282 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



in time are all quite applicable. We shall find it most convenient to adopt 

 the last of these, the mode or the maximum value of the signal, as the 

 primary indicator of the signal's location. This definition gives a straight- 

 forward development which parallels that of Paragraph 5-10 and which to a 

 first approximation leads to results in accord with more elaborate analyses. 

 Even so, we must recognize that since the choice of a definition for the signal 

 location is purely arbitrary, we are optimizing the tracking process only 

 relative to that definition and not in an absolute sense. 



In order to use the peak value of the filtered signal plus noise as an un- 

 ambiguous estimate of the signal location, we shall make several assump- 

 tions about the form of the signal and signal plus noise. First, we assume 

 that the signal itself either has a single maximum or that the greatest 

 maximum is sufficiently larger than minor maxima to allow it to be un- 

 ambiguously distinguished. Second, we assume that the primary maximum 

 of the filtered signal has a finite second derivative, since we intend to locate 

 it by setting the first derivative of the signal plus noise equal to zero. Third, 

 the filtered signal is assumed to be enough greater than the noise that there 

 are no ambiguous noise maxima in the neighborhood of the primary max- 

 imum and the shift in this maximum due to the presence of the noise is 

 small enough to be approximated by the first few terms in a series expansion. 

 Suppositions of this kind are not unusual in parameter estimation problems, 

 and equivalent assumptions and approximations almost always must be 

 adopted when a specific example is worked out. 



Fig. 5-17 shows a typical example of signal plus noise in the neighborhood 

 of the signal maximum and illustrates how the addition of noise acts to 



APPARENT SIGNAL 

 LOCATION 



ERROR IN 

 SIGNAL LOCATION 



TRUE SIGNAL 

 LOCATION 



NOISE- ^ ^ 



Fig. 5-17 Generation of Signal Location Error. 



shift the maximum slightly from its former value. The magnitude of the 

 shift can be determined approximately by differentiating the signal plus 

 noise and setting it equal to zero. The resulting expression will be in the 

 form of a quotient very similar to that given in Equation 5-118 for the 

 signal-to-noise power ratio. Schwarz's inequality can also be applied to 



