(Re)Introduction to Integral

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Nov 24, 2013 (3 years and 4 months ago)

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(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
Where do you don’t want to be?

Next Lecture

Introduction to control theory