Harris (1974) applied the Fourier transform in equations (9) and (10) to a 

 simple sinusoidal wave given by 



y(nAt) = H cos(anAt - <}>) (14) 



where a is an arbitrary frequency, a can be expressed in a form comparable 

 to equation (8) as 



2ir(m + 5) 1 . 1 ..^n 



'= NAt ' -2 < ^ ^2 ^^^^ 



where m and 6 are arbitrary constants defining the frequency of the 

 sinusoidal wave. Then the coefficients are given by 



H sin it6 cos(Tr6 - <f) ( 1 



a_ = 



N \tan[TT(m - m + 6)/N] tan[ir(m + m + 6)/N] , ^ 



H sm iro sin(iro - ^) 

 b_ = 



N 



( ^ ^ 1 



(tan[Ti(m - m + 6)/N] tan[ii(m + m + 6)/N] | 



If it is further specified that m is not near zero or N/2, the expression 

 for a and b for analysis frequencies near the true frequency can be 

 simplified to 



( --^- 1 



\tan[Tv(m - m + 6)/N] | 



( ^ 1 



ltan[iT(m - m + 6)/N] ) 



H sin ir6 cos(Tr6 - (})) 



N , _ . , , , , 



H sxn it6 sin(iro - <()) 

 N 



It is evident in both equations (16) and (17) that if 6 ;^ 0, a part of 

 the wave energy will be assigned to all analysis frequencies. Steps can be 

 taken to concentrate this spillover in a narrow band of frequencies containing 

 the true frequency as follows. The function y(nAt) is obtained from the time 

 series as 



y(nAt) = w(nAt) y(nAt) (18) 



where 



w(nAt) =1 (l - cos ^j (19) 



The data window in equation (19) is often called the Tukey window or cosine 

 bell data window. Harris (1974) and others have shown that the spillover is 

 reduced considerably when the function y(nAt) in equations (18) and (19) is 

 analyzed instead of y(nAt). 



Substitution of equations (14), (15), and (19) into equation (18) gives 



H i r2ir(m + 6) n 1 r2Tr(m - 1 + 6) n "I 

 y(nAt) = - { 2 cos <)) - cos <^ 



t2Ti(m + 1 + 6) n 1) 

 i^ *J 1 



29 



(20) 



