1. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.1 (p. 343) FS and FT representation of a periodic continuous-time signal.

2. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.2 (p. 343) FT of a cosine.

3. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.3 (p. 344) An impulse train and its FT.

4. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.4 (p. 345) Square wave for Problem 4.1.

5. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.5 (p. 346) Infinite series of frequency-shifted impulses that is 2  periodic in frequency .

6. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.6 (p. 346) DTFS and DTFT representations of a periodic discrete-time signal.

7. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.7 (p. 347) DTFT of periodic signal for Example 4.3.

8. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.8 (p. 449) Convolution property for mixture of periodic and nonperiodic signals.

9. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.9 (p. 350) Application of convolution property in Example 4.4.

10. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.10 (p. 350) Signal x(t) for Problem 4.4.

11. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.11 (p. 352) Multiplication of periodic and nonperiodic time-domain signals corresponds to convolution of the corresponding FT representations.

12. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.12 (p. 353) Solution for Example 4.5 (b).

13. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.13 (p. 353) (a) Simplified AM radio transmitter and receiver. (b) Spectrum of message signal.

14. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.14 (p. 354) Signals in the AM transmitter and receiver. (a) Transmitted signal r(t) and spectrum R(j  ). (b) Spectrum of q(t) in the receiver. (c) Spectrum of receiver output y(t).

15. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.15 (p. 357) Effect of windowing a data record. Y(e j  ) for different values of M, assuming that  1 = 7  /16 and  2 = 9  /16. (a) M = 80, (b) M = 12, (c) M = 8.

16. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.16 (p. 358) Problem 4.7 (a) System. (b) Input spectrum.

17. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.17 (p. 358) Solutions to Problem 4.7

18. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.18 (p. 359) Relationship between FT and DRFT representations of a discrete-time signal.

19. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.19 (p. 361) Solution to Problem 4.8.

20. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.20 (p. 362) Relationship between FT and DTFS representations of a discrete-time periodic signal.

21. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.21 (p. 363) Mathematical representation of sampling as the product of a given time signal and an impulse train.

22. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.22 (p. 364) The FT of a sampled signal for different sampling frequencies. (a) Spectrum of continuous-time signal. (b) Spectrum of sampled signal when  s = 3W. (c) Spectrum of sampled signal when  s = 2W. (d) Spectrum of sampled signal when  s = 1.5W.

23. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.23 (p. 365) The DTFTs corresponding to the FTs depicted in Fig. 4-22 (b)-(d). (a)  s = 3W. (b)  s = 2W. (c)  s = 1.5W.

24. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.24 (p. 367) The effect of sampling a sinusoid at different rates (Example 4.9). (a) Original signal and FT. (b) Original signal, impulse sampled representation and FT for T s = ¼. (c) Original signal, impulse sampled representation and FT for cT = 1. (d) Original signal, impulse sampled representation and FT for T s = 3/2. A cosine of frequency  /3 is shown as the dashed line.

25. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.25 (p. 368) Aliasing in a movie. (a) Wheel rotating at  radians per second and moving from right to left at meters per second. b) Sequence of movie frames, assuming that the wheel rotates less than one-half turn between frames. (c) Sequences of movie frames, assuming that the wheel rotates between one-half and one turn between frames. (d) Sequence of movie frames, assuming that the wheel rotates one turn between frames.

26. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.26 (p. 368) Spectrum of original signal for Problem 4.10.

27. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.27 (p. 369) Solution to Problem 4.10.

28. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.28 (p. 370) Factor determining the quality of the discrete-time model for the two-path communication channel.

29. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.29 (p. 372) Effect of subsampling on the DTFT. (a) Original signal spectrum. (b) m = 0 term, Xq(e j  ), in Eq. (4.27) (c) m = 1 term in Eq. (4.27). (d m = q – 1 term in Eq. (4.27). (e) Y(e j  ), assuming that W  /q.

30. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.30 (p. 373) Solution to Problem 4.11.

31. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.31 (p. 373) Block diagram illustrating conversion of a discrete-time signal to a continuous-time signal.

32. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.32 (p. 373) Two continuous-time signals x 1 (t) (dashed line) and x 2 (t) (solid line) that have the same set of samples.

33. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.33 (p. 374) FT of continuous-time signal for Example 4.12.

34. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.34 (p. 375) FT of x(t) for Problem 4.12(c).

35. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.35 (p. 376) Ideal reconstruction. (a) Spectrum of original signal. (b) Spectrum of sampled signal. (c) Frequency response of reconstruction filter.

36. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.36 (p. 377) Ideal reconstruction in the time domain.

37. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.37 (p. 377) Reconstruction via a zero-order hold.

38. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.38 (p. 378) Rectangular pulse used to analyze zero-order hold reconstruction.

39. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.39 (p. 379) Effect of the zero-order hold in the frequency domain. (a) Spectrum of original continuous-time signal. (b) FT of sampled signal. (c) Magnitude and phase of Ho(j  ). (d) Magnitude spectrum of signal reconstructed using zero-order hold.

40. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.40 (p. 380) Frequency response of a compensation filter used to eliminate some of the distortion introduced by the zero-order hold.

41. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.41 (p. 380) Block diagram of a practical reconstruction system.

42. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.42 (p. 382) Anti-imaging filter design with and without oversampling. (a) Magnitude of H o (jf) for 44.1-kHz sampling rate. Dashed lines denote signal passband and images. (b) Magnitude of H o (jf) for eight-times oversampling (352.8-kHz sampling rate. Dashed lines denote signal passband and images. (c) Normalized constraints on passband response of anti-imaging filter. Solid lines assume a 44.1-kHz sampling rate; dashed lines assume eight- times oversampling. The normalized filter response must lie between each pair of lines.

43. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.43 (p.383) Block diagram for discrete-time processing of continuous-time signals. (a) A basic system. (b) Equivalent continuous-time system.

44. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.44 (p. 385) Effect of oversampling on anti-aliasing filter specifications. (a) Spectrum of original signal. (b) Anti-aliasing filter frequency response magnitude. (c) Spectrum of signal at the anti-aliasing filter output. (d) Spectrum of the anti-aliasing filter output after sampling. The graph depicts the case of  s > 2W s.

45. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.45 (p. 387) Effect of changing the sampling rate. (a) Underlying continuous-time signal FT. (b) DTFT of sampled data at sampling interval T s1. (c) DTFT of sampled data at sampling interval T s2.

46. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.46 (p. 387) The spectrum that results from subsampling the DTFT X 2 (ej  ) depicted in Fig. 4.45 by a factor of q.

47. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.47 (p. 388) Frequency-domain interpretation of decimation. (a) Block diagram of decimation system. (b) Spectrum of oversampled input signal. Noise is depicted as the shaded portions of the spectrum. (c) Filter frequency response. (d) Spectrum of filter output. (e) Spectrum after subsampling.

48. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.48 (p. 389) Symbol for decimation by a factor of q.

49. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.49 (p. 390) Frequency-domain interpretation of interpolation. (a) Spectrum of original sequence. (b) Spectrum after inserting q – 1 zeros in between every value of the original sequence. (c) Frequency response of a filter for removing undesired replicates located at  2  /q,  4  /q, …,  (q – 1)2  /q. (d) Spectrum of interpolated sequence.

50. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.50 (p. 390) (a) Block diagram of an interpolation system. (b) Symbol denoting interpolation by a factor of q.

51. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.51 (p. 391) Block diagram of a system for discrete-time processing of continuous-time signals including decimation and interpolation.

52. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.52 (p. 392) The DTFS of a finite-duration nonperiodic signal.

53. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.53 (p. 394) The DTFT and length-N DTFS of a 32-point cosine. The dashed line denotes |X(e j  )|, while the stems represent N|X[k]|. (a) N = 32, (b) N = 60, (c) N = 120.

54. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.54 (p. 396) Block diagram depicting the sequence of operations involved in approximating the FT with the DTFS.

55. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.55 (p. 397) Effect of aliasing.

56. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.56 (p. 398) Magnitude response of M-point window.

57. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.57 (p. 400) The DTFS approximation to the FT of x(t) = e -1/10 u(t)(cos(10t) + cos(12t). The solid line is the FT |X(j  )|, and the stems denote the DTFS approximation NT s |Y[k]|. Both |X(j  ) and NT s |Y[k]| have even symmetry, so only 0 <  < 20 is displayed. (a) M = 100, N = 4000. (b) M = 500, N = 4000. (c) M = 2500, N = 4000. (d) M = 2500, N = 16,0000 for 9 <  < 13.

58. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.58 (p. 404) The DTFS approximation to the FT of x(t) = cos(2  (0.4)t) + cos(2  (0.45)t). The stems denote |Y[k]|, while the solid lines denote (1/M|Y  (j  )|. The frequency axis is displayed in units of Hz for convenience, and only positive frequencies are illustrated. (a) M = 40. (b) M = 2000. Only the stems with nonzero amplitude are depicted. (c) Behavior in the vicinity of the sinusoidal frequencies for M = 2000. (d) Behavior in the vicinity of the sinusoidal frequencies for M = 2010.

59. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.59 (p. 406) Block diagrams depicting the decomposition of an inverse DTFS as a combination of lower order inverse DTFS’s. (a) Eight-point inverse DTFS represented in terms of two four-point inverse DTFS’s. (b) four-point inverse DTFS represented in terms of two-point inverse DTFS’s. (c) Two-point inverse DTFS.

60. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.60 (p. 407) Diagram of the FFT algorithm for computing x[n] from X[k] for N = 8.

61. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.61 (p. 409) Original and resampled signals obtained using MATLAB.

62. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure 4.62 (p. 411) The use of the MATLAB command p l o t for displaying the DTFS coefficients in case (b) of Example 4.16.

63. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.16 (p. 413)

64. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.17 (p. 413)

65. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.18 (p. 414)

66. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.19 (p. 414)

67. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.20 (p. 414)

68. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.21 (p. 414)

69. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.23 (p. 415)

70. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.26 (p. 415)

71. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.28 (p. 416)

72. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.29 (p. 416)

73. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.30 (p. 416)

74. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.33 (p. 416)

75. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.34 (p. 417)

76. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.35 (p. 417)

77. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.41 (p. 418)

78. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.45 (p. 419)

79. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.47 (p. 419)

80. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.48 (p. 420)

81. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.49 (p. 420)

82. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.50 (p. 420)

83. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. Figure P4.51 (p. 421)