1. Leo Lam © 2010-2012 Signals and Systems EE235 http://www.youtube.com/v/iv6fo KX5BQY

2. Leo Lam © 2010-2012 Today’s menu Good weekend? System properties –Linearity –Superposition!

3. System properties Leo Lam © 2010-2012 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

4. Linearity Leo Lam © 2010-2012 Order of addition and multiplication doesn’t matter. = System T System T Linear combination System 1 st Combo 1 st Linear combination

5. Linearity Leo Lam © 2010-2012 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

6. Linearity Proof Leo Lam © 2010-2012 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

7. Linearity Example Leo Lam © 2010-2012 Is T linear? T x(t)y(t)=cx(t) Equal  Linear

8. Linearity Example Leo Lam © 2010-2012 Is T linear? Not equal  non-linear T x(t)y(t)=(x(t)) 2

9. Linearity Example Leo Lam © 2010-2012 Is T linear? Not equal  non-linear T x(t)y(t)=x(t)+5

10. Linearity Example Leo Lam © 2010-2012 Is T linear? =

11. Linearity unique case Leo Lam © 2010-2012 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”

12. Non-Linearity Rules of thumbs Leo Lam © 2010-2012 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!

13. Superposition Leo Lam © 2010-2012 Superposition is… Weighted sum of inputs  weighted sum of outputs “Divide & conquer”

14. Superposition example Leo Lam © 2010-2012 Graphically 14 x 1 (t) T 1 1 y 1 (t) 1 1 2 x 2 (t) T 1 1 y 2 (t) 1 1 32 T 1 ? 2 y 1 (t) 1 -y 2 (t)

15. Superposition example Leo Lam © 2010-2012 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) 1 1 2 x 2 (t) T 1 1 y 2 (t) 1 1 32

16. Superposition example Leo Lam © 2010-2012 Unique case can be used negatively 16 x 1 (t) T 1 1 y 1 (t) 1 1 2 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

17. 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 © 2010-2012

18. 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 © 2010-2012 Impulse Response h(t)

19. Briefly: recall superposition Leo Lam © 2010-2012 Superposition is… Weighted sum of inputs  weighted sum of outputs

20. Using superposition Leo Lam © 2010-2012 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

21. Briefly: recall Dirac Delta Function Leo Lam © 2010-2012 3t t x(t)  t-3) 3 t x  t-3) Got a gut feeling here?

22. Building x(t) with δ(t) Leo Lam © 2010-2012 Using the sifting properties: Change of variable: t   t0  tt0  t From a constant to a variable =