In this module, we will derive an expansion for any arbitrary
continuous-time function, and in doing so, derive the Continuous Time Fourier Transform (CTFT).
Since complex
exponentials are
eigenfunctions of linear time-invariant (LTI)
systems, calculating the output of an LTI system
ℋℋ given
est
s
t
as an input amounts to simple multiplication, where
Hs∈C
H
s
is the eigenvalue corresponding to s. As shown in the figure, a simple exponential input would yield the output
yt=Hsest
y
t
H
s
s
t
(1)
Using this and the fact that ℋℋ
is linear, calculating
yt
y
t
for combinations of complex exponentials is also
straightforward.
c
1
e
s
1
t+
c
2
e
s
2
t→
c
1
H
s
1
e
s
1
t+
c
2
H
s
2
e
s
2
t
c
1
s
1
t
c
2
s
2
t
c
1
H
s
1
s
1
t
c
2
H
s
2
s
2
t
∑n
c
n
e
s
n
t→∑n
c
n
H
s
n
e
s
n
t
n
c
n
s
n
t
n
c
n
H
s
n
s
n
t
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
e
s
n
t
s
n
t
by a different complex number
H
s
n
∈C
H
s
n
. As such, if we can write a function
ft
f
t
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
Continuous-Time Fourier Transform (CTFT), commonly
referred to as just the Fourier Transform (FT). Because the
CTFT deals with nonperiodic signals, we must find a
way to include all real frequencies in the
general equations.
For the CTFT we simply utilize integration over real numbers rather than
summation over integers in order to express the aperiodic signals.
Joseph
Fourier demonstrated that an arbitrary
st
s
t
can be written as a linear combination of harmonic
complex sinusoids
st=∑
n
=−∞∞
c
n
ej
ω
0
nt
s
t
n
c
n
j
ω
0
n
t
(2)
where
ω
0
=2πT
ω
0
2
T
is the fundamental frequency. For almost all
st
s
t
of practical interest, there exists
c
n
c
n
to make
Equation 2 true. If
st
s
t
is finite energy (
st∈L20T
s
t
L
0
T
2
), then the equality in
Equation 2
holds in the sense of energy convergence; if
st
s
t
is continuous, then
Equation 2 holds
pointwise. Also, if
st
s
t
meets some mild conditions (the Dirichlet
conditions), then
Equation 2 holds
pointwise everywhere except at points of discontinuity.
The
c
n
c
n
- called the Fourier coefficients -
tell us "how much" of the sinusoid
ej
ω
0
nt
j
ω
0
n
t
is in
st
s
t
.
The formula shows
st
s
t
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
∀
n
,n∈Z:ej
ω
0
nt
n
n
j
ω
0
n
t
form a basis for the space of T-periodic continuous
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=∫−∞∞fte−(iΩt)d
t
ℱ
Ω
t
f
t
Ω
t
(8)
ft=12π∫−∞∞ℱΩeiΩtd
Ω
f
t
1
2
Ω
ℱ
Ω
Ω
t
(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.
Click here for an
overview of the notation used in Connexion's DSP modules.
We know from Euler's formula that cos(ωt)+sin(ωt)=1-j2ejωt+1+j2e-jωt.cos(ωt)+sin(ωt)=1-j2ejωt+1+j2e-jωt.
Find the Fourier Transform (CTFT) of the function
ft={e−(αt) if t≥00 otherwise
f
t
α
t
t
0
0
(10)
In order to calculate the Fourier transform, all we need
to use is Equation 8, complex exponentials,
and basic calculus.
ℱΩ=∫−∞∞fte−(iΩt)d
t
=∫0∞e−(αt)e−(iΩt)d
t
=∫0∞e(−t)(α+iΩ)d
t
=0−-1α+iΩ
ℱ
Ω
t
f
t
Ω
t
t
0
α
t
Ω
t
t
0
t
α
Ω
0
-1
α
Ω
(11)
ℱΩ=1α+iΩ
ℱ
Ω
1
α
Ω
(12)
Find the inverse Fourier transform of the ideal lowpass filter
defined by
XΩ={1 if |Ω|≤M0 otherwise
X
Ω
1
Ω
M
0
(13)
Here we will use Equation 9 to
find the inverse FT given that
t≠0
t
0
.
xt=12π∫−MMeiΩtd
Ω
=12πeiΩt|
Ω
,
Ω
=
eiw
=1πtsinMt
x
t
1
2
Ω
M
M
Ω
t
Ω
w
1
2
Ω
t
1
t
M
t
(14)
xt=Mπ(sincMtπ)
x
t
M
sinc
M
t
(15)
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 continuous time Fourier series synthesis formula expresses a continuous time, periodic function as the sum of continuous time, discrete frequency complex exponentials.
ft=∑
n
=−∞∞
c
n
ej
ω
0
nt
f
t
n
c
n
j
ω
0
n
t
(16)
The continuous time Fourier series analysis formula gives the coefficients of the Fourier series expansion.
c
n
=1T∫0Tfte−(j
ω
0
nt)d
t
c
n
1
T
t
T
0
f
t
j
ω
0
n
t
(17)
In both of these equations
ω
0
=2πT
ω
0
2
T
is the fundamental frequency.
Properties of the Fourier Transform
Table of Common Fourier Transforms
"My introduction to signal processing course at Rice University."