Leo Lam © Signals and Systems EE235

Leo Lam © 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 © Today’s menu Lab 3 this week Convolution!

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

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!

y(t) at specific time t 0 Leo Lam © Flip Shift Multiply Integrate Here t 0 =3/4 y(t 0 =3/4)= ?3/4

y(t) at all t Leo Lam © At all t t<0 The product of these two signals is zero where they don’t overlap ShiftMultiplyIntegrate

y(t) at all t Leo Lam © At all t 0≤t<1 ShiftMultiplyIntegrate

y(t) at all t Leo Lam © At all t 1≤t<2 y(t)=2-t for 1≤t<2 ShiftMultiplyIntegrate

y(t) at all t Leo Lam © At all t t≥2 y(t)=0 for t≥2 (same as t<0, no overlap) ShiftMultiplyIntegrate

y(t) at all t Leo Lam © Combine it all –y(t)=0 for t 2 –y(t)=t for 0≤t<1 –y(t)=2-t for 1≤t<2

Leo Lam © Summary Convolution and first few examples