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Leo Lam © Signals and Systems EE235 Lecture 16

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Leo Lam © Merry Christmas! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).

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Leo Lam © Today’s scary menu Wrap up LTI system properties (Midterm) Midterm Wednesday! Onto Fourier Series!

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System properties testing given h(t) Leo Lam © Impulse response h(t) fully specifies an LTI system Gives additional tools to test system properties for LTI systems Additional ways to manipulate/simplify problems, too

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Causality for LTI Leo Lam © A system is causal if the output does not depend on future times of the input An LTI system is causal if h(t)=0 for t<0 Generally: If LTI system is causal:

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Causality for LTI Leo Lam © An LTI system is causal if h(t)=0 for t<0 If h(t) is causal, h( t- )=0 for all ( t- )<0 or all t < Only Integrate to t for causal systems

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Convolution of two causal signals Leo Lam © A signal x(t) is a causal signal if x(t)=0 for all t<0 Consider: If x 2 (t) is causal then x 2 ( t- )=0 for all ( t- )<0 i.e. x 1 ( )x 2 ( t- )=0 for all t< If x 1 (t) is causal then x 1 ( )=0 for all <0 i.e. x 1 ( )x 2 ( t- )=0 for all <0 Only Integrate from 0 to t for 2 causal signals

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Step response of LTI system Leo Lam © Impulse response h(t) Step response s(t) For a causal system: T u(t)*h(t) u(t) T h(t) (t) Only Integrate from 0 to t = Causal! (Proof for causality)

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Step response example for LTI system Leo Lam © If the impulse response to an LTI system is: First: is it causal? Find s(t)

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Stability of LTI System Leo Lam © An LTI system – BIBO stable Impulse response must be finite Bounded input system Bounded output B 1, B 2, B 3 are constants

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Stability of LTI System Leo Lam © Is this condition sufficient for stability? Prove it: abs(sum)≤sum(abs) abs(prod)=prod(abs) bounded input if Q.E.D.

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Stability of LTI System Leo Lam © Is h(t)=u(t) stable? Need to prove that

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Invertibility of LTI System Leo Lam © A system is invertible if you can find the input, given the output (undo-ing possible) You can prove invertibility of the system with impulse response h(t) by finding the impulse response of the inverse system h i (t) Often hard to do…don’t worry for now unless it’s obvious

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LTI System Properties Leo Lam © Example –Causal? –Stable? –Invertible? YES

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LTI System Properties Leo Lam © Example –Causal? –Stable? YES

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LTI System Properties Leo Lam © How about these? Causal/Stable? Stable, not causal Causal, not stable Stable and causal

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LTI System Properties Summary Leo Lam © For ALL systems y(t)=T{x(t)} x-y equation describes system Property tests in terms of basic definitions –Causal: Find time region of x() used in y(t) –Stable: BIBO test or counter-example For LTI systems ONLY y(t)=x(t)*h(t) h(t) =impulse response Property tests on h(t) –Causal: h(t)=0 t<0 –Stable:

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Leo Lam © Summary LTI system properties

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Review: Faces of exponentials Leo Lam © Constants for with s=0+j0 Real exponentials for with s=a+j0 Sine/Cosine for with s=0+j and a=1/2 Complex exponentials for s=a+j

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Exponential response of LTI system Leo Lam © What is y(t) if ? Given a specific s, H(s) is a constant S Output is just a constant times the input

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Exponential response of LTI system Leo Lam © LTI Varying s, then H(s) is a function of s H(s) becomes a Transfer Function of the input If s is “frequency”… Working toward the frequency domain

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Eigenfunctions Leo Lam © Definition: An eigenfunction of a system S is any non-zero x(t) such that Where is called an eigenvalue. Example: What is the y(t) for x(t)=e at for e at is an eigenfunction; a is the eigenvalue S{x(t)}

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Eigenfunctions Leo Lam © Definition: An eigenfunction of a system S is any non-zero x(t) such that Where is called an eigenvalue. Example: What is the y(t) for x(t)=e at for e at is an eigenfunction; 0 is the eigenvalue S{x(t)}

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Eigenfunctions Leo Lam © Definition: An eigenfunction of a system S is any non-zero x(t) such that Where is called an eigenvalue. Example: What is the y(t) for x(t)=u(t) u(t) is not an eigenfunction for S

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Recall Linear Algebra Leo Lam © Given nxn matrix A, vector x, scalar x is an eigenvector of A, corresponding to eigenvalue if Ax=x Physically: Scale, but no direction change Up to n eigenvalue-eigenvector pairs (x i, i )

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Exponential response of LTI system Leo Lam © Complex exponentials are eigenfunctions of LTI systems For any fixed s (complex valued), the output is just a constant H(s), times the input Preview: if we know H(s) and input is e st, no convolution needed! S

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LTI system transfer function Leo Lam © LTI e st H(s)e st s is complex H(s): two-sided Laplace Transform of h(t)

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LTI system transfer function Leo Lam © Let s=j LTI systems preserve frequency Complex exponential output has same frequency as the complex exponential input LTI e st H(s)e st LTI

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LTI system transfer function Leo Lam © Example: For real systems (h(t) is real): where and LTI systems preserve frequency LTI

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Importance of exponentials Leo Lam © Makes life easier Convolving with e st is the same as multiplication Because e st are eigenfunctions of LTI systems cos(t) and sin(t) are real Linked to e st

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Quick note Leo Lam © LTI e st H(s)e st LTI e st u(t) H(s)e st u(t)

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Which systems are not LTI? Leo Lam © NOT LTI

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Leo Lam © Summary Eigenfunctions/values of LTI System

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