Theory of Stochastic Signals/Stochastic System Theory - Revision history

Guenter at 09:17, 22 December 2022

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Hwang at 18:29, 21 December 2022

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Hwang at 18:18, 21 December 2022

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Guenter at 10:03, 25 February 2022

2022-02-25T10:03:59Z

Guenter at 15:03, 26 January 2022

2022-01-26T15:03:44Z

Guenter at 15:02, 26 January 2022

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Guenter at 14:17, 26 January 2022

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Guenter at 14:15, 26 January 2022

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Guenter at 14:10, 26 January 2022

2022-01-26T14:10:59Z

Guenter at 13:54, 26 January 2022

2022-01-26T13:54:34Z

@@ Line 139: / Line 139: @@
 * The Fourier retransforms are to be inserted in each case, namely
 :$${{\it \varphi}_y(\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\,{{\it \Phi}_y(f)}, \hspace{0.5cm}{{\it \varphi}_x(\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\,{{\it \Phi}_x(f)}, \hspace{0.5cm}{h(\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\,{H(f)}, \hspace{0.5cm}{h(-\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\,{H^{\star}(f)}.$$
-*Moreover,&nbsp; $\rm (PSD)$&nbsp; '''KORREKTUR: PSD?''' each multiplication becomes a convolution operation.
+*Moreover,&nbsp; each multiplication becomes a convolution operation.

@@ Line 139: / Line 139: @@
 * The Fourier retransforms are to be inserted in each case, namely
 :$${{\it \varphi}_y(\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\,{{\it \Phi}_y(f)}, \hspace{0.5cm}{{\it \varphi}_x(\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\,{{\it \Phi}_x(f)}, \hspace{0.5cm}{h(\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\,{H(f)}, \hspace{0.5cm}{h(-\tau)} \circ\hspace{0.05cm}\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\,{H^{\star}(f)}.$$
-*Moreover,&nbsp; $\rm (PSD)$&nbsp; each multiplication becomes a convolution operation.
+*Moreover,&nbsp; $\rm (PSD)$&nbsp; '''KORREKTUR: PSD?''' each multiplication becomes a convolution operation.

@@ Line 24: / Line 24: @@
 ==System model and problem definition==
 <br>
-As in the book&nbsp; [[Linear_and_Time_Invariant_Systems| Linear and Time Invariant Systems]],&nbsp; we consider the setup sketched on the right, where the system characterized both
+As in the book&nbsp; [[Linear_and_Time_Invariant_Systems|"Linear and Time Invariant Systems"]],&nbsp; we consider the setup sketched on the right, where the system characterized both
 *by the impulse response&nbsp; $h(t)$
 *as well as by its frequency response&nbsp; $H(f)$
-is described unambiguously.&nbsp; The relationship between these descriptive quantities in the time and frequency domain is given by the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#Properties_of_aperiodic_signals|Fourier transformation]].&nbsp;
+is described unambiguously.&nbsp; The relationship between these descriptive quantities in the time and frequency domain is given by the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#Properties_of_aperiodic_signals|$\text{Fourier transformation}$]].&nbsp;
 [[File:EN_Sto_T_5_1_S1.png |right| 300px|frame|Filter influence on&nbsp; "spectrum"&nbsp; and&nbsp; <br>"power-spectral density"]]
 <br>If we apply the signal&nbsp; $x(t)$&nbsp; to the input and denote the output signal by&nbsp; $y(t)$,&nbsp; the classical system theory provides the following statements:
-*The output signal&nbsp; $y(t)$&nbsp; results from the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation|convolution]]&nbsp; between the input signal&nbsp; $x(t)$&nbsp; and the impulse response&nbsp; $h(t)$.&nbsp; The following equation is equally valid for deterministic and stochastic signals:
+*The output signal&nbsp; $y(t)$&nbsp; results from the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation|$\text{convolution}$]]&nbsp; between the input signal&nbsp; $x(t)$&nbsp; and the impulse response&nbsp; $h(t)$.&nbsp; The following equation is equally valid for deterministic and stochastic signals:
 :$$y(t) = x(t) \ast h(t) = \int_{-\infty}^{+\infty} x(\tau)\cdot h ( t - \tau) \,\,{\rm d}\tau.$$
 *For deterministic signals,&nbsp; one usually takes a roundabout route using the spectral functions.&nbsp; The spectrum&nbsp; $X(f)$&nbsp; is the Fourier transform of&nbsp; $x(t)$.&nbsp; The multiplication with the frequency response&nbsp; $H(f)$&nbsp; leads to the output spectrum&nbsp; $Y(f)$.&nbsp; From this,&nbsp; the signal&nbsp; $y(t)$&nbsp; can be obtained by the Fourier inverse transformation.
-*In the case of stochastic signals this procedure fails,&nbsp; because then the time functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are not predictable for all times&nbsp; from $–∞$&nbsp; to&nbsp; $+∞$&nbsp; and thus,&nbsp; the corresponding amplitude spectra&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$&nbsp; do not exist at all.&nbsp; In this case,&nbsp; we have to switch to the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|power-spectral density]]&nbsp; defined in the last chapter.
+*In the case of stochastic signals this procedure fails,&nbsp; because then the time functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are not predictable for all times&nbsp; from $–∞$&nbsp; to&nbsp; $+∞$&nbsp; and thus,&nbsp; the corresponding amplitude spectra&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$&nbsp; do not exist at all.&nbsp; In this case,&nbsp; we have to switch to the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|$\text{power-spectral density}$]]&nbsp; defined in the last chapter.
 ==Amplitude spectrum and power-spectral density==
@@ Line 44: / Line 44: @@
 :[[File:P_ID467__Sto_T_5_1_S2_neu.png|right| |frame| For the ACF and PSD calculation of a random signal]]
 <br>
-#The&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|power-spectral density]]&nbsp; ${\it Φ}_x(f)$&nbsp; can be given – as well as the auto-correlation function&nbsp; $φ_x(τ)$ – for each individual pattern function of the stationary and ergodic random process&nbsp; $\{x(t)\}$,&nbsp; even if the specific course of&nbsp; $x(t)$&nbsp; is explicitly unknown.<br><br>
+#The&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|$\text{power-spectral density}$]]&nbsp; ${\it Φ}_x(f)$&nbsp; can be given – as well as the auto-correlation function&nbsp; $φ_x(τ)$ – for each individual pattern function of the stationary and ergodic random process&nbsp; $\{x(t)\}$,&nbsp; even if the specific course of&nbsp; $x(t)$&nbsp; is explicitly unknown.<br><br>
-#The&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|amplitude spectrum]]&nbsp; $X(f)$,&nbsp; on the other hand,&nbsp; is undefined because if the spectral function&nbsp; $X(f)$&nbsp; is known, the entire time function&nbsp; $x(t)$&nbsp; from&nbsp; $–∞$&nbsp; to&nbsp; $+∞$&nbsp; would also have to be known via the Fourier inverse transform,&nbsp; which clearly cannot be the case for a stochastic signal.<br><br>
+#The&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|$\text{amplitude spectrum}$]]&nbsp; $X(f)$,&nbsp; on the other hand,&nbsp; is undefined because if the spectral function&nbsp; $X(f)$&nbsp; is known, the entire time function&nbsp; $x(t)$&nbsp; from&nbsp; $–∞$&nbsp; to&nbsp; $+∞$&nbsp; would also have to be known via the Fourier inverse transform,&nbsp; which clearly cannot be the case for a stochastic signal.<br><br>
 #If a time section of finite time duration&nbsp; $T_{\rm M}$&nbsp; is known according to the sketch on the right,&nbsp; the Fourier transform can of course be applied to it again.
 <br clear=all>
@@ Line 56: / Line 56: @@
 {{BlaueBox|TEXT=
 $\text{Proof:}$&nbsp; Previously,&nbsp; the&nbsp;
-[[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|auto-correlation function]]&nbsp; of an ergodic process with the random signal&nbsp; $x(t)$&nbsp; was given as follows:
+[[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|$\text{auto-correlation function}$]]&nbsp; of an ergodic process with the random signal&nbsp; $x(t)$&nbsp; was given as follows:
 :$${ {\it \varphi}_x(\tau)} = \lim_{T_{\rm M}\to\infty}\hspace{0.2cm}
 \frac{1}{ T_{\rm M} }\cdot\int^{+T_{\rm M}/2}_{-T_{\rm
 M}/2}x(t)\cdot x(t + \tau)\hspace{0.1cm} \rm d \it t.$$
-*It is permissible to replace the function&nbsp; $x(t)$,&nbsp; which is unbounded in time, by the function&nbsp; $x_{\rm T}(t)$,&nbsp; which is bounded on the time range&nbsp; $-T_{\rm M}/2$&nbsp; to&nbsp; $+T_{\rm M}/2$.&nbsp; &nbsp; $x_{\rm T}(t)$&nbsp; corresponds to the spectrum&nbsp; $X_{\rm T}(f)$,&nbsp; and by applying the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|first Fourier integral]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]]:
+*It is permissible to replace the function&nbsp; $x(t)$,&nbsp; which is unbounded in time, by the function&nbsp; $x_{\rm T}(t)$,&nbsp; which is bounded on the time range&nbsp; $-T_{\rm M}/2$&nbsp; to&nbsp; $+T_{\rm M}/2$.&nbsp; &nbsp; $x_{\rm T}(t)$&nbsp; corresponds to the spectrum&nbsp; $X_{\rm T}(f)$,&nbsp; and by applying the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|$\text{first Fourier integral}$]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|$\text{shifting theorem}$]]:
 :$${ {\it \varphi}_x(\tau)} = \lim_{T_{\rm M}\to\infty}\hspace{0.2cm}
 \frac{1}{ T_{\rm M} }\cdot \int^{+T_{\rm M}/2}_{-T_{\rm
@@ Line 77: / Line 77: @@
 T}(f)\vert^2 \cdot {\rm e}^{ {\rm j}2 \pi f \tau} \hspace{0.1cm} \rm d
 \it f.$$
-*A comparison with the theorem from&nbsp; [https://en.wikipedia.org/wiki/Norbert_Wiener Wiener]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Aleksandr_Khinchin Chintchin]&nbsp; which is always valid in ergodicity,
+*A comparison with the theorem from&nbsp; [https://en.wikipedia.org/wiki/Norbert_Wiener $\text{Wiener}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Aleksandr_Khinchin $\text{Chintchin}$]&nbsp; which is always valid in ergodicity,
 :$${ {\it \varphi}_x(\tau)} = \int^{+\infty}_{-\infty}{\it \Phi}_x(f)
 \cdot {\rm e}^{ {\rm j}2 \pi f \tau} \hspace{0.1cm} \rm d \it f ,$$
@@ Line 131: / Line 131: @@
 {{BlaueBox|TEXT=
-$\text{Theorem:}$&nbsp;  The corresponding auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is then obtained according to the&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Fourier transform laws]]&nbsp; and by applying the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_the_time_domain|convolution theorem]]:
+$\text{Theorem:}$&nbsp;  The corresponding auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is then obtained according to the&nbsp; [[Signal_Representation/Fourier_Transform_Laws|$\text{Fourier transform laws}$]]&nbsp; and by applying the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_the_time_domain|$\text{convolution theorem}$]]:
 :$${ {\it \varphi}_y(\tau)} = { {\it \varphi}_x(\tau)} \ast h(\tau)\ast h(-
 \tau).$$}}
@@ Line 172: / Line 172: @@
 <br clear=all>
 {{BlaueBox|TEXT=
-$\text{Theorem:}$&nbsp;  For the&nbsp; '''cross-correlation function'''&nbsp; $\rm (CCF)$&nbsp; between the input&nbsp; and the output signal holds:
+$\text{Theorem:}$&nbsp;  For the&nbsp; &raquo;'''cross-correlation function'''&laquo;&nbsp; $\rm (CCF)$&nbsp; between the input&nbsp; and the output signal holds:
 :$${ {\it \varphi}_{xy}(\tau)} = h(\tau)\ast { {\it \varphi}_x(\tau)}  .$$
 Here,&nbsp;  $h(τ)$&nbsp; denotes the impulse response of the filter&nbsp; $($with the time variable&nbsp; $τ$&nbsp; instead of&nbsp; $t)$&nbsp; and&nbsp; ${ {\it \varphi}_{x}(\tau)}$&nbsp; denotes the ACF of the input signal.}}

@@ Line 206: / Line 206: @@
 [[Aufgaben:Exercise_5.1:_Gaussian_ACF_and_Gaussian_Low-Pass|Exercise 5.1: Gaussian ACF and Gaussian Low-Pass]]
-[[Aufgaben:Aufgabe_5.1Z:_cos²_-Rauschbegrenzung|Exercise 5.1Z: $\cos^2$ Noise Limitation]]
+[[Aufgaben:Exercise_5.1Z:_Cosine_Square_Noise_Limitation|Exercise 5.1Z: Cosine Square Noise Limitation]]
 [[Aufgaben:Aufgabe_5.2:_Bestimmung_des_Frequenzgangs|Exercise 5.2: Determination of the Frequency Response]]

@@ Line 37: / Line 37: @@
 *For deterministic signals,&nbsp; one usually takes a roundabout route using the spectral functions.&nbsp; The spectrum&nbsp; $X(f)$&nbsp; is the Fourier transform of&nbsp; $x(t)$.&nbsp; The multiplication with the frequency response&nbsp; $H(f)$&nbsp; leads to the output spectrum&nbsp; $Y(f)$.&nbsp; From this,&nbsp; the signal&nbsp; $y(t)$&nbsp; can be obtained by the Fourier inverse transformation.
-*In the case of stochastic signals this procedure fails,&nbsp; because then the time functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are not predictable for all times&nbsp; from $–∞$&nbsp; to&nbsp; $+∞$&nbsp; and thus,&nbsp; the corresponding amplitude spectra&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$&nbsp; do not exist at all.&nbsp; In this case,&nbsp; we have to switch to the&nbsp; [[Theory_of_Stochastic_Signals/Leistungsdichtespektrum_(LDS)|power-spectral densities]]&nbsp; defined in the last chapter.
+*In the case of stochastic signals this procedure fails,&nbsp; because then the time functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are not predictable for all times&nbsp; from $–∞$&nbsp; to&nbsp; $+∞$&nbsp; and thus,&nbsp; the corresponding amplitude spectra&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$&nbsp; do not exist at all.&nbsp; In this case,&nbsp; we have to switch to the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|power-spectral density]]&nbsp; defined in the last chapter.
 ==Amplitude spectrum and power-spectral density==

@@ Line 146: / Line 146: @@
 We consider again the same scenario as&nbsp; in&nbsp; [[Theory_of_Stochastic_Signals/Stochastic_System_Theory#Power-spectral_density_of_the_filter_output_signal| $\text{Example 1}$]],&nbsp; but this time in the time domain.&nbsp; It holds:
 [[File:P_ID591__Sto_T_5_1_S4_neu.png |right|frame| Filter influence in the time domain]]
-*white noise power density:&nbsp; ${ {\it \Phi}_x(f)} =N_0/2$,
+*Two-sided white noise power density:&nbsp; ${ {\it \Phi}_x(f)} =N_0/2$,
 *Gaussian filter: &nbsp; $H(f) = {\rm e}^{- \pi \hspace{0.03cm}\cdot \hspace{0.03cm}(f/\Delta f)^2}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
   h(t) = \Delta f \cdot {\rm e}^{- \pi \hspace{0.03cm}\cdot \hspace{0.03cm}(\Delta f \hspace{0.03cm}\cdot \hspace{0.03cm}t)^2}.$

@@ Line 56: / Line 56: @@
 {{BlaueBox|TEXT=
 $\text{Proof:}$&nbsp; Previously,&nbsp; the&nbsp;
-[[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|auto-correlation function]]&nbsp; of an ergodic process with the sample function&nbsp; $x(t)$&nbsp; was given as follows:
+[[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|auto-correlation function]]&nbsp; of an ergodic process with the random signal&nbsp; $x(t)$&nbsp; was given as follows:
 :$${ {\it \varphi}_x(\tau)} = \lim_{T_{\rm M}\to\infty}\hspace{0.2cm}
 \frac{1}{ T_{\rm M} }\cdot\int^{+T_{\rm M}/2}_{-T_{\rm
@@ Line 165: / Line 165: @@
 #The stochastic input signal&nbsp; $x(t)$&nbsp; is a sample function of the ergodic random process&nbsp;  $\{x(t)\}$.<br><br>
 #The corresponding auto-correlation function&nbsp; $\rm  (ACF)$&nbsp;  at the filter input is thus&nbsp; $φ_x(τ)$,&nbsp; while the power-spectral density&nbsp; $\rm  (PSD)$&nbsp;  is denoted by&nbsp;  ${\it Φ}_x(f)$.&nbsp;<br><br>
 #The corresponding descriptors of the ergodic random process&nbsp;  $\{y(t)\}$&nbsp; at the filter output are
-:*the sample function&nbsp; $y(t)$, the auto-correlation function&nbsp; $φ_y(τ)$&nbsp; and the conductance density spectrum&nbsp;  ${\it Φ}_y(f)$.
 <br clear=all>
 {{BlaueBox|TEXT=
-$\text{Theorem:}$&nbsp;  For the&nbsp; '''cross correlation function'''&nbsp; (CCF) between the input and the output signal holds:
+$\text{Theorem:}$&nbsp;  For the&nbsp; '''cross-correlation function'''&nbsp; $\rm (CCF)$&nbsp; between the input&nbsp; and the output signal holds:
 :$${ {\it \varphi}_{xy}(\tau)} = h(\tau)\ast { {\it \varphi}_x(\tau)}  .$$
-Here,&nbsp;  $h(τ)$&nbsp;denotes the impulse response of the filter&nbsp; $($with the time variable&nbsp; $τ$&nbsp; instead of&nbsp; $t)$&nbsp; and&nbsp; ${ {\it \varphi}_{x}(\tau)}$&nbsp; denotes the ACF of the input signal.}}
+Here,&nbsp;  $h(τ)$&nbsp; denotes the impulse response of the filter&nbsp; $($with the time variable&nbsp; $τ$&nbsp; instead of&nbsp; $t)$&nbsp; and&nbsp; ${ {\it \varphi}_{x}(\tau)}$&nbsp; denotes the ACF of the input signal.}}
 {{BlaueBox|TEXT=
-$\text{Proof:}$&nbsp;  In general, for the cross-correlation function between two signals&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$:
+$\text{Proof:}$&nbsp;  In general,&nbsp; for the cross-correlation function between two signals&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$:
 :$${ {\it \varphi}_{xy}(\tau)} = \lim_{T_{\rm M}\to\infty}\hspace{0.2cm}\frac{1}{ T_{\rm M} }\cdot\int^{+T_{\rm M}/2}_{-T_{\rm M}/2}x(t)\cdot y(t + \tau)\hspace{0.1cm} \rm d \it t.$$
 *With the generally valid relation&nbsp; $y(t) = h(t) \ast x(t)$&nbsp; and the formal integration variable&nbsp; $θ$,&nbsp; we can also write for this:
@@ Line 187: / Line 190: @@
 *The expression in the square brackets gives the ACF value at the input at time&nbsp; $τ - θ$:
 :$${ {\it \varphi}_{xy}(\tau)} = \int^{+\infty}_{-\infty}h(\theta) \cdot  \varphi_x(\tau - \theta)\hspace{0.1cm}\hspace{0.1cm}  {\rm d}\theta = h(\tau)\ast { {\it \varphi}_x(\tau)}  .$$
-*However, the remaining integral describes the convolution operation in detailed notation.
+*However,&nbsp; the remaining integral describes the convolution operation in detailed notation.
 <div align="right">'''q.e.d.'''</div>}}
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; In the frequency domain, the corresponding equation is:
+$\text{Conclusion:}$&nbsp; In the frequency domain,&nbsp; the corresponding equation is:
 :$${ {\it \Phi}_{xy}(f)} =   H(f)\cdot{ {\it \Phi}_x(f)} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} H(f) = \frac{ {\it \Phi}_{xy}(f)}{ {\it \Phi}_{x}(f)}.$$
-This equation shows that the filter frequency response&nbsp; $H(f)$&nbsp; from a measurement with stochastic excitation can be calculated completely – i.e., both magnitude and phase – if the following descriptive quantities are determined:
+This equation shows that the filter frequency response&nbsp; $H(f)$&nbsp; from a measurement with stochastic excitation can be calculated completely&nbsp; &ndash; i.e., both magnitude and phase &ndash;&nbsp; if the following descriptive quantities are determined:
-*the statistical characteristics at the input, either the ACF&nbsp; $φ_x(τ)$&nbsp; or the&nbsp; PSD ${\it Φ}_x(f)$,
+*the statistical characteristics at the input, either the ACF&nbsp; $φ_x(τ)$&nbsp; or the &nbsp; PSD ${\it Φ}_x(f)$,
-*as well as the cross correlation function&nbsp; $φ_{xy}(τ)$&nbsp; or its Fourier transform&nbsp; ${\it Φ}_{xy}(f)$. }}
+*as well as the cross-correlation function&nbsp; $φ_{xy}(τ)$&nbsp; or its Fourier transform&nbsp; ${\it Φ}_{xy}(f)$. }}
 ==Exercises for the chapter==

@@ Line 144: / Line 144: @@
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;
-We consider again the same scenario as&nbsp; in $\text{Example 1}$, but this time in the time domain:
+We consider again the same scenario as&nbsp; in&nbsp; [[Theory_of_Stochastic_Signals/Stochastic_System_Theory#Power-spectral_density_of_the_filter_output_signal| $\text{Example 1}$]],&nbsp; but this time in the time domain.&nbsp; It holds:
 [[File:P_ID591__Sto_T_5_1_S4_neu.png |right|frame| Filter influence in the time domain]]
-*white noise&nbsp; ${ {\it \Phi}_x(f)} =N_0/2$,
+*white noise power density:&nbsp; ${ {\it \Phi}_x(f)} =N_0/2$,
 *Gaussian filter: &nbsp; $H(f) = {\rm e}^{- \pi \hspace{0.03cm}\cdot \hspace{0.03cm}(f/\Delta f)^2}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
   h(t) = \Delta f \cdot {\rm e}^{- \pi \hspace{0.03cm}\cdot \hspace{0.03cm}(\Delta f \hspace{0.03cm}\cdot \hspace{0.03cm}t)^2}.$
@@ Line 152: / Line 152: @@
 One can see from this diagram:
-#&nbsp; The ACF of the input signal is now a Dirac function with weight&nbsp; $N_0/2$.
+#The ACF of the input signal is now a Dirac delta function with weight&nbsp; $N_0/2$.
-#&nbsp; By convolution twice with the (here also Gaussian) impulse response&nbsp; $h(t)$&nbsp; or&nbsp; $h(–t)$&nbsp; one obtains the ACF&nbsp; $φ_y(τ)$&nbsp; of the output signal.
+#By convolution twice with the&nbsp;  (here also Gaussian)&nbsp;  impulse response&nbsp; $h(t)$&nbsp; or&nbsp; $h(–t)$&nbsp; one obtains the ACF&nbsp; $φ_y(τ)$&nbsp; of the output signal.
-#&nbsp; Thus, the ACF&nbsp; $φ_y(τ)$&nbsp; of the output signal is also Gaussian.
+#Thus,&nbsp;  the ACF&nbsp; $φ_y(τ)$&nbsp; of the output signal is also Gaussian.
-#&nbsp; The ACF value at&nbsp; $τ = 0$&nbsp; is identical to the area of the power-spectral density&nbsp; ${\it Φ}_y(f)$&nbsp; and characterizes the signal power (variance)&nbsp; $σ_y^2$.
+#The ACF value at&nbsp; $τ = 0$&nbsp; is identical to the area of the power-spectral density&nbsp; ${\it Φ}_y(f)$&nbsp; and characterizes the signal power (variance)&nbsp; $σ_y^2$.
-#&nbsp; In contrast, the area at&nbsp; $φ_y(τ)$&nbsp; gives the PSD value&nbsp; ${\it Φ}_y(f = \rm 0)$, i.e.,&nbsp; $N_0/2$. }}
+#In contrast, the area at&nbsp; $φ_y(τ)$&nbsp; gives the PSD value:&nbsp; ${\it Φ}_y(f = \rm 0)=N_0/2$. }}
-==Cross correlation function between input and output signal==
+==Cross-correlation function between input and output signal==
 <br>
-[[File:EN_Sto_T_5_1_S5.png |frame| Calculating the cross correlation function |right]]
+[[File:EN_Sto_T_5_1_S5.png |frame| Calculating the cross-correlation function |right]]
 We again consider a filter with the frequency response&nbsp; $H(f)$&nbsp; and the impulse response&nbsp; $h(t)$.&nbsp; Further applies:
-*The stochastic input signal&nbsp; $x(t)$&nbsp; is a sample function of the ergodic random process&nbsp;  $\{x(t)\}$.
-*The corresponding auto-correlation function (ACF) at the filter input is thus&nbsp; $φ_x(τ)$, while the power-spectral density (PSD) is denoted by&nbsp;  ${\it Φ}_x(f)$.&nbsp;
+#The stochastic input signal&nbsp; $x(t)$&nbsp; is a sample function of the ergodic random process&nbsp;  $\{x(t)\}$.<br><br>
-*The corresponding descriptors of the ergodic random process&nbsp;  $\{y(t)\}$&nbsp; at the filter output are the sample function&nbsp; $y(t)$, the auto-correlation function&nbsp; $φ_y(τ)$&nbsp; and the conductance density spectrum&nbsp;  ${\it Φ}_y(f)$.
+#The corresponding auto-correlation function&nbsp; $\rm  (ACF)$&nbsp;  at the filter input is thus&nbsp; $φ_x(τ)$,&nbsp; while the power-spectral density&nbsp; $\rm  (PSD)$&nbsp;  is denoted by&nbsp;  ${\it Φ}_x(f)$.&nbsp;<br><br>
 <br clear=all>
 {{BlaueBox|TEXT=