Leo Lam © Signals and Systems EE235 Lecture 21

Leo Lam © Today’s menu Fourier Series (periodic signals)

Leo Lam © It’s here! Solve Given Solve

Reminder from last week Leo Lam © We want to write periodic signals as a series: And d n : Need T and  0, the rest is mechanical

Harmonic Series Leo Lam © Example: Fundamental frequency: –   =GCF(1,2,5)=1 or Re-writing: d n = 0 for all other n

Harmonic Series Leo Lam © Example (your turn): Write it in an exponential series: d 0 =-5, d 2 =d -2 =1, d 3 =1/2j, d -3 =-1/2j, d 4 =1

Harmonic Series Leo Lam © Graphically: (zoomed out in time) One period: t 1 to t 2 All time

Harmonic Series (example) Leo Lam © Example with (t) (a “delta train”): Write it in an exponential series: Signal is periodic: only need to do one period The rest just repeats in time t T

Harmonic Series (example) Leo Lam © One period: Turn it to: Fundamental frequency: Coefficients: t T * All basis function equally weighted and real! No phase shift! Complex conj.

Harmonic Series (example) Leo Lam © From: To: Width between “spikes” is: t T Fourier spectra 0 1/T  Time domain Frequency domain

Exponential Fourier Series: formulas Leo Lam © Analysis: Breaking signal down to building blocks: Synthesis: Creating signals from building blocks

Example: Shifted delta-train Leo Lam © A shifted “delta-train” In this form: For one period: Find d n : time T0 T/2 *