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The Fourier Transform
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Contents Complex numbers etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Introduction Jean Baptiste Joseph Fourier (*1768-†1830)
French Mathematician La Théorie Analitique de la Chaleur (1822)
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Fourier Series Fourier Series
Any periodic function can be expressed as a sum of sines and/or cosines Fourier Series
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Fourier Transform Even functions that
are not periodic have a finite area under curve can be expressed as an integral of sines and cosines multiplied by a weighing function Both the Fourier Series and the Fourier Transform have an inverse operation: Original Domain Fourier Domain
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Contents Complex numbers etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Complex numbers Complex number Its complex conjugate
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Complex numbers polar Complex number in polar coordinates
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Euler’s formula ? Sin (θ) ? Cos (θ)
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Vector Im Re
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Complex math Complex (vector) addition Multiplication with I
is rotation by 90 degrees
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Unit impulse (Dirac delta function)
Definition Constraint Sifting property Specifically for t=0
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Discrete unit impulse Definition Constraint Sifting property
Specifically for x=0
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Impulse train What does this look like? ΔT = 1
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Series of sines and cosines, see Euler’s formula
Fourier Series Periodic with period T with Series of sines and cosines, see Euler’s formula
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Fourier Transform Continuous Fourier Transform (1D)
Inverse Continuous Fourier Transform (1D)
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Symmetry: The only difference between the Fourier Transform and its inverse is the sign of the exponential.
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Fourier and Euler Fourier Euler
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If f(t) is real, then F(μ) is complex
F(μ) is expansion of f(t) multiplied by sinusoidal terms t is integrated over, disappears F(μ) is a function of only μ, which determines the frequency of sinusoidals Fourier transform frequency domain
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Examples – Block 1 A -W/2 W/2
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Examples – Block 2
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Examples – Block 3 ?
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Examples – Impulse constant
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Examples – Shifted impulse
Euler
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Examples – Shifted impulse 2
constant Real part Imaginary part
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Examples - Impulse train
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Examples - Impulse train 2
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Intermezzo: Symmetry in the FT
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Fourier + Convolution What is the Fourier domain equivalent of convolution?
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What is
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Intermezzo 1 What is ? Let , so
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Intermezzo 2 Property of Fourier Transform
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Fourier + Convolution cont’d
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Recapitulation 1 Convolution in one domain is multiplication in the other domain And (see book)
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Recapitulation 2 Shift in one domain is multiplication with complex exponential in the other domain And (see book)
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Sampling Sampled function can be written as
Obtain value of arbitrary sample k as
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Sampling - 2
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Sampling - 3
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Sampling - 4
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FT of sampled functions
Fourier transform of sampled function Convolution theorem From FT of impulse train (who?)
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FT of sampled functions
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Sifting property
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Sampling theorem Band-limited function Sampled function
lower value of 1/ΔT would cause triangles to merge
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Sampling theorem 2 Sampling 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/ΔT, one complete period from is enough to reconstruct the signal. This can be done via the Inverse FT.
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Sampling theorem 3 Extracting a single period from that is equal to is possible if Nyquist frequency
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Aliasing If , aliasing can occur
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Discrete Fourier Transform
Fourier Transform of a sampled function is an infinite periodic sequence of copies of the transform of the original continuous function
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Discrete Fourier Transform
Continuous transform of sampled function
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is discrete function is continuous and infinitely periodic with period /ΔT
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We need only one period to characterise
If we want to take M equally spaced samples from in the period μ = 0 to μ = 1/Δ, this can be done thus
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Substituting Into yields
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Contents Complex number etc. Impulses Fourier Transform (+examples)
Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples
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Fourier Transform Table
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Formulation in 2D spatial coordinates
Continuous Fourier Transform (2D) Inverse Continuous Fourier Transform (2D) with angular frequencies
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Contents Fourier Transform of sine and cosine 2D Fourier Transform
Properties of the Discrete Fourier Transform
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Euler’s formula
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Recall
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Recall
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1/2 1/2i Cos(ωt) Sin(ωt)
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Contents Fourier Transform of sine and cosine 2D Fourier Transform
Properties of the Discrete Fourier Transform
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Formulation in 2D spatial coordinates
Continuous Fourier Transform (2D) Inverse Continuous Fourier Transform (2D) with angular frequencies
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Discrete Fourier Transform
Forward Inverse
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Formulation in 2D spatial coordinates
Discrete Fourier Transform (2D) Inverse Discrete Transform (2D)
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Spatial and Frequency intervals
Inverse proportionality (Smallest) Frequency step depends on largest distance covered in spatial domain Suppose function is sampled M times in x, with step , distance is covered, which is related to the lowest frequency that can be measured
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Examples
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Series of sines and cosines, see Euler’s formula
Fourier Series Periodic with period T with Series of sines and cosines, see Euler’s formula
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Examples
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Periodicity 2D Fourier Transform is periodic in both directions
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Periodicity 2D Inverse Fourier Transform is periodic in both directions
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Fourier Domain
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Inverse Fourier Domain
Periodic! Periodic?
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Contents Fourier Transform of sine and cosine 2D Fourier Transform
Properties of the Discrete Fourier Transform
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Properties of the 2D DFT
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Real Real Imaginary Sin (x) Sin (x + π/2)
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Even Real Imaginary F(Cos(x)) F(Cos(x)+k)
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Odd Real Imaginary Sin (x)Sin(y) Sin (x)
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Real Imaginary (Sin (x)+1)(Sin(y)+1)
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Symmetry: even and odd Any real or complex function w(x,y) can be expressed as the sum of an even and an odd part (either real or complex)
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Properties Even function Odd function
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Properties - 2
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Consequences for the Fourier Transform
FT of real function is conjugate symmetric FT of imaginary function is conjugate antisymmetric
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Im
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Re
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Re
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Im
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FT of even and odd functions
FT of even function is real FT of odd function is imaginary
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Even Real Imaginary Cos (x)
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Odd Real Imaginary Sin (x)
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