Signals and Systems EE235 Leo Lam © 2010-2012

Pet Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's mostly been working on PET projects. Leo Lam © 2010-2012

Today’s menu System properties examples Stability Time invariance Linearity Leo Lam © 2010-2012

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 © 2010-2012

Stability test Is it stable? Bounded input results in a bounded output  STABLE! Leo Lam © 2010-2012

Stability test How about this? Let for all t Stable Leo Lam © 2010-2012

Stability test How about this, your turn? 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. Not BIBO stable Leo Lam © 2010-2012

Stability test How about this, your turn? Stable Stable NOT Stable Tx(t) is a ramp 1/x(t) blows up at x(t) = 0 Stable NOT Stable Leo Lam © 2010-2012

“System Response is the same no matter when you run the system.” System properties Time-invariance: A System is Time-Invariant if it meets this criterion “System Response is the same no matter when you run the system.” Leo Lam © 2010-2012

Time invariance The system behaves the same no matter when you use it Input is delayed by t0 seconds, output is the same but delayed t0 seconds If then System T Delay t0 x(t) x(t-t0) y(t) y(t-t0) T[x(t-t0)] System 1st Delay 1st = If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Leo Lam © 2010-2012

Time invariance example T{x(t)}=2x(t) x(t) y(t)= 2x(t) y(t-t0) T Delay x(t-t0) 2x(t-t0) Delay T Amplification of 2 is time invariant Identical  time invariant! Leo Lam © 2010-2012

Time invariance test Test steps: Find y(t) Find y(t-t0) Find T{x(t-t0)} Compare! IIf y(t-t0) = T{x(t-t0)} Time invariant! Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2012

Time invariance example T(x(t)) = x2(t) y(t) = x2(t) y(t-t0) =x2(t-t0) T(x(t-t0)) = x2(t-t0) y(t-t0) = T(x(t-t0)) Time invariant! KEY: In step 2 you replace t by t-t0. In step 3 you replace x(t) by x(t-t0). Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2012

Time invariance example Your turn! T{(x(t)} = t x(t) y(t) = t*x(t) y(t-t0) =(t-t0) x(t-t0) T(x(t-t0)) = t x(t-t0) y(t-t0)) != T(x(t-t0)) Not time invariant! KEY: In step 2 you replace t by t-t0. In step 3 you replace x(t) by x(t-t0). Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2012

Time invariance example Still you… T(x(t)) = 3x(t - 5) y(t) = 3x(t-5) y(t – t0) = 3x(t-t0-5) T(x(t – t0)) = 3x(t-t0-5) y(t-t0)) = T(x(t-t0)) Time invariant! KEY: In step 2 you replace t by t-t0. In step 3 you replace x(t) by x(t-t0). Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2012

Time invariance example Still you… T(x(t)) = x(5t) y(t) = x(5t) y(t – 3) = x(5(t-3)) = x(5t – 15) T(x(t-3)) = x(5t- 3) Oops… Not time invariant! Does it make sense? KEY: In step 2 you replace t by t-t0. In step 3 you replace x(t) by x(t-t0). Shift then scale Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2012

Time invariance example system output y(t) = x(5t) 1 Graphically: T(x(t)) = x(5t) y(t) = x(5t) y(t – 3) = x(5(t-3)) = x(5t – 15) T(x(t-3)) = x(5t- 3) t 0 3 4 shifted system output y(t-3) = x(5(t-3)) t system input x(t) 5 0.6 1.6 t system output for shifted system input T(x(t-3)) = x(5t-3) t 0 3 8 shifted system input x(t-3) Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2012

Time invariance example Integral First: Second: Third: Lastly: Time invariant! KEY: In step 2 you replace t by t-t0. In step 3 you replace x(t) by x(t-t0). Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2012