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

Today’s menu From yesterday (Signals x and y relationships) More: Describing Common Signals Periodicity Leo Lam © 2010-2011

Your turn! Find the period of: No LCM exists! Why? No LCM because the period of the second term is NOT rational. It is NOT periodic. No LCM exists! Why? Leo Lam © 2010-2011

A few more Not rational, so not periodic Decaying term means pattern does not repeat exactly, so not periodic Leo Lam © 2010-2011

Summary Description of common signals Periodicity Leo Lam © 2010-2011

Playing with signals Operations with signals add, subtract, multiply, divide signals pointwise time delay, scaling, reversal Properties of signals (cont.) even and odd Leo Lam © 2010-2011

Adding signals 1 2 3 t + = ?? x(t) y(t) x(t)+y(t) Leo Lam © 2010-2011

Delay signals 1 t 1 3 4 unit pulse signal P(t) t 1 What does y(t)=p(t-3) look like? 3 4 Leo Lam © 2010-2011

Multiply signals Leo Lam © 2010-2011

Scaling time Speed-up: y(t)=x(2t) is x(t) sped up by a factor of 2 y(t)=x(t) y(t)=x(2t) 1 1 t t 1 .5 How could you slow x(t) down by a factor of 2? Leo Lam © 2010-2011

Scaling time y(t)=x(t/2) is x(t) slowed down by a factor of 2 t 1 -1 y(t)=x(t/2) 2 -2 y(t)=x(t) Leo Lam © 2010-2011

Playing with signals 8 t 3 5 1 t 1 What is y(t) in terms of the unit pulse p(t)? Need: Wider (x-axis) factor of 2 Taller (y-axis) factor of 8 Delayed (x-axis) 3 seconds Not to scale! But the need should be clear, the pulse needs to be wider (factor of 2) and taller (factor of 8), and then “shifted/delayed” to the right by 3 seconds. 1 t 1 Leo Lam © 2010-2011

Playing with signals 8 t 3 5 8 t 2 8 t 3 5 first step: second step: in terms of unit pulse p(t) 8 t 3 5 first step: 8 t 2 t 8 second step: 3 5 Leo Lam © 2010-2011

Playing with signals 8 t 3 5 8 t 2 8 t 3 5 first step: in terms of unit pulse p(t) 8 t 3 5 first step: 8 t 2 replace t by t-3: second step: t 8 3 5 Is it correct? Leo Lam © 2010-2011

Playing with signals 8 t 3 5 Double-check: pulse starts: pulse ends: Leo Lam © 2010-2011

Do it in reverse t Sketch 1 Leo Lam © 2010-2011

Do it in reverse 3 4 t Let then Double-check pulse starts: 3t-3 = 0 that is, y(t) is a delayed pulse p(t-3) sped up by 3. 1 1 4/3 3 4 Double-check pulse starts: 3t-3 = 0 pulse ends: 3t-3=1 Leo Lam © 2010-2011

Order matters With time operations, order matters y(t)=x(at+b) can be found by: Shift by b then scale by a (delay signal by b, then speed it up by a) w(t)=x(t+b)  y(t)=w(at)=x(at+b) Scale by a then shift by b/a w(t)=x(at)  y(t)=w(t+b/a)=x(a(t+b/a))=x(at+b) Leo Lam © 2010-2011

Playing with time 1 t 2 look like? What does 1 -2 Time reverse of speech: Also a form of time scaling, only with a negative number Leo Lam © 2010-2011

Playing with time t 1 2 1 -2 3 t Describe z(t) in terms of w(t) Leo Lam © 2010-2011

Playing with time 1 t 2 x(t) 1 -2 3 time reverse it: x(t) = w(-t) you replaced the t in x(t) by t-3. so replace the t in w(t) by t-3: x(t-3) = w(-(t-3)) time reverse it: x(t) = w(-t) delay it by 3: z(t) = x(t-3) so z(t) = w(-(t-3)) = w(-t + 3) Leo Lam © 2010-2011

Playing with time 1 t 2 x(t) 1 -2 3 Doublecheck: w(t) starts at 0 so -t+3 = 0 gives t= 3, this is the start (tip) of the triangle z(t). w(t) ends at 2 So -t+3=2 gives t=1, z(t) ends there 1 -2 3 x(t) z(t) = w(-t + 3) Leo Lam © 2010-2011

Summary: Arithmetic: Add, subtract, multiple Time: delay, scaling, shift, mirror/reverse And combination of those Leo Lam © 2010-2011