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1Digital Signal Processing A.S.Kayhan

DIGITAL SIGNAL

PROCESSING

Part 4

J.S. Lim, Two Dimensional Signal Processing, in Advanced Topics In Signal Processing, Prentice-Hall.

Digital Signal Processing A.S.Kayhan

Signals:Impulses: The 2-D impulse is,

otherwise,0

0,1, 2121

nnnn


Any sequence may be represented as a linear combination of shifted impulses:

].,[],[, 221121211 2

knknkkxnnxk k

21, nnx


Line impulses are:

otherwise,0

0,1, 1121

nnnnx T


or

otherwise,0

0,1, 2221

nnnnx T


or

otherwise,0

,1, 212121

nnnnnnx T


Step sequence:

otherwise,0

0,,1, 2121

nnnnu

].,[,1

1

2

2

2121

n

k

n

k

kknnu

or

1,11,,1,, 2121212121 nnunnunnunnunn


Separable sequences:A separable sequence can be written as

221121, nfnfnnx The impulse is separable

2121, nnnn Periodic sequences:

is periodic with period N1xN2 if 21,~ nnx 22121121 ,~,~,~ NnnxnNnxnnx

Example: Periodic with 2x4

2121 )2/(cos,~ nnnnx


Linearity: A system is linear if

Systems:The relation between the input and the response of the system is given by

.,, 2121 nnxFnny

21 , nnx

.,,,, 212211212211 nnybnnyannxbnnxaF where .2,1,,, 2121 innxFnny ii

Time Invariance: A system is time-invariant if

.,, 22112211 mnmnymnmnxF


Convolution:For a LSI system with impulse response, , the input/output relation is given by

Where * denotes 2-D convolution.

Convolution with a delayed impulse :

21 , nnh

].,[],[

,*,,

221121

212121

1 2

knknhkkx

nnhnnxnny

k k

].,[,*, 2211221121 mnmnxmnmnnnx


Example:


].,[],[, 221121211 2

knknhkkxnnyk k


If the impulse response is separable

221121, nhnhnnh then

1

1 2

1 2

21111

22221111

2221112121

,][

][],[][

][][],[,

k

k k

k k

nkfknh

knhkkxknh

knhknhkkxnny


Stability: A system is BIBO stable iff a bounded input leads to a bounded output. If

then the system is BIBO stable.

1 2

],[ 21n n

nnh


Discrete Space Fourier Transform (2D F.T.):

.),()2(

1,

,),(

2121221

2121

2211

1 2

2211

1 2

ddeeXnnx

eennxX

njnj

njnj

n n

is periodic with 2 in both variables.),( 21 X

).2,(),2(),( 212121 XXX

2D Fourier Transform and its inverse are defined as


Example:


Frequency Response:

If the input to Linear Shift Invariant (LSI) system is

2211],[ 21njnj eennx

then 2211),(],[ 2121

njnj eeHnny

where

22111 2

2121 ,),(kjkj

k k

eekkhH

is the frequency response function of the system.


Example:Given LPF

otherwise,0

and,1),( 2121

baH

This function can be written as

).()(),( 221121 HHH

10


.)sin()sin(

][][],[2

2

1

1221121

n

bn

n

annhnhnnh

then


Example:Given impulse response and magnitude response of a 2D LPF as

11


Original LP-Filtered HP-Filtered


Example:Given frequency response of a 2D LPF as

2122

21

22

21

21,and,0

,1),(

c

cH

12


where J1(x) is the Bessel function.

2221122

21

212

],[ nnJnn

nnh cc

then the impulse response sequence is


2D Z-Transform:

211 2

212121 ,),(nn

n n

zznnxzzX

2D Z-Transform is defined as

where z1 and z2 are complex variables. The space represented by (z1, z2) is four-dimesional (4-D).

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Example:Given 2D sequence 2121 ,],[ 21 nnubannx nn

then

bzazbzazz

b

z

a

zznnubazzX

n n

nn

nn

n n

nn

2112

110 0 21

212121

and,1

1

1

1

,),(

1 2

21

21

1 2

21


Inverse Z-Transform:2-D polynomials, in general, can not be factored as a product of lower order polynomials, therefore the partial fraction expansion is not a general procedure for 2-D signals.

Linear, Constant Coefficient Difference Eq.:LCCDE is given in the following form

221121),(

221121),(

,,,,2121

knknxkkbknknykkakk Rkk R BA

where a and b are known, with boundary conditions.

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Example:Given an IIR filter with a first quadrant impulse response as

12

11

21 5.01

1),(

zzzzH

then

12

1121

2121 5.01

1

),(

),(),(

zzzzX

zzYzzH

),(),(5.0),( 211

21

12121 zzXzzzzYzzY

],[]1,1[5.0],[ 212121 nnxnnynny


End of Part 4

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