leo lam © 2010-2011 signals and systems ee235 october 14 th friday online version
TRANSCRIPT
- Slide 1
- Slide 2
- Leo Lam 2010-2011 Signals and Systems EE235 October 14 th Friday Online version
- Slide 3
- Leo Lam 2010-2011 Todays menu Superposition (Quick recap) System Properties Summary LTI System Impulse response
- Slide 4
- Superposition Leo Lam 2010-2011 Superposition is Weighted sum of inputs weighted sum of outputs Divide & conquer
- Slide 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)
- Slide 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
- Slide 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
- Slide 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
- Slide 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)
- Slide 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
- Slide 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?
- Slide 12
- Building x(t) with (t) Leo Lam 2010-2011 Using the sifting properties: Change of variable: t t0 tt0 t From a constant to a variable =
- Slide 13
- Building x(t) with (t) Leo Lam 2010-2011 Jumped a few steps
- Slide 14
- Building x(t) with (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
- Slide 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)
- Slide 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
- Slide 17
- Finding Impulse Response (examples) Leo Lam 2010-2011 Let x(t)=(t) What is h(t)?
- Slide 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
- Slide 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.
- Slide 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
- Slide 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!
- Slide 22
- Convolution Integral Leo Lam 2010-2011 21 Standard Notation The output of a system is its input convolved with its impulse response
- Slide 23
- Leo Lam 2010-2011 Summary LTI System Impulse response Leading into Convolution!