Impulse Response Properties of an LTI System **new** 2004/07/15 10:58:01.994 GMT-5 2004/07/15 10:59:17.864 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Jeffrey Silverman jsilv@rice.edu time invariance linearity An introduction to the impulse response properties of an LTI system. LTI is linear and time-invariant: , , .

Time Invariance

Linearity

Bottom Line For an LTI system, it is easy to compute output due to the sum of shifted and scaled impulses. By the sifting property of impulses, any signal can be decomposed in terms of an infinite sum (integral) of shifted, scaled impulses.

Sifting f t τ f τ δ t τ where the integral is an "infinite sum" ().

Peaks up where t τ .

This can be derived from the properties of δ , but it also has a nice physical interpretation.

Interpret f(t) . Recall δ t Δ 0 (). . Sum over n (). Take Δ 0 n f n Δ δ Δ t n Δ Δ τ f τ δ t τ .

Call this δ Δ t .

The height is f n Δ .