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Tutorials 12,13 discrete signals and systems. Technion, CS department, SIPC 236327 Spring 2014. Discrete LSI system. Linear Space invariant. Discrete LSI system. Linear Space invariant. Example.

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tutorials 12 13 discrete signals and systems
Tutorials 12,13discrete signals and systems

Technion, CS department, SIPC 236327

Spring 2014

discrete lsi system
Discrete LSI system
  • Linear
  • Space invariant
discrete lsi system1
Discrete LSI system
  • Linear
  • Space invariant
example
Example
  • For compression, a rule to predict the pixel value is used:Is the system linear? Space invariant?
discrete lsi system2
Discrete LSI system
  • System is defined with its impulse response
cyclic convolution
Cyclic convolution

Convolution

  • Infinite support
  • DTFT
  • Finite support
  • DFT
  • Efficient implementation
exercise
Exercise

Q: How can we use this system to calculate a linear convolution?

A: Zero padding, and truncation of the result.

H

Q: If both signals are of length N, how many zeros will we add?

A: N-1 zeros

exercise1
Exercise

Q: How can we use this system to calculate a cyclic convolution?

A: Duplicate one signal, and truncation of the result.

H

Q: If both signals are of length N, how much should we duplicate

A: N-1 cells

discrete fourier transform dft fft
Discrete Fourier Transform (DFT, FFT)

Infinite support Infinite support

Continiuous Continuous

Finite support Finite support

Discrete Discrete

slide10
DFT
  • התמרות הDFT וDFT-1מתבצעות בדרך הרגילה
  • המקדמים מחזוריים:
  • לכן במקום להתייחס לתחום [0,N-1] בד"כ מסתכלים על התחום [-N/2,N/2-1].
summary fourier transforms
Summary – Fourier Transforms
  • Fourier transform
    • Time domain – non-periodic infinite signals
    • Continuous time (t)
    • Continuous frequency (f)
    • Formulas
summary fourier transforms1
Summary – Fourier Transforms
  • DTFT: Discrete Time Fourier Transform
    • Time domain – non-periodic infinite signals
    • Discrete time (n)
    • Continuous frequency (f)
    • Formulas

לא נלמד

בקורס

summary fourier transforms2
Summary – Fourier Transforms
  • Fourier series
    • Time domain – periodic infinite signals
    • Continuous time (t)
    • Discrete frequency (f)
    • Formulas
summary fourier transforms3
Summary – Fourier Transforms
  • DFTor Discrete Time Fourier Series
    • Time domain – periodic infinite signals
    • Discrete time (n)
    • Discrete frequency (f)
    • Formulas
exercise2
Exercise
  • We have an N-length filter with impulse response h[n].We create a new filter as follows:

Express F[k] with H[k], where H[k]=DFT{h[n]},F[k]=DFT{f[n]}

  • Instructions: calculate
example discrete frequency filtration
Example – discrete frequency filtration
  • Noisy image of size 256X256

Im_out[m,n]=Im_in[m,n]+noise[m,n]

      • Harmonic noise:
      • f = 1/(8 pixels)
      • Amplitude A and phase φ are random and independent for each line.
example discrete frequency filtration smoothing vs median 8 pixels
Example – discrete frequency filtration– smoothing vs median (8 pixels)

No noisebut image is blurred

example discrete frequency filtration3
Example – discrete frequency filtration
  • Design an LSI filter
    • Such filter multiplies each frequency with a complex number
    • Can handle each frequency separately
  • In this example, we want to handle frequencies 32 and -32.
    • Notch filter – attenuates specific frequency
example discrete frequency filtration4
Example – discrete frequency filtration

Original signal in

frequency domain

Filtered signal in

frequency domain

example discrete frequency filtration5
Example – discrete frequency filtration
  • Noise removed completely
  • Original image not fully restored
    • We cannot restore the attenuated frequencies
example discrete frequency filtration6
Example – discrete frequency filtration

Smoothing filter of 8 pixels

Notch filter

example frequency filtration implementation
Example –frequency filtration - implementation

Notch filter in freq. domain

  • Filter in freq. domain:

Filter=ones(1,256);

Filter(32+1)=0;

Filter(224+1)=0;

  • Filtration:

For k=1:size(I,1),

Y=fft(I(k,:)).*Filter;

I(k,:)=ifft(Y);

end

tutorials 12 13 discrete signals and systems part ii 2d
Tutorials 12,13discrete signals and systemsPart II: 2D

Technion, CS department, SIPC 236327

Spring 2014

2d definitions
2D - definitions

2Dconvolution:

2d definitions1
2D - definitions
  • Cyclic 2D-convolution:
  • 2D DFT:
2d notes
2D - notes
  • DFT is linear, we have an operation matrix:
  • 2D-DFT can be implemented as:
  • If the input is separable:
example1
Example
  • Noisy image 512X512
  • The noise:Add 100 gray levels for all 16i lines
example2
Example

Noisy image

Average filter

example3
Example

Noisy image

Average filter

example4
Example
  • How does the noise look like in the frequency domain?
example5
Example

After freq. filtration

  • Filter implementation in the freq. domain:

H=ones(512,512);

for n=1:32:512

H(n,1) = H(1,n) = 0;

end

H(1,1) = 1;

  • Image filtration:

out = ifft( fft(img).*H );

edge detection of image a
Edge detection of Image A
  • Roberts
  • Prewitt
  • Sobel
edge detection of image a1
Edge detection of Image A

Original

Roberts

Prewitt

Sobel