5-111 



DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 



285 



is a measure of the width of that function. Thus the radius of gyration of 

 the energy spectrum is a measure of the bandwidth of that spectrum. We 

 shall adopt this definition for the bandwidth of the signal spectrum: 



Signal bandwidth (rad/sec) = B 



U 



\S(co)\"(^(ji} 



T 



{rjl !^MP^")"" 



(5-143) 



A question may arise in connection with the application of this definition 

 to spectra which are not centered at zero frequency. It is clear that the 

 radius of gyration of an energy spectrum which is concentrated at low 

 frequencies is a good measure of the bandwidth of the spectrum. The 

 radius of gyration of a spectrum whose center is displaced to some high 

 frequency, though, will be very large, and it does not correspond to the 

 conventional idea of bandwidth. Such a signal can be represented as a 

 function, denoted by/(/), whose spectrum is concentrated at low frequen- 

 cies and which is modulated by a high-frequency carrier: 



High frequency signal = f(t) cos {coj + (p). (5-144) 



This signal contains information about its time location with an accuracy 

 on the order of 1 /coc- But when the ratio of the carrier frequency to the 

 bandwidth of/(/) is large, this information is useless because it is ambiguous. 

 This is illustrated in Fig. 5-18 where a relatively smooth low-frequency 



AMBIGUOUS MAXIMA 

 DUE TO CARRIER 

 FREQUENCY 

 LOW - FREQUENCY // 1 MODULATION 



SIGNAL=f(f) 



HIGH-FREQUENCY SIGNAL^ f(f)cos(Wcf+0) 

 Fig. 5-18 Typical High-Frequency Signal. 



signal /(/) is modulated by many carrier cycles. It can be seen in this 

 figure that the carrier frequency modulation produces a number of signal 

 maxima in the neighborhood of the maximum of/(/) . Because these maxima 

 are all of about the same magnitude, there is no way of distinguishing one 

 from another, and signal location information provided by the carrier- 

 frequency modulation is ambiguous. In such a case, the low-frequency 

 signal /(/) is normally regenerated by a demodulating operation, and the 



