ContentsComplex numbers etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

IntroductionJean Baptiste Joseph Fourier (*1768-1830)French MathematicianLa Thorie Analitique de la Chaleur (1822) *

Fourier SeriesAny periodic function can be expressed as a sum of sines and/or cosinesFourier Series*

Fourier TransformEven functions that are not periodic have a finite area under curvecan be expressed as an integral of sines and cosines multiplied by a weighing functionBoth the Fourier Series and the Fourier Transform have an inverse operation:Original Domain Fourier Domain*

ContentsComplex numbers etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Complex numbersComplex number

Its complex conjugate*

Complex numbers polarComplex number in polar coordinates*

Eulers formula*Sin ()Cos ()??

*ReImVector

Complex mathComplex (vector) addition

Multiplication with I is rotation by 90 degrees*

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Unit impulse (Dirac delta function)Definition

Constraint

Sifting property

Specifically for t=0*

Discrete unit impulseDefinition

Constraint

Sifting property

Specifically for x=0*

What does this look like?Impulse train*T = 1

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Fourier Series

with*Series of sines and cosines, see Eulers formula

Fourier TransformContinuous Fourier Transform (1D) Inverse Continuous Fourier Transform (1D)

*

*Symmetry: The only difference between the Fourier Transform and its inverse is the sign of the exponential.

Fourier

Euler

Fourier and Euler*

If f(t) is real, then F() is complexF() is expansion of f(t) multiplied by sinusoidal termst is integrated over, disappearsF() is a function of only , which determines the frequency of sinusoidalsFourier transform frequency domain*

Examples Block 1*-W/2W/2A

Examples Block 2*

Examples Block 3*?

Examples Impulse*

Examples Shifted impulse*

Examples Shifted impulse 2*Real partImaginary partimpulseconstant

Examples - Impulse train*

Examples - Impulse train 2*

Intermezzo: Symmetry in the FT*

*

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Fourier + ConvolutionWhat is the Fourier domain equivalent of convolution?*

What is*

Intermezzo 1What is ?

Let , so *

Intermezzo 2

Property of Fourier Transform*

Fourier + Convolution contd*

Recapitulation 1Convolution in one domain is multiplication in the other domain

And (see book)

*

Recapitulation 2Shift in one domain is multiplication with complex exponential in the other domain

And (see book)*

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

SamplingSampled function can be written as

Obtain value of arbitrary sample k as*

Sampling - 2*

Sampling - 3*

Sampling - 4*

FT of sampled functionsFourier transform of sampled function

Convolution theorem

From FT of impulse train*(who?)

FT of sampled functions

*

Sifting property*

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Sampling theoremBand-limited function

Sampled functionlower value of 1/T would cause triangles to merge*

Sampling theorem 2Sampling theorem:If copy of can be isolated from the periodic sequence of copies contained in , can be completely recovered from the sampled version.Since is a continuous, periodic function with period 1/T, one complete period from is enough to reconstruct the signal.This can be done via the Inverse FT. *

Extracting a single period from that is equal to is possible if Sampling theorem 3*

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Aliasing

If , aliasing can occur *

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Discrete Fourier TransformFourier Transform of a sampled function is an infinite periodic sequence of copies of the transform of the original continuous function*

Discrete Fourier TransformContinuous transform of sampled function*

is discrete function is continuous and infinitely periodic with period 1/T *

We need only one period to characteriseIf we want to take M equally spaced samples from in the period = 0 to = 1/, this can be done thus *

Substituting

Into

yields*

ContentsComplex number etc.ImpulsesFourier Transform (+examples)Convolution theoremFourier Transform of sampled functionsSampling theoremAliasingDiscrete Fourier TransformApplication Examples*

Fourier Transform Table*

*

Formulation in 2D spatial coordinatesContinuous Fourier Transform (2D) Inverse Continuous Fourier Transform (2D) with angular frequencies*

ContentsFourier Transform of sine and cosine2D Fourier TransformProperties of the Discrete Fourier Transform

*

Eulers formula*

*Recall

*Recall

*Cos(t)Sin(t)1/21/2i

ContentsFourier Transform of sine and cosine2D Fourier TransformProperties of the Discrete Fourier Transform*

Formulation in 2D spatial coordinatesContinuous Fourier Transform (2D) Inverse Continuous Fourier Transform (2D) with angular frequencies*

Discrete Fourier TransformForward

Inverse*

Formulation in 2D spatial coordinatesDiscrete Fourier Transform (2D) Inverse Discrete Transform (2D) *

Spatial and Frequency intervalsInverse proportionality(Smallest) Frequency step depends on largest distance covered in spatial domainSuppose function is sampled M times in x, with step , distance is covered, which is related to the lowest frequency that can be measured *

Examples*

Fourier Series

with*Series of sines and cosines, see Eulers formula

Examples*

Periodicity2D Fourier Transform is periodic in both directions*

Periodicity2D Inverse Fourier Transform is periodic in both directions*

Fourier Domain*

Inverse Fourier DomainPeriodic?*Periodic!

ContentsFourier Transform of sine and cosine2D Fourier TransformProperties of the Discrete Fourier Transform*

Properties of the 2D DFT*

*RealImaginarySin (x)Sin (x + /2)Real

*RealImaginaryF(Cos(x))F(Cos(x)+k)Even

*RealOddSin (x)Sin(y)Sin (x)Imaginary

*RealImaginary(Sin (x)+1)(Sin(y)+1)

Symmetry: even and oddAny real or complex function w(x,y) can be expressed as the sum of an even and an odd part (either real or complex)*

PropertiesEven function

Odd function*

Properties - 2*

Consequences for the Fourier TransformFT of real function is conjugate symmetric

FT of imaginary function is conjugate antisymmetric*

*Im

*Re

*Re

*Im

FT of even and odd functionsFT of even function is real

FT of odd function is imaginary*

*RealImaginaryCos (x)Even

*RealImaginarySin (x)Odd

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