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Interpolation by a Factor I 487
Solution The upsampled signal is v(m) ={1, 0, 2, 0, 3, 0, 4, 0}. Ifwenow delay x(n) by
↑
, 1, 2, 3, 4}. The corresponding upsampled signal
one sample, we get x(n−1) = {0
↑
is v 1(m) ={0
↑
, 0, 1, 0, 2, 0, 3, 0, 4, 0} = v(m − 2) and not v(m − 1). □
MATLAB Implementation MATLAB provides the function [v] =
upsample(x,I) that upsamples input array x into output v by inserting
(I-1) zeros between input samples. An optional third parameter,
“phase,” specifies the sample offset, which must be an integer between
0 and (I-1). For example,
>> x = [1,2,3,4]; v = upsample(x,3)
v =
1 0 0 2 0 0 3 0 0 4 0 0
upsamples by a factor of 2 starting with the first sample. However,
>> v = upsample(x,3,1)
v =
0 1 0 0 2 0 0 3 0 0 4 0
>> v = upsample(x,3,2)
v =
0 0 1 0 0 2 0 0 3 0 0 4
produces two different signals by upsampling, starting with the second
and the third sample (i.e., offset by 1), respectively. Note that the lengths
of the upsampled signals are I times the length of the original signal.
The frequency-domain representation of the upsampled signal
y(m) The sequence v(m) has a z-transform
∞∑
∞∑
V (z) = v(m)z −m = v(m)z −mI = X(z I ) (9.27)
m=−∞
m=−∞
The corresponding spectrum of v(m) isobtained by evaluating (9.27) on
the unit circle. Thus
V (ω y )=X(ω y I) (9.28)
where ω y denotes the frequency variable relative to the new sampling rate
F y (i.e., ω y =2πF/F y ). Now the relationship between sampling rates
is F y = IF x , and hence the frequency variables ω x and ω y are related
according to the formula
ω y = ω x
(9.29)
I
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