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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849701.png" /> is the output of the system at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849702.png" /> resulting from a single pulse applied at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849703.png" />, the shot effect may be described by a [[Stochastic process|stochastic process]]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849704.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849705.png" /> are the arrival moments of pulses, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849706.png" /> are random variables characterizing the magnitudes of the intensities of the pulses. In the particular case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s0849709.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s08497010.png" /> are independent, uniformly-distributed random variables with finite variance, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s08497011.png" /> forms a [[Poisson flow|Poisson flow]] of events with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s08497012.png" />, the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s08497013.png" /> is a [[Stationary stochastic process|stationary stochastic process]] in the narrow sense, with
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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 ) $
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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]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s08497014.png" /></td> </tr></table>
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$$
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X ( t)  = \sum _ {\{ {k } : {\tau _ {k} \leq  t } \}
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} \alpha _ {k} W (
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t , \tau _ {k} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084970/s08497015.png" /></td> </tr></table>
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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 ) $,
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$  W ( s) = 0 $,
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$  s \leq  0 $,
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the  $  \alpha _ {k} $
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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) $
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is a [[Stationary stochastic process|stationary stochastic process]] in the narrow sense, with
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$$
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{\mathsf E} X ( t)  = \lambda {\mathsf E} \alpha _ {1} \int\limits _ { 0 } ^  \infty  W ( s)
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d s ,
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$$
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$$
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{\mathsf D} X ( t)  = \lambda {\mathsf E} \alpha _ {1}  ^ {2} \int\limits _ { 0 } ^  \infty  W  ^ {2} ( s)  d s .
 +
$$
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 09:42, 20 July 2021


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)

Comments

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)
How to Cite This Entry:
Shot effect. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shot_effect&oldid=15483
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article