5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 283 



this expression, and we can determine the optimum filter for tracking which 

 gives the minimum tracking error. This error will be interpreted as a 

 simple relation between the signal bandwidth and the signal-to-noise ratio. 

 The filtered signal and noise are denoted by So{t) and no{t) as in the 

 preceding paragraph. We suppose that the maximum value of the signal 

 occurs at the observation time io- We suppose further that the output signal 

 at the time /o + A/ can be represented by a series expansion about the time 

 to for small values of the interval A/. 



So(lo + A/) = soito) + s:'(to)Ar-/2 + -. (5-133) 



The first derivative of So{t) at /« is zero, of course, because it has a maximum 

 at that time. We assume that the shift in the maximum value of the signal 

 plus noise is small enough that all terms beyond the second in the expansion 

 above can be neglected. The derivative of signal plus noise in the neighbor- 

 hood of /o is thus given approximately by 



j^[so{t) + nom = to+At = s:'{to)M + n'^to + A/). (5-134) 



Setting this expression equal to zero and solving for A/ gives an approximate 

 value for the apparent shift in signal location due to noise: 



A/= -^''(!;\^'\ (5.135) 



^o \to) 



The variance or mean square value of the signal location error is thus given 

 approximately by the average of the square of this expression: 



If we denote the transfer function of the filter-receiver by Y(o}) and the 

 signal spectrum by S(cci) as in Paragraph (5-10), the following representation 

 of s"o{to) can be obtained by differentiating Equation 5-116 twice: 



s'o'ito) = ^ / co2y(co)^(co)^^'-'«^a;. (5-137) 



Also assuming, as in Paragraph 5-10, that the input noise has a uniform 

 spectrum of density D, the power spectrum of the output noise is D|y(w)|^ 

 while the power spectrum of the derivative of the output noise is Do}^\Y(p})\^. 

 The integral of this last spectrum gives the mean square value or variance 

 of the derivative of the output noise: 



KWP = ^/'_ Da)2|y(co)|Vco. (5-138) 



