02A Discretization of CT system - SIGNAL SYSTEM...

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SIGNAL & SYSTEM REPRESENTATION Laplace Transform: 0 ( ) that varies with time ( ) ( ) s x t function X s e x d τ =⇔ = Signals & their Laplace transform (examples) 2 2 ( ) ( ) ( ) 1 1 ( ) 1 ( ) 1 ( ) ( ) ( ) xt t X s s t s t δ = = = = 3 22 1 ( ) 1 ( ) ( ) ( ) sin ( ) ( ) cos at Xs s e sa t s t ω ⇔= = + = + = 2 2 ( ) ( ) sin 1 ( ) 2 so on n t n n nn s s e t ss and ζω ωζ ζ ωω + =− ++ Calculus operations & their Laplace transform representation 2 2 2 0 ( ) ( ) () s ( ) (0) (0 ) s ( ) (0) 1 ( ) ( ) ( ) t dx t Xs x dt dxt d x Xs s x dt dt xtd t s td ⇔− ∫∫ M 2 1 ( ) s M Linear differential equations and their Laplace transform (an example) ( ) 2 12 2 2 1 2 ( ) ( ) (0) s ( ) (0) s ( ) (0) ( ) ( ) (0) ( ) dyt d yt d ut aa y t bb u t dt dt dt dy Y s sy a Y s x a Y s b sU s u b U s dt =+ −− + −+ = σ j
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Transfer function . If all the initial conditions are zero, then the Laplace transform of the above differential equation reduces to a polynomial equation that can be interpreted in the form of a transfer function. Polynomial equation ()() 2 12 1 2 2 s () () s s Y s a Y s aY s bsU s bU s aa Y s b s b U s ++ = + = + Transfer function 2 2 ss () () Ys Gs Us bs b GsUs == + + Laplace transform is extremely useful for representing signals, and analyzing and designing control systems. Examples : Spring damper mass suspension R-L-C circuit Note to self: Physical (mechanical, electrical,… , electronics, thermal, hydraulics, pneumatics, chemical) processes can be mathematically modeled and described by differential equations. Linear models can be represented by Laplace transform.
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02A Discretization of CT system - SIGNAL SYSTEM...

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