University of Maryland at College ParkDepartment of Electrical and Computer EngineeringENEE 624 Advanced Digital Signal ProcessingProblem Set 4Spring 2003Issued: Wednesday, March 5, 2003 Due: Wednesday, March 12, 2003Reading Assignment: Lectures 9–11 (optional: Vaidyanathan, Chapter 5, Sections5.0-5.7.1, 5.8).Problem 4.1Consider the two-channel QMF bank in Fig. 4.1–1 with analysis and synthesis filters given byH0(z) = 3 + 4 z−1− 3.5 z−2+ z−3+ z−4, H1(z) = H0(−z) .Find a set of causal and stable synthesis filters that result in perfect reconstruction.x[ n]H (z)0x[ n]∧F (z)0F (z)12↑2↑2↑2↑H (z)1Figure 4.1–1Problem 4.2Consider the QMF filter bank shown in Fig. 4.1–1 where H1(z) = H0(−z). The filter H0(z)can be FIR or IIR. However, assume that it is causal and stable.(a) Assume that the polyphase components E0(z) and E1(z) of H0(z) have all their zerosoutside the unit circle. Find a set of stable synthesis filters so that aliasing and amplitudedistortion are both eliminated.(b) Repeat part (a) in the case that E0(z) and E1(z) have some zeros inside and the rest oftheir zeros outside the unit circle (i.e., no zeros on the unit circle).Problem Set 4 2Problem 4.3Consider the modified QMF bank shown in Fig. 4.3–1.x[ n]H (z)0x[ n]∧F (z)0F (z)12↑2↑2↑2↑z−1z−1H (z)1Figure 4.3–1(a) ExpressˆX(z) in terms of X(z), the Hi(z)’s, and the Fi(z)’s.In parts (b), (c), and (d) assume that the analysis filters satisfy H1(z) = H0(−z).(b) Show that the choice F0(z) = H0(z) and F1(z) = H1(z) results in an alias-free system.For this choice of synthesis filters express the distortion T (z) in terms of H0(z).(c) Let H0(z) be a real-coefficient linear-phase FIR lowpass filter of order N . Simplify T (z)and show that there is no phase distortion. Also show that N has to be even to ensurethat T (ej π/2) 6= 0.(d) For the system in part (c) with N even, what is the number of MPUs required to implementthe analysis bank?Hint: Try to exploit as many of the following facts as possible:(i) the relationship H1(z) = H0(−z);(ii) the linear-phase property;(iii) the presence of decimators in the system.Problem Set 4 3Problem 4.4Consider the analysis/synthesis bank cascade in Fig. 4.4–1.x[ n]z−1z−1z−1W*Wz−1z−1z−1x[ n]∧M↑M↑M↑M↑M↑M↑E (z )M−1ME (z )0ME (z )1MR (z )M−1MR (z )0MR (z )1MFigure 4.4–1Assume that all the analysis bank polyphase components Ei(z) are causal FIR filters. Let Ndenote the maximum filter order among the Ei(z)’s (i.e., ei[n] = 0 for all n > N and all i).Also assume that the Ei(z)’s have no zeros on the unit circle, but can have zeros inside andoutside the unit circle.(a) Determine a choice of causal FIR synthesis bank polyphase components Ri(z) so that theoverall system is free from aliasing.(b) Determine a choice of causal FIR synthesis bank polyphase components Ri(z) so that theoverall system is free from aliasing and phase distortion.(c) Determine a choice of stable and right-sided synthesis bank polyphase components Ri(z)so that the overall system is free from aliasing and amplitude distortion.Problem Set 4 4Problem 4.5Consider the three-channel QMF bank shown in Fig. 4.5–1, in the following three cases wherethe analysis filters are given by(i) H0(z) = 1, H1(z) = 2 + z−1, H2(z) = 3 + 2 z−1+ z−2.(ii) H0(z) = 1, H1(z) = 2 + z−1+ z−5, H2(z) = 3 + 2 z−1+ z−2.(iii) H0(z) = 1, H1(z) = 2 + z−1+ z−5, H2(z) = 3 + z−1+ z−2.In each case determine whether it is possible to obtain a perfect reconstruction QMF bankwith a set of FIR synthesis filters. If not, determine a set of stable IIR filters for perfectreconstruction.x[ n]H (z)0x[ n]∧F (z)03↑3↑F (z)13↑3↑H (z)1F (z)23↑3↑H (z)2Figure 4.5–1Problem 4.6 (optional)Prove that the cascade of the analysis and synthesis banks shown in Fig. 4.6–1 is a perfectreconstruction system provided that M and N are relatively prime. The matrices W and W∗are the M × M DFT and IDFT matrices, respectively.x[ n]W*z−Nz−Nz−NWz−Nz−Nz−Nx[ n]∧M↑M↑M↑M↑M↑M↑Figure 4.6–1Problem Set 4 5Problem 4.7Tree structures can be designed to obtain nonuniform QMF banks, that is, QMF banks forwhich the decimation factor is not the same for all channels. Consider the filter bank shown inFig. 4.7–2 (on page 7). The overall system can be described as in Fig. 4.7–1, where Gk(z) andPk(z) are given in terms of the filters Hk(z) and Fk(z).(a) Determine the filters Gk(z) and Pk(z) in terms of the filters Hk(z) and Fk(z). Showthat, in the case of interest, where H0(z) and H1(z) are lowpass and highpass filters,respectively, each with bandwidth ss, the filters Gk(z) and Pk(z) have unequal bandwidths.(b) Suppose that Fk(z) and Hk(z) in Fig. 4.7–2 are selected so that the 2-channel QMF bank(e.g., the one shown in Fig. 4.1–1) is a perfect reconstruction system, with distortionfunction T (z) = 1. Show that the system in Fig. 4.7–2 is also a p erfect reconstructionsystem.(c) Suppose that Fk(z) and Hk(z) in Fig. 4.7–2 are selected so that the 2-channel QMF bank(e.g., the one shown in Fig.4.1–1) is free from aliasing. Does this imply that the system inFig. 4.7–2 is free from aliasing? If yes, justify your reasoning. If no, modify the structureso that it is free from aliasing.x[n]- -----G0(z)G1(z)G2(z)G3(z)G4(z)-----↓ 8↓ 8↓ 4↓ 4↓ 4-----↑ 8↑ 8↑ 4↑ 4↑ 4------P0(z)P1(z)P2(z)P3(z)P4(z)----- -????ˆx[n]Figure 4.7–1 Nonuniformly Decimated Filter BankProblem Set 4 6x[ n] x[ n]∧H (z)02↑H (z)12↑F (z)0F (z)12↑2↑F (z)0F (z)12↑2↑H (z)02↑H (z)12↑F (z)0F (z)12↑2↑H (z)02↑H (z)12↑F (z)0F (z)12↑2↑H (z)02↑H (z)12↑Figure