Minimum-Bias Estimate with the Least Mean-Square Error 



In the present example of two measurements made with two array headings, specifying minimum 

 bias fully determines the processor. When more observations are made, there are many possible mini- 

 mum-bias estimates, and we can further specify the one with the least mean-square error. That is the 

 approach adopted in Ref. 1, and the estimator is the analog of Eq. (13). Reference 1 proves that the 

 estimator given has minimum biases, and only claims, without proof, that among minimum bias estima- 

 tors it has the least-square error. 



Estimate Giving Best Mean-Square Fit to the Measured Values 



Rather than to minimize each component of the error separately, let us attempt to minimize the 

 sum of the squEires, by choosing estimate a and b which minimizes 



E{[c - (a cos n(t> - b sin n0)] ^ + [d - (a cos n<t> + b sin «(/>)] ^}. (46) 



This is the criterion adopted in Ref. 4. 



The minimum is achieved if 



- '''' (47) 



and 



2 cos n(t> 



(48) 



2 sin n0 ' 



provided that neither cos «(/> nor sin n0 equals zero. Furthermore, the error as defined in Eq. (46) is 

 identically zero. If either cos rnp or sin n(j) is zero, then one or the other of the two estimates is inde- 

 terminate, and the error is finite and is equal to 2a2- 



This example can be used to illustrate a real danger in this error criterion. The same danger is 

 illustrated by an old anecdote about a light sleeper who had a clock which chimed the hours. On many 

 occasions he woke up during the night hearing the clock chiming. On looking at his watch, he observed 

 that the hour was invariably one hour later than the number of chimes he counted. J^'rom this he in- 

 ferred that the first chime served to wake him from his slumbers, but faded before he was consciously 

 aware of it. He also formed the further inference that if he woke up suddenly during the night and 

 heard nothing, the hour was 1:00 a.m. Although this estimate fits the observation perfectly, there is a 

 danger that the estimate may fail to be related to the true value in any causal sense. The same is true 

 of one of the estimators of Eqs. (47) and (48) if one of the factors cosine n0 or sine n<p is nearly zero. 

 This form of processing generates a very large and spurious value for one or the other of the terms a or 

 b in such a way that the observations fit the estimate perfectly, but reference to Eqs. (38) and (39) shows 

 that the estimator diverges from the true value of the quantity which it purports to estimate. 



In homelier words, "the observations fit the estimate" and "the estimate fits the true value" are not 

 necessarily equivalent statements, and, as just seen, it is possible for the observations to fit the estimate 

 perfectly while the estimate fits the true value extremely badly. We try to avoid such a pathologic situa- 

 tion, but in measuring the intensity distribution, 1(6), wdth a linear array it cannot be avoided completely. 

 No matter what observation angles are chosen, there are always some values of n for which cos n0 and 



E. M. Wilson, Directional Noise Measurements with Line Arrays Admiralty Research Laboratory, Teddington, 



England (June 1973). 



900 



