Signals and Systems EE235 Lecture 14 Leo Lam © 2010-2012

UNIX UNIX is basically a simple operating system but you have to be a genius to understand the simplicity. 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) 3 Leo Lam © 2010-2012

One more example For all t: x(t) t 4 2 1 -1 Flip Shift Can you guess the “width” of y(t)? 4 Leo Lam © 2010-2012

One more example For all t: x(t) t 5 2 1 -1 Multiply & integrate Leo Lam © 2010-2012

* Convolution examples 6 Approach? How to break it down? 1 t x(t) * 2 h(t) -1 y(t)=x(t)*h(t) Approach? How to break it down? System will start having non-zero output at time t = -1 The signal y(t) can be expressed in terms of 3 time regions: t<-1 (where y(t)=0), -1<t<1, t>1 6 Leo Lam © 2010-2012

* Convolution examples 7 Two non-zero regions: 1 t x(t) * 2 h(t) -1 y(t)=x(t)*h(t) Two non-zero regions: If you flip x(t), you’d get: If you flip h(t), you’d get: Identical? 7 Leo Lam © 2010-2012

Another example (Mathematic method) Approach? What does each part “look” like? y(t)= = 1 if 3 - > 0 = 1 if t - > 0 8 Leo Lam © 2010-2012

Another example (complicated) y(t)= = 1 if 3 - > 0 = 1 if t - > 0 Need to satisfy both: That is & Two cases to consider then: or 9 Leo Lam © 2010-2012

Another example y(t)= 10 For = 1 if t - > 0 = 1 if 3 - > 0 Leo Lam © 2010-2012

Another example 11 Combining two, with only one active at each t For Then integrate… 11 Leo Lam © 2010-2012

Summary Convolution examples Leo Lam © 2010-2012