276 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



Thus the right-hand side of this inequality is independent of the filter. Since 

 the signal-to-noise ratio is never greater than the right-hand side of the 

 expression above, and this expression does not contain y(w) at all, it must 

 give the maximum value for 2^ for the optimum choice of y(co). Referring 

 to Equation 5-112, it is apparent that the denominator in that equation 

 will be canceled and the maximum value of z^ achieved if the filter transfer 

 function is made the complex conjugate of the signal spectrum: 



y(co) = ^*(co)^-'"'«. (5-122) 



A receiver-filter which is designed on the basis of this principle, where the 

 receiver transfer function is matched to the signal waveform, is often 

 referred to as a matched filter system. Another general term which is also 

 used in reference to such systems is correlation radar. This terminology 

 originates in the observation that the ideal filtering operation is equivalent 

 to a cross correlation of the signal plus noise with an image of the signal 

 waveform. In order to see this, the impulse response of the matched filter 

 is found by taking the inverse Fourier transform of Y{oo) : 



1 /"" 

 Impulse response of matched filter = :r— / S*{ii))e '"^"'^'"^dco 



(5-123) 



= sUo - /). 



Denoting the input noise process by n{t) and the output noise process by 

 no(t) and using Equation 5-16 to relate the time histories of the input and 

 output signal plus noise gives for the filter output: 



Soil) + noil) = l_^ [sir) + nir)]sito - t + r)dT. (5-124) 



In particular, the output at the observation time to is simply 



soito) + rioito) = J_^ \sir) + niT)\siT)dr. (5-125) 



Thus, from this relation it is clear that the optimum receiver could consist 

 of taking the cross correlation of the received signal plus noise and the pure 

 signal waveform and that a matched filter receiver and a cross correlation 

 receiver are equivalent. 



Going back to Equation 5-121 for a moment, we might note an interesting 

 basic feature of radar systems which are theoretically optimum in the sense 

 of this paragraph. The maximum signal-to-noise ratio is equal to the ratio of 

 the received signal energy to the power density of the noise. That is, the 

 maximum signal-to-noise ratio does not depend upon the waveform of the 

 signal. This is not to say that the waveform is not important. Resolution, 

 tracking accuracy, and many other system characteristics are closely related 



