Centura Web Developer

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Fourier Transformation
Fourier Transformasjon
f(x)
F(u)
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Continuous Fourier TransformDef
The Fourier transform of a one-dimentional
function f(x)
The Inverse Fourier Transform
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Continuous Fourier TransformDef - Notation
The Fourier transform of a one-dimentional
function f(x)
The inverse Fourier Transform of F(u)
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Continuous Fourier TransformAlternative Def
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Continuous Fourier TransformExample -
cos(2?ft)
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Continuous Fourier TransformExample - cos(?t)
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Continuous Fourier TransformExample - sin(?t)
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Continuous Fourier TransformExample -
Delta-function
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Continuous Fourier TransformExample - Gauss
function
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Signals and Fourier TransformFrequency
Information
FT
FT
FT
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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.
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Transient SignalFrequency Information
Constant function in -3,3. Dominating frequency
? 0 and some freequency because of edges.
Transient signal resulting in extra frequencies gt
0.
Narrower transient signal resulting in extra
higher frequencies pushed away from origin.
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Transient SignalNo 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.
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Inverse Fourier Transform 1/3
Theorem
Proof
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Inverse Fourier Transform 2/3
Theorem
Proof
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Inverse Fourier Transform 3/3
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Properties
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Fourier Transforms of Harmonic and Constant
Function
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Fourier Transforms of Some Common Functions
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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
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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
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Even and Odd Functions 3/3
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The Shift Theorem
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The Similarity Theorem
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The Convolution Theorem
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ConvolutionEdge detection
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The Adjoint of the Fourier Transform
Theorem Suppose f and g er are square
integrable. Then
Proof
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Plancherel Formel - The Parselvals Theorem
Theorem Suppose f and g are square integrable.
Then
Proof
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The Rayleighs TheoremConservation of Energy
The energy of a signal in the time domain is the
same as the energy in the frequency domain
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The Fourier Series Expansionu 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.
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The Fourier Series Expansionu a discrete
variable - Inverse transform
The inverse transform becomes
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The Fourier Series Expansioncn coefficients
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The Fourier Series Expansionzn, an, bn
coefficients
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The Fourier Series Expansionan,bn coefficients
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Fourier SeriesPulse train
Pulse train approximated by Fourier Serie
N 1
N 2
N 5
N 10
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Fourier SeriesPulse train Java program
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Pulse Train approximated by Fourier Serie
f(x) square wave (T2)
N1
N2
N10
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Fourier SeriesZig tag
Zig tag approximated by Fourier Serie
N 1
N 2
N 5
N 10
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Fourier SeriesNegative sinus function
Negative sinus function approximated by Fourier
Serie
N 1
N 2
N 5
N 10
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Fourier SeriesTruncated sinus function
Truncated sinus function approximated by Fourier
Serie
N 1
N 2
N 5
N 10
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Fourier SeriesLine
Line approximated by Fourier Serie
N 1
N 2
N 5
N 10
N 50
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Fourier SeriesJava program for approximating
Fourier coefficients
Approximate functions by adjusting Fourier
coefficients (Java program)
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The Discrete Fourier Transform - DFTDiscrete
Fourier Transform - Discretize both time and
frequency
Continuous Fourier transform
Discrete frequency Fourier Serie
Discrete frequency and time Discrete Fourier
Transform
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The Discrete Fourier Transform - DFTDiscrete
Fourier Transform - Discretize both time and
frequency
fi sequence of length N, taking samples of
a continuous function at equal intervals
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Continuous Fourier Transform in two DimensionsDef
The Fourier transform of a two-dimentional
function f(x,y)
The Inverse Fourier Transform
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The Two-Dimensional DFT and Its Inverse
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Fourier Transform in Two DimensionsExample 1
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Fourier Transform in Two DimensionsExample 2
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End