What is the relationship of the CTFT of
ft
f
t
to the DTFT of
f
s
n=ft
f
s
n
f
t
sampled at
nT
n
T
?

We will use the following notation. Note that we will use
different variables, ΩΩ
and ωω, so that we do not
get confused.
ft
CTFT
FiΩ
f
t
CTFT
F
Ω
fnT=
f
s
n
CTFT
F
s
eiω
f
n
T
f
s
n
CTFT
F
s
ω
ft=12π∫−∞∞FiΩeiωnd
n
f
t
1
2
n
F
Ω
ω
n

F
s
eiω=∑n=−∞∞
f
s
ne−(iωn)=∑n=−∞∞fnTe−(iωn)=∑n=−∞∞12π∫−∞∞FiΩeiΩnTdΩe−(iωn)=12π∫−∞∞FiΩ∑n=−∞∞ein(ΩT−ω)dΩ=12π∫−∞∞FiΩ2π∑r=−∞∞δω−(ΩT−2πr)dΩ=∫−∞∞FiΩ∑r=−∞∞δω−(ΩT−2πr)dΩ=∫−∞∞FiΩ∑r=−∞∞δ(−T)(Ω−ω−2πrT)dΩ
F
s
ω
n
f
s
n
ω
n
n
f
n
T
ω
n
n
1
2
Ω
F
Ω
Ω
n
T
ω
n
1
2
Ω
F
Ω
n
n
Ω
T
ω
1
2
Ω
F
Ω
2
r
δ
ω
Ω
T
2
r
Ω
F
Ω
r
δ
ω
Ω
T
2
r
Ω
F
Ω
r
δ
T
Ω
ω
2
r
T

(1)
Recall, property of

δt
δ
t
δaΩ=1|a|δΩ
δ
a
Ω
1
a
δ
Ω

(2)
F
s
eiω=∫−∞∞FiΩ∑r=−∞∞1TδΩ−ω−2πrTdΩ=1T∑r=−∞∞∫−∞∞FiΩδΩ−ω−2πrTdΩ=1T∑r=−∞∞Fiω−2πrT
F
s
ω
Ω
F
Ω
r
1
T
δ
Ω
ω
2
r
T
1
T
r
Ω
F
Ω
δ
Ω
ω
2
r
T
1
T
r
F
ω
2
r
T

(3)
F
s
eiω
F
s
ω
is a sum of shifted (and scaled)

Fiω
F
ω
's. In order to get to our final results in

Equation 3, we used the

sifting
property.