the signals functional processing

7/24/2019 THE SIGNALS FUNCTIONAL PROCESSING

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THE SIGNALS FUNCTIONALTHE SIGNALS FUNCTIONAL

PROCESSINGPROCESSING

1.1. The signal processingThe signal processing

linear systemslinear systems

2. The singular operator2. The singular operator

3. The integral of super position3. The integral of super position4. The integral of convolution4. The integral of convolution


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The signal processingThe signal processing

linear systemslinear systemsLetLetFF11((x, yx, y),), FF22((x, yx, y),...,),..., FFNN((x, yx, y) input signals) input signals

G1(x, y), G2(x, y),..., GM(x, y) output signals


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The signal processingThe signal processing

linear systemslinear systemsThe system can be named as linear if:O{a1F1(x, y) +...+ aNFN(x, y)}=

a1O{F1(x, y)}+...+aNO{FN(x, y)}

The one-to-one mapping is defined as

(x,y) = O{F(x,y)}


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The singular operatorThe singular operator

The t!"#dimensi"nal Dirac delta function

$$, if x=y=%, if x=y=%

&&

(x,y)={(x,y)={

% ' i. a. c.% ' i. a. c.

"r"r

$ $, if x=, if x= y=y=

&&(x#(x#,y#,y#)={)={ % ' i. a. c.% ' i. a. c.

*ere*ere ,, ' arameters "f translati"n' arameters "f translati"n


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The fncti"nThe fncti"n &&can be calclated in the nextcan be calclated in the nextm"de:m"de:

+=00

2 )]}dudv-v(y)-exp{j[u(x

41),,,(

yx

= 1dxdy),( yx


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The integral of super positionThe integral of super position

A function (!, y)can "e represente# asa sum of amplitu#e$eighte# %irac #elta

functions "y the sifting integral

+

= d)d-y,-(x),F(y)F(x,

*here F(,) is the !ei-htin- fact"r "f the imlsel"cated at c""rdinates in thex'y lane


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f the "tt "f a -eneral linear "ne#t"#"ne

system is defined t" be

&'!,y'(!, y) '(**,++)(!-(!-**,y-,y-++)#)#**##++

rr

&'!, y(**,++)'(!-(!-**,y-,y-++)#)#**##++

The secon# term in the integral/

'(!-(!-**,y-,y-++)0(!,y,)0(!,y,**,++))

is calle# the impulse response of the t$o-#imensional system


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n this casen this case

&'!,y(**,++)'(!-(!-**,y-,y-++)#)#**##++

(**,++)0(!,y,0(!,y,**,++)#)#**##++

!ill reresent inte-ral "f

ser"siti"n


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The integral of convolutionThe integral of convolution

An a##itive linear t$o-#imensional systemis calle# space invariant if its impulse

response #o not #epen#s from #ifferences

!-!-**, y-, y-++..

or an invariant systemor an invariant system

0(0(!,y,!,y,**,,++) ) 0(0(!-!-**,y-,y-++))


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The superposition integral re#uces to the

special case calle# the convolution integral

'!,y(**,++)0(0(!-!-**,y-,y-++)#)#**##++

ym"olically,

'!,y (!,y)40(0(!,y)!,y)


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The convolution integral is symmetric

'!,y(**,++)0(0(!-!-**,y-,y-++)#)#**##++

((!-!-**,y-,y-++))0(0(**,,++)#)#**##++


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2D Convolution2D Convolution

the signals functional processing

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