c + d I ^"i ^°^ ""^ 

 a 



2 cos n<l> I 2a2 cos2 „0 + a| 



■a„\l — 1 (40) 



and 



b= -' 



2a^ cos"^ n0 + a^ 



+ ^ / 2ct^ sin^ m 



The mean-square errors are 



and 



2sinn0 \2af sin^ n4> + al J 

 2a f sin"' n4> + a^ 



4 

 E{(i-af} = (42) 



2 cos^ n(p+ • — 

 of 



£{(6 - &)2} = . (43) 



2 sin^ n0 + — 



It is easy to see that these are in fact smaller than the quantities in Eqs. (38) and (39), respectively. 

 This improvement is achieved, however, only at the expense of introducing biases, in general zonzero: 



-i 



2a'- cos^ n0 + a^ 



and 



-1 



2a\ sm"^ n0 + a^ 



It is interesting to see what happens in this c£ise if cosine of m0 approaches zero. Looking at Eq. 

 40, we see that the estimator approaches zero independent of the data. Looking at Eq. 42, we see that 

 the mean-square error approaches a^, independent of a^. The bias, given in Eq. 44, approaches - a„. 

 The optimum processing according to this error criterion in effect ignores the measurement and gives 

 back the a priori estimate. 



899 



