Leo Lam © Signals and Systems EE235 KX5BQY

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

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”

Non-Linearity Rules of thumbs 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 14 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 15 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 16 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 ©

Impulse response (Definition) Any signal can be built out of impulses Impulse response is the response of any Linear Time Invariant system when the input is a unit impulse Leo Lam © Impulse Response h(t)

Briefly: recall superposition Leo Lam © Superposition is… Weighted sum of inputs  weighted sum of outputs

Using superposition Leo Lam © Easiest when: x k (t) are simple signals (easy to find y k (t)) x k (t) are similar for different k Two different building blocks: –Impulses with different time shifts –Complex exponentials (or sinusoids) of different frequencies

Briefly: recall Dirac Delta Function Leo Lam © t t x(t)  t-3) 3 t x  t-3) Got a gut feeling here?

Building x(t) with δ(t) Leo Lam © Using the sifting properties: Change of variable: t   t0  tt0  t From a constant to a variable =