Leo Lam © Signals and Systems EE235 Leo Lam

Leo Lam © Today’s menu Midterm questions? Exponential response of LTI system

Leo Lam © Why do we care? Convolution = complicated Leading to frequency etc.

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 

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

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

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)}

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)}

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

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 )

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

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)

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

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

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

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

Which systems are not LTI? Leo Lam © NOT LTI

Leo Lam © Summary Eigenfunctions/values of LTI System