Use of L' Hospital Rule in equation (C-3) when 6 ->- 0, shows that: 



a^ = A cos (j) , b^ = A sin (f) , m = in . (C-5) 

 Application of the cosine bell data window, 



f (nAt) = J {[l - cos ^]} f(nAt) , n = 1, . . . N , (C-6) 



is equivalent to replacing the original sinusoid f(nAt) with the sum 

 of three sinusoids (Harris, 1974), where 



f (nAt) = 2 



2 cos 



2iT (in 



2Trrm + 6)n 



. ^ J 



1 + 6)n 1 r2TT(ni +1 + 6) 1 



*J - ^°^ L — ^ *Jj 



(C-7) 



For 6=0, and in view of equation (C-5) , the Fourier Transform of this 

 modified function will be given by: 



*m-l 



^w+1 



A 



= - -r COS 

 4 



'm-l 



= b 



m+l 



sin ()) 



and 



^ 



cos (j) , 



sin cf) 



with a;72 = bm = for all other values of m. Thus, energy appears at 

 three adjacent m values. 



Harris (1974) shows that after application of the cosine bell data 

 window, the approximate values of the coefficients are given by: 



^ A sin tt6 cos((f) -'tt6) 



bv„ = 



2^(m - m + 6) [(m - m + 6)2 - i] ' 



A sin tt6 sin((() - 7r6) 

 2Tr(in - m + 6) [(in - m + 6)^ - i] 



m = 1, 



^. (C-8) 



Thus, convergence increases rapidly, and values of the coefficients for 

 (ill - m) ^ 3 may be disregarded. 



Equations (C-3) , CC-4) , and (C-8) imply that 



tan((() - ^6) = r« 

 b 



m 



(C-9) 



49 



