Leo Lam © Signals and Systems EE235 Lecture 21

Leo Lam © It’s here! Solve Given Solve

Leo Lam © Today’s menu Fourier Series

Trigonometric Fourier Series Leo Lam © Set of sinusoids: fundamental frequency  0 Note a change in index

Trigonometric Fourier Series Leo Lam © Orthogonality check: for m,n>0

Trigonometric Fourier Series Leo Lam © Similarly: Also true: prove it to yourself at home:

Trigonometric Fourier Series Leo Lam © Find coefficients: The average value of f(t) over one period (DC offset!)

Trigonometric Fourier Series Leo Lam © Similarly for:

Compact Trigonometric Fourier Series Leo Lam © Compact Trigonometric: Instead of having both cos and sin: Recall: Expand and equate to the LHS

Compact Trigonometric to e st Leo Lam © In compact trig. form: Remember goal: Approx. f(t)  Sum of e st Re-writing: And finally:

Compact Trigonometric to e st Leo Lam © Most common form Fourier Series Orthonormal:, Coefficient relationship: d n is complex: Angle of d n : Angle of d -n :

So for d n 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 © Building periodic signals with complex exp. Obvious case: sums of sines and cosines 1.Find fundamental frequency 2.Expand sinusoids into complex exponentials (“CE’s”) 3.Write CEs in terms of n times the fundamental frequency 4.Read off c n or d n

Harmonic Series Leo Lam © Example: Expand: Fundamental freq.

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