Theory of Stochastic Signals/Auto-Correlation Function - Revision history

Guenter at 09:07, 22 December 2022

2022-12-22T09:07:42Z

Hwang at 15:06, 21 December 2022

2022-12-21T15:06:13Z

Hwang at 15:02, 21 December 2022

2022-12-21T15:02:19Z

Hwang at 14:59, 21 December 2022

2022-12-21T14:59:18Z

Hwang at 14:54, 21 December 2022

2022-12-21T14:54:36Z

Hwang at 14:49, 21 December 2022

2022-12-21T14:49:22Z

Hwang at 14:48, 21 December 2022

2022-12-21T14:48:03Z

Hwang at 14:42, 21 December 2022

2022-12-21T14:42:45Z

Hwang at 14:42, 21 December 2022

2022-12-21T14:42:20Z

Hwang at 14:19, 21 December 2022

2022-12-21T14:19:47Z

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 $\text{Example 3:}$&nbsp; The above definition of the auto-correlation function applies in general,&nbsp; i.e. also to non-stationary and non-ergodic processes.
-*An example of a non-stationary process is the occurrence of&nbsp; "pulse interference '''KORREKTUR: intersymbol interference'''"&nbsp; in the telephone network caused by dial pulses in adjacent lines.
+*An example of a non-stationary process is the occurrence of&nbsp; "intersymbol interference"&nbsp; in the telephone network caused by dial pulses in adjacent lines.
 *In digital signal transmission,&nbsp; such non-stationary interference processes usually lead to trunking errors.}}

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 Justification of the worse result according to this&nbsp; $\text{Example 6}$:
-*Due to the intrinsic statistical bindings,&nbsp; not all samples now provide full information about the underlying random process.
+*Due to the internal statistical bindings,&nbsp; not all samples now provide full information about the underlying random process.
 *Furthermore,&nbsp; the images reveal that in numerical ACF calculation of a random variable with statistical bindings,&nbsp; also the errors are correlated.
 *If,&nbsp; for example,&nbsp; as seen in the upper image,&nbsp; the ACF value&nbsp; $φ_x({\rm 26})$&nbsp; is spuriously positive and large,&nbsp; the neighboring ACF values&nbsp; $φ_x({\rm 25})$&nbsp; and&nbsp; $φ_x({\rm 27})$&nbsp; will also be positive with similar numerical values.&nbsp; This area is marked by the rectangle in the upper graph.}}

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 #You define the field&nbsp; $\rm ACF\big[\hspace{0.05cm}0 : {\it l}\hspace{0.1cm}\big]$&nbsp; of type&nbsp; "float"&nbsp; and preallocate zeros to all elements.
 #On each loop pass&nbsp; $($indexed with variable&nbsp; $k)$,&nbsp; the&nbsp; $l + 1$&nbsp; field elements&nbsp; ${\rm ACF}\big[\hspace{0.03cm}k\hspace{0.03cm}\big]$&nbsp; are incremented by the contribution &nbsp;$x_ν \cdot x_{ν+k}$.
-#All&nbsp; $l+1$&nbsp; field elements are processed,&nbsp; however,&nbsp; only as long as the controln variable&nbsp; $k$&nbsp; is not greater than&nbsp; $N - l$.
+#All&nbsp; $l+1$&nbsp; field elements are processed,&nbsp; however,&nbsp; only as long as the control variable&nbsp; $k$&nbsp; is not greater than&nbsp; $N - l$.
 #It must always be considered that&nbsp; $ν + k ≤ N$&nbsp; must hold.&nbsp; This means that the averaging in the different fields&nbsp; $\rm ACF\big[\hspace{0.03cm}0\hspace{0.03cm}\big]$ ... ${\rm ACF}\big[\hspace{0.03cm}l\hspace{0.03cm}\big]$&nbsp; must be done over a different number of summands.
 #If,&nbsp; at the end of the calculation,&nbsp; the values stored in&nbsp; ${\rm ACF}\big[\hspace{0.03cm}k\hspace{0.03cm}\big]$&nbsp; are still divided by the number of summands&nbsp; $(N - k)$&nbsp; this field contains the discrete ACF values we are looking for:

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 Following this scheme results in the following algorithm:
 #You define the field&nbsp; $\rm ACF\big[\hspace{0.05cm}0 : {\it l}\hspace{0.1cm}\big]$&nbsp; of type&nbsp; "float"&nbsp; and preallocate zeros to all elements.
-#On each loop pass&nbsp; $($indexed with variable&nbsp; $k)$&nbsp; the&nbsp; $l + 1$&nbsp; field elements&nbsp; ${\rm ACF}\big[\hspace{0.03cm}k\hspace{0.03cm}\big]$&nbsp; are incremented by the contribution &nbsp;$x_ν \cdot x_{ν+k}$.
+#On each loop pass&nbsp; $($indexed with variable&nbsp; $k)$,&nbsp; the&nbsp; $l + 1$&nbsp; field elements&nbsp; ${\rm ACF}\big[\hspace{0.03cm}k\hspace{0.03cm}\big]$&nbsp; are incremented by the contribution &nbsp;$x_ν \cdot x_{ν+k}$.
 #All&nbsp; $l+1$&nbsp; field elements are processed,&nbsp; however,&nbsp; only as long as the controln variable&nbsp; $k$&nbsp; is not greater than&nbsp; $N - l$.
 #It must always be considered that&nbsp; $ν + k ≤ N$&nbsp; must hold.&nbsp; This means that the averaging in the different fields&nbsp; $\rm ACF\big[\hspace{0.03cm}0\hspace{0.03cm}\big]$ ... ${\rm ACF}\big[\hspace{0.03cm}l\hspace{0.03cm}\big]$&nbsp; must be done over a different number of summands.

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 *For the processes considered here&nbsp; (with Gaussian-like ACF)&nbsp; holds according to the above sketch: &nbsp; $∇τ_x = 0.33 \hspace{0.05cm} \rm &micro; s$ &nbsp; resp. &nbsp; $∇τ_y = 1 \hspace{0.05cm} \rm &micro;s$.
-*As another measure of the strength of the statistical bindings, the&nbsp; [[Digital_Signal_Transmission/Bündelfehlerkanäle#Fehlerkorrelationsfunktion_des_GE.E2.80.93Modells|$\text{correlation duration}$]]&nbsp; $T_{\rm K}$&nbsp; is often used in the literature.&nbsp;
+*As another measure of the strength of the statistical bindings, the&nbsp; [[Digital_Signal_Transmission/Burst_Error_Channels#Error_correlation_function_of_the_Gilbert-Elliott_model|$\text{correlation duration}$]]&nbsp; $T_{\rm K}$&nbsp; is often used in the literature.&nbsp;
 *This indicates the time duration at which the auto-correlation function has dropped to half of its maximum value.

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 Here we compile important properties of the auto-correlation function&nbsp;  $\rm (ACF)$,&nbsp;  starting from the ergodic ACF&ndash;form&nbsp; $φ_x(τ)$:
 *If the random process under consideration is real, so is its auto-correlation function is also real.
-*The ACF has the unit of a power,is  e.g.&nbsp;  "Watt"&nbsp; $\rm (W)$.&nbsp; Often one relates it to the resistance&nbsp; $1\hspace{0.03cm} Ω$;&nbsp; $φ_x(τ)$&nbsp; then with the unit&nbsp; $\rm V^2$&nbsp; resp.&nbsp; $\rm A^2$.
+*The ACF has the unit of a power, e.g.&nbsp;  "Watt"&nbsp; $\rm (W)$.&nbsp; Often one relates it to the resistance&nbsp; $1\hspace{0.03cm} Ω$;&nbsp; $φ_x(τ)$&nbsp; then with the unit&nbsp; $\rm V^2$&nbsp; resp.&nbsp; $\rm A^2$.
 *The ACF is always an even function&nbsp; ⇒ &nbsp; $φ_x(-τ) = φ_x(τ)$. &nbsp; All phase relations of the random process are lost in the ACF.
 *The ACF at the point&nbsp; $τ = 0$&nbsp; gives the second order moment&nbsp; $m_2$&nbsp; and thus the total signal power of the random process&nbsp; (DC and AC components):

← Older revision		Revision as of 14:42, 21 December 2022
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	*An example of a non-stationary process is the occurrence of  "pulse interference '''KORREKTUR: intersymbol interference'''"  in the telephone network caused by dial pulses in adjacent lines.		*An example of a non-stationary process is the occurrence of  "pulse interference '''KORREKTUR: intersymbol interference'''"  in the telephone network caused by dial pulses in adjacent lines.
−	*In digital signal transmission,  such non-stationary interference processes usually lead to trunking errors}}.	+	*In digital signal transmission,  such non-stationary interference processes usually lead to trunking errors.}}

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 $\text{Definitions:}$&nbsp; By a&nbsp; &raquo;'''random process'''&laquo;&nbsp; $\{x_i(t)\}$&nbsp; we understand a mathematical model&nbsp; for an ensemble of (many) random signals,&nbsp; which can and will differ from each other in detail,&nbsp; but nevertheless they have certain common properties.
 *To describe a random process&nbsp; $\{x_i(t)\}$&nbsp; we start from the notion that there are any number of random generators,&nbsp; completely identical in their physical and statistical properties,&nbsp; each of which yields a random signal&nbsp; $x_i(t)$.
-*Each random generator,&nbsp; despite having the same physical realization,&nbsp; outputs a different time signal&nbsp; $x_i(t)$&nbsp; that exists for all times from&nbsp; $-∞$&nbsp; to&nbsp; $+∞$&nbsp; This specific random signal is called the&nbsp; &raquo;'''$i$-th pattern signal'''&laquo;.
+*Each random generator,&nbsp; despite having the same physical realization,&nbsp; outputs a different time signal&nbsp; $x_i(t)$&nbsp; that exists for all times from&nbsp; $-∞$&nbsp; to&nbsp; $+∞$.&nbsp; This specific random signal is called the&nbsp; &raquo;'''$i$-th pattern signal'''&laquo;.
 *Every random process involves at least one stochastic component - for example:&nbsp; the amplitude,&nbsp; frequency,&nbsp; or phase of a message signal - and therefore cannot be accurately predicted by an observer.
 *The random process differs from the usual random experiments in probability theory or statistics in that the result is not an&nbsp; "event"&nbsp; but a&nbsp; "function"&nbsp; (time signal).