Shot effect

A mathematical description of voltage fluctuations at the output of a linear system at the input of which there are random perturbations produced at random moments of time. If $ W ( t , \tau ) $ is the output of the system at time $ t $ resulting from a single pulse applied at time $ \tau \leq t $, the shot effect may be described by a stochastic process

$$ X ( t) = \sum _ {\{ {k } : {\tau _ {k} \leq t } \} } \alpha _ {k} W ( t , \tau _ {k} ) , $$

where $ \dots < \tau _ {-} 1 < \tau _ {0} < \tau _ {1} < \dots < \tau _ {k} < \dots $ are the arrival moments of pulses, while $ \alpha _ {k} $ are random variables characterizing the magnitudes of the intensities of the pulses. In the particular case when $ W ( t , \tau ) = W ( t - \tau ) $, $ W ( s) = 0 $, $ s \leq 0 $, the $ \alpha _ {k} $ are independent, uniformly-distributed random variables with finite variance, while $ \dots < \tau _ {-} 1 < \tau _ {0} < \tau _ {1} < {} \dots $ forms a Poisson flow of events with parameter $ \lambda $, the process $ X ( t) $ is a stationary stochastic process in the narrow sense, with

$$ {\mathsf E} X ( t) = \lambda {\mathsf E} \alpha _ {1} \int\limits _ { 0 } ^ \infty W ( s) d s , $$

$$ {\mathsf D} X ( t) = \lambda {\mathsf E} \alpha _ {1} ^ {2} \int\limits _ { 0 } ^ \infty W ^ {2} ( s) d s . $$

References

[1]	J.H. Laning, R.G. Battin, "Random processes in automatic control" , McGraw-Hill (1956)

References

[a1a]	S.O. Rice, "Mathematical analysis of random noise" Bell Systems Techn. J. , 23 (1944) pp. 283–332
[a1b]	S.O. Rice, "Mathematical analysis of random noise" Bell Systems Techn. J. , 24 (1945) pp. 46–156
[a2]	N. Wax (ed.) , Selected papers on noise and stochastic processes , Dover, reprint (1953)
[a3]	E. Parzen, "Stochastic processes" , Holden-Day (1962)
[a4]	E. Wong, "Stochastic processes in information and dynamical systems" , McGraw-Hill (1971)