Aliasing in Time
We conclude that an original discrete time sequence with finite
duration
L
can be exactly recovered from its spectrum samples at
frequencies
ω
k
=
2
π
k
N
if
N
≥
L
.
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
20 / 193
Aliasing in Time
Example (7.1)
Consider the following signal
x
(
n
) =
a
n
u
(
n
)
,
0
<
a
<
1
(16)
Sample the spectrum at
ω
k
=
2
π
k
N
. Look at the effect of time
domain aliasing as N varies
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
21 / 193
Aliasing in Time
Example (7.1)
Recall that the DTFT of
x
(
n
)
is written as
X
(
ω
) =
1
1

ae
j
ω
(17)
Sampling the frequency domain at
ω
k
=
2
π
k
N
yields
X
(
k
) =
X
(
ω
)

ω
=
2
π
k
N
=
1
1

ae

j
2
π
k
N
(18)
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
22 / 193
Aliasing in Time
Example (7.1)
The periodic inverse of
X
(
k
)
,
˜
x
(
n
)
, is written as
˜
x
(
n
)
=
∞
X
s
=
∞
x
(
n

sN
)
=
∞
X
s
=
∞
a
n

sN
u
(
n

sN
)
(19)
It is clear that
u
(
n

sN
) =
0 for
s
>
n
/
N
and
u
(
n

sN
) =
1 for
s
≤
n
/
N
.
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
23 / 193
Aliasing in Time
Example (7.1)
We can rewrite (19) as
˜
x
(
n
)
=
0
X
s
=
∞
a
n

sN
=
a
n
0
X
s
=
∞
a

sN
=
a
n
∞
X
s
=
0
a
sN
=
a
n
1

a
N
,
0
≤
n
≤
N

1
(20)
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
24 / 193
Aliasing in Time
Example (7.1)
Where the factor
1
1

a
N
represents the effect of aliasing.
Below, we show the change in the aliasing factor for various
values of
N
,
a
=
0
.
8.
N
=
5
N
=
10
N
=
50
N
=
100
1
1

a
N
1
.
4874
1
.
1203
1
.
000014
1
(21)
Notice that the effect of aliasing tends toward zero as
N
→ ∞
.
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
25 / 193
The DFT
Consider again a finite sequence
x
(
n
)
with nonzero values at
0
≤
n
≤
L

1
Let
˜
x
(
n
)
be a periodic repetition of
x
(
n
)
with period
N
≥
L
such
that
˜
x
(
n
) =
∞
X
s
=
∞
x
(
n

sN
)
(22)
over a single period of
˜
x
(
n
)
we can write
˜
x
(
n
) =
x
(
n
)
,
0
≤
n
≤
L

1
0
,
L
≤
n
≤
N

1
(23)
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
26 / 193
(25)
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
27 / 193
The DFT
We can redefine
x
(
n
)
≡
˜
x
(
n
)
over a single period
This will require that the original signal of length
L
is padded by
N

L
zeros
We can then rewrite the DFT as
X
(
k
) =
N

1
X
n
=
0
x
(
n
)
e

j
2
π
kn
N
,
k
=
0
,
1
, . . . ,
N

1
(26)
where increasing the length does not change the information
contained in
X
(
k
)
because
x
(
n
) =
0 for
n
≥
L
This is call the Discrete Fourier Transform (DFT)
C. Williams & W. Alexander (NCSU)
THE DISCRETE FOURIER TRANSFORM
ECE 513, Fall 2019
28 / 193