Digital Signal Processing Reference
In-Depth Information
X(e
j
ω
)
(a)
−
π
π
ω
Y(e
j
ω
)
(b)
−
π
π
ω
Fig. 5.7
Up-sampling by a factor of L = 3
By letting z = e
jx
,(
5.14
) becomes Y
ð
e
jw
Þ¼
X
ð
e
jwL
Þ;
which implies that the up-
sampled spectrum is compressed and repeated L times in the frequency range
½
p
;
p
:
This is illustrated in Fig.
5.7
for L = 3.
The spectrum repetition phenomenon which occurs in upsampling is called
''imaging'', as it contains L - 1 undesired replicas in the baseband. These images
need to be removed with an anti-image low-pass filter or ''interpolation filter''.
This low-pass filtering has the effect of ''interpolating'' the zero-valued samples
that have been inserted during the up-sampling process.
Fractional changes L/M of the sampling rate can be achieved by combining a
down-sampler with factor M, followed by an up-sampler with factor L. as will be
discussed later.
5.3 Sampling Rate Conversion Using Multirate Structures
5.3.1 Decimation
A decimator includes both a down-sampler and a lowpass filter to ensure that no
aliasing occurs when the sampling rate is changed. A block diagram of the deci-
mator is shown in Fig.
5.8
. The output of the decimator is at a lower sampling rate
Fig. 5.8
Decimator
u
1
n
)
H
(
z
)
M
x
(
n
)
y
(
n
)
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