Th preceding discussions used traditional Fourier techniques to develop
sampling tools. If distributions or delta functions are allowed, the Fourier
transform will exist for a much larger class of signals. One should take care
when using distributions as if they were functions but it is a very powerful
extension.
There are several functions which have equally spaced sequences of impulses
that can be used as tools in deriving a sampling formula. These are called
``pitch fork" functions, picket fence functions, comb functions and shah
functions. We start first with a finite length sequence to be used with the
DFT. We define
⨿
M
(
n
)
=
∑
m
=
0
L
−
1
δ
(
n
−
M
m
)
⨿
M
(
n
)
=
∑
m
=
0
L
−
1
δ
(
n
−
M
m
)
(1)
where
N
=
L
M
N
=
L
M
.
D
F
T
{
⨿
M
(
n
)
}
=
∑
n
=
0
N
−
1
[
∑
m
=
0
L
−
1
δ
(
n
−
M
m
)
]
e
−
j
2
π
n
k
/
N
D
F
T
{
⨿
M
(
n
)
}
=
∑
n
=
0
N
−
1
[
∑
m
=
0
L
−
1
δ
(
n
−
M
m
)
]
e
−
j
2
π
n
k
/
N
=
∑
m
=
0
L
−
1
[
∑
n
=
0
N
−
1
δ
(
n
−
M
m
)
e
−
j
2
π
n
k
/
N
]
=
∑
m
=
0
L
−
1
[
∑
n
=
0
N
−
1
δ
(
n
−
M
m
)
e
−
j
2
π
n
k
/
N
]
=
∑
m
=
0
L
−
1
e
−
j
2
π
M
m
k
/
N
=
∑
m
=
0
L
−
1
e
−
j
2
π
m
k
/
L
=
∑
m
=
0
L
−
1
e
−
j
2
π
M
m
k
/
N
=
∑
m
=
0
L
−
1
e
−
j
2
π
m
k
/
L
=
{
L
<
k
>
L
=
0
0
otherwise
=
{
L
<
k
>
L
=
0
0
otherwise
=
L
∑
l
=
0
M
−
1
δ
(
k
−
L
l
)
=
L
⨿
L
(
k
)
=
L
∑
l
=
0
M
−
1
δ
(
k
−
L
l
)
=
L
⨿
L
(
k
)
(2)
For the DTFT we have a similar derivation:
D
T
F
T
{
⨿
M
(
n
)
}
=
∑
n
=
−
∞
∞
[
∑
m
=
0
L
−
1
δ
(
n
−
M
m
)
]
e
−
j
ω
n
D
T
F
T
{
⨿
M
(
n
)
}
=
∑
n
=
−
∞
∞
[
∑
m
=
0
L
−
1
δ
(
n
−
M
m
)
]
e
−
j
ω
n
=
∑
m
=
0
L
−
1
[
∑
n
=
−
∞
∞
δ
(
n
−
M
m
)
e
−
j
ω
n
]
=
∑
m
=
0
L
−
1
[
∑
n
=
−
∞
∞
δ
(
n
−
M
m
)
e
−
j
ω
n
]
=
∑
m
=
0
L
−
1
e
−
j
ω
M
m
=
∑
m
=
0
L
−
1
e
−
j
ω
M
m
=
{
∞
ω
=
k
2
π
/
M
0
otherwise
=
{
∞
ω
=
k
2
π
/
M
0
otherwise
=
K
∑
l
=
0
M
−
1
δ
(
ω
−
2
π
l
/
M
l
)
=
K
⨿
2
π
/
M
(
ω
)
=
K
∑
l
=
0
M
−
1
δ
(
ω
−
2
π
l
/
M
l
)
=
K
⨿
2
π
/
M
(
ω
)
(3)
where
K
K
is a constant.
An alternate derivation for the DTFT uses the inverse DTFT.
I
D
T
F
T
{
⨿
2
π
/
M
(
ω
)
}
=
1
2
π
∫
−
π
π
⨿
2
π
/
M
(
ω
)
e
j
ω
n
ⅆ
ω
I
D
T
F
T
{
⨿
2
π
/
M
(
ω
)
}
=
1
2
π
∫
−
π
π
⨿
2
π
/
M
(
ω
)
e
j
ω
n
ⅆ
ω
=
1
2
π
∫
−
π
π
∑
l
δ
(
ω
−
2
π
l
/
M
)
e
j
ω
n
ⅆ
ω
=
1
2
π
∫
−
π
π
∑
l
δ
(
ω
−
2
π
l
/
M
)
e
j
ω
n
ⅆ
ω
(4)
=
1
2
π
∑
l
∫
−
π
π
δ
(
ω
−
2
π
l
/
M
)
e
j
ω
n
ⅆ
ω
=
1
2
π
∑
l
∫
−
π
π
δ
(
ω
−
2
π
l
/
M
)
e
j
ω
n
ⅆ
ω
=
1
2
π
∑
l
=
0
M
−
1
e
2
π
l
n
/
M
=
{
M
/
2
π
n
=
M
0
otherwise
=
1
2
π
∑
l
=
0
M
−
1
e
2
π
l
n
/
M
=
{
M
/
2
π
n
=
M
0
otherwise
=
(
M
2
π
)
⨿
M
(
n
)
=
(
M
2
π
)
⨿
M
(
n
)
(5)
Therefore,
⨿
M
(
n
)
→
(
2
π
M
)
⨿
2
π
/
M
(
ω
)
⨿
M
(
n
)
→
(
2
π
M
)
⨿
2
π
/
M
(
ω
)
(6)
For regular Fourier transform, we have a string of impulse functions in both
the time and frequency. This we see from:
F
T
{
⨿
T
(
t
)
}
=
∫
−
∞
∞
∑
n
δ
(
t
−
n
T
)
e
−
j
ω
t
ⅆ
t
=
∑
n
∫
δ
(
t
−
n
T
)
e
−
j
ω
t
ⅆ
t
F
T
{
⨿
T
(
t
)
}
=
∫
−
∞
∞
∑
n
δ
(
t
−
n
T
)
e
−
j
ω
t
ⅆ
t
=
∑
n
∫
δ
(
t
−
n
T
)
e
−
j
ω
t
ⅆ
t
(7)
=
∑
n
e
−
j
ω
n
T
=
{
∞
ω
=
2
π
/
T
0
otherwise
=
∑
n
e
−
j
ω
n
T
=
{
∞
ω
=
2
π
/
T
0
otherwise
(8)
=
2
π
T
⨿
2
π
/
T
(
ω
)
=
2
π
T
⨿
2
π
/
T
(
ω
)
(9)
The multiplicative constant is found from knowing the result for a single
delta function.
These ``shah functions" will be useful in sampling signals in both the
continuous time and discrete time cases.
