288 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



The peak pulse power will be a'^ jl. We assume that the individual pulses 

 are rectangular and of width b. The total energy in a pulse, then, is a^B jl. 

 Assuming noise with a uniform power spectrum of density D, the maximum 

 signal-to-noise ratio which can be obtained with a filter matched to a pulse 

 is given by the ratio of the pulse energy to the power density of the noise 

 as was derived in Equation 5-127: 



Peak signal to noise ratio = S/N = a^/lD. (5-147) 



The noise at the output of the IF amplifier corresponds to a narrow band 

 noise process similar to those discussed in Paragraph 5-8. Since the IF 

 amplifier is matched to the envelope of the pulse signal, the autocorrela- 

 tion function and power spectrum of the low-frequency in-phase and 

 quadrature components of the noise, x{t) and yif) in Equation 5-73, will 

 be of the same form as that of a rectangular pulse. This autocorrelation is 

 a triangular function of width 25 and height equal to the noise power. 



The properties of the video signal and noise after a square law detector 

 can be determined from Equation 5-87 which gives the autocorrelation 

 function of the output of a square-law second detector: 



7^2 = (202(1 + S/NY + (2cr2)2[p2 + 2{S/N)p]. (5-87) 



The first term in this expression corresponds to the video d-c level during a 

 pulse while the term involving p and p^ corresponds to the video noise. In 

 order to exclude the possibility of ambiguities incident to a noise maximum 

 in the neighborhood of the signal maximum, we assumed that the signal-to- 

 noise ratio was large in the development of Equation 5-145. It will simplify 

 the present analysis if we approximate Equation 5-87 for large S jN by 

 considering only the dominant d-c and noise terms in that equation: 



77^- = {2<rr~{S/NY + (:lcj^-Y2{S/N)p, S/N»\. (5-148) 



The normalized autocorrelation function p in this expression corresponds to 

 the triangular function noted above of width ITd but of unit height. 



The shape of the video spectrum will not be exactly uniform as is re- 

 quired for the developments of this paragraph to be rigorous. The total 

 width of the spectrum, though, will normally be greater than the spectrum 

 of the scan modulation by a factor on the order of 10^. Variation of the 

 noise spectrum over the scan modulation bandwidth, then, will be quite 

 small, and we are justified in approximating the noise spectrum by a 

 spectrum with a uniform power density. The power density of the noise 

 spectrum at zero frequency, which we shall assume to be extended to all 

 frequencies, is found by integrating the autocorrelation of the noise. The 

 integral of the triangular function p{t) is 5. When this value is substituted 

 for p in Equation 5-148, the term involving this factor gives the power 

 density of the noise at zero frequency during a pulse or with a c-w signal. 



