1. Leo Lam © 2010-2012 Signals and Systems EE235

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

3. Time invariance Leo Lam © 2010-2012 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 =

4. Time invariance example Leo Lam © 2010-2012 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 ).

5. Time invariance example Leo Lam © 2010-2012 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

6. Time invariance example Leo Lam © 2010-2012 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 0 3 4 shifted system output y(t-3) = x(5(t-3)) t 0 3 8 shifted system input x(t-3) 0.6 1.6 t system output for shifted system input T(x(t-3)) = x(5t-3)

7. Time invariance example Leo Lam © 2010-2012 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 ).

8. System properties Leo Lam © 2010-2011 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

9. Linearity Leo Lam © 2010-2011 Order of addition and multiplication doesn’t matter. = System T System T Linear combination System 1 st Combo 1 st Linear combination

10. Linearity Leo Lam © 2010-2011 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

11. Linearity Proof Leo Lam © 2010-2011 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

12. Linearity Example Leo Lam © 2010-2011 Is T linear? T x(t)y(t)=cx(t) Equal  Linear

13. Linearity Example Leo Lam © 2010-2011 Is T linear? Not equal  non-linear T x(t)y(t)=(x(t)) 2

14. Linearity Example Leo Lam © 2010-2011 Is T linear? Not equal  non-linear T x(t)y(t)=x(t)+5

15. Linearity Example Leo Lam © 2010-2011 Is T linear? =

16. Linearity unique case Leo Lam © 2010-2011 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”

17. Spotting non-linearity Leo Lam © 2010-2011 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!

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

19. Superposition example Leo Lam © 2010-2012 Graphically 19 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)

20. 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 20 x 1 (t) T 1 1 y 1 (t) 1 1 2 x 2 (t) T 1 1 y 2 (t) 1 1 32

21. Superposition example Leo Lam © 2010-2012 Unique case can be used negatively 21 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

22. 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