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Decimation by a Factor D 481
|X(ω x )|
A
−π
−π3
0
π3
π
ω x
|Y(ω y )|
A3
−3π −2π −π 0 π 2π 3π
ω y
FIGURE 9.5
Spectra of x(n) and y(m) in no-aliasing case
are related by
ω y = Dω x (9.16)
Thus, as expected, the frequency range 0 ≤|ω x |≤π/D is stretched into
the corresponding frequency range 0 ≤|ω y |≤π by the downsampling
process.
We conclude that the spectrum Y (ω y ), which is obtained by evaluating
(9.12) on the unit circle, can be expressed as 1
Y (ω y )= 1 D
D−1
∑
k=0
( )
ωy − 2πk
X
D
(9.17)
which is an aliased version of the spectrum X(ω x )ofx(n). To avoid aliasing
error, one needs the spectrum X(ω x )tobeless than full band or
bandlimited (note that this bandlimitedness is in the digital frequency
domain). In fact we must have
π
X(ω x )=0 for
D ≤|ω x|≤π (9.18)
Then,
Y (ω y )= 1 (
D X ωy
)
, |ω y |≤π (9.19)
D
and no aliasing error is present. An example of this for D =3isshown in
Figure 9.5.
1 In this chapter, we will make a slight change in our notation for the DTFT. We will use
X(ω) todenote the spectrum of x(n) instead of the previously used notation X(e jω ).
Although this change does conflict with the z-transform notation, the meaning should
be clear from the context. This change is made for the sake of clarity and visibility of
variables.
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