Tutorials 12,13 discrete signals and systems

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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 12,13discrete signals and systems

Technion, CS department, SIPC 236327

Spring 2014

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

Convolution

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

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)

Infinite support Infinite support

Continiuous Continuous

Finite support Finite support

Discrete Discrete

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
• Fourier transform
• Time domain – non-periodic infinite signals
• Continuous time (t)
• Continuous frequency (f)
• Formulas
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 Transforms
• Fourier series
• Time domain – periodic infinite signals
• Continuous time (t)
• Discrete frequency (f)
• Formulas
Summary – Fourier Transforms
• DFTor Discrete Time Fourier Series
• Time domain – periodic infinite signals
• Discrete time (n)
• Discrete frequency (f)
• Formulas
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
• 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)

No noisebut image is blurred

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 filtration

Original signal in

frequency domain

Filtered signal in

frequency domain

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

Smoothing filter of 8 pixels

Notch filter

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 12,13discrete signals and systemsPart II: 2D

Technion, CS department, SIPC 236327

Spring 2014

2D - definitions

2Dconvolution:

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

Noisy image

Average filter

Example

Noisy image

Average filter

Example
• How does the noise look like in the frequency domain?
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
• Roberts
• Prewitt
• Sobel
Edge detection of Image A

Original

Roberts

Prewitt

Sobel