Leo Lam © Signals and Systems EE235

Courtesy of Phillip Leo Lam ©

Today’s menu Fourier Series

Leo Lam © Fourier Series/Transform: Build signals out of complex exponentials Established “orthogonality” x(t) to X(j  ) Oppenheim Ch Schaum’s Ch. 5

Fourier Series: Orthogonality Leo Lam © Vectors as a sum of orthogonal unit vectors Signals as a sum of orthogonal unit signals How much of x and of y to add? x and y are orthonormal (orthogonal and normalized with unit of 1) x y a = 2x + y of x of y a

Fourier Series: Orthogonality in signals Leo Lam © Signals as a sum of orthogonal unit signals For a signal f(t) from t 1 to t 2 Orthonormal set of signals x 1 (t), x 2 (t), x 3 (t) … x N (t) of Does it equal f(t)?

Fourier Series: Signal representation Leo Lam © For a signal f(t) from t 1 to t 2 Orthonormal set of signals x 1 (t), x 2 (t), x 3 (t) … x N (t) Let Error: of

Fourier Series: Signal representation Leo Lam © For a signal f(t) from t 1 to t 2 Error: Let {x n } be a complete orthonormal basis Then: Summation series is an approximation Depends on the completeness of basis Does it equal f(t)? of Kind of!

Fourier Series: Parseval’s Theorem Leo Lam © Compare to Pythagoras Theorem Parseval’s Theorem Generally: c a b Energy of vector Energy of each of orthogonal basis vectors All x n are orthonormal vectors with energy = 1

Fourier Series: Orthonormal basis Leo Lam © x n (t) – orthonormal basis: –Trigonometric functions (sinusoids) –Exponentials –Wavelets, Walsh, Bessel, Legendre etc... Fourier Series functions

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.