162 



INTENSITY FLUCTUATIONS 



example of Rayleigh distribution, the intensity which 

 will result if a very large number of randomly located 

 scatterers return a single-frequency signal to an echo- 

 ranging transducer. This situation is significant, be- 

 cause it is probably the most realistic model of volume 

 reverberation. Each of these scatterers will return a 

 weak echo, with definite amplitude but random 

 phase. The resultant of all these individual echoes 

 interfering with each other in a random manner will 

 be the reverberation recorded. We shall not give a 

 rigorous derivation of the resulting distribution func- 

 tion but shall sketch the argument leading to it. 



Figure 4. Reverberation amplitude produced by 

 many individual echoes. 



In Chapter 2, it was explained how the amplitude 

 plus phase may be combined into a single quantity, 

 the "complex amplitude," which is designated hyA to 

 distinguish it from the real amplitude a. Obviously, 

 a is the absolute value oi A. In an interference prob- 

 lem the complex amplitude of the resultant is the sum 

 of the complex amplitudes of the interfering com- 

 ponents. If we have a large number of interfering 

 components with random phases, we may plot the 

 individual complex ampHtudes Ai, A2, ■ ■ ■ , An and 

 the resultant complex amplitude At in the complex 

 A plane as illustrated in Figure 4. The direction of 

 each individual component is completely random, 

 while its magnitude is fixed. Obviously, the direction 

 (phase) of the resultant Ar will be random. As for its 

 magnitude, it is well to consider at first only its 

 component in one direction, say the x axis. This 



component of Ar will be the algebraic sum of the 

 X components of the individual complex amplitudes 

 Ai, A2, ■ ■ ■. In the mathematical theory of proba- 

 bility it is shown that the distribution function for 

 the sum of a large number of random terms is usually 

 a Gaussian distribution, centered in this case about 

 the zero point. In other words, the probability of the 

 X component oi Ar having a value between x and 

 X + dxis (1 /-\/2x6^)e~^''^*dx, and the combined prob- 

 ability of having the x component and the y compo- 

 nentof Arin specified brackets of infinitesimalwidth is 



V{x,y)dxdy = ^/-"^''^^'''+''-''dxdy. (8) 



It is convenient to introduce the polar coordinates a 

 and d in the complex A plane; 



a? = x^ -\- y''- , 

 tan 6 = - ■ 



X 



Equation (8) then assumes the form 



p(a,e)deda = —e-^'"^^''^adeda- (10) 



27r6^ 



If we wish to disregard the dependence on the phase 

 angle d, we may integrate over 6 from zero to 27r, 

 with the result 



pia)da = -e-^'"'''\da = '-e-^^'^'^dl. (11) 



This last expression can be simplified by the intro- 

 duction of the average intensity 7. By definition, this 

 average intensity is given by the formula 



J-«co 

 Iq{I)dI, 

 1=0 



in which q{I) is, according to equation (11), 



Carrying out the integration, we find for I 



52 



(12) 



(13) 



/ = 



pc 



(14) 



which means that we have for q(I) and for Q{I) 

 1 



qil) 



-a 1 1) 



and 



Q{I) = l-e 



-(.i/'i) 



(15) 



(16) 



respectively. Whenever a signal is the resultant of a 

 large number of components with random phase re- 



