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494 Chapter 9 SAMPLING RATE CONVERSION
FIGURE 9.15
Ideal Resampler
x(n)
↑I
v(k) IDEAL
LPF
w(k)
↓D y(m)
h(k)
Rate: F x IF x IF x
I
F x = F y
D
Method for sampling rate conversion by a factor I/D
H(ω v ) other than to extract the fundamental period of W (ω v ). In this
respect, H(ω v ) acts as a lowpass filter as in the ideal interpolator. On
the other hand, if D/I > 1, then we have net decimation. Hence it is
necessary to first truncate even the fundamental period of W (ω v )toget
the frequency band down to [−π/D, π/D] and to avoid aliasing in the
decimation that follows. In this respect, H(ω v ) acts as a smoothing filter
in the ideal decimator. When D or I is equal to 1, the general decimator/interpolator
in Figure 9.15 along with (9.36) reduces to the ideal
interpolator or decimator as special case, respectively.
In the time domain, the output of the upsampler is the sequence
{
x(k/I), k =0, ±I,±2I,...
v(k) =
(9.37)
0, otherwise
and the output of the linear time-invariant filter is
∞∑
∞∑
w(k) = h(k − l)v(l) = h(k − lI)x(l) (9.38)
l=−∞
l=−∞
Finally, the output of the sampling rate converter is the sequence {y(m)},
which is obtained by downsampling the sequence {w(k)} by afactor of
D. Thus
∞∑
y(m) =w(mD) = h(mD − lI)x(l) (9.39)
l=−∞
It is illuminating to express (9.39) in a different form by making a
change in variable. Let ⌊ ⌋ mD
l = − n (9.40)
I
where the notation⌊r⌋ denotes the largest integer contained in r. With
this change in variable, (9.39) becomes
∞∑
( ⌊ ⌋ ) (⌊ ⌋ )
mD
mD
y(m) = h mD − I + nI x − n (9.41)
I
I
n=−∞
We note that
⌊ ⌋ mD
mD − I =(mD) modulo I =((mD)) I
I
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