lecture 21

# lecture 21 - 1 1 Summary of Lecture#21 • Convolution...

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Unformatted text preview: 1 1 Summary of Lecture #21 10/13/2010 • Convolution integral • Flip, shift, multiply and integrate • Properties of Delta function, δ (t), a review • Convolution with Delta function • Graphical convolution- examples • Convolution algebra ∫ ∫ ∞ ∞- ∞ ∞- τ τ- τ = τ τ τ- = = d ) t ( h ) ( f d ) ( h ) t ( f ) t ( h ) t ( f ) t ( y 4 t h (1) (t) Ramp, unit step and delta functions 5 h d (t) r(t) tu(t) 1, t u(t) 0, t u(t) u(t) u(t h) dt h lim → δ = ≥ = - = = <- Unit step function Ramp function Delta function t t r(t) u( )d tu(t) (12. u(t) ( ) d (12.9 8) ) ∞ ∞-- = τ τ = = δ ∫ ∫ q q dt ) t ( r d ) t ( u = 6 Delta function, δ (t) A pulse of infinite height, zero width and unit area. Key properties: ) t ( | a | 1 ) at ( ) a T ( f | a | 1 dt ) T at ( ) t ( f ) T ( f dt ) T t ( ) t ( f dt ) T t ( ) t ( f 1 dt ) t ( dt ) t ( T T δ = δ =- δ =- δ =- δ = δ = δ ∫ ∫ ∫ ∫ ∫ ∞ ∞- ∞ ∞- ∞ ∞- +- +- 7 Ex. 4. Given f (t) = sin(2 π t + θ ) and h(t) = δ (t - 5), Find f*h . Ans: y(t) = f(t)*h(t) = sin(2 π (t - 5) + θ ) y(t) f (t) h(t) f (t )h( )d sin(2 ( ) ) ( ) d si t 5 5 n(2 (t ) ) ∞-∞ ∞-∞ τ - = ≡- τ τ τ = π + θ δ τ- τ = π- + θ ∫ ∫ 8 Ex. 5. Given f (t) = u(t) + u(t- 1)- 2u(t- 2) and h(t) =- 2 δ (t- 3), find f*h....
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lecture 21 - 1 1 Summary of Lecture#21 • Convolution...

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