Digital communication systems and filter banks - Signal Processing and Optimization for Transceiver Systems - page 84

(a) input

x(n)

n

01

(b)

expander
output

s(n)

n

-6 -3 0123456 7

(c)

filter

h(n)

6

-6

n

-3 01 23

(d)

interpolated
output

y(n)

n

0123456 7

-6 -3

Figure 3.14 . Operation of the interpolation filter in the time domain. (a) Input
signal, (b) output of the expander, (c) impulse response of the interpolation filter, and
(d) the final interpolated output.

3.4.2 The Nyquist(M) property

Note that the sinc impulse response satisfies the property

h ( Mn )= δ ( n ) (Nyquist( M ) property). (3 . 20)

That is, it has regular zero crossings at nonzero multiples of M ,andmoreover
h (0) = 1 . Afilter H ( z ) satisfying this property is called a Nyquist( M )filteror
just a Nyquist filter . We will show that this property ensures that the original
samples x ( n ) of the input are retained by the interpolation filter without dis-
tortion, as shown by the thick lines in Fig. 3.14(d). More precisely, we will
show

y ( Mn )= x ( n ) .