Convolution Integral **new** 2004/07/15 13:50:05.295 GMT-5 2004/07/15 13:51:15.471 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Jeffrey Silverman jsilv@rice.edu An introduction to the convolution integral. Combine these facts Signals can be represented as infinite sums (integrals) of scaled and shifted impulses. Time invariance of system implies . Linearity of the system implies .

Let's do it!

Derivation of Convolution .

h is LTI.

Step 1 Signal representation in terms of impulses: f t τ f t δ t τ

Step 2 Use the time invariance and scaling part of linearity ().

f τ is a constant with respect to the variable t , which the system operates on.

Step 3 Use the additivity part of linearity to add up an infinite number of these terms, one for each τ ().

f t is the input and y t is the output.

Alternative Derivation in Terms of Limits and Sums .

Call this δ Δ t .

δ t put through h yields h t Δ 0 δ Δ t put through h yields h t Δ 0 δ Δ t n Δ put through h yields Δ 0 h t n Δ Δ 0 f n Δ δ t n Δ Δ put through h yields Δ 0 f n Δ h t n Δ Δ 0 n f n Δ δ t n Δ Δ put through h yields Δ 0 n f n Δ h t n Δ Δ τ f τ δ t τ put through h yields τ f τ h t τ f t put through h yields y t

Any way you look at it, we have .

h is LTI.

Convolution Integral for LTI Systems y t τ f τ h t τ You will need to memorize . f t h t y t Interpretation y t is an infinite sum of shifted "shock responses" each weighted by a signal value. This equation holds for all LTI systems!!!

Pictorially: Konvolution (Graphical Interpretation I) . .

h t is the impulse response.

y t is the convolution integral.