Fourier Transformation fx Fourier Transformasjon Fu 1 Continuous

Fourier Transformation f(x) Fourier Transformasjon F(u) 1

Continuous Fourier Transform Def The Fourier transform of a one-dimentional function f(x) The Inverse Fourier Transform 2

Continuous Fourier Transform Def - Notation The Fourier transform of a one-dimentional function f(x) The inverse Fourier Transform of F(u) 3

Continuous Fourier Transform Alternative Def 4

Continuous Fourier Transform Example - cos(2 ft) 5

Continuous Fourier Transform Example - cos( t) 6

Continuous Fourier Transform Example - sin( t) 7

Continuous Fourier Transform Example - Delta-function 8

Continuous Fourier Transform Example - Gauss function 9

Signals and Fourier Transform Frequency Information FT FT FT 10

Stationary / Non-stationary signals Stationary FT Non stationary FT The stationary and the non-stationary signal both have the same FT. FT is not suitable to take care of non-stationary signals to give information about time. 11

Transient Signal Frequency Information Constant function in [-3, 3]. Dominating frequency = 0 and some freequency because of edges. Transient signal resulting in extra frequencies > 0. Narrower transient signal resulting in extra higher frequencies pushed away from origin. 12

Transient Signal No Information about Position Moving the transient part of the signal to a new position does not result in any change in the transformed signal. Conclusion: The Fourier transformation contains information of a transient part of a signal, but only the frequency not the position. 13

Inverse Fourier Transform [1/3] Theorem: Proof: 14

Inverse Fourier Transform [2/3] Theorem: Proof: 15

Inverse Fourier Transform [3/3] 16

Properties 17

Fourier Transforms of Harmonic and Constant Function 18

Fourier Transforms of Some Common Functions 19

Even and Odd Functions [1/3] Def Every function can be split in an even and an odd part and each of this can in turn be split in a real and an imaginary part 20

Even and Odd Functions [2/3] 1. 2. 3. Even component in f produces an even component in F Odd component in f produces an odd component in F Oddcomponent in f produces an coefficient -j 21

Even and Odd Functions [3/3] 22

The Shift Theorem 23

The Similarity Theorem 24

The Convolution Theorem 25

Convolution Edge detection 26

The Adjoint of the Fourier Transform Theorem: Suppose f and g er are square integrable. Then: Proof: 27

Plancherel Formel - The Parselval’s Theorem: Suppose f and g are square integrable. Then: Proof: 28

The Rayleigh’s Theorem Conservation of Energy The energy of a signal in the time domain is the same as the energy in the frequency domain 29

The Fourier Series Expansion u a discrete variable - Forward transform Suppose f(t) is a transient function that is zero outside the interval [-T/2, T/2] or is considered to be one cycle of a periodic function. We can obtain a sequence of coefficients by making a discrete variable and integrating only over the interval. 30

The Fourier Series Expansion u a discrete variable - Inverse transform The inverse transform becomes: 31

The Fourier Series Expansion cn coefficients 32

The Fourier Series Expansion zn, an, bn coefficients 33

The Fourier Series Expansion an, bn coefficients 34

Fourier Series Pulse train approximated by Fourier Serie N=1 N=2 N=5 N = 10 35

Fourier Series Pulse train – Java program 36

Pulse Train approximated by Fourier Serie f(x) square wave (T=2) N=1 N=2 N=10 37

Fourier Series Zig tag approximated by Fourier Serie N=1 N=2 N=5 N = 10 38

Fourier Series Negative sinus function N=1 Negative sinus function approximated by Fourier Serie N=2 N=5 N = 10 39

Fourier Series Truncated sinus function N=1 Truncated sinus function approximated by Fourier Serie N=2 N=5 N = 10 40

Fourier Series Line approximated by Fourier Serie N=1 N=2 N=5 41 N = 10 N = 50

Fourier Series Java program for approximating Fourier coefficients Approximate functions by adjusting Fourier coefficients (Java program) 42

The Discrete Fourier Transform - DFT Discrete Fourier Transform - Discretize both time and frequency Continuous Fourier transform Discrete frequency Fourier Serie Discrete frequency and time Discrete Fourier Transform 43

The Discrete Fourier Transform - DFT Discrete Fourier Transform - Discretize both time and frequency { fi } sequence of length N, taking samples of a continuous function at equal intervals 44

Continuous Fourier Transform in two Dimensions Def The Fourier transform of a two-dimentional function f(x, y) The Inverse Fourier Transform 45

The Two-Dimensional DFT and Its Inverse 46

Fourier Transform in Two Dimensions Example 1 47

Fourier Transform in Two Dimensions Example 2 48

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