With some initial discussion of the process illustrated in Figure 1 complete, the relationship between the continuous time, linear time invariant filter H1H1 and the discrete time, linear time invariant filter H2H2 can be explored. We will assume the use of ideal, infinite precision ADCs and DACs that perform sampling and perfect reconstruction, respectively, using a sampling rate ωs=2π/Ts≥2Bωs=2π/Ts≥2B where the input signal xx is bandlimited to (B,B)(B,B). Note that these arguments fail if this condition is not met and aliasing occurs. In that case, preapplication of an antialiasing filter is necessary for these arguments to hold.
Recall that we have already calculated the spectrum XsXs of the samples xsxs given an input xx with spectrum XX as
X
s
(
ω
)
=
1
T
s
∑
k
=

∞
∞
X
ω

2
π
k
T
s
.
X
s
(
ω
)
=
1
T
s
∑
k
=

∞
∞
X
ω

2
π
k
T
s
.
(1)Likewise, the spectrum YsYs of the samples ysys given an output yy with spectrum YY is
Y
s
(
ω
)
=
1
T
s
∑
k
=

∞
∞
Y
ω

2
π
k
T
s
.
Y
s
(
ω
)
=
1
T
s
∑
k
=

∞
∞
Y
ω

2
π
k
T
s
.
(2)From the knowledge that ys=(H1x)s=H2(xs)ys=(H1x)s=H2(xs), it follows that
∑
k
=

∞
∞
H
1
ω

2
π
k
T
s
X
ω

2
π
k
T
s
=
H
2
(
ω
)
∑
k
=

∞
∞
X
ω

2
π
k
T
s
.
∑
k
=

∞
∞
H
1
ω

2
π
k
T
s
X
ω

2
π
k
T
s
=
H
2
(
ω
)
∑
k
=

∞
∞
X
ω

2
π
k
T
s
.
(3)Because XX is bandlimited to (π/Ts,π/Ts)(π/Ts,π/Ts), we may conclude that
H
2
(
ω
)
=
∑
k
=

∞
∞
H
1
ω

2
π
k
T
s
u
ω

(
2
k

1
)
π

u
ω

(
2
k
+
1
)
π
.
H
2
(
ω
)
=
∑
k
=

∞
∞
H
1
ω

2
π
k
T
s
u
ω

(
2
k

1
)
π

u
ω

(
2
k
+
1
)
π
.
(4)More simply stated, H2H2 is 2π2π periodic and H2(ω)=H1(ω/Ts)H2(ω)=H1(ω/Ts) for ω∈[π,π)ω∈[π,π).
Given a specific continuous time, linear time invariant filter H1H1, the above equation solves the system design problem provided we know how to implement H2H2. The filter H2H2 must be chosen such that it has a frequency response where each period has the same shape as the frequency response of H1H1 on (π/Ts,π/Ts)(π/Ts,π/Ts). This is illustrated in the frequency responses shown in Figure 2.
We might also want to consider the system analysis problem in which a specific discrete time, linear time invariant filter H2H2 is given, and we wish to describe the filter H1H1. There are many such filters, but we can describe their frequency responses on (π/Ts,π/Ts)(π/Ts,π/Ts) using the above equation. Isolating one period of H2(ω)H2(ω) yields
the conclusion that H1(ω)=H2(ωTs)H1(ω)=H2(ωTs) for ω∈(π/Ts,π/Ts)ω∈(π/Ts,π/Ts). Because xx was assumed to be bandlimited to (π/T,π/T)(π/T,π/T), the value of the frequency response elsewhere is irrelevant.
"My introduction to signal processing course at Rice University."