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EE 4780. 2D Fourier Transform. Fourier Transform. What is ahead? 1D Fourier Transform of continuous signals 2D Fourier Transform of continuous signals 2D Fourier Transform of discrete signals 2D Discrete Fourier Transform (DFT). Fourier Transform: Concept.

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Ee 4780 l.jpg

EE 4780

2D Fourier Transform


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Fourier Transform

  • What is ahead?

    • 1D Fourier Transform of continuous signals

    • 2D Fourier Transform of continuous signals

    • 2D Fourier Transform of discrete signals

    • 2D Discrete Fourier Transform (DFT)


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Fourier Transform: Concept

  • A signal can be represented as a weighted sum of sinusoids.

  • Fourier Transform is a change of basis, where the basis functions consist of sines and cosines (complex exponentials).


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Fourier Transform

  • Cosine/sine signals are easy to define and interpret.

  • However, it turns out that the analysis and manipulation of sinusoidal signals is greatly simplified by dealing with related signals called complex exponential signals.

  • A complex number has real and imaginary parts: z = x + j*y

  • A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a)


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Fourier Transform: 1D Cont. Signals

  • Fourier Transform of a 1D continuous signal

“Euler’s formula”

  • Inverse Fourier Transform


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Fourier Transform: 2D Cont. Signals

  • Fourier Transform of a 2D continuous signal

  • Inverse Fourier Transform

  • Fandfare two different representations of the same signal.


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Fourier Transform: Properties

  • Remember the impulse function (Dirac delta function) definition

  • Fourier Transform of the impulse function


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Fourier Transform: Properties

  • Fourier Transform of 1

Take the inverse Fourier Transform of the impulse function


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Fourier Transform: Properties

  • Fourier Transform of cosine


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Examples

Magnitudes are shown


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Examples


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Fourier Transform: Properties

  • Linearity

  • Shifting

  • Modulation

  • Convolution

  • Multiplication

  • Separable functions


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Fourier Transform: Properties

  • Separability

2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.


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Fourier Transform: Properties

  • Energy conservation


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Fourier Transform: 2D Discrete Signals

  • Fourier Transform of a 2D discrete signal is defined as

where

  • Inverse Fourier Transform


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Fourier Transform: Properties

  • Periodicity: Fourier Transform of a discrete signal is periodic with period 1.

1

1

Arbitrary integers


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Fourier Transform: Properties

  • Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.


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Fourier Transform: Properties

  • Linearity

  • Shifting

  • Modulation

  • Convolution

  • Multiplication

  • Separable functions

  • Energy conservation


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Fourier Transform: Properties

  • Define Kronecker delta function

  • Fourier Transform of the Kronecker delta function


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Fourier Transform: Properties

  • Fourier Transform of 1

To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1.


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Impulse Train

  • Define a comb function (impulse train) as follows

where M and N are integers


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Impulse Train

  • Fourier Transform of an impulse train is also an impulse train:


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Impulse Train


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Impulse Train

  • In the case of continuous signals:


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Impulse Train


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Sampling


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Sampling

No aliasing if


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Sampling

If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.


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Sampling

Aliased


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Sampling

Anti-aliasing filter


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Sampling

  • Without anti-aliasing filter:

  • With anti-aliasing filter:


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Anti-Aliasing

a=imread(‘barbara.tif’);


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Anti-Aliasing

a=imread(‘barbara.tif’);

b=imresize(a,0.25);

c=imresize(b,4);


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Anti-Aliasing

a=imread(‘barbara.tif’);

b=imresize(a,0.25);

c=imresize(b,4);

H=zeros(512,512);

H(256-64:256+64, 256-64:256+64)=1;

Da=fft2(a);

Da=fftshift(Da);

Dd=Da.*H;

Dd=fftshift(Dd);

d=real(ifft2(Dd));


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Sampling


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Sampling

No aliasing if and


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Interpolation

Ideal reconstruction filter:


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Ideal Reconstruction Filter


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