The effect of sampling is that it extends the spectrum of X( f ) repeatedly on both sides
of the f-axis,as shown in Figure 4.12. When the sampling rate f
s
is selected to be greater
than 2 f
M
,i.e.,if f
M
f
s
=2,the spectrum X( f ) is preserved in X(F ) as shown in Figure
4.12(b). Therefore when f
s
! 2f
M
,we have
XF
1
T
X f for jFj
1
2
or j f j f
N
,
4:4:20
where f
N
f
s
=2 is called the Nyquist frequency. In this case,there is no aliasing,and the
spectrum of the discrete-time signal is identical (within the scale factor 1/T) to the
spectrum of the analog signal within the fundamental frequency range j f j f
N
or
jFj 1=2. The analog signal x(t) can be recovered from the discrete-time signal x(n)
by passing it through an ideal lowpass filter with bandwidth f
M
and gain T. This
fundamental result is the sampling theorem defined in (1.2.3). This sampling theorem
states that a bandlimited analog signal x(t) with its highest frequency (bandwidth) being
f
M
can be uniquely recovered from its digital samples, x(n),provided that the sampling
rate f
s
! 2f
M
.
However,if the sampling rate is selected such that f
s
< 2f
M
,the shifted replicas of
X( f ) will overlap in X(F),as shown in Figure 4.12(c). This phenomenon is called
aliasing,since the frequency components in the overlapped region are corrupted when
the signal is converted back to the analog form. As discussed in Section 1.1,we used an
analog lowpass filter with cut-off frequency less than f
N
before the A/D converter in
order to prevent aliasing. The goal of filtering is to remove signal components that may
corrupt desired signal components below f
N
. Thus the lowpass filter is called the
antialiasing filter.
Consider two sinewaves of frequencies f
1
1 Hz and f
2
5 Hz that are sampled at
f
s
4 Hz,rather than at 10 Hz according to the sampling theorem. The analog wave-
forms are illustrated in Figure 4.13(a),while their digital samples and reconstructed
waveforms are illustrated in Figure 4.13(b). As shown in the figures,we can reconstruct
the original waveform from the digital samples for the sinewave of frequency f
1
1 Hz.
However,for the original sinewave of frequency f
2
5 Hz,the reconstructed signal
is identical to the sinewave of frequency 1 Hz. Therefore f
1
and f
2
are said to be aliased to
one another,i.e.,they cannot be distinguished by their discrete-time samples.
In general,the aliasing frequency f
2
related to f
1
for a given sampling frequency f
s
can
be expressed as
f
2
if
s
Æ f
1
, i ! 1:
4:4:21
For example,if f
1
1 Hz and f
s
4 Hz,the set of aliased f
2
corresponding to f
1
is given
by
f
2
i Á 4 Æ 1, i 1,2,3, F F F
3,5,7,9, F F F
4:4:22
The folding phenomenon can be illustrated as the aliasing diagram shown in Figure
4.14. From the aliasing diagram,it is apparent that when aliasing occurs,aliasing
frequencies in x(t) that are higher than f
N
will fold over into the region 0 f f
N
.
156
FREQUENCY ANALYSIS