In this module, we will derive an expansion for
arbitrary discrete-time functions, and in doing so, derive the Discrete Time Fourier Transform (DTFT).
Since complex
exponentials are
eigenfunctions of linear time-invariant (LTI)
systems, calculating the output of an LTI system
ℋℋ given
eiωn
ω
n
as an input amounts to simple multiplication, where
ω
0
=2πkN
ω
0
2
k
N
, and where
Hk∈C
H
k
is the eigenvalue corresponding to k. As shown in the figure, a simple exponential input would yield the output
yn=Hkeiωn
y
n
H
k
ω
n
(1)
Using this and the fact that ℋℋ
is linear, calculating
yn
y
n
for combinations of complex exponentials is also
straightforward.
c
1
ei
ω
1
n+
c
2
ei
ω
2
n→
c
1
H
k
1
ei
ω
1
n+
c
2
H
k
2
ei
ω
1
n
c
1
ω
1
n
c
2
ω
2
n
c
1
H
k
1
ω
1
n
c
2
H
k
2
ω
1
n
∑l
c
l
ei
ω
l
n→∑l
c
l
H
k
l
ei
ω
l
n
l
c
l
ω
l
n
l
c
l
H
k
l
ω
l
n
The action of HH on an input such
as those in the two equations above is easy to explain.
ℋℋ independently
scales each exponential component
ei
ω
l
n
ω
l
n
by a different complex number
H
k
l
∈C
H
k
l
. As such, if we can write a function
yn
y
n
as a combination of complex exponentials it allows us to easily calculate the output of a system.
Now, we will look to use the power of complex exponentials to see how we may
represent arbitrary signals in terms of a set of simpler functions by
superposition of a number of complex exponentials. Below we will present the
Discrete-Time Fourier Transform (DTFT). Because the
DTFT deals with nonperiodic signals, we must find a
way to include all real frequencies in the
general equations.
For the DTFT we simply utilize summation over all real numbers rather than
summation over integers in order to express the aperiodic signals.
It can be demonstrated that an arbitrary
Discrete Time-periodic function
fn
f
n
can be written as a linear combination of harmonic
complex sinusoids
fn=∑
k
=0N−1
c
k
ei
ω
0
kn
f
n
k
N
1
0
c
k
ω
0
k
n
(2)
where
ω
0
=2πN
ω
0
2
N
is the fundamental frequency. For almost all
fn
f
n
of practical interest, there exists
c
n
c
n
to make
Equation 2 true. If
fn
f
n
is finite energy (
fn∈L20N
f
n
L
0
N
2
), then the equality in
Equation 2
holds in the sense of energy convergence; with discrete-time signals, there are no concerns for divergence as there are with continuous-time signals.
The
c
n
c
n
- called the Fourier coefficients -
tell us "how much" of the sinusoid
ej
ω
0
kn
j
ω
0
k
n
is in
fn
f
n
.
The formula shows
fn
f
n
as a sum of complex exponentials, each of which is easily processed by an
LTI system (since it is an eigenfunction of
every LTI system). Mathematically,
it tells us that the set of
complex exponentials
∀
k
,k∈Z:ej
ω
0
kn
k
k
j
ω
0
k
n
form a basis for the space of N-periodic discrete
time functions.
Now, in order to take this useful tool and apply it to arbitrary non-periodic signals, we will have to delve deeper into the use of the superposition principle. Let
sT(t)sT(t)
be a periodic signal having period
TT.
We want to consider what happens to this signal's spectrum as the period goes to infinity. We denote the spectrum for any assumed value of the period by
cn(T)cn(T).
We calculate the spectrum according to the Fourier formula for a periodic signal, known as the Fourier Series (for more on this derivation, see the section on Fourier Series.)
c
n
=
1
T
∫
0
T
s
(
t
)
exp
(
-
ı
ω
0
t
)
d
t
c
n
=
1
T
∫
0
T
s
(
t
)
exp
(
-
ı
ω
0
t
)
d
t
(3)
where
ω0=2πTω0=2πT and where we have used a symmetric placement of the integration interval about the origin for subsequent derivational convenience. We vary the frequency index
nn proportionally as we increase the period. Define
S
T
(
f
)
≡
T
c
n
=
1
T
∫
0
T
(
S
T
(
f
)
exp
(
ı
ω
0
t
)
d
t
S
T
(
f
)
≡
T
c
n
=
1
T
∫
0
T
(
S
T
(
f
)
exp
(
ı
ω
0
t
)
d
t
(4)
making the corresponding Fourier Series
s
T
(
t
)
=
∑
-
∞
∞
f
(
t
)
exp
(
ı
ω
0
t
)
1
T
)
s
T
(
t
)
=
∑
-
∞
∞
f
(
t
)
exp
(
ı
ω
0
t
)
1
T
)
(5)
As the period increases, the spectral lines become closer together, becoming a continuum. Therefore,
lim
T
→
∞
s
T
(
t
)
≡
s
(
t
)
=
∫
-
∞
∞
S
(
f
)
exp
(
ı
ω
0
t
)
d
f
lim
T
→
∞
s
T
(
t
)
≡
s
(
t
)
=
∫
-
∞
∞
S
(
f
)
exp
(
ı
ω
0
t
)
d
f
(6)
with
S
(
f
)
=
∫
-
∞
∞
s
(
t
)
exp
(
-
ı
ω
0
t
)
d
t
S
(
f
)
=
∫
-
∞
∞
s
(
t
)
exp
(
-
ı
ω
0
t
)
d
t
(7)
ℱω=∑
n
=−∞∞fne−(iωn)
ℱ
ω
n
f
n
ω
n
(8)
fn=12π∫−ππℱωeiωnd
ω
f
n
1
2
ω
ℱ
ω
ω
n
(9)
It is not uncommon to see the above formula written slightly different. One of the most common differences is the way that the exponential is written. The above equations use the radial frequency variable
ω
ω in the exponential, where
ω=2πf
ω
2
f
, but it is also common to include the more explicit expression,
i2πft
2
f
t
, in the exponential.
Sometimes DTFT notation is expressed as
Feiω
F
ω
, to make it clear that it is not a CTFT (which is denoted as
FΩ
F
Ω
).
Click here for an
overview of the notation used in Connexion's DSP modules.
Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The discrete time Fourier transform synthesis formula expresses a discrete time, aperiodic function as the infinite sum of continuous frequency complex exponentials.
ℱω=∑
n
=−∞∞fne−(iωn)
ℱ
ω
n
f
n
ω
n
(10)
The discrete time Fourier transform analysis formula takes the same discrete time domain signal and represents the signal in the continuous frequency domain.
fn=12π∫−ππℱωeiωnd
ω
f
n
1
2
ω
ℱ
ω
ω
n
(11)
"My introduction to signal processing course at Rice University."