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

Today’s menu Good weekend? System properties Time invariance (cont’) Linearity Superposition! Leo Lam © 2010-2011

“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 © 2010-2011

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 © 2010-2011

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 the t by t-t0. In step 3 you replace the x(t) by x(t-t0). Shift then scale Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2011

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 © 2010-2011

Time invariance example Integral First: Second: Third: Lastly: Time invariant! KEY: In step 2 you replace the t by t-t0. In step 3 you replace the x(t) by x(t-t0). Basically taking BOTH routes should come to the same answer. Leo Lam © 2010-2011

System properties Linearity: A System is Linear if it meets the following two criteria: Together…superposition If and Additivity Then If Then Scaling “System Response to a linear combination of inputs is the linear combination of the outputs.” Leo Lam © 2010-2011

Linearity Order of addition and multiplication doesn’t matter. System T Linear combination System 1st Combo 1st = If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Linear combination Leo Lam © 2010-2011

Linearity Positive proof Negative proof Prove both scaling & additivity separately Prove them together with combined formula Negative proof Show either scaling OR additivity fail (mathematically, or with a counter example) Show combined formula doesn’t hold If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Leo Lam © 2010-2011

Linearity Proof Combo Proof System 1st Combo 1st Step 1: find yi(t) Step 2: find y_combo Step 3: find T{x_combo} Step 4: If y_combo = T{x_combo} Linear System T Linear combination System 1st Combo 1st If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Leo Lam © 2010-2011

Linearity Example Is T linear? T x(t) y(t)=cx(t) Equal  Linear If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Equal  Linear Leo Lam © 2010-2011

Linearity Example Is T linear? T x(t) y(t)=(x(t))2 If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Not equal  non-linear Leo Lam © 2010-2011

Linearity Example Is T linear? T x(t) y(t)=x(t)+5 If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Not equal  non-linear Leo Lam © 2010-2011

Linearity Example Is T linear? = If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds = Leo Lam © 2010-2011

Linearity unique case How about scaling with 0? If T{x(t)} is a linear system, then zero input must give a zero output A great “negative test” If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds Leo Lam © 2010-2011

Linearity Rules of thumbs multiplying x(t) by another x() y(t)=g[x(t)] where g() is nonlinear piecewise definition of y(t) in terms of values of x, e.g. NOT Formal Proofs! (although sometimes ok) Leo Lam © 2010-2011

Superposition Superposition is… Weighted sum of inputs  weighted sum of outputs “Divide & conquer” Leo Lam © 2010-2011

Superposition example Graphically x1(t) T 1 y1(t) 2 x2(t) y2(t) 3 -1 ? -y2(t) 19 Leo Lam © 2010-2011

Superposition example Slightly aside (same system) Is it time-invariant? No idea: not enough information Single input-output pair cannot test positively x1(t) T 1 y1(t) 2 x2(t) y2(t) 3 Specific input-output pair (or pairs), you CANNOT test positively for system properties. 20 Leo Lam © 2010-2011

Superposition example Unique case can be used negatively x1(t) T 1 y1(t) 2 x2(t) -1 y2(t) -2 NOT Time Invariant: Shift by 1  shift by 2 x1(t)=u(t) S y1(t)=tu(t) NOT Stable: Bounded input gives unbounded output 21 Leo Lam © 2010-2011

Summary: System properties Causal: output does not depend on future input times Invertible: can uniquely find system input for any output Stable: bounded input gives bounded output Time-invariant: Time-shifted input gives a time-shifted output Linear: response to linear combo of inputs is the linear combo of corresponding outputs Leo Lam © 2010-2011