1. Leo Lam © 2010-2013 Signals and Systems EE235

2. Leo Lam © 2010-2013 Today’s menu LTI System – Impulse response Lead in to Convolution

3. 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 © 2010-2013 Impulse Response h(t)

4. Briefly: recall superposition Leo Lam © 2010-2013 Superposition is… Weighted sum of inputs  weighted sum of outputs

5. Using superposition Leo Lam © 2010-2013 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

6. Building x(t) with δ(t) Leo Lam © 2010-2013 Using the sifting properties: Change of variable: t   t0  tt0  t From a constant to a variable =

7. Building x(t) with δ(t) Leo Lam © 2010-2013 Jumped a few steps…

8. Building x(t) with δ(t) Leo Lam © 2010-2013 Another way to see… x(t) t   (t) t  1/  Compensate for the height of the “unit pulse” Value at the “tip”

9. So what? Leo Lam © 2010-2013 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)

10. h(t) Leo Lam © 2010-2013 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

11. Finding Impulse Response (examples) Leo Lam © 2010-2013 Let x(t)=(t) What is h(t)?

12. Finding Impulse Response Leo Lam © 2010-2013 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

13. Finding Impulse Response Leo Lam © 2010-2013 Knowing T, and let x(t)=(t) What is h(t)? 13 This system is not linear –impulse response not useful.

14. Summary: Impulse response for LTI Systems Leo Lam © 2010-2013 14 T  (t-  )h(t-  ) Time Invariant T Linear Weighted “sum” of impulses in Weighted “sum” of impulse responses out First we had Superposition

15. Summary: another vantage point Leo Lam © 2010-2013 15 LINEARITY TIME INVARIANCE Output! An LTI system can be completely described by its impulse response! And with this, you have learned Convolution!

16. Convolution Integral Leo Lam © 2010-2013 16 Standard Notation The output of a system is its input convolved with its impulse response

17. Convolution Integral Leo Lam © 2010-2013 17 Standard Notation The output of a system is its input convolved with its impulse response

18. Leo Lam © 2010-2013 Summary Convolution!