(Re)Introduction to Integral
Transforms:
Fourier and Laplace
COMP417
Instructor: Philippe Giguère
Outline
•
Transforms in context of problem solving
•
Convolution
–
Dirac delta function
d
(t)
•
Fourier Transform
•
Sampling
•
Laplace Transform
Why use Transforms
?
•
Transforms are not simply math curiosity
sketched at the corner of a woodstove by
ol’ Frenchmen.
•
Way to reframe a problem in a way that
makes it easier to understand, analyze
and solve.
General Scheme using Transforms
Problem
Equation
of the problem
Solution
of the equation
Result
Transformation
Inverse
transformation
Transformed
equation
Solution of the
transformed equation
= HARD
= EASY
Which Transform to Use?
Application
Continuous
Domain
Discrete
Domain
Signal
Processing
Fourier T.
Discrete F.T.
(DFT/FFT)
Control Theory
Laplace T.
z

Transform
Typical Problem
•
Given an input signal
x(t)
, what is the output
signal
y(t)
after going through the system?
•
To solve it in the
time domain
(
t
) is
cumbersome!
System
/
Filter
t
x
(
t
)
y
(
t
)
?
Integrating Differential Equation?
•
Let’s have a simple first order low

pass filter
with resistor R and capacitor C:
•
The system is described by diff. eq.:
•
To find a solution, we can integrate.
Ugh!
)
(
)
(
)
(
'
t
x
t
y
t
RCy
Convolution
•
Math operator (symbol *) that takes two input
functions (
x(t)
and
h(t)
) and produces a third (
y(t)
)
•
Expresses the amount of overlap of one function
x(t)
as it is shifted over another function
h(t)
.
•
Way of «
blending
» one function with another.
d
h
t
x
t
h
t
x
t
y
)
(
)
(
)
(
)
(
)
(
“Frame

by

frame” convolution
movie
•
Can be visualized as flipping one function (
x
),
sliding it, and doing the dot product.
More convolution
examples
t
t
*
=
t
Smoothing
t
t
*
=
t
(Moving Average of GOOG)
(Gaussian Blur in 2D)
*
=
•
Convolution is heavily used in image
processing
Dirac Delta function
d
(t)
t
1
/
l
l
Area
=
1
t
Area
=
1
l
→
∞
Convolution
with Dirac delta
d
(t)
•
Convolving a signal with Dirac delta
d
(t)
simply yields the same signal.
movie
t
x
(
t
)
t
x
(
t
)
t
d
(
t
)
*
=
•
Convolving with a shifted delta
d
(t

)
shifts the
original signal: delay.
movie
t
x
(
t
)
t
x
(
t

)
t
d
(
t

)
*
=
Delay operator as convolution
with
d
(t

)
Convolution Properties
•
Convolution is a linear operation and therefore
has the typical linear properties:
–
Commutativity
–
Associativity
–
Distributivity
–
Scalar multiplication
Using Convolution to Solve
•
Again same first order low

pass filter:
•
The system is described by its impulse response:
•
Solution is convolution impulse resp. with
x(t)
t
t
*
=
t
t
Use a Convolution to Solve
•
Convolution is expensive to compute.
•
Little intuition about output signal
y(t).
t
t
*
=
?
t
t
*
=
?
t
t
*
=
?
Fourier Transform
Jean

Baptiste Fourier had crazy
idea (1807):
Any
periodic function
can be rewritten as a
weighted sum of
sines and cosines of
different frequencies.
Called Fourier Series
Square

Wave Deconstruction
Other examples
FT expands this idea
•
Take
any
signal (periodic and non

periodic) in
time
domain and decompose it in sines + cosines to have a
representation in the
frequency
domain.
t
f
f
Real
:
Cosine
Coefficients
+
Imaginary
:
Sine
Coefficients
FT
FT
Time Domain
Frequency Domain
FT: Formal Definition
•
Convention: Upper

case to describe transformed
variables:
•
Transform:
F
{ x(t) } = X(
w
)
or
X(f) (
w
=2
p
f)
•
Inverse:
F

1
{
Y(
w
)
or
Y(f)
}= y(t)
FT gives complex numbers
•
You get complex numbers
–
Cosine coefficients are real
–
Sine coefficients are imaginary
t
f
f
Real
:
Cosine
Coefficients
+
Imaginary
:
Sine
Coefficients
FT
Complex plane
•
Complex number can
be represented:
–
Combination of real +
imaginary value:
x +iy
–
Amplitude + Phase
A
and
j
Alternative representation of FT
•
Complex numbers can be represented also as
amplitude
+
phase
.
t
f
f
Real
+
Imaginary
OR
FT
f
f
Amplitude
+
Phase
p
p
Example Fourier Transform
t
FT
f
Amplitude
Spectrum
t
FT
f
Amplitude
Spectrum
Fast moving vs slow moving signals
Example Fourier Transform
Time Domain
t
Frequency Domain
w
Real
Real
Real
Example Fourier Transform
Example Fourier Transform
Example Fourier Transform
Example Fourier Transform
Note: FT is imaginary for sine
Example Fourier Transform
Time Domain
t
Frequency Domain
w
Real
Real
«
DC component
»
FT of Delay
d(
t
)
•
Amplitude + phase is easier to understand:
(click
movie
)
•
Amplitude:
–
Gives you information about frequencies/tones in a
signal.
•
Phase:
–
More about when it happens in time.
Important FT Properties
•
Addition
•
Scalar Multiplication
•
Convolution in time
t
•
Convolution in frequency
w
)
(
)
(
)}
(
)
(
{
w
w
B
A
t
b
t
a
F
)
(
)}
(
{
w
kA
t
ka
F
)
(
)
(
)}
(
{
)}
(
{
)}
(
)
(
{
w
w
H
X
t
h
F
t
x
F
t
h
t
x
F
)
(
)
(
2
)}
(
)
(
{
1
t
h
t
x
H
t
X
F
p
w
FT time
frequency duality
Time Domain
Frequency Domain
“narrow”
“wide”
“wide”
“narrow”
Multiplication
Convolution
Convolution
Multiplication
Box
Sinc
Sinc
Box
Gauss
Gauss
Real + Even
Real+Even (just cosine)
Real + Odd
Im + Odd (just sine)
Etc..
Etc..
FT: Reframing the problem in
Frequency Domain
Problem
x(t),h(t)
Solution
of the equation
Result
*
Fourier Transform
Inverse
Fourier
Transform
X(
w
), H(
w
)
X(
w
)H(
w
)
x
= HARD
= EASY
Completely sidesteps the convolution!
FT: Another Example
f
15
k
Hz
oscillator
Y
(
f
)

X
(
f
)

Multiplier
5
kHz

5
kHz
(
Voice
)
(
Carrier
)
What is the amplitude spectrum 
Y(f)
of a voice signal
(
bandlimited
to 5 kHz) when multiplied by a cosine
f
=15
kHz
?
(Note: this is Amplitude Modulation
AM radio)
FT: Solution

5
kHz
5
kHz
t
x
t
f

X
(
f
)

*
f
x
(
t
)
=
?
=
Remember
!
Convolving
with
d
(
f

f’
)
==
Shifting signal
f

Y
(
f
)


15
kHz

10
kHz

20
kHz
20
kHz
10
kHz
15
kHz
Time
Domain
Frequency
Domain

15
kHz
15
kHz
(Look Ma! No Algebra!)
FT Gaussian Blur
*
=
x
=
f
f
f
Frequency
Space
Sampling Theorem
•
In order to be used within a digital system, a
continuous
signal must be converted into a
stream of values.
•
Done by
sampling
the
continuous
signal at
regular intervals.
•
But at which interval?
Sampling Theorem
•
Sampling can be
thought of
multiplying a signal
by a
d
pulse train:
t
x
(
t
)
t
...
...
x
=
...
...
t
Aliasing
•
If sampling rate is
too small
compared with
frequency of signal,
aliasing
WILL occur:
t
t
...
...
=
t
...
...
=
t
...
t
...
...
t
x
x
Fourier Analysis of Sampling
•
The FT of a pulse train with frequency
f
s
is
another pulse train with interval
1/f
s
:
...
...
t
...
...
f
f
s
FT
1
/
f
s
Fourier Analysis of Sampling
•
Aliasing
will happen
if
f
s
<2
f
max
–
Nyquist frequency =
f
s
/2
...
t
1
/
f
s
...
f
f
f
max
t

f
max
x
*
=
f
f
f
f
s
>
2
f
max
f
s
f
s
=
2
f
max
f
s
<
2
f
max
Time
Domain
Frequency
Domain
A few sampling frequencies
•
Telephone systems: 8 kHz
•
CD music: 44.1 kHz
•
DVD

audio: 96 or 192 kHz
•
Aqua robot: 1 kHz
•
Digital Thermostat (HMTD84) : 0.2 Hz
Laplace Transform
•
Formal definition:
•
Compare this to FT:
•
Small differences:
–
Integral from 0 to
to for Laplace
•
f(t) for t<0 is not taken into account
–

s instead of

i
w
0
)
(
)
(
)]
(
[
dt
e
t
f
s
F
t
f
st
L
dt
e
t
f
F
t
i
w
w
)
(
)
(
Common Laplace Transfom
Name
f
(
t
)
F
(
s
)
Impulse
d
Step
Ramp
Exponential
Sine
1
s
1
2
1
s
a
s
1
2
2
w
w
s
1
)
(
t
f
t
t
f
)
(
at
e
t
f
)
(
)
sin(
)
(
t
t
f
w
0
0
0
1
)
(
t
t
t
f
Damped Sine
2
2
)
(
w
w
a
s
)
sin(
)
(
t
e
t
f
at
w
Laplace Transform Properties
•
Similar to Fourier transform:
–
Addition/Scaling
–
Convolution
•
Derivation
t
s
F
s
F
d
τ
(
τ
τ)f
(t
f
0
2
1
2
1
)
(
)
(
)
)
(
)
(
)]
(
)
(
[
2
1
2
1
s
bF
s
aF
t
bf
t
af
L
)
0
(
)
(
)
(
f
s
sF
t
f
dt
d
L
Transfer Function H(s)
•
Definition
–
H(s) = Y(s) / X(s)
•
Relates the output of a linear system (or
component) to its input.
•
Describes how a linear system responds
to an impulse.
•
All linear operations allowed
–
Scaling, addition, multiplication.
H
(
s
)
X
(
s
)
Y
(
s
)
RC Circuit Revisited
=
t
t
“step”
function
x
y
dt
dy
RC
RCs
1
1
s
1
*
x
s
RC
s
RCs
s
1
1
1
)
1
(
1
t

t
Time Domain
Laplace
Domain
Poles and Zeros
m
b
s
b
s
b
s
B
a
s
a
s
a
s
A
s
B
s
A
s
F
m
m
n
n
poles
#
system
of
Order
complex
are
zeros
and
Poles
0
A(s)
for which
s
of
values
the
are
Zeros
0
B(s)
for which
s
of
values
the
are
Poles
...
)
(
...
)
(
)
(
)
(
)
(
Given
0
1
0
1
Poles and Zeros
Zeros
No
,
are
Poles
)]
[sin(
:
sine
For
Zeros
No
is
Pole
1
]
[
example,
For
2
2
w
w
w
w
w
i
i
s
s
t
L
a
s
a
s
e
L
at
Poles and Zeros
Name
f
(
t
)
F
(
s
)
Impulse
d
Step
Ramp
Exponential
Sine
1
s
1
2
1
s
a
s
1
2
2
w
w
s
1
)
(
t
f
t
t
f
)
(
at
e
t
f
)
(
)
sin(
)
(
t
t
f
w
0
0
0
1
)
(
t
t
t
f
Damped Sine
2
2
)
(
w
w
a
s
)
sin(
)
(
t
e
t
f
at
w
Poles
0
0 (double)
n/a

a

i
w
,i
w

a

i
w
,

a+i
w
Poles and Zeros
•
If pole has:
•
Real negative: exponential decay
•
Real positive: exponential growth
•
If imaginary
0
:
oscillation of frequency
w
Effect of Poles Location
Im
(
s
)
Re
(
s
)
Growing
Sine
Damped
Sine
Exponential
Decay
Increased
Damping
Increased
Blow

up
Increasing
Frequency
Constant
Constant
Sine
Exponential
Increase
Im
(
s
)
Re
(
s
)
Growing
Sine
Damped
Sine
Exponential
Decay
Increased
Damping
Increased
Blow

up
Increasing
Frequency
Constant
Constant
Sine
Exponential
Increase
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