1. 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

2. 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

3. 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

4. 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)}

5. 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

6. The function δ can be calculated in the next mode:    00 2 )]}dudv-v(y)-exp{i[u(x 4 1 ),,,(    yx

7. 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

9. 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

10. 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

11. 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)

13. 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)

14. 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

15. 2D Convolution