3-3] 



DETECTION PROBABILITY FOR A PULSE RADAR 



145 



with a peak power of S. We are now interested in finding analytical 

 expressions for the square-law detector output under the conditions of 

 noise-only inputs and signal-plus-noise inputs. 



It is conventional and convenient to approximate the video voltage in the 

 absence of the signal by a series of independent samples which are spaced 

 at intervals equal to the reciprocal of the predetection bandwidth. Such 

 an approximation is shown in Fig. 3-2. This approximation is based upon 



TIME 



Fig. 3-2 Representation of Continuous Video Voltage by a Sequence of Sample 



the famous sampling theorem which states: If a function /{i) contains no 

 frequencies higher than W j2 cps, it is completely determined by its ordinates at 

 a series of points spaced 1 IW seconds apart. ^ In this connection, the envelope 

 of the noise in a predetection band of width W cps can be shown to be 

 equivalent to a low- frequency function limited to frequencies less than W 11 

 cps, and it can be represented by a series of samples spaced by T = 1 IW 

 seconds. 



Since the spectrum of the predetection filter is matched to the spectrum 

 of the pulse envelope, it will have a width approximately equal to the 

 reciprocal of the pulse length, r. In this case, the samples will be spaced by 

 intervals equal to t. It can also be shown by an appropriate application of 

 the material in Chapter 5 that the statistical fluctuations in these samples 

 are independent. In this case, each sample can be considered a separate 

 detection trial, and the input to the decision element during an observation 

 period can be regarded as a series of independent trials for which methods 

 of analysis are well known. For instance, if the probability of exceeding the 

 threshold is ^, the probability of exceeding the threshold at least once in 

 m trials is 



Probability of at least one success in m trials 



1 - (1 - pY. (3-12) 



Further, the average number of trials between successes is the same as the 

 average number of trials per success, which is equal to the reciprocal of the 

 probability on a single trial, 1 \p. When there is no signal present so that 

 any exceeding of the threshold represents a false alarm, the number 1 jp 

 is called t\i& false -alarm number. 



^C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE 37, 10-21 (1949). 



