SOLUTIONS MANUAL to accompany Digital Signal Processing: A Computer-Based Approach Second Edition Sanjit K. Mitra Prepared by Rajeev Gandhi, Serkan Hatipoglu, Zhihai He, Luca Lucchese, Michael Moore, and Mylene Queiroz de Farias 1 Errata List of "Digital Signal Processing: A Computer-Based Approach", Second Edition Chapter 2 1. Page 48, Eq. (2.17): Replace "y[n]" with "x [n]". u 1 2. Page, 51: Eq. (2.24a): Delete " (x[n]+x*[N- n])". Eq. (2.24b): Delete 2 1 " (x[n]- x*[N- n])". 2 ( ) 3. Page 59, Line 5 from top and line 2 from bottom: Replace "- cos (w +w - p )n " with 1 2 ( ) "cos (2p –w –w )n ". 1 2 ( ) ( ) 4. Page 61, Eq. (2.52): Replace "Acos (W +kW )t+f " with "Acos – (W t+f )+kW t ". o T o T 5. Page 62, line 11 from bottom: Replace "W >2W " with "W >2W ". T o T o 6. Page 62, line 8 from bottom: Replace "2p W /w " with "2p W /W ". o T o T 7. Page 65, Program 2_4, line 7: Replace "x = s + d;" with "x = s + d';". ¥ ¥ 8. Page 79, line 5 below Eq. (2.76): Replace " (cid:229) a n " with " (cid:229) a n". n=0 n=0 9. Page 81, Eq. (2.88): Replace "a L+1l n2 +a Nl nN- L" with " a L+1l n2 +L+a Nl nN- L". 10. Page 93, Eq. (2.116): Replace the lower limit "n=–M+1" on all summation signs with "n=0". 11. Page 100, line below Eq. (2.140) and caption of Figure 2.38: Replace "w =0.03" with o "w =0.06p ". o 12. Page 110, Problem 2.44: Replace "{y[n]}= {- 1, - 1, 11, - 3, - 10, 20, - 16}" with "{y[n]}= {- 1, - 1, 11, - 3, 30, 28, 48}", and "{y[n]}= {- 14- j5, - 3- j17, - 2+ j5, - 26+ j22, 9+ j12}" with "{y[n]}= {- 14- j5, - 3- j17, - 2+ j5, - 9.73+ j12.5, 5.8+ j5.67}". 13. Page 116, Exercise M2.15: Replace "randn" with "rand". Chapter 3 1. Page 118, line 10 below Eq. (3.4): Replace "real" with "even". 2 ¥ ¥ 2. Page 121, Line 5 below Eq. (3.9): Replace " (cid:229) a n " with " (cid:229) a n". n=0 n=0 3. Page 125, Eq. (3.16): Delete the stray a . 4. Page 138, line 2 below Eq. (3.48): Replace "frequency response" with "discrete-time Fourier transform". 5. Page 139, Eq. (3.53): Replace "x(n + mN)" with "x[n + mN]". 6. Page 139, lin2 2, Example 3.14: Replace "x[n] ={0 1 2 3 4 5}" with "{x[n]} ={0 1 2 3 4 5}". 7. Page 139, line 3, Example 3.14: Replace "x[n]" with "{x[n]}", and "p k/4" with "2p k/4". 8. Page 139, line 6 from bottom: Replace "y[n] ={4 6 2 3 4 6}" with "{y[n]} ={4 6 2 3}". 9. Page 141, Table 3.5: Replace "N[g<- k > ]" with "Ng[<- k> ]". N N 10. Page 142, Table 3.7: Replace "argX[<- k > ]" with "– argX[<- k > ]". N N Ø 1 1 1 1ø Ø 1 1 1 1ø Œ œ Œ œ 1 j - 1 - j 1 - j - 1 j Œ œ Œ œ 11. Page 147, Eq. (3.86): Replace " " with " ". Œ 1 - 1 1 - 1œ Œ 1 - 1 1 - 1œ Œ œ Œ œ º 1 - j - 1 j ß º 1 j - 1 - jß - 1 - 1 12. Page 158, Eq.(3.112): Replace " (cid:229) a nz- n" with "- (cid:229) a nz- n". n=-¥ n=-¥ 13. Page 165, line 4 above Eq. (3.125); Replace "0.0667" with "0.6667". 14. Page 165, line 3 above Eq. (3.125): Replace "10.0000" with "1.000", and "20.0000" with "2.0000". 15. Page 165, line above Eq. (3.125): Replace "0.0667" with "0.6667", "10.0" with "1.0", and "20.0" with "2.0". 16. Page 165, Eq. (3.125): Replace "0.667" with "0.6667". 17. Page 168, line below Eq. (3.132): Replace "z > l " with "z > l ". l l 18. Page 176, line below Eq. (3.143): Replace "R " with "1/R ". h h 19. Page 182, Problem 3.18: Replace "X(e- jw /2)" with "X(- ejw /2)". 3 20. Page 186, Problem 3.42, Part (e): Replace "argX[<- k > ]" with "– argX[<- k > ]". N N 21. Page 187, Problem 3.53: Replace "N-point DFT" with "MN-point DFT", replace "0£ k£ N- 1" with "0£ k£ MN- 1", and replace "x[<n> ]" with "x[<n> ]". M N 22. Page 191, Problem 3.83: Replace " lim " with " lim ". nfi ¥ zfi ¥ P(z) P(z) 23. Page 193, Problem 3.100: Replace " " with "- l ". l D'(z) D' (z) 24. Page 194, Problem 3.106, Parts (b) and (d): Replace " z < a " with "z >1/ a ". 25. page 199, Problem 3.128: Replace "(0.6)m [n]" with "(0.6)nm [n]", and replace "(0.8)m [n]" with "(0.8)nm [n]". 26. Page 199, Exercise M3.5: Delete "following". Chapter 4 1. Page 217, first line: Replace "x " with "x ". N M 2. Page 230, line 2 below Eq. (4.88): Replace "q (w )" with "q (w )". g 3. Page 236, line 2 below Eq. (4.109): Replace "decreases" with "increases". 4. Page 246, line 4 below Eq. (4.132): Replace "q (ejw )" with "q (w )". c c 5. Page 265, Eq. (4.202): Replace "1,2,K,3" with "1,2,3". j0 jp /4 6. Page 279, Problem 4.18: Replace " H(e ) " with "H(e ) ". 7. Page 286, Problem 4.71: Replace "z = - j0.3" with "z = - 0.3". 3 3 8. Page 291, Problem 4.102: Replace 0.4+0.5z- 1+1.2z- 2 +1.2z- 3+0.5z- 4 +0.4z- 5 "H(z) = " with 1+0.9z- 2 +0.2z- 4 0.1+0.5z- 1+ 0.45z- 2 +0.45z- 3+0.5z- 4+0.1z- 5 "H(z) = ". 1+0.9z- 2 +0.2z- 4 9. Page 295, Problem 4.125: Insert a comma "," before "the autocorrelation". Chapter 5 1. Page 302, line 7 below Eq. (5.9): Replace "response" with "spectrum". 4 2. Page 309, Example 5.2, line 4: Replace "10 Hz to 20 Hz" with "5 Hz to 10 Hz". Line 6: Replace "5k + 15" with "5k + 5". Line 7: Replace "10k + 6" with "5k + 3", and replace "10k – 6" with "5(k+1) – 3". 3. Page 311, Eq. (5.24): Replace "G (jW - 2k(DW ))" with "G (j(W - 2k(DW )))". a a 4. Page 318, Eq. (5.40): Replace "H(s)" with "H (s)", and replace "" with "". a 5. Page 321, Eq. (5.54): Replace "" with "". 6. Page 333, first line: Replace "W " with "W ˆ ", and "W " with "W ˆ ". p1 p1 p2 p2 7. Page 349, line 9 from bottom: Replace "1/T" with "2p /T". 8. Page 354, Problem 5.8: Interchange "W " and "W ". 1 2 9. Page 355, Problem 5.23: Replace "1 Hz" in the first line with "0.2 Hz". 10. Page 355, Problem 5.24: Replace "1 Hz" in the first line with "0.16 Hz". Chapter 6 1. Page 394, line 4 from bottom: Replace "alpha1" with "fliplr(alpha1)". 2. Page 413, Problem 6.16: Replace ( ( )) "H(z) = b0 +b1(cid:230) Ł z- 1+b2z- 1 z- 1+ b3z- 1 1+L+ bN- 1z- 1(1+bNz- 1) (cid:246) ł " with ( ( )) "H(z) = b0 +b1z- 1(cid:230) Ł 1+b2z- 1 z- 1+b3z- 11+L+bN- 1z- 1(1+bNz- 1) (cid:246) ł ". 3z2+18.5z+17.5 3. Page 415, Problem 6.27: Replace "H(z) = " with (2z+1)(z+2) 3z2+18.5z+17.5 "H(z) = ". (z+0.5)(z+2) 4. Page 415, Problem 6.28: Replace the multiplier value "0.4" in Figure P6.12 with "–9.75". 5. Page 421, Exercise M6.1: Replace "- 7.6185z- 3" with "- 71.6185z- 3". 6. Page 422, Exercise M6.4: Replace "Program 6_3" with "Program 6_4". 7. Page 422, Exercise M6.5: Replace "Program 6_3" with "Program 6_4". 8. Page 422, Exercise M6.6: Replace "Program 6_4" with "Program 6_6". 5 Chapter 7 1. Page 426, Eq. (7.11): Replace "h[n – N]" with "h[N – n]". 2. Page 436, line 14 from top: Replace "(5.32b)" with "(5.32a)". 3. Page 438, line 17 from bottom: Replace "(5.60)" with "(5.59)". 4. Page 439, line 7 from bottom: Replace "50" with "40". - 1 5. Page 442, line below Eq. (7.42): Replace "F (ˆz ) " with "1/ F(ˆz ) ". - 1 6. Page 442, line above Eq. (7.43): Replace "F (ˆz ) " with "F(ˆz ) ". (cid:230) (cid:246) L zˆ - a 7. Page 442, Eq. (7.43): Replace it with "F(ˆz ) = – (cid:213) (cid:231) l (cid:247) ". l=1Ł 1- a l*ˆz ł 8. Page 442, line below Eq. (7.43): Replace "where a " with "where a ". l l 9. Page 446, Eq. (7.51): Replace "b (1- a )" with "b (1+a )". 10. Page 448, Eq. (7.58): Replace "w <w £ p " with "w < w £ p ". c c 11. Page 453, line 6 from bottom: Replace "w - w " with "w - w ". p s s p 12. Page 457, line 8 from bottom: Replace "length" with "order". 13. Page 465, line 5 from top: Add "at w =w " before "or in". i 14. Page 500, Problem 7.15: Replace "2 kHz" in the second line with "0.5 kHz". Bs 15. Page 502, Problem 7.22: Replace Eq. (7.158) with "H (s) = ". a s2 +Bs+W 2 0 16. Page 502, Problem 7.25: Replace "7.2" with "7.1". 17. Page 504, Problem 7.41: Replace "H (ejw ) = e- jw " in Eq. (7.161) with int "H (ejw ) = 1 ". int jw 18. Page 505, Problem 7.46: Replace "16" with "9" in the third line from bottom. 19. Page 505, Problem 7.49: Replace "16" with "9" in the second line. 20. Page 510, Exercise M7.3: Replace "Program 7_5" with "Program 7_3". 21. Page 510, Exercise M7.4: Replace "Program 7_7" with "M-file impinvar". 22. Page 510, Exercise M7.6: Replace "Program 7_4" with "Program 7_2". 23. Page 511, Exercise M7.16: Replace "length" with "order". 6 24. Page 512, Exercise M7.24: Replace "length" with "order". Chapter 8 1. Page 518, line 4 below Eq. (6.7): Delete "set" before "digital". 2. Page 540, line 3 above Eq. (8.39): Replace "G[k]" with "X [k]" and "H[k]" with "X [k]". 0 1 Chapter 9 1. Page 595, line 2 below Eq. (9.30c): Replace "this vector has" with "these vectors have". b - b 2. Page 601, line 2 below Eq. (9.63): Replace "2 " with "2 ". (cid:230) a z+1(cid:246) (cid:230) a z+1(cid:246) 3. Page 651, Problem 9.10, line 2 from bottom: Replace "(cid:231) k (cid:247) " with "(cid:231) k (cid:247) ". Ł 1+a zł Ł z+a ł k k 4. Page 653, Problem 9.15, line 7: Replace "two cascade" with "four cascade". d d + d z- 1+z- 2 5. Page 653, Problem 9.17: Replace "A (z) = 1 2 1 " with 2 1+d z- 1+ d d z- 2 1 1 2 d +d z- 1+z- 2 "A (z) = 2 1 ". 2 1+d z- 1+ d z- 2 1 2 6. Page 654, Problem 9.27: Replace "structure" with "structures". 7. Page 658, Exercise M9.9: Replace "alpha" with "a ". Chapter 10 1. Page 692, Eq. (10.57b): Replace "P (a ) = 0.2469" with "P (a ) = 0.7407". 0 1 0 1 2. Page 693, Eq. (10.58b): Replace "P (a ) = –0.4321" with "P (a ) = –1.2963". 0 2 0 2 3. Page 694, Figure 10.38(c): Replace "P- 2(a 0)" with "P1(a 0)", "P- 1(a 0)" with "P0(a 0)", "P0(a 0)" with "P- 1(a 0)", "P1(a 0)" with "P- 2(a 0)", "P- 2(a 1)" with "P1(a 1)", "P- 1(a 1)" with "P0(a 1)", "P0(a 1)" with "P- 1(a 1)", "P1(a 1)" with "P- 2(a 1)", "P- 2(a 2)" with "P1(a 2)", "P- 1(a 2)" with "P0(a 2)", "P0(a 2)" with "P- 1(a 2)", and "P- 1(a 2)" with "P- 2(a 2)". 4. Page 741, Problem 10.13: Replace "2.5 kHz" with "1.25 kHz". (cid:229) N i (cid:229) N- 1 i 5. Page 741, Problem 10.20: Replace " z " with " z ". i=0 i=0 7 6. Page 743, Problem 10.28: Replace "half-band filter" with a "lowpass half-band filter with a zero at z = –1". 7. Page 747, Problem 10.50: Interchange "Y " and "the output sequence y[n]". k 8. Page 747, Problem 10.51: Replace the unit delays "z- 1" on the right-hand side of the structure of Figure P10.8 with unit advance operators "z". [ ] 9. Page 749, Eq. (10.215): Replace "3H2(z)- 2H2(z)" with "z- 2 3H2(z)- 2H2(z) ". 10. Page 751, Exercise M10.9: Replace "60" with "61". 11. Page 751, Exercise M10.10: Replace the problem statement with "Design a fifth-order IIR half-band Butterworth lowpass filter and realize it with 2 multipliers". 12. Page 751, Exercise M10.11: Replace the problem statement with "Design a seventh-order IIR half-band Butterworth lowpass filter and realize it with 3 multipliers". Chapter 11 1. Page 758, line 4 below Figure 11.2 caption: Replace "grid" with "grid;". 2. age 830, Problem 11.5: Insert "g (t) = cos(200p t)" after "signal" and delete a "= cos(200p n)". 3. Page 831, Problem 11.11: Replace "has to be a power-of-2" with " = 2l, where l is an integer". 8 Chapter 2 (2e) 2.1 (a) u[n]=x[n]+y[n]={3 5 1 - 2 8 14 0} (b) v[n]=x[n](cid:215) w[n]={- 15 - 8 0 6 - 20 0 2} (c) s[n]=y[n]- w[n]={5 3 - 2 - 9 9 9 - 3} (d) r[n]=4.5y[n]={0 31.5 4.5 - 13.5 18 40.5 - 9} 2.2 (a) From the figure shown below we obtain v[n] v[n–1] x[n] –1 z a b y[n] g v[n]=x[n]+a v[n- 1] and y[n]=b v[n- 1]+g v[n- 1]=(b +g )v[n- 1]. Hence, v[n- 1]=x[n- 1]+a v[n- 2] and y[n- 1]=(b +g )v[n- 2]. Therefore, y[n- 1] y[n]=(b +g )v[n- 1]=(b +g )x[n- 1]+a (b +g )v[n- 2] =(b +g )x[n- 1]+a (b +g ) (b +g ) =(b +g )x[n- 1]+a y[n- 1]. (b) From the figure shown below we obtain x[n–1] x[n–2] x[n] z–1 z–1 –1 –1 z z x[n–4] x[n–3] a b g y[n] y[n]=g x[n- 2]+b (x[n- 1]+x[n- 3])+a (x[n]+x[n- 4]). (c) From the figure shown below we obtain v[n] v[n–1] –1 x[n] z y[n] –1 d –1 1 z 6 v[n]=x[n]- d v[n- 1] and y[n]=d v[n]+v[n- 1]. Therefore we can rewrite the second 1 1 ( ) ( ) equation as y[n]=d x[n]- d v[n- 1] +v[n- 1]=d x[n]+ 1- d2 v[n- 1] (1) 1 1 1 1 ( )( ) ( ) ( ) =d x[n]+ 1- d2 x[n- 1]- d v[n- 2] =d x[n]+ 1- d2 x[n- 1]- d 1- d2 v[n- 2] 1 1 1 1 1 1 1 ( ) From Eq. (1), y[n- 1]=d x[n- 1]+ 1- d2 v[n- 2], or equivalently, 1 1 ( ) d y[n- 1]=d2x[n- 1]+d 1- d2 v[n- 2]. Therefore, 1 1 1 1 ( ) ( ) ( ) y[n]+d y[n- 1]=d x[n]+ 1- d2 x[n- 1]- d 1- d2 v[n- 2]+d2x[n- 1]+d 1- d2 v[n- 2] 1 1 1 1 1 1 1 1 =d x[n]+x[n- 1], or y[n]=d x[n]+x[n- 1]- d y[n- 1]. 1 1 1 (d) From the figure shown below we obtain v[n] v[n–1] v[n–2] x[n] z–1 z–1 y[n] –1 d 1 w[n] u[n] d 2 v[n]=x[n]- w[n], w[n]=d v[n- 1]+d u[n], and u[n]=v[n- 2]+x[n]. From these equations 1 2 we get w[n]=d x[n]+d x[n- 1]+d x[n- 2]- d w[n- 1]- d w[n- 2]. From the figure we also 2 1 2 1 2 obtain y[n]=v[n- 2]+w[n]=x[n- 2]+w[n]- w[n- 2], which yields d y[n- 1]=d x[n- 3]+d w[n- 1]- d w[n- 3], and 1 1 1 1 d y[n- 2]=d x[n- 4]+d w[n- 2]- d w[n- 4], Therefore, 2 2 2 2 y[n]+d y[n- 1]+d y[n- 2]=x[n- 2]+d x[n- 3]+d x[n- 4] 1 2 1 2 ( ) ( ) + w[n]+d w[n- 1]+d w[n- 2] - w[n- 2]+d w[n- 3]+d w[n- 4] 1 2 1 2 =x[n- 2]+d x[n]+d x[n- 1] or equivalently, 2 1 y[n]=d x[n]+d x[n- 1]+x[n- 2]- d y[n- 1]- d y[n- 2]. 2 1 1 2 2.3 (a) x[n]={3 - 2 0 1 4 5 2}, Hence, x[- n]={2 5 4 1 0 - 2 3}, - 3£ n£ 3. 1 Thus, x [n]= (x[n]+x[- n])={5/2 3/2 2 1 2 3/2 5/2}, - 3£ n£ 3, and ev 2 1 x [n]= (x[n]- x[- n])={1/2 - 7/2 - 2 0 2 7/2 - 1/2}, - 3£ n£ 3. od 2 (b) y[n]={0 7 1 - 3 4 9 - 2}. Hence, y[- n]={- 2 9 4 - 3 1 7 0}, - 3£ n£ 3. 1 Thus, y [n]= (y[n]+y[- n])={- 1 8 5/2 - 3 5/2 8 - 1}, - 3£ n£ 3, and ev 2 1 y [n]= (y[n]- y[- n])={1 - 1 - 3/2 0 1 3/2 - 1}, - 3£ n£ 3. od 2 7