Signals and Systems EE235 Leo Lam © 2010-2011

Chicken Why did the chicken cross the Möbius Strip? To get to the other…er…um… Leo Lam © 2010-2011

Today’s menu System properties Lots of examples! Linearity Time invariance Stability Invertibility Causality Lots of examples! Leo Lam © 2010-2011

“System Response is the same no matter when you run the system.” System properties Linearity: A System is Linear if it meets the following two criteria: Time-invariance: A System is Time-Invariant if it meets this criterion If and Then If Then “System Response is the same no matter when you run the system.” “System Response to a linear combination of inputs is the linear combination of the outputs.” If Then Leo Lam © 2010-2011

“The system doesn’t blow up if given reasonable inputs.” System properties Stability: A System is BIBO Stable if it meets this criterion Invertibility: A System is Invertible if it meets this criterion: BIBO = “Bounded input, bounded output” If Then “If you know the output signal, then you know exactly what the input signal was.” “The system doesn’t blow up if given reasonable inputs.” If You can undo the effects of the system. Leo Lam © 2010-2011

System properties Causality: A System is Causal if it meets this criterion If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a The output depends only on current or past values of the input. “The system does not anticipate the input.” (It does not laugh before it’s tickled!) Leo Lam © 2010-2011

Test for Causality y(t) = 4x(t) y(t) = x(t –3) y(t) = x(t + 5) System is causal if output depends only on past and present input signal y(t) = 4x(t) y(t) = x(t –3) y(t) = x(t + 5) y(t) = x(3t) y(t) = (t + 5)x(t) y(t) = x(-t) causal (amplification) causal (delay) non-causal (time-shift forward, y(0)=x(5)) non-causal (speed-up, y(1)=x(3)) causal (ramp times x(t)) non-causal (time reverse, negative time needs future, y(-1)=x(1)) Leo Lam © 2010-2011

causal if Causality Example What values of t0 would make T causal? Leo Lam © 2010-2011

Causality Example Is T causal? YES Depends only on past and present signals Leo Lam © 2010-2011

Causality Example What values of a would make T causal? Leo Lam © 2010-2011

Causality Example NOT causal: x(t)’s include t =t+1 Causal: Change variable, y(t) does not depend on future t. NOT causal: x(t)’s include t =2t Leo Lam © 2010-2011

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 © 2010-2011

Invertibility Example Is T invertible? NOT Invertible Leo Lam © 2010-2011

Invertibility Example Is T invertible? YES Leo Lam © 2010-2011

Invertibility Example y(t) = 4x(t) y(t) = x(t –3) y(t) = x2(t) y(t) = x(3t) y(t) = (t + 5)x(t) y(t) = cos(x(t)) invertible: Ti{y(t)}=y(t)/4 invertible: Ti{y(t)}=y(t+3) NOT invertible: don’t know sign of x(t) invertible: Ti{y(t)}=y(t/3) NOT invertible: can’t find x(-5) NOT invertible: x=0,2 π,4 π,… all give cos(x)=1 Leo Lam © 2010-2011

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 © 2010-2011

Stability test Is it stable? Bounded input results in a bounded output  STABLE! Leo Lam © 2010-2011

Stability test How about this? Let for all t Stable Leo Lam © 2010-2011

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 © 2010-2011

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 © 2010-2011