ZeroState - Zero State Response Linear Systems and Signals...

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Zero State Response Linear Systems and Signals – Lecture 5 Dr. J. K. Aggarwal The University of Texas at Austin
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Zero State Response - Linearity ± Let us assume that the system is in zero state. Consider: with zero initial conditions Let us express Correspondingly, due to Linearity ( ) () QD y tP D x t = 12 1 () m m i i xt x t = = ++ + = " 1 m m i i yt y t = = + = "
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Step and Impulse Functions 1 0 u ( t ) t Step function A representation A visualization ε 1 2 0 ε→ t 0 t Delta Function
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Step function Revisited t 1 a () ut a 0 t 1 b ut b 0 t ut a ut b −− 1 a b a>0 b>a a
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Another look 0 u ( t ) t Step function Impulse function 1/ Δ τ 0 t aa + Δ τ Choose a=n Δ τ 1/ Δ τ
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Impulse function 1/ Δ τ 0 t a a+ Δ τ Choose a=n Δ τ A use of delta function
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Zero State Response . .contd Let us consider a particular decomposition of x(t) Pulse is: () [ ( ) ( ( 1 ) ) ] xn ut n τ ττ Δ −Δ + Δ ( ( ( 1 ) ) ( ) ( )( ) ( ) ) xt x t n tx d t τδ τ τ −∞ Δ− + Δ Δ Δ = ≅Δ Δ Δ Δ x(t) tn t x(t) t
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Zero State Response . .contd ± The contribution to the output due to this pulse is or ± This is called the Convolution Integral i.e. the output is the convolution of inputs and the impulse response () . ( ) xn ht n τ ττ ≈Δ Δ Δ 0 l im ( ) ( ) yt Δ→ Δ Δ ( ) ( ) y tx h t d −∞ =− Fundamental result sought!
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The Convolution Integral ±
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ZeroState - Zero State Response Linear Systems and Signals...

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