Leo Lam © Signals and Systems EE235

Leo Lam © Today’s menu LTI System – Impulse response Lead in to Convolution

Impulse response (Definition) Any signal can be built out of impulses Impulse response is the response of any Linear Time Invariant system when the input is a unit impulse Leo Lam © Impulse Response h(t)

Briefly: recall superposition Leo Lam © Superposition is… Weighted sum of inputs  weighted sum of outputs

Using superposition Leo Lam © Easiest when: x k (t) are simple signals (easy to find y k (t)) x k (t) are similar for different k Two different building blocks: –Impulses with different time shifts –Complex exponentials (or sinusoids) of different frequencies

Building x(t) with δ(t) Leo Lam © Using the sifting properties: Change of variable: t   t0  tt0  t From a constant to a variable =

Building x(t) with δ(t) Leo Lam © Jumped a few steps…

Building x(t) with δ(t) Leo Lam © Another way to see… x(t) t   (t) t  1/  Compensate for the height of the “unit pulse” Value at the “tip”

So what? Leo Lam © Two things we have learned If the system is LTI, we can completely characterize the system by how it responds to an input impulse. Impulse Response h(t)

h(t) Leo Lam © For LTI system T x(t)y(t) T (t) h(t) Impulse  Impulse response T (t-t 0 ) h(t-t 0 ) Shifted Impulse  Shifted Impulse response

Finding Impulse Response (examples) Leo Lam © Let x(t)=(t) What is h(t)?

Finding Impulse Response Leo Lam © For an LTI system, if –x(t)=(t-1)  y(t)=u(t)-u(t-2) –What is h(t)? h(t)  (t-1) u(t)-u(t-2) h(t)=u(t+1)-u(t-1) An impulse turns into two unit steps shifted in time Remember the definition, and that this is time invariant

Finding Impulse Response Leo Lam © Knowing T, and let x(t)=(t) What is h(t)? 13 This system is not linear –impulse response not useful.

Summary: Impulse response for LTI Systems Leo Lam © T  (t-  )h(t-  ) Time Invariant T Linear Weighted “sum” of impulses in Weighted “sum” of impulse responses out First we had Superposition

Summary: another vantage point Leo Lam © LINEARITY TIME INVARIANCE Output! An LTI system can be completely described by its impulse response! And with this, you have learned Convolution!

Convolution Integral Leo Lam © Standard Notation The output of a system is its input convolved with its impulse response

Convolution Integral Leo Lam © Standard Notation The output of a system is its input convolved with its impulse response

Leo Lam © Summary Convolution!