Signals and Systems EE235 Lecture 14 Leo Lam © 2010-2011

UNIX UNIX is basically a simple operating system but you have to be a genius to understand the simplicity. 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) 3 Leo Lam © 2010-2011

Another example At all t t<0 4 Shift Multiply Integrate The product of these two signals is zero where they don’t overlap 4 Leo Lam © 2010-2011

Another example At all t 0≤t<0.5 5 h(t) moving right Shift Multiply Integrate h(t) moving right 5 Leo Lam © 2010-2011

Another example At all t 0.5≤t<1 6 h(t) moving right Shift Multiply Integrate h(t) moving right 6 Leo Lam © 2010-2011

Another example At all t 1≤t<1.5 7 h(t) moving right Shift Multiply Integrate h(t) moving right 7 Leo Lam © 2010-2011

Another example At all t 1.5≤t? 8 Shift Multiply Integrate y(t)=0 because there is no more overlapping 8 Leo Lam © 2010-2011

Another example At all t Combining Can you plot and formulate it? 9 Leo Lam © 2010-2011

Another example At all t 10 Leo Lam © 2010-2011

Few things to note Three things: Stretching the thinking 11 Width of y(t) = Width of x(t)+Width of h(t) Start time adds End time adds y(t) is smoother than x(t) and h(t) (mostly) Stretching the thinking What if one signal has infinite width? 11 Leo Lam © 2010-2011

From yesterday Stretching the thinking What if one signal has infinite width? Width = infinite (infinite overlapping) Start time and end time all infinite 12 Leo Lam © 2010-2011

One more example For all t: x(t) t 13 2 1 -1 Flip Shift Can you guess the “width” of y(t)? 13 Leo Lam © 2010-2011

One more example For all t: x(t) t 14 2 1 -1 Multiply & integrate Leo Lam © 2010-2011

Summary Convolution examples Leo Lam © 2010-2011