The Fourier Transform

Contents Complex numbers etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Introduction Jean Baptiste Joseph Fourier (*1768-†1830) French Mathematician La Théorie Analitique de la Chaleur (1822)

Fourier Series Fourier Series Any periodic function can be expressed as a sum of sines and/or cosines Fourier Series

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

Contents Complex numbers etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Complex numbers Complex number Its complex conjugate

Complex numbers polar Complex number in polar coordinates

Euler’s formula ? Sin (θ) ? Cos (θ)

Vector Im Re

Complex math Complex (vector) addition Multiplication with I is rotation by 90 degrees

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Unit impulse (Dirac delta function) Definition Constraint Sifting property Specifically for t=0

Discrete unit impulse Definition Constraint Sifting property Specifically for x=0

Impulse train What does this look like? ΔT = 1

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

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

Fourier Transform Continuous 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 and Euler Fourier Euler

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

Examples – Block 1 A -W/2 W/2

Examples – Block 2

Examples – Block 3 ?

Examples – Impulse constant

Examples – Shifted impulse Euler

Examples – Shifted impulse 2 constant Real part Imaginary part

Examples - Impulse train

Examples - Impulse train 2

Intermezzo: Symmetry in the FT

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Fourier + Convolution What is the Fourier domain equivalent of convolution?

What is

Intermezzo 1 What is ? Let , so

Intermezzo 2 Property of Fourier Transform

Fourier + Convolution cont’d

Recapitulation 1 Convolution in one domain is multiplication in the other domain And (see book)

Recapitulation 2 Shift in one domain is multiplication with complex exponential in the other domain And (see book)

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Sampling Sampled function can be written as Obtain value of arbitrary sample k as

Sampling - 2

Sampling - 3

Sampling - 4

FT of sampled functions Fourier transform of sampled function Convolution theorem From FT of impulse train (who?)

FT of sampled functions

Sifting property

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Sampling theorem Band-limited function Sampled function lower value of 1/ΔT would cause triangles to merge

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.

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

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Aliasing If , aliasing can occur

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

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

Discrete Fourier Transform Continuous transform of sampled function

is discrete function is continuous and infinitely periodic with period 1/ΔT

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

Substituting Into yields

Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

Fourier Transform Table

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

Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform

Euler’s formula

Recall

Recall

1/2 1/2i Cos(ωt) Sin(ωt)

Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform

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

Discrete Fourier Transform Forward Inverse

Formulation in 2D spatial coordinates Discrete Fourier Transform (2D) Inverse Discrete Transform (2D)

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

Examples

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

Examples

Periodicity 2D Fourier Transform is periodic in both directions

Periodicity 2D Inverse Fourier Transform is periodic in both directions

Fourier Domain

Inverse Fourier Domain Periodic! Periodic?

Contents Fourier Transform of sine and cosine 2D Fourier Transform Properties of the Discrete Fourier Transform

Properties of the 2D DFT

Real Real Imaginary Sin (x) Sin (x + π/2)

Even Real Imaginary F(Cos(x)) F(Cos(x)+k)

Odd Real Imaginary Sin (x)Sin(y) Sin (x)

Real Imaginary (Sin (x)+1)(Sin(y)+1)

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)

Properties Even function Odd function

Properties - 2

Consequences for the Fourier Transform FT of real function is conjugate symmetric FT of imaginary function is conjugate antisymmetric

Im

Re

Re

Im

FT of even and odd functions FT of even function is real FT of odd function is imaginary

Even Real Imaginary Cos (x)

Odd Real Imaginary Sin (x)