calclec9

calclec9 - tom.h.wilson tom.wilson@mail.wvu.edu Dept....

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1 tom.h.wilson tom.wilson@mail.wvu.edu Tom Wilson, Department of Geology and Geography Dept. Geology and Geography West Virginia University Special topics … Fourier series  0 1 () cos s in nn n f ta a n t b n t  We were able to create this step using a sum of sines.
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2 The Fourier Transform • A transform takes one function (or signal) in time and turns it into another function (or signal) in frequency • This can be done with continuous functions or discrete functions  0 1 () cos s in nn n f ta a n t b n t  The Fourier Transform • The general problem is to find the coefficients: a 0 , a 1 , b 1 , etc.   0 1 s n f a n t b n t Take the integral of f(t) from 0 to T (where T is 1/f ). Note = 2 /T 1 T ftd t 0 T What do you get? Looks like an average!
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3 The average turns out to be a o 1 () T o a f td t 0 T There’s a little more work involved in getting the other coefficients. Tom Wilson, Department of Geology and Geography To get the other Fourier coefficients multiply by cos i t or sin i t what happens when you multiply the terms in the series by terms like cos( i t ) or sin( i t )? 01 1 22 33 ( )cos cos cos cos sin cos cos2 cos sin 2 cos cos3 cos sin3 cos ... +... ft it a t it b t it at i t b t i t i t b t i t   cos i a cos sin cos ... +... i
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4 integrate the sum of all terms x cos i t 01 1 00 22 ( )cos ( cos sin cos cos cos cos cos c 2s i n 2 os TT ft it d t a a t b t tb t    33 cos3 sin3 ... +... cos c o s a at t b i t cos sin .. cos cos . +... ) ii ai t b i t dt 0 0 2 cos 0 T t d t T This is just the average of i periods of the cosine Look at the first term Now integrate f(t) cos( i t ) 1 0 2 cos cos ? T i t d t T 11 cos cos cos( ) cos( ) A BA B A B  Use the identity If i=1 then the a 1 term = 1 cos cos (cos2 cos0) a t t  1 2 1 0 2 cos cos cos 2 cos0 T aa a t tdt tdt dt T
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5 What does this give us?
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calclec9 - tom.h.wilson tom.wilson@mail.wvu.edu Dept....

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