Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 395

used in recording music on a compact disc (CD). Finally, Section 9.5 con-
cludes our discussion with a summary of the key concepts introduced in the
chapter.

9.1 Ideal impulse-t rain sampling

In this section, we consider sampling of a CT signal x ( t ) with a bounded CTFT
X ( ω ) such that

X ( ω ) = 0 for ω > 2 πβ. (9.1)

A CT signal x ( t ) satisfying Eq. (9.1) is referred to as a baseband signal, which
is band-limited to 2 πβ radians/s or β Hz. In the following discussion, we prove
that a baseband signal x ( t ) can be transformed into a DT sequence x [ k ] with
no loss of information if the sampling interval T s satisfies the criterion that
T s ≤ 1 / 2 β .
To derive the DT version of the baseband signal x ( t ), we multiply x ( t )byan
impulse train:

k =−∞ δ ( t − kT s ) ,

∞

s ( t ) = (9.2)

where T s denotes the separation between two consecutive impulses and is called
the sampling interval. Another related parameter is the sampling rate ω s , with
units of radians/s, which is defined as follows:

= 2 π
T s

ω s . (9.3)

Mathematically, the resulting sampled signal, x s ( t ) = x ( t ) s ( t ), is given by

k =−∞ δ ( t − kT s ) =

∞

k =−∞ x ( kT s ) δ ( t − kT s ) .

∞

x s ( t ) = x ( t ) (9.4)

Figure 9.2 illustrates the time-domain representation of the process of the
impulse-train sampling. Figure 9.2(a) shows the time-varying waveform repre-
senting the baseband signal x ( t ). In Figs. 9.2(b) and (c), we plot the sampled
signal x s ( t ) for two different values of the sampling interval. In Fig. 9.2(b), the
sampling interval T s = T and the sampled signal x s ( t ) provides a fairly good
approximation of x ( t ). In Fig. 9.2(c), the sampling interval T s is increased to
2 T . With T s set to a larger value, the separation between the adjacent samples
in x s ( t ) increases. Compared to Fig. 9.2(b), the sampled signal in Fig. 9.2(c)
provides a coarser representation of x ( t ). The choice of T s therefore determines
how accurately the sampled signal x s ( t ) represents the original CT signal x ( t ).
To determine the optimal value of T s , we consider the effect of sampling in the
frequency domain.