Lecture#3 - BIM 108 Lecture #3 Convolution Discrete-time...

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1 BIM 108 Lecture #3 Convolution Discrete-time Example Time invariant ?
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2 Discrete-time Convolution 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 [ ] [0] [ ] [1] [ 1] [2] [ 2] . .. [ ] [ ], 0,1,2,. .. i xn x n x n x n xi n i n δ δδ = =+ + + =− = 0 [ ] [0] [ ] [2] [ 2] . .. [][ ] [ ] [ ], 0,1,2,. .. i yn x hn xihn i xn hn n = = +− + + ∗= ± Matlab command: y=conv(x,h); Recall Delta Function Recall impulse function: ) ( t = ε λ d ) ( = = τ d t x d x ) ( ) ( ) ( ) ( 0 t n x(t)
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3 Continuous-time Convolution If we know an LTI system's response to the impulse function, we can compute the response of the system to x(t) using LTI properties. Specifically, ) ( ) ( t h t H ⎯→ δ we denote the impulse response as h(t) () H t δτ −⎯ Time invariance. ()( ) H xt τδ τ Linearity ) H d τ τ −∞ Linearity = d t h x t y t x H ) ( ) ( ) ( ) ( = = d t h x t h t x t y ) ( ) ( ) ( ) ( ) ( Convolution Integral: Evaluation of Convolution Integrals = = d t h x t h t x t y ) ( ) ( ) ( ) ( ) ( Ex1: (i) Fold h( τ ) (ii) Shift h(- τ
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This note was uploaded on 10/15/2010 for the course BIM BIM108 taught by Professor Qiu during the Winter '09 term at UC Davis.

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Lecture#3 - BIM 108 Lecture #3 Convolution Discrete-time...

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