Fourier Transformation Transformasjon f(x) F(u)

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

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

Continuous Fourier Transform Alternative Def

Continuous Fourier Transform Example - cos(2ft)

Continuous Fourier Transform Example - cos(t)

Continuous Fourier Transform Example - sin(t)

Continuous Fourier Transform Example - Delta-function

Continuous Fourier Transform Example - Gauss function

Signals and Fourier Transform Frequency Information FT FT Øverst vises funksjonen y1 = sin(w1*t), dvs en funksjon med en gitt frekvens w1. Den Fourier-transformerte funksjonen viser en enkelt topp svarende til denne ene frekvensen. I midten vises funksjonen y2 = sin(w2*t) med en gitt frekvens w2 hvor w2 > w1. Den Fourier-transformerte funksjonen viser en enkelt topp svarende til denne ene frekvensen, men vi ser at denne toppen er plassert lenger til høyre enn den toppen i den forrige Fourier-transformerte, svarende til at vi nå har en høyere frekvens. Nederst vises en funksjon y3 = sin(w1*t) + sin(w2*t), dvs en funksjon som inneholder to gitte frekvenser. Dette gjenspeiles i den Fourier-transformerte som to topper i diagrammet, de to toppene fra de to foregående Fourier-transformerte. FT

Stationary / Non-stationary signals FT Non stationary Øverst vises en funksjon y3 = sin(w1*t) + sin(w2*t), dvs en funksjon som inneholder to gitte frekvenser (samme som siste figur på foregående slide), samt den Fourier-transformerte som inneholder to topper svarende til de to frekvensene. Legg merke til at de to frekvensene opptrer samtidig hele tiden i tid-rommet. Nederst vises en grein-funksjon y4 hvor de to frekvensene ikke lengre opptrer samtidig i tid. Først opptrer frekvensen w1, deretter overtar frekvensen w2. Legg merke til at De to Fourier-transformerte funksjonene er like, dvs en Fourier-transformasjon kan plukke ut de enkelte frekvensene, men klarer ikke å plassere disse i tid. 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.

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.

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.

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

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

Inverse Fourier Transform [3/3]

Properties

Fourier Transforms of Harmonic and Constant Function

Fourier Transforms of Some Common Functions

Even and Odd Functions [1/3] Def Every function can be split in an even and an odd part 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

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

Even and Odd Functions [3/3]

The Shift Theorem

The Similarity Theorem

The Convolution Theorem

Convolution Edge detection

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

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

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

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.

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

The Fourier Series Expansion cn coefficients

The Fourier Series Expansion zn, an, bn coefficients

The Fourier Series Expansion an,bn coefficients

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

Fourier Series Pulse train – Java program

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

Fourier Series Zig tag Zig tag approximated by Fourier Serie N = 1

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

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

Fourier Series Line Line approximated by Fourier Serie N = 1 N = 2

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

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

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

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

The Two-Dimensional DFT and Its Inverse

Fourier Transform in Two Dimensions Example 1

Fourier Transform in Two Dimensions Example 2

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