5-4] RANDOM NOISE PROCESSES 247 



In this expression Xi and X2 are values of the noise process at times /i and /2, 

 0-2 is the variance of Xi and X2, and p is a factor indicating the degree of 

 correlation between Xi and X2. This factor is called the normalized autocorre- 

 lation function. It is defined in this case, where the mean is zero, as the 

 average value of the product XiX^, divided by the average value o{ x"^ which 

 normalizes it so that its range is from +1 to — 1. When /i and t^. are close 

 together so that x-i and X2 have about the same values, the value of p will be 

 close to unity, indicating a high degree of correlation. That is, when Xx is 

 high, Xi is also very likely high; and when Xi is low, x^ is probably low. On 

 the other hand, when /i and ti are sufficiently far apart for several oscil- 

 lations of the noise functions to occur between them, Xi and x^ will tend to 

 be uncorrelated and p will be close to zero. When the process is stationary, 

 the autocorrelation function will be independent of the particular times 

 /i and /2 and depend only upon their difference, which we denote by 



T = ti — /o. 



The significant and meaningful attributes of noise processes must be 

 expressed as average values. The notation we shall adopt to indicate the 

 average value of some function of the process is to simply bar that function. 

 Thus the average value of the process itself is denoted by x- If the process 

 represents voltages or currents, then x can be interpreted as the d-c level. 

 For the process whose probability density is given by Equation 5-22, the 

 average value corresponding to the d-c is zero. The mean square value of 

 the process about the mean or the variance can similarly be regarded as the 

 average power in a unit resistance. As previously noted, this quantity is 

 denoted by cr^. 



a^ = {^x -xY -= x^ - x\ (5-24) 



Actually, the term power will often be used very generally to refer to the 

 square of arbitrarily measured variables so that sometimes it cannot be 

 identified with physical power, although the electrical terminology has been 

 retained. As an example, suppose that an angle 6 is found to be oscillating 

 with an amplitude A and a frequency co or 6 = A cos cot. In this case, we 

 might say that the angle d has a power of A^ /2 although the dimensions 

 of this quantity are certainly not watts. 



The average value of the product X1X2 is very significant in signal and 

 noise studies. This quantity is called the autocorrelation function, and we 

 shall denote it by ^(r), where r is the time difference ti — t\. For a station- 

 ary process, X\ and Xi are uncorrelated when t is very large except for the 

 average value or d-c component: 



^(±00)= p. (5-25) 



When T = 0, the autocorrelation function simply equals the average value 

 of x^. Thus, the variance of the process is given by 



