5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 273 



guish the target echo from random noise which tends to obscure it and 

 render detection a matter of chance. This is the problem that we shall 

 discuss in this paragraph. We shall determine the characteristics of an 

 optimum receiver which will provide the most reliable detection of target 

 echoes obscured by random noise. 



There are several possible approaches to this problem, depending upon 

 the generality desired, the definition of most reliable detection adopted, and 

 various assumptions made about the signal. We shall adopt the simplest 

 possible approach, although the receiver design criterion which will be 

 derived is operationally equivalent to the results of more sophisticated 

 analyses in most cases. 



We suppose that in the general radar situation a signal is received as an 

 echo from the target. During the process of reception, noise is added to the 

 signal. The question we consider is, "What function must the receiver 

 perform in order that the most reliable detection of the signal may be 

 obtained?" We shall limit our study to receivers which are linear. That is, 

 the effect of the receiver on the signal and noise is that of a linear filter. 

 The output of the receiver-filter will consist of a filtered signal and filtered 

 noise. Thus a ratio of the output signal and noise powers can be formed. 

 We shall choose the optimum receiver-filter as that which maximizes this 

 signal-to-noise ratio. We shall subsequently indicate how a maximum 

 signal-to-noise ratio gives a maximum probability of detection for a fixed 

 false-alarm rate and thus provides the most reliable detection in this sense. 

 It will turn out, interestingly enough, that the receiver-filter which is 

 optimum in the sense described above has a transfer function which is the 

 conjugate of the target echo spectrum,^ and for this reason such a radar is 

 often called a matched filter system. That is, the filter transfer function is 

 matched to the target echo spectrum. We shall also demonstrate that such 

 a system is equivalent to a cross correlation of the signal plus noise with an 

 image of the signal waveform which is the origin of the term correlation radar 

 sometimes used in reference to such systems. 



We adopt the following notation for this analysis: 



sit) = signal input to receiver-filter 



S(o}) = spectrum of s(t) 



So{t) = signal output of receiver-filter 



So{co) = spectrum of So(,t) 



^This result is sometimes called the Fourier transform criterion and is attributed to a number 

 of authors: namely, D. O. North, W. W. Hansen, N. Weiner, J. H. Van Vleck, and D. Middle- 

 ton. See particularly Van Vleck and Middleton, "A Theoretical Comparison of Visual, Aural, 

 and Meter Reception of Pulsed Signals in the Presence of Noise," J. Appl. Phy. 17, 940-971 

 (1946^. 



