lect31

lect31 - UCF Physics: AST 5765/4762: (Advanced)...

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UCF Physics: AST 5765/4762: (Advanced) Astronomical Data Analysis Fall 2009 Lecture Notes: 31. Fourier Transforms 1 Check In: 10:30 — 10:35, 5 min Questions? How are projects going? Who will try to write the paper in LaTeX? 2 Basis Functions: 10:35 — 10:45, 10 min Decompose an arbitrary function h ( t ) into its amplitudes in a set of basis functions b j ( t ) h ( t ) = n s j =0 A j b j ( t ) (1) A j is the j th coefficient (amplitude) (of basis function b j ( t ) ) Basis function b j ( t ) could be sin( jt ) , Hermitian polynomials, Chebychev polynomials, etc. Useful if the functions have properties that make them easy to calculate with Basis functions must be orthogonal functions: 0 = i -∞ b j ( t ) b k ( t ) dt, j n = k (2) Any even function is orthogonal to any odd function Show that sin( jt ) are orthogonal for integer j on board Need both sin and cos, or can only make an even or odd function That is, need a complete set of functions The coefficient of a particular basis function, b j ( t ) : A j = i -∞ h ( t ) b j ( t ) dt (3) 1
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3 Transforms: 10:45 — 10:50, 5 min Make the coefficient a continuous function Make the basis functions a continuous set, like e iωt Do one integral: H ( f ) = i -∞ h ( t ) b ( f, t ) dt (4) H ( f ) and h ( t ) are a transform pair 4 Sound Example: 10:50 — 11:00, 10 min Want amplitude vs. pitch: power spectrum Air pressure is h ( t ) , a continuous function Do transform with oscillation as basis function H ( f ) is amplitude vs. pitch
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lect31 - UCF Physics: AST 5765/4762: (Advanced)...

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