3.4.1 Mathematical representation of linear systems 5. Frequency preservability f →)y(t and for x(t=xejot then the output j(at+)
5. Frequency preservability If and for then the output 3.4.1 Mathematical representation of linear systems x(t)→ y(t) ( ) j t x t x e = 0 ( ) ( + ) = j t y t y e0
3.4.1 Mathematical representation of linear systems Proof: According to the proportionality property 2 x(t)→>O (38) According to the differentiation property (3.9) dt d x(t oxt+ oy (3.10) Since x(t J d2x(o) Jo)xoe
Proof: According to the proportionality property According to the differentiation property Since 3.4.1 Mathematical representation of linear systems x(t) y(t) 2 2 → (3.8) ( ) ( ) 2 2 2 2 dt dy t dt d x t → (3.9) ( ) ( ) ( ) ( ) → + + 2 2 2 2 2 2 dt dy t y t dt d x t x t (3.10) ( ) j t x t x e = 0 ( ) ( ) x(t) x e j x e dt d x t j t j t 2 0 2 0 2 2 2 = − = − =
3.4.1 Mathematical representation of linear systems Letting the left-hand side of eq. 3.10) be zero dolt 2x(t)+ then the right-hand side of eq ( 3.10) must also be zero 0 y(t)+ t Solving the equation yields yoe J(at+o where is the phase shift
Letting the left-hand side of Eq. (3.10) be zero, then the right-hand side of Eq. (3.10) must also be zero, Solving the equation yields: where φ is the phase shift. 3.4.1 Mathematical representation of linear systems ( ) ( ) 0 2 2 2 + = dt d x t x t ( ) ( ) 0 2 2 2 + = dt d y t y t ( ) ( + ) = j t y t y e0
3.4.2 Representation of system's characteristics in terms of transfer function or frequency response 1。 Transfer function 日 Definition: For tso, y(t=0, the Laplace transform Y(s) (3.1 of y(t) is defined as y(o e where s is the laplace operator: s=a+jb for a>0
1. Transfer function ❑ Definition: For t0, y(t)=0, the Laplace transform Y(s) of y(t) is defined as where s is the Laplace operator: s=a+jb for a>0. 3.4.2 Representation of system’s characteristics in terms of transfer function or frequency response ( ) ( ) − = 0 Y s y t e dt st (3.11)
3.4.2 Representation of system 's characteristics in terms of transfer function or frequency response If all the systems initial conditions are zero, making Laplace transform of Eg (3.3) gives then the expression y(slans"+amS+.+a,s+ao X((bns"+bnsm+…+b1s+b。) The transfer function H(s) y(s)bm5+bm-ISm-++b,5+b H 1 (3.12) )as"+as 十a1S+a 0 The transfer function H(s) represents the transfer characteristics of a system
If all the system’s initial conditions are zero, making Laplace transform of Eq. (3.3) gives then the expression The transfer function H(s): ❖The transfer function H(s) represents the transfer characteristics of a system. 3.4.2 Representation of system’s characteristics in terms of transfer function or frequency response ( )( ) ( )( ) 1 0 1 1 1 0 1 1 X s b s b s b s b Y s a s a s a s a m m m m n n n n = + + + + + + + + − − − − ( ) ( ) ( ) 1 0 1 1 1 0 1 1 a s a s a s a b s b s b s b X s Y s H s n n n n m m m m + + + + + + + + = = − − − − (3.12)