Signals and Systems - A Software-Defined GPS and Galileo Receiver: A Single-Frequency Approach - page 9

S X (ω)

T

ω

FIGURE 1.6. Power spectral density S X (ω) of a random sequence of pulses.

where x is the sampled signal that con-

sists of a sequence of impulses separated in time by T s .Theterm

( t ) is the signal being sampled, and x δ ( t )

δ( t − nT s )

represents a delta function positioned at time t = nT s . The Fourier transform of

x δ ( t ) is

∞

X δ ( f ) = f s X ( f − nf s ).

n =−∞

Figures 1.7 and 1.8 show the sampling process in the time and frequency domain,
respectively.
Figure 1.8 reveals that if the sampling rate f s is lower that 2 B , then the fre-
quency-shifted components of X

overlap and the spectrum of the sampled
signal is not similar to the spectrum of the original signal x

( f )

( t ) . The spectral over-

lap effect is known as aliasing , and the sampling rate f s = 2 B is called the

Nyquist rate .
To avoid the effects of aliasing, we may use a lowpass anti-aliasing filter to
attenuate frequency components above B , (see Figure 1.8), and sample the signal
with a rate higher than the Nyquist rate, i.e., f s

> 2 B . We return to the issue of

sampling in Chapter 4.

1.3 Characterization of Systems

In the continuous-time domain, a system is a functional relationship between the
input signal x

( ) ( )

t and the output signal y t . The input-output relation of a system

may be denoted as
y f x ) , ( t 0 ) = ( t where −∞ < t , t 0 < ∞ . (1.17)