View source for Shot effect

<!--
s0849701.png
$#A+1 = 15 n = 0
$#C+1 = 15 : ~/encyclopedia/old_files/data/S084/S.0804970 Shot effect
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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|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|Poisson flow]] of events with parameter  $  \lambda $, 
the process  $  X ( t) $
is a [[Stationary stochastic process|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.H. Laning,   R.G. Battin,   "Random processes in automatic control" , McGraw-Hill  (1956)</TD></TR></table>

====Comments====

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