APPENDIX C 

 FOURIER COEFFICIENTS FOR A MIXTURE OF THREE SINUSOIDS 



The Fourier Transform of the function: 



f(t) = A cos (at - (}>) (C-1) 



with 



2iT(in +6) 1-1 1 



NAt '1-1-2 



(C-2) 



where 1 << m << N, computed from N values of f(t) evaluated at 

 equal increments of t; At is given by the set of coefficients (Harris, 

 1974) : 



2A sin ir6 cos(ir6 - <}>) 

 ^m = ^ 



2A sin 176 sin(7r6 - (t) 



r 1 , 1 1 



[tan [it (in - m + 6) /NJ * tan[TT(in + m + 6)/N]J ' 



r _! ], 



ltan[Tr(m - m + 6)/N] tan[TT(m + m + 6) /NM 



N 

 m = 1, 2, . . . I . (C-3) 



Harris shows that for values of m near in, and for in far removed 

 from 1 and N/2, 



• A sin 7:6 cos ((j) - irS) 

 3m = 



TT(m - m + 6) 



, A sin .6 sin(^ - .6) , = i, 2, . . ^ (c-4) 



ir(m - m + oj 2 



are good approximations to the coefficients in equation (C-3) . Equation 

 (C-4) shows that convergence is slow. 



For the special case 6=0, substitution into equations (C-3) or 

 (C-4) gives : 



^m - ^m = ^> m ji^ in , 

 and 



a^ = hm = indeterminate for m = in . 



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