OBSERVED FLUCTUATION 



165 



expression /„/„, wiiuils by definition the double sum 

 1 N 



IJm - TTjZ- ^,''^"" 

 iV n,m=l 



which in turn can be written as the product of two 

 single sums. 



A72 ^J"^" 

 I\ n,m=l 



N-n=l " = 1 



If there is no correlation between consecutive signals, 

 then the two terms (/„ — /„)- and (/„ — /„-i)^ in 

 equation (26) are equal, and Si vanishes. If the cor- 

 relation between consecutive signals is positive, then 

 the rms difference between two signals picked at ran- 

 dom will be greater than the rms difference between 

 two consecutive signals, and Si will be positive. If Si 

 should turn out to be negative, that would mean that 

 the average difference square between consecutive 

 signals exceeds the random value; the correlation 

 between consecutive signals would then be said to be 

 negative. 



The quantity Si has the dimension of an intensity 

 squared. If it is desired to obtain a measure of correla- 

 tion which is dimensionless, it appears reasonable to 

 divide Si through by the mean squared random dif- 

 ference, 



(/n - I^r = 2{P - t). (27) 



For if the correlation were perfect (that is, if each 

 signal pulse had the same intensity as its predecessor, 

 a situation which can obviously not be realized ex- 

 actly), this ratio would equal unity, while for nega- 

 tive self-correlation, the ratio can be shown never to 

 drop below the value —1. Hence, it is customary to 

 measure the self -correlation of consecutive signals by 

 means of the quantity 



J J 72 



pi ^ '"i"-^ _/ , (28) 



P- I 



which is called the coefficient of self-correlation for 



unit step interval. In close analogy to this quantity, 



we may define the self -correlation coefficient for an 



interval of s steps, p„ by means of the expression 



J J — P 



P. - '"'--' _; • (29) 



p- I 



The averaging in the first term of the numerator is to 

 be carried out by averaging over all values of the 

 index n while keeping the step interval s fixed. For 

 s = 0, the self-correlation coefficient equals unity, by 

 definition. It can be shown that for all values of s, ps 

 lies between —1 (complete anticorrelation) and 1 



(complete correlation). Furthermore, p, is an even 

 function of its argument s, that is: 



P. = P-a- (30) 



A MEAN RANGE IIS YARDS 



1.0 



o.e 



0.6 



0.4 



0.2 



P 



-0.2 



-0.4 



-0.6 



-0.8 



-1.0 



1.0 



0.8 



0.6 



0.4 



0.2 



9 



-0.2 



-0.4 



•0.6 



-0.8 



-1.0 



O 2 4 6 8 10 12 14 16 



CORRELATION INTERVAL IN SECONDS 



8 16 24 32 40 48 56 64 



CORRELATION INTERVAL IN SIGNALS 



Figure 5. Self-correlation coefficients of two se- 

 quences of supersonic signals. 



Figure 5 shows two self-correlation coefficients 

 which were obtained at UCDWR and which were 

 computed from samples at different ranges. In both 

 cases, the receiving hydrophone was in the direct 

 sound field. The abscissa represents the step interval, 

 marked both in terms of the number of pulses s and 

 in terms of the time in seconds. These two plots, 

 which are typical of the others obtained, show that 

 there is a marked positive self-correlation for step 

 intervals of a few seconds. It appears that the longer 

 the range, the longer is the step interval of positive 

 correlation (that is, the slower is the fluctuation), al- 

 though the evidence on that point is too scanty to be 

 considered conclusive. 



For many of these plots, the self-correlation be- 

 comes negative for some step interval before it drops 

 down to zero. This anticorrelation has not yet been 

 explained, although it is observed more often than 

 not. 



