5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 279 



The energy spectrum of a typical train of short pulses is shown in Fig. 5-1 5b. 

 We note that the filter primarily acts to accentuate the harmonics of the 

 repetition frequency. Because of its distinctive appearance, such a device 

 is often called a co7nb filter. 



Some explanation on the mechanism of the detection process itself is 

 in order since the previous discussion related only to maximizing the signal- 

 to-noise ratio. The output of the matched filter characterizes a signal- 

 plus-noise situation by a single number So{t^ -\- njyt^. This number is a 

 random variable with a normal distribution and mean Jo(0- The detection 

 process will consist of a decision as to whether the observed number comes 

 from a distribution with mean sj^t^ or the distribution of noise alone 

 with a zero mean. This decision can be made by selecting a critical value or 

 threshold and deciding for or against the existence of the signal depending 

 upon whether or not the observed number exceeds the threshold. Fig. 5-16 



-Decision Bias 

 Probability Density 

 of Noise Alone 



Probability Density 

 of Signal Plus Noise 



h\ Value of Signal 



False Alarm pi^g ^^^^^ 



Probability 



Fig. 5-16 The Use of a Decision Bias for Determining Whether Noise Alone or 

 Signal Plus Noise Is Present. 



shows the probability densities of the filtered signal plus noise and noise 

 alone and a decision bias b for distinguishing the two cases. Because the 

 two probability densities overlap, mistakes will be made. On some occasion 

 a target will be thought present when there is none, while at other times 

 the signal plus noise will be thought to be noise alone. The probability 

 of making an error of the first kind is equal to the crosshatched area 

 under the curve of noise alone and to the right of b in the diagram. This 

 probability is normally called the false-alarm probability by radar system 

 designers. The shaded area under the probability density curve of signal 

 plus noise and to the right of ^ is the probability of detection. The difference 

 between this probability and unity is, of course, the probability of making 

 an error of the second kind or not seeing a target that is actually present. 

 When human operators make a detection, the situation is not nearly so 

 clear-cut, but some similar mechanism must take place. The decision bias 

 might be visualized as diffuse, and it will vary with operators, time, and 

 other conditions. 



A basic problem is the choice of the false-alarm probability at which 

 the system is to operate. Most often this operating parameter is chosen 



