EE 4780 2 D Fourier Transform Bahadir K

EE 4780 2 D Fourier Transform Bahadir K. Gunturk

Fourier Transform n What is ahead? q q 1 D Fourier Transform of continuous signals 2 D Fourier Transform of discrete signals 2 D Discrete Fourier Transform (DFT) Bahadir K. Gunturk 2

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). Bahadir K. Gunturk 3

Fourier Transform n n 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. n A complex number has real and imaginary parts: z = x + j*y n A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a) Bahadir K. Gunturk 4

Fourier Transform: 1 D Cont. Signals ■ Fourier Transform of a 1 D continuous signal “Euler’s formula” ■ Inverse Fourier Transform Bahadir K. Gunturk 5

Fourier Transform: 2 D Cont. Signals ■ Fourier Transform of a 2 D continuous signal ■ Inverse Fourier Transform ■ F and f are two different representations of the same signal. Bahadir K. Gunturk 6

Fourier Transform: Properties ■ Remember the impulse function (Dirac delta function) definition ■ Fourier Transform of the impulse function Bahadir K. Gunturk 7

Fourier Transform: Properties ■ Fourier Transform of 1 Take the inverse Fourier Transform of the impulse function Bahadir K. Gunturk 8

Fourier Transform: Properties ■ Fourier Transform of cosine Bahadir K. Gunturk 9

Examples Magnitudes are shown Bahadir K. Gunturk 10

Examples Bahadir K. Gunturk 11

Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions Bahadir K. Gunturk 12

Fourier Transform: Properties ■ Separability 2 D Fourier Transform can be implemented as a sequence of 1 D Fourier Transform operations. Bahadir K. Gunturk 13

Fourier Transform: Properties ■ Energy conservation Bahadir K. Gunturk 14

Fourier Transform: 2 D Discrete Signals ■ Fourier Transform of a 2 D discrete signal is defined as where ■ Inverse Fourier Transform Bahadir K. Gunturk 15

Fourier Transform: Properties ■ Periodicity: Fourier Transform of a discrete signal is periodic with period 1. 1 1 Arbitrary integers Bahadir K. Gunturk 16

Fourier Transform: Properties ■ Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2 D Fourier Transform of discrete signals. Bahadir K. Gunturk 17

Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions ■ Energy conservation Bahadir K. Gunturk 18

Fourier Transform: Properties ■ Define Kronecker delta function ■ Fourier Transform of the Kronecker delta function Bahadir K. Gunturk 19

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. Bahadir K. Gunturk 20

Impulse Train ■ Define a comb function (impulse train) as follows where M and N are integers Bahadir K. Gunturk 21

Impulse Train n Fourier Transform of an impulse train is also an impulse train: Bahadir K. Gunturk 22

Impulse Train Bahadir K. Gunturk 23

Impulse Train n In the case of continuous signals: Bahadir K. Gunturk 24

Impulse Train Bahadir K. Gunturk 25

Sampling Bahadir K. Gunturk 26

Sampling No aliasing if Bahadir K. Gunturk 27

Sampling If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering. Bahadir K. Gunturk 28

Sampling Aliased Bahadir K. Gunturk 29

Sampling Anti-aliasing filter Bahadir K. Gunturk 30

Sampling ■ Without anti-aliasing filter: ■ With anti-aliasing filter: Bahadir K. Gunturk 31

Anti-Aliasing a=imread(‘barbara. tif’); Bahadir K. Gunturk 32

Anti-Aliasing a=imread(‘barbara. tif’); b=imresize(a, 0. 25); c=imresize(b, 4); Bahadir K. Gunturk 33

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=fft 2(a); Da=fftshift(Da); Dd=Da. *H; Dd=fftshift(Dd); d=real(ifft 2(Dd)); Bahadir K. Gunturk 34

Sampling Bahadir K. Gunturk 35

Sampling No aliasing if Bahadir K. Gunturk and 36

Interpolation Ideal reconstruction filter: Bahadir K. Gunturk 37

Ideal Reconstruction Filter Bahadir K. Gunturk 38