ECE 3250
THE DFT
Fall 2008
Realworld discretetime signal processing deals exclusively with finiteduration signals.
Infiniteduration signals are useful mathematically, but nobody has ever watched such a
signal play out in its entirety. When addressing signals and systems problems involving
infiniteduration signals, one’s goal is often to generate useful approximate results by
manipulating finiteduration signals in a computationally efficient way.
The DFT is a
tool that facilitates such manipulations. In essence, the DFT reduces a variety of signal
processing calculations to finitedimensional linear algebra.
For ease of exposition, I’ll assume that all the signals we’re dealing with here are
complexvalued.
This doesn’t cost us any generality, since a realvalued signal is just a
special kind of complexvalued signal.
Definition 1:
Given a positive natural number
N
, an
N
point signal
is a discretetime
signal
x
∈
C
Z
that satisfies
x
(
n
) = 0 for
n <
0 and
x
(
n
) = 0 for
n
≥
N
.
An
N
point signal is simply a finiteduration signal
x
whose “duration interval” is
contained in the range 0
≤
n < N
.
Observe that
x
(
n
) could still be zero for some
n

values in that range. In particular, if
M > N
, we can regard any
N
point signal
x
as an
M
point signal that just happens to satisfy
x
(
n
) = 0 for
N
≤
n < M
.
The important
thing to recognize is that an
N
point signal
x
is specified completely by
N
numbers
x
(0),
x
(1), . . . ,
x
(
N

1). In this way, the set of all
C
valued
N
point signals is in onetoone
correspondence with
C
N
, the set of all
N
vectors with entries in
C
.
If
x
is an
N
point
signal, I’ll denote by
x
the column
N
vector
ˆ
x
(0)
x
(1)
x
(2)
.
.
.
x
(
N

1)
˜
T
.
The set of all
N
point signals is closed under the taking of linear combinations in
C
Z
,
so it forms a vector space under the usual vector operations on
C
Z
.
It is easy to check
that those vector operations map nicely to the usual vector operations on
C
N
under the
correspondence I alluded to in the preceding paragraph. In other words, if
x
1
and
x
2
are
the vectors corresponding to
N
point signals
x
1
and
x
2
, then for any
c
1
and
c
2
in
C
the
vector
c
1
x
1
+
c
2
x
2
corresponds to the
N
point signal
c
1
x
1
+
c
2
x
2
.
I’ll be making considerable use of the notation
l
N
for
l
∈
Z
and nonzero
N
∈
N
.
Read that notation as “
l
mod
N
.”
It denotes the unique natural number
m
such that
0
≤
m < N
and
m
=
pN
+
l
for some
p
∈
Z
. Technically,
l
N
is the remainder you
obtain after dividing
l
by
N
.
For example,
7
5
= 2 whereas
15
5
= 0.
Note that
l
N
=
l
if and only if 0
≤
l < N
.
Definition 2:
Given a positive
N
∈
Z
and an
N
point signal
x
and any
n
o
with
0
≤
n
o
< N
, the
cyclic shift of
x
by
n
o
is the
N
point signal CShift
n
o
(
x
) with specification
CShift
n
o
(
x
)(
n
) =
x
(
n

n
o
N
) =
x
(
n

n
o
)
if
n
o
≤
n < N
x
(
N
+
n

n
o
)
if 0
≤
n < n
o
.
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 '04
 LIPSON
 Digital Signal Processing, Signal Processing, DFT, point signal, point signals

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