Digital communication systems and filter banks - Signal Processing and Optimization for Transceiver Systems - page 76

3.2.2.B Notations for decimation and expansion

Since the transform-domain expressions for decimation are rather complicated,
we often abbreviate them as follows:
X ( e jω ) ↓M

or X ( e jω )

M− 1

1
M

=

X ( e j ( ω− 2 π ) /M ) ,

(3 . 14)

↓M

=0

X ( z ) ↓M

or X ( z )

M− 1

1
M

=

X ( z 1 /M W ) .

(3 . 15)

↓M

=0

j ω

X ( e )

(a)

ω

−π 0 π

images

j ω

Y ( e ) (b)

e

ω

π

−π 0 π/3

Figure 3.5 . Fourier transforms of (a) input to the expander and (b) output of the
expander for M =3 . In general there are M − 1 images.

j ω

X ( e )

(a)

ω

−2π 0 2π

shifted
version

stretched
version

shifted
version

j ω

Y ( e )

d

(b)

ω

−2π 0 2π

overlap or
aliasing

Figure 3.6 . Fourier transforms of (a) input to the decimator and (b) output of the
decimator for M =3 . In general there are M − 1 shifted versions of the stretched
version.