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Fourier andFourier andFourier TransformFourier Transform
Why should we learn Fourier Why should we learn Fourier Transform?Transform?
Born: 21 March 1768 in Auxerre, Bourgogne, FranceDied: 16 May 1830 in Paris, France
Joseph FourierJoseph Fourier
Joseph’s father was a tailor in AuxerreJoseph was the ninth of twelve childrenHis mother died when he was nine andhis father died the following year
Fourier demonstrated talent on mathat the age of 14.In 1787 Fourier decided to train for the priesthood - a religious life or a mathematical life?In 1793, Fourier joined the local Revolutionary Committee
Fourier’s “Controversy” WorkFourier’s “Controversy” Work
Fourier did his important mathematical work Fourier did his important mathematical work on the theory of heat (highly regarded on the theory of heat (highly regarded memoir memoir On the Propagation of Heat in Solid On the Propagation of Heat in Solid BodiesBodies ) from 1804 to 1807 ) from 1804 to 1807
This memoir received objection from This memoir received objection from Fourier’s mentors (Laplace and Lagrange) Fourier’s mentors (Laplace and Lagrange) and not able to be published until 1815and not able to be published until 1815
Napoleon awarded him a pension of 6000 francs, payable from 1 July, 1815. Napoleon awarded him a pension of 6000 francs, payable from 1 July, 1815. However Napoleon was defeated on 1 July and Fourier did not receive any moneyHowever Napoleon was defeated on 1 July and Fourier did not receive any money
Expansion of a FunctionExpansion of a Function
Example (Taylor Series)
constant
first-orderterm
second-orderterm
…
Fourier SeriesFourier Series
Fourier series make use of the orthogonality relationships of the sine and cosine functions
ExamplesExamples
Fourier TransformFourier Transform
The Fourier transform is a generalization of the The Fourier transform is a generalization of the complex Fourier series in the limit complex Fourier series in the limit
Fourier analysis = frequency domain analysis Fourier analysis = frequency domain analysis – Low frequency: sin(nx),cos(nx) with a small nLow frequency: sin(nx),cos(nx) with a small n– High frequency: sin(nx),cos(nx) with a large nHigh frequency: sin(nx),cos(nx) with a large n
Note that sine and cosine waves are infinitely long Note that sine and cosine waves are infinitely long – this is a shortcoming of Fourier analysis, which – this is a shortcoming of Fourier analysis, which explains why a more advanced tool, wavelet explains why a more advanced tool, wavelet analysis, is more appropriate for certain signalsanalysis, is more appropriate for certain signals
Applications of Fourier TransformApplications of Fourier Transform
PhysicsPhysics– Solve linear PDEs (heat conduction, Laplace, Solve linear PDEs (heat conduction, Laplace,
wave propagation)wave propagation)
Antenna designAntenna design– Seismic arrays, side scan sonar, GPS, SARSeismic arrays, side scan sonar, GPS, SAR
Signal processingSignal processing– 1D: speech analysis, enhancement …1D: speech analysis, enhancement …– 2D: image restoration, enhancement …2D: image restoration, enhancement …
Not Just for EENot Just for EE
Just like Calculus invented by Newton, Just like Calculus invented by Newton, Fourier analysis is another mathematical Fourier analysis is another mathematical tooltool
BIOM: fake iris detectionBIOM: fake iris detection CS: anti-aliasing in computer graphicsCS: anti-aliasing in computer graphics CpE: hardware and software systemsCpE: hardware and software systems
FT in BiometricsFT in Biometrics
natural fake
FT in CSFT in CS
Anti-aliasing in 3D graphic display
FT in CpEFT in CpE
Computer Engineering: The creative Computer Engineering: The creative application of engineering principles and application of engineering principles and methods to the design and development of methods to the design and development of hardware and softwarehardware and software systems systems
If the goal is to build faster computer alone If the goal is to build faster computer alone (e.g., Intel), you might not need FT; but as (e.g., Intel), you might not need FT; but as long as applications are involved, there is a long as applications are involved, there is a place for FT (e.g., Texas Instrument) place for FT (e.g., Texas Instrument)
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Frequency-Domain Analysis of InterpolationFrequency-Domain Analysis of Interpolation
Step-I: UpsamplingStep-I: Upsampling
Step-II: Low-pass filteringStep-II: Low-pass filtering Different interpolation schemes correspond Different interpolation schemes correspond
to different low-pass filtersto different low-pass filters
L/nTxL/nxnx ci
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Frequency Domain Representation of UpsamplingFrequency Domain Representation of Upsampling
k
Lkje wLXekxwX
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Frequency Domain Representation of InterpolationFrequency Domain Representation of Interpolation