Leo Lam © Signals and Systems EE235

Leo Lam © Surgery Five surgeons were taking a coffee break and were discussing their work. The first said, "I think accountants are the easiest to operate on. You open them up and everything inside is numbered.“ The second said, "I think librarians are the easiest to operate on. You open them up and everything inside is in alphabetical order.“ The third said, "I like to operate on electricians. You open them up and everything inside is color-coded.“ The fourth one said, "I like to operate on lawyers. They're heartless, spineless, gutless, and their heads and their butts are interchangeable." Fifth surgeon said, "I like Engineers...they always understand when you have a few parts left over at the end..."

Leo Lam © Today’s menu Lab 2 this week (posted last week) Convolution

Finding Impulse Response Leo Lam © Knowing T, and let x(t)=(t) What is h(t)? 4 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

Quick recap Leo Lam © x(t) is the sum of the weighted shifted impulses

Convolution integral Leo Lam © Function of  =h(- (-t)) Function of  shifted by t , not t

Convolution integral Leo Lam © Two ways to evaluate: –Mathematically –Graphically For graphical convolution, see demo in Riskin interactive notes (lesson 6, lesson 7)

Convolution (mathematically) Leo Lam © Use sampling property of delta: Evaluate integral to arrive at output signal: Does this make sense physically?

Convolution (graphically) Leo Lam © τ y(t=-5) -5 t Does not move wrt t -2 Goal: Find y(t) x( τ ) and h(t- τ ) no overlap, y(t)=0

Convolution (graphically) Leo Lam © τ t 2 -2 Overlapped at τ =0 y(t=-1)

Convolution (graphically) Leo Lam © Both overlapped y(t=1)

Convolution (graphically) Leo Lam © Overlapped at τ =2 y(t=3) Does it make sense?

Convolution (mathematically) Leo Lam © Using Linearity Let’s focus on this part

Convolution (mathematically) Leo Lam © Consider this part: Recall that: And the integral becomes:

Convolution (mathematically) Leo Lam © Same answer as the graphically method Apply delta rules:

Summary: Convolution Leo Lam © Draw x() 2.Draw h() 3.Flip h() to get h(-) 4.Shift forward in time by t to get h(t-) 5.Multiply x() and h(t-) for all values of  6.Integrate (add up) the product x()h(t-) over all  to get y(t) for this particular t value (you have to do this for every t that you are interested in)

Summary: Convolution Leo Lam © Flip Shift Multiply Integrate

Leo Lam © Summary Convolution!