Leo Lam © Signals and Systems EE235 October 14 th Friday Online version

Leo Lam © Today’s menu Superposition (Quick recap) System Properties Summary LTI System – Impulse response

Superposition Leo Lam © Superposition is… Weighted sum of inputs  weighted sum of outputs “Divide & conquer”

Superposition example Leo Lam © Graphically 4 x 1 (t) T 1 1 y 1 (t) x 2 (t) T 1 1 y 2 (t) T 1 ? 2 y 1 (t) 1 -y 2 (t)

Superposition example Leo Lam © Slightly aside (same system) Is it time-invariant? No idea: not enough information Single input-output pair cannot test positively 5 x 1 (t) T 1 1 y 1 (t) x 2 (t) T 1 1 y 2 (t)

Superposition example Leo Lam © Unique case can be used negatively 6 x 1 (t) T 1 1 y 1 (t) x 2 (t) T 1 y 2 (t) 1 -2 NOT Time Invariant: Shift by 1  shift by 2 x 1 (t)=u(t) S y 1 (t)=tu(t) NOT Stable: Bounded input gives unbounded output

Summary: System properties –Causal: output does not depend on future input times –Invertible: can uniquely find system input for any output –Stable: bounded input gives bounded output –Time-invariant: Time-shifted input gives a time-shifted output –Linear: response to linear combo of inputs is the linear combo of corresponding outputs Leo Lam ©

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)

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

Briefly: recall Dirac Delta Function Leo Lam © t t x(t)  t-3) 3 t x  t-3) Got a gut feeling here?

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)? 18 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

Leo Lam © Summary LTI System – Impulse response Leading into Convolution!