Ee 4780
Download
1 / 38

EE 4780 - PowerPoint PPT Presentation


  • 157 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'EE 4780' - eugenia


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Ee 4780 l.jpg

EE 4780

2D Fourier Transform


Fourier transform l.jpg
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 l.jpg
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).


Fourier transform4 l.jpg
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)


Fourier transform 1d cont signals l.jpg
Fourier Transform: 1D Cont. Signals

  • Fourier Transform of a 1D continuous signal

“Euler’s formula”

  • Inverse Fourier Transform


Fourier transform 2d cont signals l.jpg
Fourier Transform: 2D Cont. Signals

  • Fourier Transform of a 2D continuous signal

  • Inverse Fourier Transform

  • Fandfare two different representations of the same signal.


Fourier transform properties l.jpg
Fourier Transform: Properties

  • Remember the impulse function (Dirac delta function) definition

  • Fourier Transform of the impulse function


Fourier transform properties8 l.jpg
Fourier Transform: Properties

  • Fourier Transform of 1

Take the inverse Fourier Transform of the impulse function


Fourier transform properties9 l.jpg
Fourier Transform: Properties

  • Fourier Transform of cosine


Examples l.jpg
Examples

Magnitudes are shown



Fourier transform properties12 l.jpg
Fourier Transform: Properties

  • Linearity

  • Shifting

  • Modulation

  • Convolution

  • Multiplication

  • Separable functions


Fourier transform properties13 l.jpg
Fourier Transform: Properties

  • Separability

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


Fourier transform properties14 l.jpg
Fourier Transform: Properties

  • Energy conservation


Fourier transform 2d discrete signals l.jpg
Fourier Transform: 2D Discrete Signals

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

where

  • Inverse Fourier Transform


Fourier transform properties16 l.jpg
Fourier Transform: Properties

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

1

1

Arbitrary integers


Fourier transform properties17 l.jpg
Fourier Transform: Properties

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


Fourier transform properties18 l.jpg
Fourier Transform: Properties

  • Linearity

  • Shifting

  • Modulation

  • Convolution

  • Multiplication

  • Separable functions

  • Energy conservation


Fourier transform properties19 l.jpg
Fourier Transform: Properties

  • Define Kronecker delta function

  • Fourier Transform of the Kronecker delta function


Fourier transform properties20 l.jpg
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.


Impulse train l.jpg
Impulse Train

  • Define a comb function (impulse train) as follows

where M and N are integers


Impulse train22 l.jpg
Impulse Train

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



Impulse train24 l.jpg
Impulse Train

  • In the case of continuous signals:




Sampling27 l.jpg
Sampling

No aliasing if


Sampling28 l.jpg
Sampling

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


Sampling29 l.jpg
Sampling

Aliased


Sampling30 l.jpg
Sampling

Anti-aliasing filter


Sampling31 l.jpg
Sampling

  • Without anti-aliasing filter:

  • With anti-aliasing filter:


Anti aliasing l.jpg
Anti-Aliasing

a=imread(‘barbara.tif’);


Anti aliasing33 l.jpg
Anti-Aliasing

a=imread(‘barbara.tif’);

b=imresize(a,0.25);

c=imresize(b,4);


Anti aliasing34 l.jpg
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));



Sampling36 l.jpg
Sampling

No aliasing if and


Interpolation l.jpg
Interpolation

Ideal reconstruction filter:



ad