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Free International University of Moldova Faculty of Informatics and Engineering DIGITAL SIGNALS PROCESSING Course of lectures Theme: THE SIGNALS FUNCTIONAL PROCESSING Veacheslav L. Perju, DSC, Acad. Chisinau 2008
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THE SIGNALS FUNCTIONAL PROCESSING 1. The signal processing linear systems 2. The singular operator 3. The integral of super position 4. The integral of convolution
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The signal processing linear systems Let F 1 (x, y), F 2 (x, y),..., F N (x, y) – input signals G 1 (x, y), G 2 (x, y),..., G M (x, y) – output signals
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The signal processing linear systems The system can be named as linear if: ψ{a 1 F 1 (x, y) +...+ a N F N (x, y)}= a 1 ψ {F 1 (x, y)}+...+a N ψ {F N (x, y)} The one-to-one mapping is defined as G(x,y) = ψ {F(x,y)}
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The singular operator The two-dimensional Dirac delta function ∞, if x=y=0 ∞, if x=y=0 δ(x,y)={ 0 – i. a. c. 0 – i. a. c.Or ∞, if x= ξ y= η ∞, if x= ξ y= η δ(x-ξ,y-η)={ 0 – i. a. c. 0 – i. a. c. Were ξ, η – parameters of translation
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The function δ can be calculated in the next mode: 00 2 )]}dudv-v(y)-exp{i[u(x 4 1 ),,,( yx
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The integral of super position A function F(x,y) can be represented as a sum of amplitude weighted Dirac delta functions by the shifting integral Where F(,) is the weighting factor of the impulse located at coordinates in the x–y plane
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If the output of a general linear one-to- one system is defined to be ξηδ(x-ξ,y-η)dξdη P{x,y)={F(x, y)}= F(ξ,η)δ(x-ξ,y-η)dξdηOr ξηδ(x-ξ,y-η)}dξdη P{x, y)=F(ξ,η){δ(x-ξ,y-η)}dξdη The second term in the integral: δ(x-ξ,y-η)}=H(x,y,ξη) {δ(x-ξ,y-η)}=H(x,y,ξ,η) is called the impulse response of the two- dimensional system
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In this case ξηδ(x-ξ,y-η)}dξdη= P{x,y)=F(ξ,η){δ(x-ξ,y-η)}dξdη= ξηH(x,y,ξη)dξdη =F(ξ,η)H(x,y,ξ,η)dξdη will represent integral of superposition
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The integral of convolution x-ξ, y-η. An additive linear two-dimensional system is called space invariant if its impulse response do not depends from differences x-ξ, y-η. For an invariant system H( x,y,ξ,η) = H( x-ξ,y-η)
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The superposition integral reduces to the special case called the convolution integral ξη H( x-ξ,y-η)dξdη C{x,y}=F(ξ,η) H( x-ξ,y-η)dξdη Symbolically, H( x,y) C{x,y}= F(x,y)* H( x,y)
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The convolution integral is symmetric ξη H( x-ξ,y-η)dξdη= C{x,y}=F(ξ,η) H( x-ξ,y-η)dξdη= ( x-ξ,y-η) H( ξ,η)dξdη =F ( x-ξ,y-η) H( ξ,η)dξdη
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2D Convolution
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