation and \(1 \mathrm{~dB}\) of passband
Depict in detail the structure of the cosine-modulated filter bank utilizing \(M=2\) bands.
Design a cosine-modulated filter bank with \(M=5\) sub-bands and at least \(40 \mathrm{~dB}\) of stopband attenuation.
Design a cosine-modulated filter bank with \(M=15\) sub-bands with at least \(20 \mathrm{~dB}\) of stopband attenuation.
Express the DFT as an \(M\)-band filter bank. Plot the magnitude responses of the analysis filters for \(M=8\).
Show that \(\hat{\mathbf{C}}_{1}\) and \(\hat{\mathbf{C}}_{2}\) defined in Equations (9.237) and (9.238) satisfy Equations (9.233) and (9.234). C + CC = 1 T (9.233) 0. (9.234)
Design a fast LOT-based filter bank with at least eight sub-bands.
Prove the relationship in Equation (9.210). CC2=CC = 0, (9.210)
Show that the relations in Equations (9.256)-(9.258) are valid. E(z) = [C3+2 (I - 3)]4. (9.256)
For Example 9.12, is an orthogonal solution possible? Compute \(\mathbf{E}^{-1}(z)\) for the proof. Example 9.12. Show the two-band lapped-transform structure that realizes the filter bank with
In Example 9.12, we could attempt to generalize the orthogonal realization of the LOT by allowing the matrix \(\mathbf{L}_{1}\) of Figure 9.45 to be a full matrix and design a simple biorthogonal
Propose an alternative and simpler structure to that of Figure 9.46 in Example 9.12. The simplified structure should be based on Equation (9.267). Example 9.12. Show the two-band lapped-transform
Design a linear-phase lapped biorthogonal transform with \(M=6\) sub-bands aiming at minimizing the stopband ripple of the sub-filters.
Design a linear-phase lapped biorthogonal transform with \(M=10\) sub-bands.
Design a length-16 linear-phase lapped biorthogonal transform with \(M=8\) sub-bands aiming at minimizing the stopband ripple of the sub-filters.
Design a GenLOT with \(M=10\) sub-bands having at least \(25 \mathrm{~dB}\) of stopband attenuation.
Design a GenLOT with \(M=8\) sub-bands having at least \(25 \mathrm{~dB}\) of stopband attenuation.
Design a GenLOT with \(M=6\) sub-bands having at least \(20 \mathrm{~dB}\) of stopband attenuation.
Show that if a filter bank has linear-phase analysis and synthesis filters with the same lengths \(N=L M\), then the following relations for the polyphase matrices are valid:\[\begin{aligned}&
Consider an \(M\)-band linear-phase filter bank with perfect reconstruction with all analysis and synthesis filters having the same length \(N=L M\). Show that, for a polyphase matrix
Deduce Equations (10.1)-(10.4). (S) low (2) How S = Xs(z) X(z) =Ho (22) (10.1) k=0
Consider a two-band linear-phase filter bank whose product filter \(P(z)=\) \(H_{0}(z) H_{1}(-z)\) of order \((4 M-2)\) can be expressed as\[P(z)=z^{-2 M+1}+\sum_{k=0}^{M-1} a_{2 k}\left(z^{-2
Compute the analysis and synthesis wavelets and scaling functions corresponding to the analysis filters in Table 10.4. Table 10.4. Coefficients of the lowpass and highpass analysis filters
Show that the sub-bands from a two-band decomposition of a periodic signal with odd period \(N\) have \(2 N\) independent samples.
Show that \(\psi(t)\) as defined in Equation (10.64) is orthogonal to \(\phi(t-n)\) and that \(\psi(t-n)\), for \(n \in \mathbb{Z}\), is an orthonormal basis for \(W_{0}\) (Vetterli \&
For a two-band perfect reconstruction filter bank, assume the analysis filter \(H_{0}(z)\) and the synthesis filter \(G_{0}(z)\) satisfy the condition of Equation (10.142). Assume also that these
Apply the technique, known as balancing, described in Exercise 10.6, to the filters of Equations (10.204) and (10.205) and comment on the observed results.Exercise 10.6,For a two-band perfect
Develop a table similar to Table 10.1 for computing symmetric extensions of oddlength signals. Note in this case that for odd order the lowpass and highpass bands may not have the same lengths. Table
Quantization of wavelet transforms: use the signal generated in Experiment 10.1 and compute its \(N\)-stage wavelet transform with both the bior 4.4 and the \(\mathrm{db} 4\) wavelets. Use \(N=3,
Repeat Exercise 10.9, this time using as input the image cameraman. tif from the MATLAB Image Processing Toolbox. Observe how the increase in the quantization step sizes affects the reconstructed
Repeat Exercise 10.10 using the filter bank whose analysis lowpass and highpass filters are listed in Table 10.4. Compare the results with those obtained in Exercise 10.10 for the same quantization
Repeat Exercises 10.10 and 10.11, this time comparing the results obtained from the use of the periodic and symmetric extensions. Refer to the MatLab function dwtmode, which sets the type of signal
Compute the number of vanishing moments of the analysis and synthesis wavelets of Exercise 10.3, using, for instance, the MATLAB function \(t f 2 \mathrm{zp}\).Exercise 10.3,Compute the analysis and
Compute the STFT of the signal generated in Experiment 10.1 and compare it with the wavelet transform obtained in Experiment 10.1, using the MATLAB command spectrogram. Describe the differences in
If one wants to compute the wavelet transform of a continuous signal, then one has to assume that its digital representation is derived as in Equation (10.9) and Figure 10.9. However, the digital
Design a circuit that determines the two's complement of a binary number in a serial fashion.
Show, using Equation (11.20), that, if \(X\) and \(Y\) are represented in two's-complement arithmetic, then:(a) \(X-Y=X+c[Y]\), where \(c[Y]\) is the two's complement of \(Y\).(b)
Describe an architecture for the parallel multiplier where the coefficients are represented in the two's-complement format.
Describe the internal circuitry of the multiplier in Figure 11.4 for 2-bit input data. Then perform the multiplication of several combinations of data, determining the internal data obtained at each
Describe the implementation of the FIR digital filter in Figure 4.3 using the distributed arithmetic technique. Design the architecture in detail, specifying the dimensions of the memory unit as a
Determine the content of the memory unit in a distributed arithmetic implementation of the direct-form realization of a digital filter whose coefficients are given by\[\begin{aligned}&
Determine the content of the memory unit in a distributed arithmetic implementation of the cascade realization of three second-order state-space blocks whose coefficients are given in Table 11.6
Describe the number -0.832645 using one's-complement, two's-complement, and CSD formats.
Describe the number -0.00061245 in floating-point representation, with the mantissa in two's complement.
Deduce the probability density function of the quantization error for the signmagnitude and one's-complement representations. Consider the cases of rounding, truncation, and magnitude truncation.
Calculate the scaling factors for the radix-4 butterfly of Figure 3.16 using the \(L_{2}\) and \(L_{\infty}\) norms. So(K) S(K) S(K) W S(k + L/4) S(K) 2k W 3k W S(K) -1 Fig. 3.16. More efficient
Show that the \(L_{2}\) norm of the transfer function\[H(z)=\frac{b_{1} z+b_{2}}{z^{2}+a_{1} z+a_{2}}\]is\[\|H(z)\|_{2}^{2}=\frac{\left(b_{1}^{2}+b_{2}^{2}\right)\left(1+a_{2}\right)-2 b_{1} b_{2}
Given the transfer function\[H(z)=\frac{1-\left(a z^{-1}\right)^{M+1}}{1-a z^{-1}}\]compute the scaling factors using the \(L_{2}\) and \(L_{\infty}\) norms assuming \(|a|
Calculate the scaling factor for the digital filter in Figure 11.39, using the \(L_{2}\) and \(L_{\infty}\) norms. Fig. 11.39. x(n) o + -m (X) ( + -o y(n) -m2 () Second-order digital filter: m =
Show that, for the Type 2 canonic direct-form realization for IIR filters (as depicted in Figure 4.13), the scaling coefficient is given by\[\lambda=\frac{1}{\max
Calculate the relative output-noise variance in decibels for the filter of Figure 11.39 using fixed-point and floating-point arithmetics. Fig. 11.39. x(n) o + -m (X) ( + -o y(n) -m2 (X) Second-order
Calculate the output-noise RPSD for the filter in Figure 11.40 using fixed-point and floating-point arithmetics. x(n) + + -mi (X -m2 + (X Fig. 11.40. Second-order digital filter. y(n)
Derive Equations (11.133)-(11.135). ISH (e) 'mi = 2 SH (ei) | (ejo)) + |H (ejo) | ( 2 a(w) (11.133) mi |H(ej)| a(w) = (ej + (11.134) mi ami 2 = mi mi *+ m (30) 2 (11.135) ami
Calculate the maximum expected value of the transfer function deviation, given by the maximum value of the tolerance function, with a confidence factor of \(95 \%\), for the digital filter in Figure
Determine the minimum number of bits a multiplier should have to keep a signalto-noise ratio above \(80 \mathrm{~dB}\) at its output. Consider that the type of quantization is rounding.
Plot the pole grid for the structure of Figure 11.30 when the coefficients are implemented in two's complement with 6 bits, including the sign bit. Fig. 11.30. P3 Co + 0+ (+ YLP(n) + -o y(n) YN(n) +
Discuss whether granular limit cycles can be eliminated in the filter of Figure 11.40 by using magnitude truncation quantizers at the state variables. x(n) + -mi + (X) -m2 + (X) Fig. 11.40.
Verify whether it is possible to eliminate granular and overflow limit cycles in the structure of Figure 11.17b. em (+ -m (X y(n) -m2 ( em2 + (b) Fig. 11.17. Second-order networks. x;(n) +
Plot the overflow characteristics of simple one's-complement arithmetic.
Suppose there is a structure that is free from zero-input limit cycles, when employing magnitude truncation, and that it is also forced-input stable when using saturation arithmetic. Discuss whether
Prove Equations (12.6) and (12.7). Ni(z) = Ni-1(2) + kiz'Ni-1 (z) (12.6) (z) = z Ni(z) (12.7)
Write down a MatLab command that determines the FIR lattice coefficients from the FIR direct-form coefficients.
For \(\mathbf{E}(z)\) and \(\mathbf{R}(z)\) defined as in Equations (12.20) and (12.22), show that:(a) \(\mathbf{E}(z) \mathbf{R}(z)=z^{-M}\).(b) The synthesis filter bank has linear phase. E(z) = KM
Synthesize a two-band lattice filter bank for the case such that\[H_{0}(z)=z^{-5}+2 z^{-4}+4 z^{-3}+4 z^{-2}+2 z^{-1}+1 .\]Determine the corresponding \(H_{1}(z)\).
Design a two-band QMF filter bank satisfying the following specifications:\[\begin{aligned}\delta_{\mathrm{p}} & =0.5 \mathrm{~dB} \\\delta_{\mathrm{r}} & =40 \mathrm{~dB} \\\Omega_{\mathrm{p}} &
Design a two-band perfect-reconstruction filter bank with linear-phase subfilters satisfying the following specifications:\[\begin{aligned}\delta_{\mathrm{p}} & =1 \mathrm{~dB} \\\delta_{\mathrm{r}}
A lattice-like realization with second-order section allows the design of linear-phase two-band filter banks with even order where both \(H_{0}(z)\) and \(H_{1}(z)\) are symmetric. In this
Design a highpass filter using the minimax method satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =0.8 \mathrm{~dB} \\A_{\mathrm{r}} & =40 \mathrm{~dB} \\\Omega_{\mathrm{r}}
Use the MATLAB commands filter and filt latt with the coefficients obtained in Exercise 12.9 to filter a given input signal. Verify that the output signals are identical. Compare the processing time
Show that the linear-phase relations in Equation (12.24) hold for the analysis filters when their polyphase components are given by Equation (12.20). Ho(z) =z-2M-1 Ho(z) H(2)=-2-2M-H(2-1) Go(z)
Use the MATLAB commands \(\mathrm{filt} \mathrm{ter}\), conv, and \(\mathrm{f} f \mathrm{t}\) to filter a given input signal, with the impulse response obtained in Exercise 12.9. Verify what must be
Discuss the distinct feature of an RRS filter with even and odd values of \(M\).
By replacing \(z\) with \(-z\) in an RRS filter, where will its pole and zeros be located? What type of magnitude response will result?
Replace \(z\) by \(z^{2}\) in an RRS filter and discuss the properties of the resulting filter.
Plot the magnitude response of the modified-sinc filter with \(M=2,4,6,8\). Choose an appropriate \(\omega_{0}\) and compare the resulting passband widths.
Plot the magnitude response of the modified-sinc filter with \(M=7\). Choose four values for \(\omega_{0}\) and discuss the effect on the magnitude response.
Given the transfer function below\[H(z)=-z^{-2 N}+\cdots-z^{-6}+z^{-4}-z^{-2}+1\]where \(M\) is an odd number:(a) Show a realization with the minimum number of adders.(b) For a fixed-point
Design a lowpass filter using the prefilter and interpolation methods satisfying the following specifications:\[\begin{aligned}A_{\mathrm{p}} & =0.8 \mathrm{~dB} \\A_{\mathrm{r}} & =40 \mathrm{~dB}
Repeat Exercise 12.19 using the modified-sinc structure as building block.Exercise 12.19 Design a lowpass filter using the prefilter and interpolation methods satisfying the following
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

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