The Fourier Transform
Lecture 5
Why Fourier Transform?
f (t )
ω0, ω1, ω2……
F(ω)
numbers
t Correlation
(similarity measuring
device)
0 ω0 ω1 ω2 ω(Hz)
“spectrum analyzer”
(DC)
For a continuous-time, periodic signal, the frequency
content (spectrum) is displayed by computing the
Fourier series.
The Fourier series for periodic functions of time
produce discrete line spectra with non-zero values only
at specific frequencies referred to as harmonics.
Why Fourier Transform?
ω0, ω1, ω2……
f (t )
F(ω)
Correlation
(similarity-measuring
device)
t “spectrum analyzer” 0 ω (Hz)
The spectrum of a non-periodic, continuous-time signal is
displayed by computing the continuous-time Fourier
transform (FT).
The frequency spectra for these functions are continuous.
Why Fourier Transform?
In general, the Laplace and Fourier transforms are used to
convert certain difficult mathematical operations into easier
ones:
Applied to linear differential equations, transform them
into algebraic equations.
The Fourier transform is a mathematical autocorrelation
function (similarity-measuring) that compares an input time
domain signal to all frequencies, and generates the complex
equivalent of the level and phase for each frequency,
creating a frequency domain representation of the input time
domain signal.
When we do Fourier transform, we convert a signal into its
frequency components, so that we can have a better
analysis of that signal such as noise removal….
Definition
For a non-periodic signal, the Fourier transform or the
Fourier integral is defined as
The Inverse Fourier transform is defined as
The following notation is used to express the Fourier
transform and its inverse
Definition
The Fourier transform is, in general, a complex
function. It can be expressed as the sum of its real
and imaginary components, or in exponential form
as:
|F (ω)|: tells us how much the frequency
component ω is present over the signal,
ɸ(ω): tells us how much the contribution of this
frequency component is phase-shifted.
What is Phase Shift?
If two identical waveforms are not
aligned to each other along the x-axis,
then there is a phase shift.
Special Forms of the Fourier
Transform
The time function is, in general, complex. It can
be expressed as the sum of its real and
imaginary parts
The real and imaginary parts of F(ω) are
The real and imaginary parts of f(t) are
Special Forms of the Fourier
Transform
The table shows that if f (t ) is real: the real part of F(ω) is
even, and the imaginary part is odd. Then
It follows that:
Special Forms of the Fourier
Transform
The previous table states that:
Properties and Theorems of the Fourier
Transform
Fourier Transform Pairs of
Common Functions
The delta function:
Proof:
The sifting theorem of the delta function states that:
At t0 = 0
From Fourier transform definition:
Fourier Transform Pairs of
Common Functions
The time-shifted delta function:
Proof:
Similarly, from the sifting theorem
Therefore,
F t t 0 e j t dt e j t 0
Fourier Transform Pairs of
Common Functions
The unity function:
Proof:
Recalling that:
Applying the symmetry property:
Since δ(-ω)=δ(ω) (an even function), therefore:
Also, using the frequency shifting property:
Fourier Transform Pairs of
Common Functions
The cosine function:
Proof:
Since
And
e j 0t 2 0
Therefore, by adding both terms and dividing by 2, we
get the transform of the cosine function.
Fourier Transform Pairs of
Common Functions
The sine function:
Proof:
Similar to the proof of the cosine function.
Fourier Transform Pairs of
Common Functions
The signum function:
Proof:
We express the sgn(t) as an exponential that approaches a
limit
Therefore,
Fourier Transform Pairs of
Common Functions
The unit step function:
Proof:
The sgn(t) can be expressed as
Therefore, u0(t) is expressed as
Recalling that and
Therefore,
Fourier Transform Pairs of
Common Functions
One-sided exponential:
Proof:
From the Fourier transform of the unit step function,
and the frequency shifting property,
we reach the above relation.
Fourier Transform Pairs of
Common Functions
One-sided cosine function:
Proof:
The cosine function is first expressed as:
Using
and
We reach the above relation.
Fourier Transform Pairs of
Common Functions
One-sided sine function:
Proof:
The sine function is expressed as:
The proof then continues similar to the one-sided cosine
function.
Fourier Transform Pairs of Common
Functions
w
Summary: Fourier Transform of
Common Functions
The delta Function
The unity function
Summary: Fourier Transform of
Common Functions
The cosine and sine functions
Summary: Fourier Transform of
Common Functions
The signum function
The unit-step function
Fourier Transform of
Common Waveforms
Derive the Fourier transform of the rectangular pulse
Solution:
sinc function
Fourier Transform of
Common Waveforms
Derive the Fourier transform of the rectangular pulse
Solution:
Using the time-shifting property:
And from the previous example:
Fourier Transform of
Common Waveforms
Derive the Fourier transform of the function
f2
Solution:
f1
This is precisely the sum of the waveforms of the two previous
examples
Fourier Transform of
Common Waveforms
Derive the Fourier transform of the cosine-modulated
pulse
Solution:
Using the frequency shifting property:
since and
Therefore, the transform of the cosine and sine-modulated
signals is
Thus
Fourier Transform of
Common Waveforms
Derive the Fourier transform of the periodic train
function
Solution:
Fourier transform of a periodic train of equidistant delta
functions in the time domain, is a periodic train of equally
spaced delta functions in the frequency domain.
where ωo = 1/T
Finding the Fourier Transform
from Laplace Transform
The Fourier transform is nothing but
the Laplace transform evaluated on
the imaginary axis,
i.e. s = σ + j ω At σ = 0 s = jω.
Finding the Fourier Transform
from Laplace Transform
Non-integrable functions are difficult to compute
its Fourier transform -σt
e-σt e
sin function
Unit-step
Computing Fourier transform of f(t)e- σt u(t) is
equivalent to F(s)
Thus, Laplace transform is a generalized Fourier
transform for one-sided, exponentially weighted
functions.
Finding the Fourier Transform
from Laplace Transform
If a time function f(t) is zero for t ≤ 0, we can obtain
its Fourier transform from the one-sided Laplace
transform by substitution of s with jω .
If the time function f(t) = 0 for t ≥ 0, and f(t) ≠ 0 for t
< 0 , we use the substitution
Finding the Fourier Transform
from Laplace Transform
Example:
Solution:
Finding the Fourier Transform
from Laplace Transform
Example:
Solution:
t
𝑡, 𝑡≥0
𝑡 =
−𝑡, 𝑡<0
Finding the Fourier Transform
from Laplace Transform
Solution, cont.
f (t) f (-t)
The System Function
The system function and the impulse response
form the Fourier transform pair
If we know the impulse response h(t), we can
compute the response g(t) of any input f(t), by
multiplication of the Fourier transforms F(ω)
and H(ω) to obtain G(ω) . Then, we take the
Inverse Fourier transform of G(ω) to obtain the
response g(t).
The System Function
Example:
In a linear system with an impulse response
h(t), use the Fourier transform method to
compute the response g(t) when the input f(t) is
as shown.
The System Function
Solution:
The System Function
Solution, cont.
The System Function
Solution, cont.
x(t) y(t)
LTI
x(t - a) y(t - a)
Replacing t with (t-3)
The overall response is
The System Function
Example:
For the linear circuit shown, the input-output
relation is
Use the Fourier transform method to find the
output voltage.
The System Function
Solution:
Taking the Fourier transform of both sides of the equation and
recalling that
Therefore
The Fourier transform of the input is
The System Function
Solution, cont.
The Fourier transform of the output voltage is
By partial fractions expansion
The output response is
