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88 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
x2(n)
1
0.8
0.6
0.4
0.2
Discrete Signal
Ts=1 msec
X2(w)
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
t in msec.
2.5
2
1.5
1
0.5
Discrete-time Fourier Transform
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Frequency in π units
FIGURE 3.13 Plots in Example 3.18b
This two-step procedure can be described mathematically using an interpolating
formula [23]
x a (t) =
sin πx
∞∑
n=−∞
x(n) sinc [F s (t − nT s )] (3.37)
where sinc(x) =
πx
is an interpolating function. The physical interpretation
of the above reconstruction (3.37) is given in Figure 3.14, from
which we observe that this ideal interpolation is not practically feasible
because the entire system is noncausal and hence not realizable.
□ EXAMPLE 3.20 Consider the sampled signal x(n) from Example 3.17. It is applied as an input
to an ideal D/A converter (that is, an ideal interpolator) to obtain the
analog signal y a(t). The ideal D/A converter is also operating at F s = 200
sam/sec. Obtain the reconstructed signal y a(t), and determine whether the sampling/reconstruction
operation resulted in any aliasing. Also plot the Fourier
transforms X a(jΩ), X(e jω ), and Y a(jΩ).
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