2. Leo Lam © 2010-2011 Signals and Systems EE235 October 14 th Friday Online version

3. Leo Lam © 2010-2011 Today’s menu Superposition (Quick recap) System Properties Summary LTI System – Impulse response

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

5. Superposition example Leo Lam © 2010-2011 Graphically 4 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)

6. Superposition example Leo Lam © 2010-2011 Slightly aside (same system) Is it time-invariant? No idea: not enough information Single input-output pair cannot test positively 5 x 1 (t) T 1 1 y 1 (t) 1 1 2 x 2 (t) T 1 1 y 2 (t) 1 1 32

7. Superposition example Leo Lam © 2010-2011 Unique case can be used negatively 6 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

8. 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

9. 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 © 2010-2011 Impulse Response h(t)

10. Using superposition Leo Lam © 2010-2011 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

11. Briefly: recall Dirac Delta Function Leo Lam © 2010-2011 3t t x(t)  t-3) 3 t x  t-3) Got a gut feeling here?

12. Building x(t) with δ(t) Leo Lam © 2010-2011 Using the sifting properties: Change of variable: t   t0  tt0  t From a constant to a variable =

13. Building x(t) with δ(t) Leo Lam © 2010-2011 Jumped a few steps…

14. Building x(t) with δ(t) Leo Lam © 2010-2011 Another way to see… x(t) t   (t) t  1/  Compensate for the height of the “unit pulse” Value at the “tip”

15. So what? Leo Lam © 2010-2011 Two things we have learned If the system is LTI, we can completely characterize the system by how it responds to an input impulse. Impulse Response h(t)

16. h(t) Leo Lam © 2010-2011 For LTI system T x(t)y(t) T (t) h(t) Impulse  Impulse response T (t-t 0 ) h(t-t 0 ) Shifted Impulse  Shifted Impulse response

17. Finding Impulse Response (examples) Leo Lam © 2010-2011 Let x(t)=(t) What is h(t)?

18. Finding Impulse Response Leo Lam © 2010-2011 For an LTI system, if –x(t)=(t-1)  y(t)=u(t)-u(t-2) –What is h(t)? h(t)  (t-1) u(t)-u(t-2) h(t)=u(t+1)-u(t-1) An impulse turns into two unit steps shifted in time Remember the definition, and that this is time invariant

19. Finding Impulse Response Leo Lam © 2010-2011 Knowing T, and let x(t)=(t) What is h(t)? 18 This system is not linear –impulse response not useful.

20. Summary: Impulse response for LTI Systems Leo Lam © 2010-2011 19 T  (t-  )h(t-  ) Time Invariant T Linear Weighted “sum” of impulses in Weighted “sum” of impulse responses out First we had Superposition

21. Summary: another vantage point Leo Lam © 2010-2011 20 LINEARITY TIME INVARIANCE Output! An LTI system can be completely described by its impulse response! And with this, you have learned Convolution!

22. Convolution Integral Leo Lam © 2010-2011 21 Standard Notation The output of a system is its input convolved with its impulse response

23. Leo Lam © 2010-2011 Summary LTI System – Impulse response Leading into Convolution!