Page 81 - DSP Integrated Circuits

66 Chapter 3 Digital Signal Processing

It is well known [6,15, 32, 40] that the Fourier transform of the sequence x(n)
that is obtained by sampling, i.e., x(n) = x a(t) for t - nT, is related to the Fourier
transform of the analog signal x a(t) by Poisson's summation formula:

Figure 3.2 illustrates
that the spectrum of the
digital signal consists of
repeated images of its ana-
log spectrum. For the sake
of simplicity we have
assumed that all spectra
are real. Figure 3.2 also
shows that these images
will overlap if the analog
signal is not bandlimited
or if the sample frequency
is too low. In that case, the
analog signal cannot be
recovered. Figure 3.2 Analog and digital spectra obtained by
sampling
If the images do not
overlap, i.e., if x n(t) is
bandlimited such that \X a(o)}\ = 0 for I col > COQ, then the analog signal can be
reconstructed by filtering with an ideal analog lowpass filter with cutoff frequency
co c = COQ < n/T. We get, after filtering,

THEOREM 3.1—The Nyquist Sampling Theorem
If an analog signal, x a(t), is bandlimited so that \X a(co) I = 0 for I co\ > COQ,
then all information is retained in the sequence x(nT), obtained by peri-
odic sampling x a(t) at t = nT where T < K/COQ. Furthermore, the analog
signal can, in principle, be reconstructed from the sequence x(nT} by lin-
ear filtering.

Information will be lost if the requirements given by the sampling theorem
are not met [15, 17]. For example, Figure 3.3 shows the spectrum for an analog
sinusoidal signal with a frequency COQ that is larger than n/T. The spectrum for the
sampled signal will appear to have come from sampling an analog signal of fre-
quency 2n/T— COQ. This phenomenon is called aliasing or folding.
Figure 3.4 shows another example of an analog signal that is not bandlimited.
Notice that the high frequency content of the analog signal is folded and that the
spectrum of the sampled signal is distorted. In this case, the original analog signal