284 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



The quotient in Equation 5-136 giving tiie variance of the signal location 

 error thus has the following form: 



_ ^f Dco2|y(co)|Vco 



A/- = , /;~'° TT (5-139) 



(^/ c,'Yio^)S{co)e''^'o^oA 



For convenience, we denote the quotient on the right-hand side by ^. The 

 denominator of this quotient is in a form to which Schwarz's inequality, 

 given by Equation 5-125, can be applied. Using this relation to split the 

 denominator into two separate integrals leads to the following inequality: 



^f Dco^iy(a;)|Vco ^ 



The integrals involving the transfer function of the filter simply cancel as 

 they did in Paragraph 5-10, where the maximum value of the signal-to-noise 

 ratio was determined. Since the quotient ^is never less than the expression 

 given on the right-hand side above, which does not contain Y(o}) at all, 

 this expression must give the minimum value of ^ for the most judicious 

 choice of y(co). Referring to Equation 5-139 above, it is apparent that this 

 minimum value of ^ will actually be achieved if the filter transfer function 

 is chosen to be the conjugate of the signal spectrum. In this case, the 

 numerator in Equation 5-139 cancels one of the factors in the denominator, 

 and we have 



A7^=^.,mi„= , ,.co ^ (5-141) 



i/. 



cjo-\S{u)\~dcjo 



The optimum filter transfer function giving this result is 



y(aj) - S*(c^)e-''''o. (5-142) 



This is exactly the transfer function determined in Paragraph 5-10 (Equa- 

 tion 5-128) to give the maximum signal-to-noise ratio. Thus to a first 

 approximation the matched filter giving the maximum signal-to-noise ratio 

 also provides the minimum error in locating the signal in time. 



The relation given by Equation 5-141 above for the minimum variance of 

 the error in locating the signal can be given an interesting and rather useful 

 physical interpretation. We note that the denominator has the form of the 

 moment of inertia of the energy spectrum of the signal. If this denominator 

 is divided by the total signal energy, we obtain the square of the radius of 

 gyration of the energy spectrum. Now the radius of gyration of a function 



