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Sampling and Reconstruction of Analog Signals 81
DAC). Using Fourier analysis, we can describe the sampling operation
from the frequency-domain viewpoint, analyze its effects, and then address
the reconstruction operation. We will also assume that the number
of quantization levels is sufficiently large that the effect of quantization
on discrete signals is negligible. We will study the effects of quantization
in Chapter 10.
3.4.1 SAMPLING
Let x a (t) beananalog (absolutely integrable) signal. Its continuous-time
Fourier transform (CTFT) is given by
X a (jΩ) △ =
∫ ∞
−∞
x a (t)e −jΩt dt (3.24)
where Ω is an analog frequency in radians/sec. The inverse continuoustime
Fourier transform is given by
x a (t) = 1
2π
∫ ∞
−∞
X a (jΩ)e jΩt dΩ (3.25)
We now sample x a (t) atsampling interval T s seconds apart to obtain the
discrete-time signal x(n).
x(n) △ = x a ( nT s )
Let X(e jω )bethe discrete-time Fourier transform of x(n). Then it can be
shown [23] that X(e jω )isacountable sum of amplitude-scaled, frequencyscaled,
and translated versions of the Fourier transform X a (jΩ).
X(e jω )= 1 ∑
∞ ( ω
X a
[j − 2π )]
l
(3.26)
T s
T s T s
l=−∞
This relation is known as the aliasing formula. The analog and digital
frequencies are related through T s
ω =ΩT s (3.27)
while the sampling frequency F s is given by
△ 1
F s = , sam/sec (3.28)
T s
The graphical illustration of (3.26) is shown in Figure 3.10, from which
we observe that, in general, the discrete signal is an aliased version of the
corresponding analog signal because higher frequencies are aliased into
lower frequencies if there is an overlap. However, it is possible to recover
the Fourier transform X a (jΩ) from X(e jω ) [or equivalently, the analog
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