280 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



on subjective grounds because the data upon which to base a rational 

 choice are not available. Factors which can be used to determine an 

 optimum false-alarm probability are the cost of a false alarm in time and 

 subsequent commitments, the gain associated with a correct decision, 

 and prior probability of a target's existence. If quantitative estimates of 

 these factors are available, the false-alarm probability can be chosen to 

 minimize the total cost of the detection operation. Or even when the 

 prior probability is not known, it is possible to operate the system at false- 

 alarm rates so as to minimize the cost for the most adverse value of the 

 prior probability. As noted above, however, data on detection costs and 

 prior detection probabilities are known only subjectively in the majority 

 of cases, and most often a rather arbitrary estimate of a desirable value 

 of the false-alarm rate is made after a thorough but subjective study of 

 the problem. 



At the beginning of our discussion of the optimum receiving system, it 

 was assumed that the waveform of the signal was known exactly, and the 

 only issue was its existence. In a practical detection situation, however, the 

 signal waveform may depend upon a number of unknown parameters. 

 Three such parameters which are of particular importance are the signal 

 amplitude, the time of arrival of the signal, and the radio frequency of 

 the carrier. The signal amplitude will vary with the range, aspect, and 

 size of the target, while the time of arrival is, of course, directly proportional 

 to range in a radar system. The RF carrier will vary because of the 

 doppler shifts proportional to the relative target velocity. An optimum 

 receiver in this case will consist of a parallel combination of optimum 

 receivers for all the possible waveforms. Luckily, this does not require a 

 duplication of equipment to cover the possibilities of amplitude and time- 

 of-arrival variations. If the signal amplitude is changed by some factor, 

 then the average value of the filter output is changed by the same factor. 

 The same filter will produce the maximum value of z- for all possible signal 

 amplitudes. A similar situation applies to variations in time-of-arrival. 

 The optimum receiver produces its maximum output at a time T after a 

 signal is received. Continuous monitoring of the receiver output, then, 

 will provide an observation of the filtered signal over a continuous range of 

 possibilities for the time of arrival. In order to account for variations in 

 the radio frequency, however, it will in general be necessary to have 

 separate receiving systems for the possible radio frequencies which may 

 occur. This situation will be recognized in the design of many doppler 

 systems where a bank of narrow band filters, each connected to its own 

 threshold, is used to cover the possible spectrum of doppler signals. 



The situation is complicated further by the fact that some of the signal 

 parameters are random variables in their own right. For example, the 

 amplitudes of echoes reflected by aircraft fluctuate owing to their motions. 



