Theory of Stochastic Signals/Power-Spectral Density - Revision history

Guenter at 15:13, 22 December 2022

2022-12-22T15:13:22Z

2022-12-22T09:12:51Z

2022-12-22T09:11:56Z

2022-12-22T09:09:15Z

2022-12-21T17:43:17Z

2022-12-21T17:37:21Z

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 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp; We consider here a zero mean process with statistically independent samples.&nbsp; Thus,&nbsp; only the ACF value&nbsp; $φ_x(0) = σ_x^2$&nbsp; should be different from zero.
-[[File:EN Sto T 4 5 S5 v1.png |450px|right|frame| Accuracy of numerical PSD calculation '''KORREKTUR: theoretisch''']]
+[[File:EN_Sto_T_4_5_S5_b_neu_v2.png |450px|right|frame| Accuracy of numerical PSD calculation ]]
 *But if one determines the ACF numerically from only&nbsp; $N = 1000$&nbsp; samples,&nbsp; one obtains finite ACF values even for&nbsp; $k ≠ 0$.

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 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;
-A limiting case of the reciprocity law '''KORREKTUR: theorem''' represents the so-called&nbsp; "White Noise":
+A limiting case of the reciprocity theorem represents the so-called&nbsp; "White Noise":
 *This includes all spectral components&nbsp; (up to infinity).
 *The equivalent PSD bandwidth&nbsp; $∇f$&nbsp; is infinite.
-The law '''KORREKTUR: theorem''' given here states that for the equivalent ACF duration&nbsp; $∇τ = 0$&nbsp; must hold &nbsp; &rArr; &nbsp; &raquo;'''white noise has a Dirac-shaped ACF'''&laquo;.
+The theorem given here states that for the equivalent ACF duration&nbsp; $∇τ = 0$&nbsp; must hold &nbsp; &rArr; &nbsp; &raquo;'''white noise has a Dirac-shaped ACF'''&laquo;.
 For more on this topic, see the three-part&nbsp; (German language)&nbsp; learning video&nbsp; [[Der_AWGN-Kanal_(Lernvideo)|"The AWGN channel"]],&nbsp; especially the second part.}}

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 <br>
 All the&nbsp; [[Signal_Representation/Fourier_Transform_Laws|$\text{Fourier transform theorems}$]]&nbsp; derived in the book&nbsp; "Signal Representation"&nbsp; for deterministic signals can also be applied to
-[[File:P_ID390__Sto_T_4_5_S3_Ganz_neu.png |frame| On the&nbsp; "Reciprocity Law '''KORREKTUR: Theorem'''"&nbsp; of ACF and PSD]]
+[[File:P_ID390__Sto_T_4_5_S3_Ganz_neu.png |frame| On the&nbsp; "Reciprocity Theorem"&nbsp; of ACF and PSD]]
 *the&nbsp; auto-correlation function&nbsp; $\rm (ACF)$,&nbsp; and

@@ Line 13: / Line 13: @@
 :$$\varphi_x(t_1,t_2)={\rm E}\big[x(t_{\rm 1})\cdot x(t_{\rm 2})\big] = \varphi_x(\tau)= \int^{+\infty}_{-\infty}x(t)\cdot x(t+\tau)\,{\rm d}t.$$
-The auto-correlation function provides quantitative information about the&nbsp; (linear)&nbsp; statistical bindings within the ergodic process&nbsp; $\{x_i(t)\}$&nbsp; in the time domain.&nbsp; The equivalent descriptor in the frequency domain is the&nbsp; "power-spectral density",&nbsp; often also referred to as the&nbsp; "power density spectrum '''KORREKTUR: power-spectral density'''".
+The auto-correlation function provides quantitative information about the&nbsp; (linear)&nbsp; statistical bindings within the ergodic process&nbsp; $\{x_i(t)\}$&nbsp; in the time domain.&nbsp; The equivalent descriptor in the frequency domain is the&nbsp; "power-spectral density",&nbsp; often also referred to as the&nbsp; "power-spectral density".
 {{BlaueBox|TEXT=

@@ Line 174: / Line 174: @@
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp; We consider here a zero mean process with statistically independent samples.&nbsp; Thus,&nbsp; only the ACF value&nbsp; $φ_x(0) = σ_x^2$&nbsp; should be different from zero.
-[[File:EN Sto T 4 5 S5 v1.png |450px|right|frame| Accuracy of numerical PSD calculation]]
+[[File:EN Sto T 4 5 S5 v1.png |450px|right|frame| Accuracy of numerical PSD calculation '''KORREKTUR: theoretisch''']]
 *But if one determines the ACF numerically from only&nbsp; $N = 1000$&nbsp; samples,&nbsp; one obtains finite ACF values even for&nbsp; $k ≠ 0$.

← Older revision		Revision as of 17:37, 21 December 2022
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−	The law given here states that for the equivalent ACF duration  $∇τ = 0$  must hold   ⇒   »'''white noise has a Dirac-shaped ACF'''«.	+	The law '''KORREKTUR: theorem''' given here states that for the equivalent ACF duration  $∇τ = 0$  must hold   ⇒   »'''white noise has a Dirac-shaped ACF'''«.

	For more on this topic, see the three-part  (German language)  learning video  [[Der_AWGN-Kanal_(Lernvideo)\|"The AWGN channel"]],  especially the second part.}}		For more on this topic, see the three-part  (German language)  learning video  [[Der_AWGN-Kanal_(Lernvideo)\|"The AWGN channel"]],  especially the second part.}}

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 *the&nbsp; auto-correlation function&nbsp; $\rm (ACF)$,&nbsp; and
-*the&nbsp; power-spectral Density&nbsp; $\rm (PSD)$.&nbsp;
+*the&nbsp; power-spectral density&nbsp; $\rm (PSD)$.&nbsp;
 <br>However,&nbsp; not all laws yield meaningful results due to the specific properties
 *of auto-correlation function&nbsp; (always real and even)

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 $\text{Definition:}$&nbsp; The&nbsp; &raquo;'''power-spectral density'''&laquo;&nbsp; $\rm (PSD)$&nbsp; of an ergodic random process&nbsp; $\{x_i(t)\}$&nbsp; is the Fourier transform of the auto-correlation function&nbsp; $\rm (ACF)$:
 :$${\it \Phi}_x(f)=\int^{+\infty}_{-\infty}\varphi_x(\tau) \cdot {\rm e}^{- {\rm j\hspace{0.05cm}\cdot \hspace{0.05cm} \pi}\hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}\tau} {\rm d} \tau. $$
-This Functional relationship is called the&nbsp; "Theorem of&nbsp; [https://en.wikipedia.org/wiki/Norbert_Wiener $\text{Wiener}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Aleksandr_Khinchin $\text{Khinchin}$]". }}
+This functional relationship is called the&nbsp; "Theorem of&nbsp; [https://en.wikipedia.org/wiki/Norbert_Wiener $\text{Wiener}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Aleksandr_Khinchin $\text{Khinchin}$]". }}