Fundamentals - The MPEG - page 25

[ 4 ] Shannon, C.E., A mathematical theory of communication. Bell Syst. Tech. J. , 27 , 379 (1948)
[ 5 ] Jerri, A.J., The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc.
IEEE , 65 , 1565-1596 (1977)
[ 6 ] Whittaker, E.T., On the functions which are represented by the expansions of the interpolation theory. Proc. R.
Soc. Edinburgh , 181-194 (1915)
2.6 Reconstruction
Perfect reconstruction was theoretically demonstrated by Shannon as shown in Figure 2.10 . The input must be
band limited by an ideal linear- phase low-pass filter with a rectangular frequency response and a bandwidth of
one-half the sampling frequency. The samples must be taken at an instant with no averaging of the waveform.
These instantaneous samples can then be passed through a second, identical filter which will perfectly reconstruct
that part of the input waveform which was within the passband.
Figure 2.10: Shannon's concept of perfect reconstruction requires the hypothetical approach shown here. The anti-
aliasing and reconstruction filters must have linear phase and rectangular frequency response. The sample period
must be infinitely short and the sample clock must be perfectly regular. Then the output and input waveforms will
be identical if the sampling frequency is twice the input bandwidth (or more).
There are some practical difficulties in implementing Figure 2.10 exactly, but well-engineered systems can
approach it and so it forms a useful performance target. The impulse response of a linear-phase ideal low-pass
filter is a sin x / x waveform as shown in Figure 2.11 (a). Such a waveform passes through zero volts periodically. If
the cut-off frequency of the filter is one-half of the sampling rate, the impulse passes through zero at the sites of all
other samples . It can be seen from Figure 2.11 ( b) that at the output of such a filter, the voltage at the centre of a
sample is due to that sample alone, since the value of all other samples is zero at that instant. In other words the
continuous output waveform must pass through the tops of the input samples. In between the sample instants, the
output of the filter is the sum of the contributions from many impulses (theoretically an infinite number), causing the
waveform to pass smoothly from sample to sample.