FIGURE 23.11 Finite-width pulse sampling signals and spectra: (a) bandlimited signal and spectrum, (b) finite- width train of pulses and its transform, (c) sampled signal and its spectra, (d) quantization process. 0066_Frame_C23 Page 20 Wednesday, January 9, 2002 1:53 PM ©2002 CRC Press LLC FIGURE 23.11 Finite-width pulse sampling signals and spectra: (a) bandlimited signal and spectrum, (b) finite- width train of pulses and its transform, (c) sampled signal and its spectra, (d) quantization process. 0066_Frame_C23 Page 20 Wednesday, January 9, 2002 1:53 PM ©2002 CRC Press LLC The Discrete Fourier Transform Consider a finite length sequence that is zero outside the interval 0 ≤ k ≤ N − 1. Evaluation of the z transform X ( z ) at N equally spaced points on the unit circle z = exp( i ω k T ) = exp[ i (2 π / NT ) kT ] for k = 0, 1,…, N − 1 defines the discrete Fourier transform (DFT) of a signal x with a sampling period h and N measurements: (23.60) Notice that the discrete Fourier transform is only defined at the discrete frequency points (23.61) In fact, the discrete Fourier transform adapts the Fourier transform and the z transform to the practical requirements of finite measurements. Similar properties hold for the discrete Laplace transform with z = exp( sT ), where s is the Laplace transform variable. The Transfer Function Consider the following discrete-time linear system with input sequence { u k } (stimulus) and output sequence { y k } (response). The dependency of the output of a linear system is characterized by the convolution- type equation and its z transform, (23.62) where the sequence { v k } represents some external input of errors and disturbances and with Y ( z ) = ᐆ { y }, U ( z ) = ᐆ { u }, V ( z ) = ᐆ { v } as output and inputs. The weighting function h ( kT ) = , which is zero for negative k and for reasons of causality is sometimes called pulse response of the digital system (compare impulse response of continuous-time systems). The pulse response and its z transform, the pulse transfer function , (23.63) determine the system’s response to an input U ( z ); see Fig. 23.18. The pulse transfer function H ( z ) is obtained as the ratio (23.64) FIGURE 23.18 Block diagram with an assumed transfer function relationship H ( z ) between input U ( z ), disturbance V ( z ), intermediate X ( z ), and output Y ( z ). {x k } k=0 N−1 X k DFT xkT(){} x l iw k lT–()exp l=0 N−1 ∑ Xe iw k T ()== = {X k } k=0 N−1 w k 2p NT k, for k 0, 1,…, N 1–== y k h m u k−m v k + m=0 ∞ ∑ h k−m u m v k , k+ m=−∞ k ∑ …, −1, 0, 1, 2,…== = Yz() Hz()Uz() Vz()+= {h k } k=0 ∞ Hz() ᐆ hkT(){} h k z k– k=0 ∞ ∑ == Hz() Xz() Uz() = U(z) X(z) Y(z) V(z) Σ H(z) 0066_Frame_C23 Page 33 Wednesday, January 9, 2002 1:55 PM ©2002 CRC Press LLC . signal and its spectra, (d) quantization process. 0066_Frame_C 23 Page 20 Wednesday, January 9, 2002 1: 53 PM 2002 CRC Press LLC FIGURE 23. 11 Finite-width pulse sampling signals and spectra: (a) bandlimited. h k z k– k=0 ∞ ∑ == Hz() Xz() Uz() = U(z) X(z) Y(z) V(z) Σ H(z) 0066_Frame_C 23 Page 33 Wednesday, January 9, 2002 1:55 PM 2002 CRC Press LLC . transfer function , ( 23. 63) determine the system’s response to an input U ( z ); see Fig. 23. 18. The pulse transfer function H ( z ) is obtained as the ratio ( 23. 64) FIGURE 23. 18 Block