Signals and Systems - Electrical Engineering

7.2 Uniform Sampling 423

n Equation (7.7) provides the relation between the continuous frequency(rad/sec) of x(t)and the discrete
frequencyω(rad) of the discrete-time signal x(nTs)or x[n]^1 :

ω=Ts [rad/sec]×[sec]=[rad]

Sampling a continuous-time signalx(t)at uniform times{nTs}gives a sampled signal

xs(t)=

∑

n

x(nTs)δ(t−nTs) (7.8)

or a sequence of samples{x(nTs)}. Sampling is equivalent to modulating the sampling signal

δTs(t)=

∑

n

δ(t−nTs) (7.9)

periodic of periodTs(the sampling period) withx(t).

IfX()is the Fourier transform ofx(t), the Fourier transform of the sampled signalxs(t)is given by the

equivalent expressions

Xs()=

1

Ts

∑

k

X(−ks)

=

∑

n

x(nTs)e−jTsn s=

2 π

Ts

(7.10)

Depending on the maximum frequency present in the spectrum of x(t)and on the chosen sampling frequency
s(or the sampling period Ts) it is possible to have overlaps when the spectrum of x(t)is shifted and added
to obtain the spectrum of the sampled signal. We have three possible situations:

n If the signal has a low-pass spectrum of finite support—that is, X()= 0 for ||> max (see
Figure 7.2(a)) wheremaxis the maximum frequency present in the signal—such a signal is called
band limited. As shown in Figure 7.2(b), for band-limited signals it is possible to choosesso that the
spectrum of the sampled signal consists of shifted nonoverlapping versions of( 1 /Ts)X(). Graphically (see
Figure 7.2(b)), this can be accomplished by lettings−max≥max, or

s≥ 2 max

which is called theNyquist sampling rate condition. As we will see later, in this case we are able
to recover X(), or x(t), from Xs()or from the sampled signal xs(t). Thus, the information in x(t)is
preserved in the sampled signal xs(t).
n On the other hand, if the signal x(t)is band limited but we lets< 2 max, then when creating Xs()
the shifted spectra of x(t)overlap (see Figure 7.2(c)). In this case, due to the overlap it will not be

(^1) To help the reader visualize the difference between a continuous-time signal, which depends on a continuous variablet, or a real
number, and a discrete-time signal, which depends on the integer variablen, we will use square brackets for these. Thus,η(t)is a
continuous-time signal, whileρ[n] is a discrete-time signal.