Leo Lam © Signals and Systems EE235 Leo Lam

Leo Lam © Today’s menu Orthogonality (last slide) Fourier Series

x 1 (t) t x 2 (t) t x 3 (t) t T T T T/2 x 1 (t)x 2 (t) t T x 2 (t)x 3 (t) t T 3 Orthogonal signals Any special observation here?

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 t1 to t2 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:

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:

Leo Lam © Summary Fourier series Periodic signals into sum of exp.