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

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	{{Display}}		{{Display}}
			[[de:Stochastische_Signaltheorie/Leistungsdichtespektrum_(LDS)]]

Guenter at 15:13, 22 December 2022

2022-12-22T15:13:22Z

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	$\text{Example 4:}$  We consider here a zero mean process with statistically independent samples.  Thus,  only the ACF value  $φ_x(0) = σ_x^2$  should be different from zero.		$\text{Example 4:}$  We consider here a zero mean process with statistically independent samples.  Thus,  only the ACF value  $φ_x(0) = σ_x^2$  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  $N = 1000$  samples,  one obtains finite ACF values even for  $k ≠ 0$.		*But if one determines the ACF numerically from only  $N = 1000$  samples,  one obtains finite ACF values even for  $k ≠ 0$.

Guenter at 09:12, 22 December 2022

2022-12-22T09:12:51Z

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	$\text{Example 2:}$		$\text{Example 2:}$
	A limiting case of the reciprocity ~~law '''KORREKTUR:~~ theorem~~'''~~ represents the so-called  "White Noise":		A limiting case of the reciprocity theorem represents the so-called  "White Noise":
	*This includes all spectral components  (up to infinity).		*This includes all spectral components  (up to infinity).
	*The equivalent PSD bandwidth  $∇f$  is infinite.		*The equivalent PSD bandwidth  $∇f$  is infinite.


	The ~~law '''KORREKTUR:~~ theorem~~'''~~ given here states that for the equivalent ACF duration  $∇τ = 0$  must hold   ⇒   »'''white noise has a Dirac-shaped ACF'''«.		The 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.}}

Guenter at 09:11, 22 December 2022

2022-12-22T09:11:56Z

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	<br>		<br>
	All the  [[Signal_Representation/Fourier_Transform_Laws\|$\text{Fourier transform theorems}$]]  derived in the book  "Signal Representation"  for deterministic signals can also be applied to		All the  [[Signal_Representation/Fourier_Transform_Laws\|$\text{Fourier transform theorems}$]]  derived in the book  "Signal Representation"  for deterministic signals can also be applied to
	[[File:P_ID390__Sto_T_4_5_S3_Ganz_neu.png \|frame\| On the  "Reciprocity ~~Law '''KORREKTUR:~~ Theorem~~'''~~"  of ACF and PSD]]		[[File:P_ID390__Sto_T_4_5_S3_Ganz_neu.png \|frame\| On the  "Reciprocity Theorem"  of ACF and PSD]]

	*the  auto-correlation function  $\rm (ACF)$,  and		*the  auto-correlation function  $\rm (ACF)$,  and

Guenter at 09:09, 22 December 2022

2022-12-22T09:09:15Z

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	:$$\varphi_x(t_1,t_2)={\rm E}\big[x(t_{\rm 1})\cdot x(t_{\rm 2})\big] = \varphi_x(\tau)= \int^{+\infty}_{-\infty}x(t)\cdot x(t+\tau)\,{\rm d}t.$$		:$$\varphi_x(t_1,t_2)={\rm E}\big[x(t_{\rm 1})\cdot x(t_{\rm 2})\big] = \varphi_x(\tau)= \int^{+\infty}_{-\infty}x(t)\cdot x(t+\tau)\,{\rm d}t.$$

	The auto-correlation function provides quantitative information about the  (linear)  statistical bindings within the ergodic process  $\{x_i(t)\}$  in the time domain.  The equivalent descriptor in the frequency domain is the  "power-spectral density",  often also referred to as the  "~~power density spectrum '''KORREKTUR:~~ power-spectral density~~'''~~".		The auto-correlation function provides quantitative information about the  (linear)  statistical bindings within the ergodic process  $\{x_i(t)\}$  in the time domain.  The equivalent descriptor in the frequency domain is the  "power-spectral density",  often also referred to as the  "power-spectral density".

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Hwang at 17:43, 21 December 2022

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	$\text{Example 4:}$  We consider here a zero mean process with statistically independent samples.  Thus,  only the ACF value  $φ_x(0) = σ_x^2$  should be different from zero.		$\text{Example 4:}$  We consider here a zero mean process with statistically independent samples.  Thus,  only the ACF value  $φ_x(0) = σ_x^2$  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  $N = 1000$  samples,  one obtains finite ACF values even for  $k ≠ 0$.		*But if one determines the ACF numerically from only  $N = 1000$  samples,  one obtains finite ACF values even for  $k ≠ 0$.

Hwang at 17:37, 21 December 2022

2022-12-21T17:37:21Z

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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.}}

Hwang at 15:35, 21 December 2022

2022-12-21T15:35:00Z

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	<br>		<br>
	All the  [[Signal_Representation/Fourier_Transform_Laws\|$\text{Fourier transform theorems}$]]  derived in the book  "Signal Representation"  for deterministic signals can also be applied to		All the  [[Signal_Representation/Fourier_Transform_Laws\|$\text{Fourier transform theorems}$]]  derived in the book  "Signal Representation"  for deterministic signals can also be applied to
	[[File:P_ID390__Sto_T_4_5_S3_Ganz_neu.png \|frame\| On the  "Reciprocity Law"  of ACF and PSD]]		[[File:P_ID390__Sto_T_4_5_S3_Ganz_neu.png \|frame\| On the  "Reciprocity Law '''KORREKTUR: Theorem'''"  of ACF and PSD]]

	*the  auto-correlation function  $\rm (ACF)$,  and		*the  auto-correlation function  $\rm (ACF)$,  and
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	$\text{Example 2:}$		$\text{Example 2:}$
	A limiting case of the reciprocity law represents the so-called  "White Noise":		A limiting case of the reciprocity law '''KORREKTUR: theorem''' represents the so-called  "White Noise":
	*This includes all spectral components  (up to infinity).		*This includes all spectral components  (up to infinity).
	*The equivalent PSD bandwidth  $∇f$  is infinite.		*The equivalent PSD bandwidth  $∇f$  is infinite.

Hwang at 15:33, 21 December 2022

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

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	Similarly,  the auto-correlation function can be computed as the inverse Fourier transform of the power-spectral density  (see section  [[Signal_Representation/~~Fourier_Transform_and_its_Inverse~~#The_second_Fourier_integral\|"Inverse Fourier transform"]]  in the book  "Signal Representation"):		Similarly,  the auto-correlation function can be computed as the inverse Fourier transform of the power-spectral density  (see section  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|"Inverse Fourier transform"]]  in the book  "Signal Representation"):
	:$$ \varphi_x(\tau)=\int^{+\infty}_{-\infty} {\it \Phi}_x \cdot {\rm e}^{- {\rm j\hspace{0.05cm}\cdot \hspace{0.05cm} \pi}\hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}\tau} {\rm d} f.$$		:$$ \varphi_x(\tau)=\int^{+\infty}_{-\infty} {\it \Phi}_x \cdot {\rm e}^{- {\rm j\hspace{0.05cm}\cdot \hspace{0.05cm} \pi}\hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}\tau} {\rm d} f.$$
	*The two equations are directly applicable only if the random process contains neither a DC component nor periodic components.		*The two equations are directly applicable only if the random process contains neither a DC component nor periodic components.
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	*The areas under the PSD curves are equal   ⇒   the processes  $\{x_i(t)\}$  and  $\{y_i(t)\}$  have the same power:		*The areas under the PSD curves are equal   ⇒   the processes  $\{x_i(t)\}$  and  $\{y_i(t)\}$  have the same power:
	:$${\varphi_x({\rm 0})}\hspace{0.05cm} =\hspace{0.05cm} \int^{+\infty}_{-\infty}{{\it \Phi}_x(f)} \hspace{0.1cm} {\rm d} f \hspace{0.2cm} = \hspace{0.2cm}{\varphi_y({\rm 0})} = \int^{+\infty}_{-\infty}{{\it \Phi}_y(f)} \hspace{0.1cm} {\rm d} f .$$		:$${\varphi_x({\rm 0})}\hspace{0.05cm} =\hspace{0.05cm} \int^{+\infty}_{-\infty}{{\it \Phi}_x(f)} \hspace{0.1cm} {\rm d} f \hspace{0.2cm} = \hspace{0.2cm}{\varphi_y({\rm 0})} = \int^{+\infty}_{-\infty}{{\it \Phi}_y(f)} \hspace{0.1cm} {\rm d} f .$$
	*The from classical  (deterministic)  system theory well known   [[Signal_Representation/~~Fourier_Transform_Theorems~~#Reciprocity_Theorem_of_time_duration_and_bandwidth\|$\text{Reciprocity Theorem of time duration and bandwidth}$]]  also applies here:   '''A narrow ACF corresponds to a broad PSD and vice versa'''.		*The from classical  (deterministic)  system theory well known   [[Signal_Representation/The_Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|$\text{Reciprocity Theorem of time duration and bandwidth}$]]  also applies here:   '''A narrow ACF corresponds to a broad PSD and vice versa'''.
	*As a descriptive quantity,  we use here the  »'''equivalent PSD bandwidth'''«   $∇f$  $($one speaks  "Nabla-f"$)$,  <br>similarly defined as the equivalent ACF duration  $∇τ$  in chapter  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Interpretation_of_the_auto-correlation_function\|"Interpretation of the auto-correlation function"]]:		*As a descriptive quantity,  we use here the  »'''equivalent PSD bandwidth'''«   $∇f$  $($one speaks  "Nabla-f"$)$,  <br>similarly defined as the equivalent ACF duration  $∇τ$  in chapter  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Interpretation_of_the_auto-correlation_function\|"Interpretation of the auto-correlation function"]]:
	:$${{\rm \nabla} f_x} = \frac {1}{{\it \Phi}_x(f = {\rm 0})} \cdot \int^{+\infty}_{-\infty}{{\it \Phi}_x(f)} \hspace{0.1cm} {\rm d} f, $$		:$${{\rm \nabla} f_x} = \frac {1}{{\it \Phi}_x(f = {\rm 0})} \cdot \int^{+\infty}_{-\infty}{{\it \Phi}_x(f)} \hspace{0.1cm} {\rm d} f, $$
	:$${ {\rm \nabla} \tau_x} = \frac {\rm 1}{ \varphi_x(\tau = \rm 0)} \cdot \int^{+\infty}_{-\infty}{\varphi_x(\tau )} \hspace{0.1cm} {\rm d} \tau.$$		:$${ {\rm \nabla} \tau_x} = \frac {\rm 1}{ \varphi_x(\tau = \rm 0)} \cdot \int^{+\infty}_{-\infty}{\varphi_x(\tau )} \hspace{0.1cm} {\rm d} \tau.$$
	*With these definitions,  the following basic relationship holds:		*With these definitions,  the following basic relationship holds:
	:$${{\rm \nabla} \tau_x} \cdot {{\rm \nabla} f_x} = 1\hspace{1cm}{\rm resp.}\hspace{1cm}		:$${{\rm \nabla} \tau_x} \cdot {{\rm \nabla} f_x} = 1\hspace{1cm}{\rm resp.}\hspace{1cm}{{\rm \nabla} \tau_y} \cdot {{\rm \nabla} f_y} = 1.$$
	{{\rm \nabla} \tau_y} \cdot {{\rm \nabla} f_y} = 1.$$

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	==Numerical PSD determination==		==Numerical PSD determination==
	<br>		<br>
	Auto-correlation function and power-spectral density are strictly related via the  [[Signal_Representation/~~Fourier_Transform_and_its_Inverse~~#Fourier_transform\|$\text{Fourier transform}$]].  This relationship also holds for discrete-time ACF representation with the sampling operator  ${\rm A} \{ \varphi_x ( \tau ) \} $,  thus for		Auto-correlation function and power-spectral density are strictly related via the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#Fourier_transform\|$\text{Fourier transform}$]].  This relationship also holds for discrete-time ACF representation with the sampling operator  ${\rm A} \{ \varphi_x ( \tau ) \} $,  thus for
	:$${\rm A} \{ \varphi_x ( \tau ) \} = \varphi_x ( \tau ) \cdot \sum_{k= - \infty}^{\infty} T_{\rm A} \cdot \delta ( \tau - k \cdot T_{\rm A}).$$		:$${\rm A} \{ \varphi_x ( \tau ) \} = \varphi_x ( \tau ) \cdot \sum_{k= - \infty}^{\infty} T_{\rm A} \cdot \delta ( \tau - k \cdot T_{\rm A}).$$