1. Leo Lam © 2010-2013 Signals and Systems EE235

2. Leo Lam © 2010-2013 Chicken Why did the chicken cross the Möbius Strip? To get to the other…er…um…

3. Leo Lam © 2010-2013 Today’s menu System properties –Linearity –Time invariance –Stability –Invertibility –Causality Lots of examples!

4. System properties Leo Lam © 2010-2013 Causality: A System is Causal if it meets this criterion “The system does not anticipate the input.” (It does not laugh before it’s tickled!) The output depends only on current or past values of the input. If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a

5. Causality Example Leo Lam © 2010-2013 What values of t 0 would make T causal? causal if

6. Causality Example Leo Lam © 2010-2013 Is T causal? YES Depends only on past and present signals

7. Causality Example Leo Lam © 2010-2013 What values of a would make T causal?

8. Causality Example Leo Lam © 2010-2013 NOT causal: x(t)’s include t =t+1 NOT causal: x(t)’s include t =2t Causal: Change variable, y(t) does not depend on future t.

9. Invertibility test Positive test: find the inverse For some systems, you need tools that we’ll learn later in the quarter… Negative test: find an output that could be generated by two different inputs (note that these two different inputs might only differ at only one time value) Each input signal results in a unique output signal, and vice versa  Invertible Leo Lam © 2010-2013

10. Invertibility Example Leo Lam © 2010-2013 Is T invertible? NOT Invertible

11. Invertibility Example Leo Lam © 2010-2013 Is T invertible? YES

12. Invertibility Example Leo Lam © 2010-2013 1)y(t) = 4x(t) 2)y(t) = x(t –3) 3)y(t) = x 2 (t) 4)y(t) = x(3t) 5)y(t) = (t + 5)x(t) 6)y(t) = cos(x(t)) invertible: T i {y(t)}=y(t)/4 invertible: T i {y(t)}=y(t/3) invertible: T i {y(t)}=y(t+3) NOT invertible: don’t know sign of x(t) NOT invertible: can’t find x(-5) NOT invertible: x=0,2 π,4 π,… all give cos(x)=1

13. Stability test For positive proof: show analytically that –a “bounded input” signal gives a “bounded output” signal (BIBO stability) For negative proof: –Find one counter example, a bounded input signal that gives an unbounded output signal –Some good things to try: 1, u(t), cos(t), 0 Leo Lam © 2010-2013

14. Stability test Is it stable? Leo Lam © 2010-2013 Bounded input results in a bounded output  STABLE!

15. Stability test How about this? Leo Lam © 2010-2013 Stable Let for all t

16. Stability test How about this, your turn? Leo Lam © 2010-2013 Not BIBO stable Counter example: x(t)=u(t)  y(t)=5tu(t)=5r(t) Input u(t) is bounded. Output y(t) is a ramp, which is unbounded.

17. Stability test How about this, your turn? Leo Lam © 2010-2013 Stable NOT Stable Stable

18. System properties Leo Lam © 2010-2013 Time-invariance: A System is Time-Invariant if it meets this criterion “System Response is the same no matter when you run the system.”

19. Time invariance Leo Lam © 2010-2013 The system behaves the same no matter when you use it Input is delayed by t 0 seconds, output is the same but delayed t 0 seconds If then System T Delay t 0 System T Delay t 0 x(t) x(t-t 0 ) y(t) y(t-t 0 ) T[x(t-t 0 )] System 1 st Delay 1 st =

20. Time invariance example Leo Lam © 2010-2013 T{x(t)}=2x(t) x(t) y(t)= 2x(t) y(t-t 0 ) T Delay x(t-t 0 ) 2x(t-t 0 ) Delay T Identical  time invariant!

21. Time invariance test Leo Lam © 2010-2013 Test steps: 1.Find y(t) 2.Find y(t-t 0 ) 3.Find T{x(t-t 0 )} 4.Compare! IIf y(t-t 0 ) = T{x(t-t 0 )} Time invariant!

22. Time invariance example Leo Lam © 2010-2013 T(x(t)) = x 2 (t) 1.y(t) = x 2 (t) 2.y(t-t 0 ) =x 2 (t-t 0 ) 3.T(x(t-t 0 )) = x 2 (t-t 0 ) 4.y(t-t 0 ) = T(x(t-t 0 )) Time invariant! KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ).

23. Time invariance example Leo Lam © 2010-2013 Your turn! T{(x(t)} = t x(t) 1.y(t) = t*x(t) 2.y(t-t 0 ) =(t-t 0 ) x(t-t 0 ) 3.T(x(t-t 0 )) = t x(t-t 0 ) 4.y(t-t 0 )) != T(x(t-t 0 )) Not time invariant! KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ).