The Fourier Transform I Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer The Fourier Transform I

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

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 (see figure 4.1 book)

Fourier Transform Even functions that are not periodic and 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 Discrete Fourier Transform

Complex numbers Complex number Its complex conjugate

Complex numbers polar Complex number in polar coordinates

Euler’s formula ? Sin (θ) ? Cos (θ)

Im Re

Complex math Complex (vector) addition Multiplication with i is rotation by 90 degrees in the complex plane

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

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

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

What does this look like? Impulse train What does this look like? ΔT = 1 Note: impulses can be continuous or discrete!

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

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 – 1D cont. case 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

Also: using the following symmetry

Examples - Impulse train Periodic with period ΔT Encompasses only one impulse, so

Examples - Impulse train 2

Intermezzo: Symmetry in the FT

So: the Fourier transform of an impulse train with period is also an impulse train with period

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

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

Convolution theorem Convolution in one domain is multiplication in the other domain: And also:

And: Shift in one domain is multiplication with complex exponential (modulation) in the other domain And:

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

Sampling Idea: convert a continuous function into a sequence of discrete values. (see figure 4.5 book)

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 of is a periodic infinite sequence of copies of , with period

Sampling Note that sampled function is discrete but its Fourier transform is continuous!

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.

Extracting a single period from that is equal to is possible if 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 Discrete Fourier Transform

Discrete Fourier Transform Continuous transform of sampled function

is continuous and infinitely periodic with period 1/ΔT

We need only one period to characterize 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 Note: separation between samples in F. domain is

By now we probably need some …