Lecture_11

# Lecture_11 - PHYS 342 Fall 2010 Lecture 11 A Fourier...

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PHYS 342 Lecture 11: A Fourier Transform Primer Fall 2010 Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 11 1

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In many endeavors, we encounter signals that periodically repeat f(x) x L f(t) T t 2
Such repeating signals can be well understood using trigonometric functions Periodic functions: sin x, cos x, sin 2x, cos 2x, ……. sin nx, cos nx Combinatorial properties: 1       11 sin sin 2sin cos 22 cos cos cos( ) cos( ) sin sin cos( ) cos( )       sin cos cos sin dd   Orthogonality: cos 0 sin 0 cos sin 0 nx dx nx dx mx nx dx     0 cos cos mn mx nx dx 3 0 sin sin mx nx dx

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Fourier’s Theorem (1822) : ANY periodic function of x (or t) that is periodic in L (or T) can be written as a sum of sines and cosines. In what follows, let’s focus on periodic signals as a function of ngth rather than time Assume that we know f(x) 22 () cos sin N n n nx nx f xa a b     length rather than time. Assume that we know f(x). (1) 1 on n LL    ) L nx fd here 0 cos ) s i n n L a f x dx f x d x where 0 1 ) n L bf f x d x (2) 0 o af L 4 Note: Eq. 1 is called a Fourier Series
Consider the derivation of the b n equation on previous slide 1 22 () c o s s i n 2 N on n n nx nx fx a a b LL x m         Fourier’s Theorem si n: sin ( ) sin o Multiply both sides by L xx L mm 11 cos sin sin sin NN nn nx x nx x ab tegr m m   : from to L Integra 00 0: sin ( ) sin o te from fxd x a d x  first term 1 0 cos sin 2 L N n n L N nx x ad x x x m second term ird term 1 0 si ns in n n nx bd x m third term 5

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First Term: 0 0 22 sin cos 0 2 L L oo xL x ad x a LL mm m      1 0 cos sin L N n n nx x x m Second Term: 1 cos ( ) cos ( ) 2( ) 2( ) L N n n nx a nm L    0 1 c 2 N n n L a  os 2 ( ) cos 2 ( ) 1 1 2( ) 2( ) 2( ) 2( ) nn  cos 2 ( ) cos 2 [( (2 )] 2( ) 2( ) c o s 2( c o s 2) s i n2( s i 2) ( ) ) ) ) rewrite and m n m m expan m d m   2( cos 2 ( 1 sin 2 ( 0 cos 2 ) () )  ) 2( ) m n 6
1 0 22 cos sin L N n n nx x ad x LL m     1 cos 2 ( ) cos 2 ( ) 1 1 2 2 () 2 2 2 1 11 N n n Ln n a nm m m       1 1 cos 2( ) 2 2 2 2 2 N n n n n L an m a 2 2 2 2 1 N L m   2 ) 0( if n m is even or odd 2 L N x x hird Term: 1 0 sin sin sin ( ) sin ( ) n n L nx bd x nx m m m

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## This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue.

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Lecture_11 - PHYS 342 Fall 2010 Lecture 11 A Fourier...

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