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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
**************