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Decimation by a Factor D 483
The output of the filter is a sequence v(n) given as
v(n) △ =
∞∑
h(k)x(n − k) (9.21)
k=0
which is then downsampled by the factor D to produce y(m). Thus,
y(m) =v(mD) =
∞∑
h(k)x(mD − k) (9.22)
k=0
Although the filtering operation on x(n) islinear and time invariant, the
downsampling operation in combination with the filtering results also in
a time-variant system.
The frequency-domain characteristics of the output sequence y(m) obtained
through the filtered signal v(n) can be determined by following the
analysis steps given before—i.e., by relating the spectrum of y(m) tothe
spectrum of the input sequence x(n). Using these steps, we can show that
or that
Y (z) = 1 D
Y (ω y )= 1 D
D−1
∑
k=0
H ( e −j2πk/D z 1/D) X ( e −j2πk/D z 1/D) (9.23)
D−1
∑
k=0
(
ωy − 2πk
H
D
)
X
( )
ωy − 2πk
D
(9.24)
With a properly designed filter H D (ω), the aliasing is eliminated and,
consequently, all but the first term in (9.24) vanish. Hence,
Y (ω y )= 1 D H D
( ωy
)
X
D
( ωy
)
D
= 1 (
D X ωy
)
D
(9.25)
for 0 ≤|ω y |≤π. The spectra for the sequences x(n), h(n), v(n), and
y(m) are illustrated in Figure 9.7.
MATLAB Implementation MATLAB provides the function y =
decimate(x,D) that resamples the sequence in array x at 1/D times
the original sampling rate. The resulting resampled array y is D times
shorter—i.e., length(y) = length(x)/D. Anideal lowpass filter given
in (9.20) is not possible in the MATLAB implementation; however, fairly
accurate approximations are used. The default lowpass filter used in the
function is an 8th-order Chebyshev type-I lowpass filter with the cutoff
frequency of 0.8π/D. Using additional optional arguments, the filter order
can be changed or an FIR filter of specified order and cutoff frequency
can be used.
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