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Decimation by a Factor D 479
given by
¯x(n) △ =
{
x(n), n =0, ±D, ±2D, ...
0, elsewhere
(9.7)
Clearly, ¯x(n) can be viewed as a sequence obtained by multiplying x(n)
with a periodic train of impulses p(n), with period D, asillustrated in
Figure 9.4. The discrete Fourier series representation of p(n) is
p(n) △ =
{
1, n =0, ±D, ±2D, . . .
0, elsewhere.
= 1 D
D−1
∑
l=0
e j 2π D ln (9.8)
Hence we can write
¯x(n) =x(n)p(n) (9.9)
and
y(m) =¯x(mD) =x(mD)p(mD) =x(mD) (9.10)
x(n)
(a)
−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9
p(n)
n
(b)
−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9
n
x(n)
(c)
−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9
n
y(m)
(d)
−3 −2 −1 0 1 2 3
FIGURE 9.4 Operation of downsampling: (a) original signal x(n), (b) periodic
impulse train p(n) with period D =3, (c) multiplication of x(n) with p(n), and
(d) downsampled signal y(n)
m
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