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Family of Transforms
Family of Transforms:
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Handout 1:
Fourier Circuit Analysis Chapter 18 W. Hayt Engineering Circuit
Analysis 8e
DSP Math 1
CH 2
Fourier Series:
Fourier Series
Periodic Signals
Signal Decomposition
(1 -> )
Domain after application: Time
Sinusoidal Components
Discrete/Line Spectrum
Family of Transforms
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HR Spring 15 DSP
sin cos
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DSP Math 1
CH 2
Fourier Series:
Family of Transforms
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HR Spring 15 DSP
sin cos
2 2 1
2 cos
2
DSP Math 1
CH 2
Fourier Series:
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Family of Transforms
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DSP Math 1
CH 2
Fourier Series:
Each of the three color channels of a picture can be realized as
a function
f(x,y) of two variables. These three functions red(x,y),
green(x,y),
blue(x,y) determine the picture. The animation shows the first
70 Fourier
approximations to a picture of dimensions 150 times 150
pixels.
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Family of Transforms
DSP Math 1
CH 2
Fourier Series:
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HR Spring 15 DSP
Family of Transforms
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DSP Math 1
CH 2
Fourier Series:
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HR Spring 15 DSP
Family of Transforms
DSP Math 1
CH 2
Fourier Series:
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HR Spring 15 DSP
Family of Transforms
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DSP Math 1
CH 2
Fourier Series:
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Family of Transforms
DSP Math 1
CH 2
Fourier Series Symmetries:
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Family of Transforms
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DSP Math 1
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Fourier Series:
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Family of Transforms
Assignment 1:
1. Compute Fourier
series of following
signals:
Due: Wednesday
28th Jan 2015,
3:00 p.m.
DSP Math 1
CH 2
Fourier Series:
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Family of Transforms
Assignment 2:
1. Write a program to plot Fourier components of a signal
similar to the animation given:
Due: Wednesday 4th Feb 2015, 03:00 p.m.
Signals time components
Resultant of component addition
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DSP Math 1
CH 2
Laplace Transform:
Laplace Transform
Aperiodic/Periodic Signals
Signal
Decomposition /Domain conversion
Domain after application: Frequency
Generalized Form of FT:
s = +j
Pole Zero Diagram
Magnitude response
Phase Plot
( ) ( ) stX s x t e dt
=
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Family of Transforms
DSP Math 1
CH 2
Laplace Transform:
( ) ( ) stX s x t e dt
=
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Family of Transforms
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DSP Math 1
CH 2
Laplace Transform:
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Family of Transforms
DSP Math 1
CH 2
Fourier Transform:
Fourier Transform
Aperiodic/Periodic Signals
Signal
Decomposition /
Domain conversion
Domain after application: Frequency
Specialized Form of LT:
s = j
Magnitude response
Phase Plot
( ) ( ) j tX x t e dt
=
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Family of Transforms
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DSP Math 1
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Fourier Transform:
( ) ( ) j tX x t e dt
=
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Family of Transforms
DSP Math 1
CH 2
DFT:
DFT
Discrete Counter Part of FT
Signal Decomposition
Domain conversion
Specialized Form of ZT: z = ej
Magnitude response
Phase Plot
FFT: Fast DFT (Computing Machines)
( ) ( ) kj nk nX x n e
=
=
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Family of Transforms
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DSP Math 1
CH 2
ZT:
ZT
Discrete Counter Part of LT
Signal Decomposition
Domain conversion
Generalized Form of DFT:
z = ej
Magnitude response
Phase Plot
Pole Zero Plot
( ) ( ) nn
X z x z z =
=
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Family of Transforms
DSP Math 1
CH 2
Transform Space:
LT - - - ZT
FT - - - DFT
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Family of Transforms
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DSP Math 1
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Fourier Transform
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DSP Math 1
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Fourier Transform:
Fourier Transform
"#$
%&'() *#%$ + , * % '-.,%/%
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Integral computation
Properties
Computation using properties
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1. Single sided exponential function
2. Double sided exponential function
3. Gate pulse
4. Periodic function: sinusoids
5. Constant
6. Unit impulse
7. Trapezoidal function
8. Train of impulses
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Fourier Transform
Fourier Transform:
DSP Math 1
CH 2
FT Computation of commonly used signals:
Single sided exponential function
Double sided exponential function
#$
#$
A
A
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Fourier Transform
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FT Computation of commonly used signals:
Gate Pulse #$
1
A
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Fourier Transform
DSP Math 1
CH 2
FT Computation of commonly used signals:
Sinusoids: #$
A
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Fourier Transform
