09-05-2012, 12:01 PM
Discrete time processing of continuous time signals
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Even though this course is primarily about the discrete time signal processing,
most signals we encounter in daily life are continuous in time such as speech,
music and images. Increasingly discrete-time signals processing algorithms
are being used to process such signals. For processing by digital systems, the
discrete time signals are represented in digital form with each discrete time
sample as binary word. Therefore we need the analog to digital and digital to
analog interface circuits to convert the continuous time signals into discrete
time digital form and vice versa. As a result it is necessary to develop the
relations between continuous time and discrete time representations.
Sampling of continuous time signals:
Let {xc(t)} be a continuous time signal that is sampled uniformly at t = nT
generating the sequence {x[n]} where
x[n] = xc(nT), −∞ < n < ∞, T>0
T is called sampling period, the reciprocal of T is called the sampling frequency
fs = 1/T . The frequency domain representation of {xc(t)} is given
by its Fourier transform
The effect of underselling: Aliasing
We have seen earlier that spectrum Xc(jΩ) is not faithfully copied when Ωs <
2Ωm. The terms in (8.4) overlap. The signal xc(t) is no longer recoverable
from xp(t). This effect, in which individual terms in equation (8.4) overlap
is called aliasing.
Discrete time processing of continuous time signal
a system for discrete time processing of continuous time
system
FIGURE 8.8
The over all system has xc(t) as input and yr(t) as output. We have the
following relations among the signals.