. Airborne radar. Airplanes; Guided missiles. 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 t
. Airborne radar. Airplanes; Guided missiles. 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.
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