262 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



17 



S 16 



or 



CO 



z 



Q 14 



5 13 



CQ 



12 



1 2 3 



RELATIVE VALUE OF THE ENVELOPE, r/a 



Fig. 5-11 Probability Density Functions of the Envelope of Narrow-Band Signal 



Plus Noise. 





^^^^^^Vlx^^^P 



^72(72 ,»1, 



(5-79) 

 (5-80) 



The first of these forms is often called a Rayleigh probability density and 

 corresponds to the case of noise alone. When the signal-to-noise ratio is 

 large, the envelope has approximately a normal distribution as is indicated 

 by Equation 5-80. 



As might be expected from the form of the probability density function 

 of r, its basic statistical properties such as its autocorrelation or spectrum 

 cannot be expressed simply in terms of elementary functions. Approximate 

 expressions valid for either large or small values of the signal-to-noise ratio 

 have been developed. '^ Instead of becoming involved with such approxi- 

 mations, however, it is often either more convenient analytically or more 

 realistic in a physical sense to assume that the second detector is a square 

 law rectifier producing the square of the envelope rather than the envelope 

 itself. In most problems where such an assumption is made, the variations 

 of many phenomena with parameters of interest are relatively independent 

 of the detector law. The statistical properties of the square of the envelope 

 can be expressed in much simpler forms than those of the envelope itself 

 because r^ is a simple second-degree polynomial function of a; and j'. Thus, 

 the autocorrelation function of r^ will involve the average values of products 

 of the form Xi^x^"^ ^.nA yi^yi^ which have already been evaluated in Para- 

 ''Uid., Chap. 7. 



