In several situations we would like to increase the data rate of a signal or,
to increase its length if it has finite length. This may be part of a multi
rate system or part of an interpolation process. Consider the process of
inserting
M
−
1
M
−
1
zeros between each sample of a discrete time signal.
y
(
n
)
=
{
x
(
n
/
M
)
<
n
>
M
=
0
(or n = kM)
0
otherwise
y
(
n
)
=
{
x
(
n
/
M
)
<
n
>
M
=
0
(or n = kM)
0
otherwise
(1)
For the finite length sequence case we calculate the DFT of the stretched or
up--sampled sequence by
C
s
(
k
)
=
∑
n
=
0
M
N
−
1
y
(
n
)
W
M
N
n
k
C
s
(
k
)
=
∑
n
=
0
M
N
−
1
y
(
n
)
W
M
N
n
k
(2)
C
s
(
k
)
=
∑
n
=
0
M
N
−
1
x
(
n
/
M
)
⨿
M
(
n
)
W
M
N
n
k
C
s
(
k
)
=
∑
n
=
0
M
N
−
1
x
(
n
/
M
)
⨿
M
(
n
)
W
M
N
n
k
(3)
where the length is now
N
M
N
M
and
k
=
0
,
1
,
⋯
,
N
M
−
1
k
=
0
,
1
,
⋯
,
N
M
−
1
.
Changing the index variable
n
=
M
m
n
=
M
m
gives:
C
s
(
k
)
=
∑
m
=
0
N
−
1
x
(
m
)
W
N
m
k
=
C
(
k
)
.
C
s
(
k
)
=
∑
m
=
0
N
−
1
x
(
m
)
W
N
m
k
=
C
(
k
)
.
(4)
which says the DFT of the stretched sequence is exactly the same as the DFT of
the original sequence but over
M
M
periods, each of length
N
N
.
For up--sampling an infinitely long sequence, we calculate the DTFT of the
modified sequence in
((Reference)) as
C
s
(
ω
)
=
∑
n
=
−
∞
∞
x
(
n
/
M
)
⨿
M
(
n
)
e
−
j
ω
n
=
∑
m
x
(
m
)
e
−
j
ω
M
m
C
s
(
ω
)
=
∑
n
=
−
∞
∞
x
(
n
/
M
)
⨿
M
(
n
)
e
−
j
ω
n
=
∑
m
x
(
m
)
e
−
j
ω
M
m
=
C
(
M
ω
)
=
C
(
M
ω
)
(5)
where
C
(
ω
)
C
(
ω
)
is the DTFT of
x
(
n
)
x
(
n
)
.
Here again the transforms of the up--sampled signal is the same as the
original signal except over
M
M
periods. This shows up here as
C
s
(
ω
)
C
s
(
ω
)
being a compressed version of
M
M
periods of
C
(
ω
)
C
(
ω
)
.
The z-transform of an up--sampled sequence is simply derived by:
Y
(
z
)
=
∑
n
=
−
∞
∞
y
(
n
)
z
−
n
=
∑
n
x
(
n
/
M
)
⨿
M
(
n
)
z
−
n
=
∑
m
x
(
m
)
z
−
M
m
Y
(
z
)
=
∑
n
=
−
∞
∞
y
(
n
)
z
−
n
=
∑
n
x
(
n
/
M
)
⨿
M
(
n
)
z
−
n
=
∑
m
x
(
m
)
z
−
M
m
(6)
=
X
(
z
M
)
=
X
(
z
M
)
(7)
which is consistent with a complex version of the DTFT in
(
Equation 5).
Notice that in all of these cases, there is no loss of information or
invertibility. In other words, there is no aliasing.
