Signals and Systems EE235 Leo Lam © 2010-2012

Today’s menu Homework 2 posted Lab 2 this week Convolution! Leo Lam © 2010-2012

Summary: Impulse response for LTI Systems First we had Superposition Weighted “sum” of impulses in Weighted “sum” of impulse responses out T Linear T d(t-t) h(t-t) Time Invariant 3 Leo Lam © 2010-2012

Summary: another vantage point LINEARITY And with this, you have learned Convolution! Output! TIME INVARIANCE An LTI system can be completely described by its impulse response! 4 Leo Lam © 2010-2012

Convolution Integral Standard Notation The output of a system is its input convolved with its impulse response 5 Leo Lam © 2010-2012

Quick recap 6 x(t) is the sum of the weighted shifted impulses Leo Lam © 2010-2012

Convolution integral Function of  , not t =h(- (-t)) Function of  shifted by t Function of  , not t 7 Leo Lam © 2010-2012

Convolution integral Two ways to evaluate: Mathematically Graphically For graphical convolution, see demo in Riskin interactive notes (lesson 6, lesson 7) 8 Leo Lam © 2010-2012

Convolution (mathematically) Use sampling property of delta: Evaluate integral to arrive at output signal: Does this make sense physically? 9 Leo Lam © 2010-2012

Convolution (graphically) x(τ) and h(t- τ) no overlap, y(t)=0 Does not move wrt t -6 τ -2 2 y(t=-5) -5 t Goal: Find y(t) 10 Leo Lam © 2010-2012

Convolution (graphically) Overlapped at τ=0 -2 2 τ y(t=-1) t -5 -1 11 Leo Lam © 2010-2012

Convolution (graphically) Both overlapped 2 y(t=1) -5 -1 1 -1 12 Leo Lam © 2010-2012

Convolution (graphically) Overlapped at τ=2 Does it make sense? 2 4 y(t=3) -1 1 3 13 Leo Lam © 2010-2012

Convolution (mathematically) Using Linearity Let’s focus on this part 14 Leo Lam © 2010-2012

Convolution (mathematically) Consider this part: Recall that: And the integral becomes: 15 Leo Lam © 2010-2012

Convolution (mathematically) Apply delta rules: Same answer as the graphically method 16 Leo Lam © 2010-2012

Summary: Convolution Draw x() Draw h() Flip h() to get h(-) Shift forward in time by t to get h(t-) Multiply x() and h(t-) for all values of  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) 17 Leo Lam © 2010-2012

Flip Shift Multiply Integrate Summary: Convolution 18 Leo Lam © 2010-2012

Summary Convolution! Leo Lam © 2010-2012