Crc Press Mechatronics Handbook 2002 By Laxxuss Episode 3 Part 4 pptx

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 qua

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.

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.

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(iωk T) = exp[i(2π/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) ={y},

U(z) ={u}, V(z) ={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 N=0−1

l=0

N−1

{X k}k N=0−1

NT

m=0

∞

m=−∞

k

Y z( ) = H z ( )U z ( ) V z+ ( )

{h k}k∞=0

k=0

∞

∑

-=

V(z) Σ H(z)

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