after the introduction of the periodicity assumed in equation (14] . It 

 follows that 



p(f^) = (A^ A„)/NAt = (U2 + v2)/NAt , [19] 



where Am denotes the complex conjugate of A^i- This is the desired con- 

 clusion and equations (5) and (17) have been shown to be equivalent. 



Returning to the question of the amount of spectral smoothing involved 



in the estimate, p(fj^), define the finite Dirac comb (Blackman and Tukey, 

 1958) as: 



c-1 



V^ (T;At) = ^ 6 (t + c At) + At ^ 6 (t - kAt) + ^ 6(t - c At) , 



'^=-'^"1 (20) 



where 6(x) is the Dirac function which has the properties: 



CO 



J 6(x)dx = 1 (21) 



-00 



6(x) = , if xj^O (22) 



and for any bounded function g(x): 



/ 6(c - x)g(x)dx = g(c) . (23) 



• _oo 



Now suppose the sampling questions involved Cj^ are ignored and C(kAt) 

 is inserted in place of C]^ in equation (16). Then, 



p(f^) - f C(T) Vj^/2 f^^^t) e -i2TTfn,T ^^ 



J _co 



(24) 



The right-hand side of this equation is the Fourier transform of 

 C(t) Vf^/2 (T;At). Thus, by Fourier transform theory p(fni) is the convo- 

 lution of the transforms of the separate functions or 



oo 



P(fm) == P(fin3 * Z Qoffm " ^) • (25) 



