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Signals and Systems EE235 Lecture 19 Leo Lam © 2010-2012 Today’s menu • Fourier Series Leo Lam © 2010-2012 Visualize dot product • In general, for d-dimensional a and b • For signals f(t) and x(t) • For signals f(t) and x(t) to be orthogonal from t1 to t2 • For complex signals Leo Lam © 2010-2012 Fancy word: What does it mean physically? 3 Orthogonal signal (example) • Are x(t) and y(t) orthogonal? Yes. Orthogonal over any timespan! Leo Lam © 2010-2012 4 Orthogonal signal (example 2) • Are a(t) and b(t) orthogonal in [0,2p]? • a(t)=cos(2t) and b(t)=cos(3t) • Do it…(2 minutes) 1 cos(x) cos(y ) (cos(x y ) cos(x y)) 2 2p 1 2p 1 1 cos(5t ) cos(t )dt sin(5t ) sin(t ) 0 2 0 2 5 0 Leo Lam © 2010-2012 5 Orthogonal signal (example 3) • • • • x(t) is some even function y(t) is some odd function Show a(t) and b(t) are orthogonal in [-1,1]? Need to show: • Equivalently: • We know the property of odd function: • And then? Leo Lam © 2010-2012 6 Orthogonal signal (example 3) • x(t) is some even function • y(t) is some odd function • Show x(t) and y(t) are orthogonal in [-1,1]? • Change in variable v=-t • Then flip and negate: Same, QED -1 Leo Lam © 2010-2012 1 7 Orthogonal signals x1(t)x2(t) x1(t) T t x (t ) x (t )dt 0 1 t 2 0 T/2 x2(t)x3(t) T x3(t) T t T x (t ) x (t )dt 0 2 0 Leo Lam © 2010-2012 t Any special observation T here? x2(t) T T 3 t Summary • Intro to Fourier Series/Transform • Orthogonality • Periodic signals are orthogonal=building blocks Leo Lam © 2010-2012 Fourier Series • Fourier Series/Transform: Build signals out of complex exponentials • Established “orthogonality” • x(t) to X(jw) • Oppenheim Ch. 3.1-3.5 • Schaum’s Ch. 5 Leo Lam © 2010-2012 10 Fourier Series: Orthogonality • Vectors as a sum of orthogonal unit vectors • Signals as a sum of orthogonal unit signals y a = 2x + y x • How much of x and of y to add? of x of y a • x and y are orthonormal (orthogonal and normalized with unit of 1) Leo Lam © 2010-2012 11 Fourier Series: Orthogonality in signals • Signals as a sum of orthogonal unit signals • For a signal f(t) from t1 to t2 • Orthonormal set of signals x1(t), x2(t), x3(t) … xN(t) of of of Does it equal f(t)? Leo Lam © 2010-2012 12 Fourier Series: Signal representation • For a signal f(t) from t1 to t2 • Orthonormal set of signals x1(t), x2(t), x3(t) … xN(t) of of of • Let • Error: Leo Lam © 2010-2012 13 Fourier Series: Signal representation • For a signal f(t) from t1 to t2 • Error: • Let {xn} be a complete orthonormal basis • Then: of of Does it equal f(t)? of Kind of! • Summation series is an approximation • Depends on the completeness of basis Leo Lam © 2010-2012 14 Fourier Series: Parseval’s Theorem • Compare to Pythagoras Theorem c Energy of vector Energy of b a • Parseval’s Theorem each of orthogonal basis vectors All xn are orthonormal vectors with energy = 1 • Generally: Leo Lam © 2010-2012 15 Fourier Series: Orthonormal basis • xn(t) – orthonormal basis: – Trigonometric functions (sinusoids) – Exponentials – Wavelets, Walsh, Bessel, Legendre etc... Fourier Series functions Leo Lam © 2010-2012 16