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
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