Signals and Systems EE235 Leo Lam © 2010-2011

Convergence Two mathematicians are studying a convergent series. The first one says: "Do you realize that the series converges even when all the terms are made positive?" The second one asks: "Are you sure?" "Absolutely!" Leo Lam © 2010-2011

Today’s menu Homework solutions posted Midterm: time to vote More convoluted convolution Leo Lam © 2010-2011

Convolution Integral The output of a system is its input convolved with its impulse response 4 Leo Lam © 2010-2011

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) 5 Leo Lam © 2010-2011

Convolution (graphically) Overlapped at τ=0 -2 2 τ y(t=-1) t -5 -1 6 Leo Lam © 2010-2011

Convolution (graphically) Both overlapped 2 y(t=1) -5 -1 1 -1 7 Leo Lam © 2010-2011

Convolution (graphically) Overlapped at τ=2 Does it make sense? 2 4 y(t=3) -1 1 3 8 Leo Lam © 2010-2011

Convolution (mathematically) Using Linearity Let’s focus on this part 9 Leo Lam © 2010-2011

Convolution (mathematically) Consider this part: Recall that: And the integral becomes: 10 Leo Lam © 2010-2011

Convolution (mathematically) Apply delta rules: Same answer as the graphically method 11 Leo Lam © 2010-2011

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) 12 Leo Lam © 2010-2011

Flip Shift Multiply Integrate Summary: Convolution 13 Leo Lam © 2010-2011

Summary Convolution! Leo Lam © 2010-2011