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Consider a continuous-time LTI system with input x ( t ) and output y(t). We wish to find a discrete-time LTI system with input x[n] and output y[n] such that if x [ n ] =x(nT,) then y [ n ] where T, is the sampling interval.
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= y(nT,)
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(6.86)
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FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
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Let H,(s) and HJz) be the system functions of the continuous-time and discrete-time systems, respectively (Fig. 6-51. Let Then from Eqs. (3.1) and (4.1) we have
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y (t ) = H,( jw ) elw'
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Y [ n ]=H ~ ( ~ J W ~ ) eJnwTs
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(6.88)
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Thus, the requirement y[n] = y(nTs) leads to the condition H,(jo) eJnwT,= Hd(ejwK) e~nwrs from which it follows that H,(jw)
= Hd(ejwT1)
In terms of the Fourier transform, Eq. (6.89) can be expressed as H A 4 = HdW) R = wTs (6.90) Note that the frequency response Hd(R) of the discrete-time system is a periodic function of w (with period 27r/Ts), but that the frequency response H,(o) of the continuous-time system is not. Therefore, Eq. (6.90) or Eq. (6.89) cannot, in general, be true for every w . If the input x(t) is band-limited [Eq. (5.9411, then it is possible, in principle, to satisfy Eq. (6.89) for every w in the frequency range (-rr/Ts,r/Ts) (Fig. 6-6). However, from Eqs. (5.85) and (6.771, we see that Hc(w) is a rational function of w, whereas Hd(R) is a rational function of eJn (R = wT,). Therefore, Eq. (6.89) is impossible to satisfy. However, there are methods for determining a discrete-time system so as to satisfy Eq. (6.89) with reasonable accuracy for every w in the band of the input (Probs. 6.43 to 6.47).
Fig. 6-5 Digital simulation of analog systems.
--2n
" -T,
2n -
Fig. 6-6
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
6.8 THE DISCRETE FOURIER TRANSFORM
In this section we introduce the technique known as the discrete Fourier transform (DFT) for finite-length sequences. It should be noted that the DFT should not be confused with the Fourier transform.
Definition: Let x [ n ] be a finite-length sequence of length N, that is, x [ n ]= O
N- l
outside the range 0 I n I N - 1
The DFT of x [ n ] ,denoted as X [ k ] ,is defined by X [ k ]=
x[n]W,kn
k = 0 , 1 , ..., N - 1
where WN is the Nth root of unity given by The inverse DFT (IDFT) is given by
The DFT pair is denoted by x b l -X[kI Important features of the DFT are the following:
1. There is a one-to-one correspondence between x [ n ] and X [ k ] . 2. There is an extremely fast algorithm, called the fast Fourier transform (FFT) for its calculation. 3. The DFT is closely related to the discrete Fourier series and the Fourier transform. 4. The DFT is the appropriate Fourier representation for digital computer realization because it is discrete and of finite length in both the time and frequency domains.
Note that the choice of N in Eq. (6.92) is not fixed. If x [ n ] has length N , < N, we want to assume that x [ n ] has length N by simply adding ( N - Nl) samples with a value of 0. This addition of dummy samples is known as zero padding. Then the resultant x [ n ] is often referred to as an N-point sequence, and X [ k ] defined in Eq. (6.92) is referred to as an N-point DFT. By a judicious choice of N, such as choosing it to be a power of 2, computational efficiencies can be gained. B. Relationship between the DET and the Discrete Fourier Series: Comparing Eqs. (6.94) and (6.92) with Eqs. (6.7) and (6.81, we see that X [ k ] of finite sequence x [ n ] can be interpreted as the coefficients c, in the discrete Fourier series representation of its periodic extension multiplied by the period N,, and NO= N. That is, X [ k ] = Nc, (6.96) Actually, the two can be made identical by including the factor 1/N with the D R rather than with the IDFT.
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