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