Leo Lam © Signals and Systems EE235

Leo Lam © Today’s menu Good weekend? System properties –Time Invariance –Linearity –Superposition!

Time invariance Leo Lam © 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 =

Time invariance example Leo Lam © Still you… T(x(t)) = 3x(t - 5) 1.y(t) = 3x(t-5) 2.y(t – t 0 ) = 3x(t-t 0 -5) 3.T(x(t – t 0 )) = 3x(t-t 0 -5) 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 ).

Time invariance example Leo Lam © Still you… T(x(t)) = x(5t) 1.y(t) = x(5t) 2.y(t – 3) = x(5(t-3)) = x(5t – 15) 3.T(x(t-3)) = x(5t- 3) 4.Oops… Not time invariant! Does it make sense? KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ). Shift then scale

Time invariance example Leo Lam © Graphically: T(x(t)) = x(5t) 1.y(t) = x(5t) 2.y(t – 3) = x(5(t-3)) = x(5t – 15) 3.T(x(t-3)) = x(5t- 3) t 0 system input x(t) 5 t 0 system output y(t) = x(5t) 1 t shifted system output y(t-3) = x(5(t-3)) t shifted system input x(t-3) t system output for shifted system input T(x(t-3)) = x(5t-3)

Time invariance example Leo Lam © Integral 1.First: 2.Second: 3.Third: 4.Lastly: 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 ).

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

Linearity Leo Lam © Order of addition and multiplication doesn’t matter. = System T System T Linear combination System 1 st Combo 1 st Linear combination

Linearity Leo Lam © Positive 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

Linearity Proof Leo Lam © Combo Proof Step 1: find y i (t) Step 2: find y_combo Step 3: find T{x_combo} Step 4: If y_combo = T{x_combo} Linear System T System T Linear combination System 1 st Combo 1 st Linear combination

Linearity Example Leo Lam © Is T linear? T x(t)y(t)=cx(t) Equal  Linear

Linearity Example Leo Lam © Is T linear? Not equal  non-linear T x(t)y(t)=(x(t)) 2

Linearity Example Leo Lam © Is T linear? Not equal  non-linear T x(t)y(t)=x(t)+5

Linearity Example Leo Lam © Is T linear? =

Linearity unique case Leo Lam © 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”

Spotting non-linearity Leo Lam © 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. (although sometimes ok) NOT Formal Proofs!

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

Superposition example Leo Lam © Graphically 19 x 1 (t) T 1 1 y 1 (t) x 2 (t) T 1 1 y 2 (t) T 1 ? 2 y 1 (t) 1 -y 2 (t)

Superposition example Leo Lam © Slightly aside (same system) Is it time-invariant? No idea: not enough information Single input-output pair cannot test positively 20 x 1 (t) T 1 1 y 1 (t) x 2 (t) T 1 1 y 2 (t)

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

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 ©