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**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

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**Contents Complex numbers etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform

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**Introduction Jean Baptiste Joseph Fourier (*1768-†1830)**

French Mathematician La Théorie Analitique de la Chaleur (1822)

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**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)

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**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

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**Contents Complex numbers etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform

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Complex numbers Complex number Its complex conjugate

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Complex numbers polar Complex number in polar coordinates

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Euler’s formula ? Sin (θ) ? Cos (θ)

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Im Re

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**Complex math Complex (vector) addition Multiplication with i**

is rotation by 90 degrees in the complex plane

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**Contents Complex number etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform

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**Unit impulse (Dirac delta function)**

Definition Constraint Sifting property Specifically for t=0

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**Discrete unit impulse Definition Constraint Sifting property**

Specifically for x=0

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**What does this look like?**

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

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**Contents Complex number etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform

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**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

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**Fourier transform – 1D cont. case**

Symmetry: The only difference between the Fourier transform and its inverse is the sign of the exponential.

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Fourier and Euler Fourier Euler

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**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

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Examples – Block 1 A -W/2 W/2

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Examples – Block 2

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Examples – Block 3 ?

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Examples – Impulse constant

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**Examples – Shifted impulse**

Euler

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**Examples – Shifted impulse 2**

constant Real part Imaginary part

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**Also: using the following symmetry**

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**Examples - Impulse train**

Periodic with period ΔT Encompasses only one impulse, so

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**Examples - Impulse train 2**

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**Intermezzo: Symmetry in the FT**

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

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**Contents Complex number etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform

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

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What is

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Intermezzo 1 What is ? Let , so

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Intermezzo 2 Property of Fourier Transform

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**Fourier + Convolution cont’d**

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

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

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**Contents Complex number etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform

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

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**Sampling Sampled function can be written as**

Obtain value of arbitrary sample k as

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Sampling - 2

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Sampling - 3

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Sampling - 4

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**FT of sampled functions**

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

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**FT of sampled functions**

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

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

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**Contents Complex number etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

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**Sampling theorem Band-limited function Sampled function**

lower value of 1/ΔT would cause triangles to merge

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**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.

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**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

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**Contents Complex number etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples

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Aliasing If , aliasing can occur

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**Contents Complex number etc. Impulses Fourier Transform (+examples)**

Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform

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**Discrete Fourier Transform**

Continuous transform of sampled function

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**is continuous and infinitely periodic with period 1/ΔT**

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**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

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**Substituting Into yields**

Note: separation between samples in F. domain is

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**By now we probably need some …**

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