References 49 The maximum likelihood estimate 1 , 2 is given by numerical minimization of the cost function K d( 1 , 2 ) = k=1 y H [k]P S( 1 , 2 ) y[k] (1.186) where S( 1 , 2 ) = [a( 1 ), a( 2 )] ^ ^ After evaluating 1 , 2 , the estimates of 1 , 2 are given by 1 1 ^ ^ ^ ^ ^ ^ ^ = (S H ( 1 , 2 )S( 1 , 2 )) -1 S H ( 1 , 2 ) 2 ^ K K (1.187) y[k] k=1 (1.188) Inserting the estimated parameters into Equations (1.183) and (1.184), we can now calculate R 0 and R 1 and the log likelihood ratio L. The GLR detector operates by comparing the log likelihood ratio to a thresh- old value Th. If L > Th, we choose H 1 (i.e., we decide that two signals are present). If L < Th, we choose H 0 (i.e., we decide that only one signal is present). The detector can make two types of error: deciding H 1 when H 0 is true, or deciding H 0 when H 1 is true. The probabilities of these events are Pr{L > Th|H 0 } and Pr{L Th|H respectively. The overall error probability P is then given