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82 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
x a (t)
1
CTFT
X a (jΩ )
A
0
t
−Ω 0 0
Ω 0
Ω
Sample
eq. (3.27)
x(n)
1
T s < π/Ω 0
DTFT
A /T s
X(e jw )
−Ω 0 /T s
Ω 0 /T s
−5
0
5
n
−2π −π 0 π 2π
w
x(n)
X(e jw )
A /T s
T s > π/Ω 0
−Ω 0 /T s
Ω 0 /T s
−2 −1 0 1 2
n
−2π
−π
0 π 2π
w
FIGURE 3.10
Sampling operation in the time and frequency domains
signal x a (t) from its samples x(n)] if the infinite “replicas” of X a (jΩ) do
not overlap with each other to form X(e jω ). This is true for band-limited
analog signals.
DEFINITION 2
[Band-limited Signal] A signal is band-limited if there exists a finite radian
frequency Ω 0 such that X a (jΩ) is zero for |Ω| > Ω 0 . The frequency
F 0= Ω 0 /2π is called the signal bandwidth in Hz.
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