which make the ratio ^rnl ^m ^ pooi' estimator of the value of the phase 

 of the sinusoid. 



The numerators of the terras inside the braces in the above equations 

 for a^j and b^- are at most of order A^, i = 1,2,3. 



Randomness in the values of 6^, i = 1,2,3 and in the phase rela- 

 tionships of the three sinusoids at the origin of coordinates might pro- 

 duce partial cancellations among the terms inside the braces to reduce 

 the resulting error. 



Assume for example: <))^ = 0, i = 1,2,3. The coefficients reduce to: 



A2 sin 2tt62 A2 sin 2-n&2 A3 sin 27163' 



V " t ''°^ ^^J 



+ sin kQ 



[1] 



At sin^ iri 



[i: 



[2] 

 A2 sin^ -ni 

 [2] 



[3] 

 A3 sin^ -ni 

 [3] 



similarly, for b 

 b 



mj ■ 



I jA]^ sin 21162 A2 sin 2tt62 A3 sin 2Tr63 



^ - - sm k^j I ~ + — + — 



cos k9, 



A, sin^ tt6 ^ 



A„ sin^ Tr6 Ag sin^ tt6 



:i] 



[2] 



[3] 



Letting: 



L^ = A^ sin 27762 ' 



Ml 



Ao sin 2nt 



L2 = A^ sin^ 776 2 , 

 M2 = A2 sin^ 7762 , 



Ni = A3 sin 27763 , 

 then : 



Aq sin^ 77^ 



^mj 



and 

 b 



1 cos k^i [k_ . !ll_ . NlJ . sin kQi r^ 



2 ^ [[1] [2] [3]J ^ [[l 



-i— + — i— + -i— - cos kQj \^— 



[[1] [2] [3]J ^ [[1] 



M2 N2 



^ - - sin kaj 



M2 N2 



CC-16) 



(C-17) 



(C-18) 



Assume further: A^ ~ A2 ~ A3 and |m^ - m| =0 for i = 2 and equal 

 for i = 1 and 3. Since \&\ < 1, the terms [1] and [3] are of 



52 



