Aliasing can occur when a signal with energy at frequencies other that (B,B)(B,B) is sampled at rate ωs<2Bωs<2B. Thus, when sampling below the Nyquist frequency, it is desirable to remove as much signal energy outside the frequency range (B,B)(B,B) as possible while keeping as much signal energy in the frequency range (B,B)(B,B) as possible. This suggests that the ideal lowpass filter with cutoff frequency ωs/2ωs/2 would be the optimal antialiasing filter to apply before sampling. While this is true, the ideal lowpass filter can only be approximated in real situations.
In order to demonstrate the importance of antialiasing filters, consider the calculation of the error energy between the original signal and its WhittakerShannon reconstruction from its samples taken with and without the use of an antialiasing filter. Let xx be the original signal and y=Gxy=Gx be the antialias filtered signal where GG is the ideal lowpass filter with cutoff frequency ωs/2ωs/2. It is easy to show that the reconstructed spectrum using no antialiasing filter is given by
X
˜
(
ω
)
=
T
s
X
s
(
T
s
ω
)

ω

<
ω
s
/
2
0
otherwise
=
∑
k
=

∞
∞
X
(
ω

k
ω
s
)

ω

<
ω
s
/
2
0
otherwise
.
X
˜
(
ω
)
=
T
s
X
s
(
T
s
ω
)

ω

<
ω
s
/
2
0
otherwise
=
∑
k
=

∞
∞
X
(
ω

k
ω
s
)

ω

<
ω
s
/
2
0
otherwise
.
(1)Thus, the reconstruction error spectrum for this case is
(
X

X
˜
)
(
ω
)
=

∑
k
=
1
∞
X
(
ω
+
k
ω
s
)
+
X
(
ω

k
ω
s
)

ω

<
ω
s
/
2
X
(
ω
)
otherwise
.
(
X

X
˜
)
(
ω
)
=

∑
k
=
1
∞
X
(
ω
+
k
ω
s
)
+
X
(
ω

k
ω
s
)

ω

<
ω
s
/
2
X
(
ω
)
otherwise
.
(2)Similarly, the reconstructed spectrum using the ideal lowpass antialiasing filter is given by
Y
˜
(
ω
)
=
Y
(
ω
)
=
X
(
ω
)

ω

<
ω
s
/
2
0
otherwise
.
Y
˜
(
ω
)
=
Y
(
ω
)
=
X
(
ω
)

ω

<
ω
s
/
2
0
otherwise
.
(3)Thus, the reconstruction error spectrum for this case is
(
X

Y
˜
)
(
ω
)
=
0

ω

<
ω
s
/
2
X
(
ω
)
otherwise
.
(
X

Y
˜
)
(
ω
)
=
0

ω

<
ω
s
/
2
X
(
ω
)
otherwise
.
(4)Hence, by Parseval's theorem, it follows that xy˜≤xx˜xy˜≤xx˜. Also note that the spectrum of Y˜Y˜ is identical to that of the original signal XX at frequencies ω∈(ωs/2,ωs/2).ω∈(ωs/2,ωs/2). This is graphically shown in Figure 1.
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