252 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



The power density spectrum of this process will thus have the following 

 form: 



A» = -r^ [t + 27rT^5(co)]. (5-42) 



The autocorrelation function, which is just the Fourier transform of the 

 power spectrum, has already been partially computed in Equation 5-9 and 

 will have the form 



<p(t) = ^ ^-H + ^2, (5_43) 



We may note that since we have used an A(/) corresponding to the impulse 

 response of the RC filter pictured in Fig. 5-1 , the noise process that has been 

 defined can be generated approximately by short pulses occurring at 

 random times which are modified by this filter. 



A physical interpretation of our model of a noise process is provided by 

 shot noise, fluctuations in the number of electrons arriving at the plate of a 

 vacuum tube per unit time. We shall use our model to show that the mean 

 square fluctuation in electron current, AP, incident to the shot eflFect is 

 given by 



(a7)2 = lelAF (5-44) 



where e = electronic charge 



/ = average current (d-c) 



AF = observation bandwidth. 



We suppose that each function h(i — tk) in the sum in Equation 5-36 

 represents the arrival of one electron at the plate. In this case, the integral 

 oi h{t) should equal the electronic charge e, and we assume this, or what is 

 equivalent, that //(O) = e. The magnitude of both the square of the direct 

 current and average of the square of the fluctuation or noise currents can be 

 determined from Equation 5-41. The square of the direct current corre- 

 sponds to the magnitude of the impulse function at zero frequency in that 

 expression and is given by 



P = \HmW' = e'y'. (5-45) 



The mean square value of the noise currents corresponds to the integral of 

 the nonsingular term |//(co)|^7, in Equation 5-41. We are unable to deter- 

 mine this exacdy without knowing the form of the spectrum of a current 

 pulse, //(co). If, however, we are interested in the output of a filter which is 

 narrow compared to //(co) we can approximate the mean square current 

 in the output of the narrow filter by the product of twice the filter band- 

 width 2AF and the low-frequency power density of the electronic pulse 



