286 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



location of the signal is determined on the basis of/(/) alone. The effective 

 bandwidth in such a case, then, is that of the spectrum of/(/), and it would 

 be determined relative to the carrier frequency rather than zero frequency 

 as is indicated in Equation 5-143. When the definition of bandwidth given 

 by Equation 5-143 is combined with Equation 5-141, the mean square time 

 error is found to be approximately equal to the ratio of D, the input noise 

 power density, to the product of the square of the signal bandwidth and 

 the signal energy. The ratio of the signal energy to D, however, was estab- 

 lished in Equation 5-129 as the greatest possible signal-to-noise ratio an^ 

 was denoted by z^. Thus we can assert that the minimum rms error in 

 measuring the time of arrival of a signal is approximately equal to the re- 

 ciprocal of the product of the signal bandwidth and the voltage signal-to-noise 

 ratio: 



(X7^)i/2 ^ iiBz (5-145) 



rms time error = 1 /(signal bandwidth) (voltage signal-to-noise ratio). 



As an example of the application of these ideas, let us consider a pulse 

 radar with a narrow antenna pattern which is scanned over the target at a 

 constant angular rate. Such a system is similar to the AEW example dis- 

 cussed in Paragraphs 2-10 to 2-20 and the results that we shall develop are 

 applicable to the design considerations in that example. 



The video signal generated by such a system would have a form similar 

 to that shown in Fig. 5-19. The time at which the envelope of the pulses 



Reference 



-.^TllrC 



•Target 



0=Beamwidth measured 

 between half - power 

 .Envelope points 



^^, 



i/'s = Scanner Angle 



Fig. 5-19 Receiver Voltage Pulse Train Return from a Point Target. 



reaches its peak value will be correlated with the angular position of the 

 target so that the problem of locating the target in angle is essentially that 

 of determining the arrival time of the signal, and the ideas and develop- 

 ments of this paragraph are applicable. 



The basic functions performed by the system are indicated in the block 

 diagram in Fig. 5-20. The received signal is amplified and filtered by an IF 

 amplifier which is matched to the envelope of an individual pulse. Noise 



