Aufgaben:Exercise 4.12Z: White Gaussian Noise - Revision history

Guenter at 14:27, 25 March 2022

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Guenter at 14:11, 25 March 2022

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Guenter at 13:50, 25 March 2022

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Guenter at 09:44, 28 February 2022

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Guenter at 09:42, 28 February 2022

2022-02-28T09:42:27Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.12Z: Weißes Rauschen to Aufgaben:Exercise 4.12Z: White Gaussian Noise

2022-02-28T09:33:25Z

Guenter moved page Aufgabe 4.12Z: Weißes Rauschen to Exercise 4.12Z: White Gaussian Noise

Javier: Text replacement - "”" to """

2021-05-28T14:37:50Z

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Javier: Text replacement - "„" to """

2021-05-28T14:36:47Z

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 The bottom graph shows the two power-spectral densities&nbsp; ${\it \Phi}_{b}(f)$&nbsp; and&nbsp; ${\it \Phi}_{b+}(f)$&nbsp; of bandlimited white noise signal&nbsp; $b(t)$.&nbsp; It holds with the one-sided bandwidth $B$:
-:$${\it \Phi}_b(f)=\left\{ {N_0/2\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad |f|\le B \atop {\rm else}}\right.,$$
+:$${\it \Phi}_b(f)=\left\{ {N_0/2\atop 0}{\hspace{0.5cm} {\rm for}\quad |f|\le B \atop {\rm else}}\right.,$$
-:$${\it \Phi}_{b+}(f)=\left\{ {N_0\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad 0 \le f\le B \atop {\rm else}}\right.$$
+:$${\it \Phi}_{b+}(f)=\left\{ {N_0\atop 0}{\hspace{0.5cm} {\rm for}\quad 0 \le f\le B \atop {\rm else}}\right.$$
 For computer simulation of noise processes,&nbsp; band-limited noise must always be assumed,&nbsp; since only discrete-time processes can be handled.&nbsp; For this,&nbsp; the&nbsp; [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|Sampling Theorem]]&nbsp; must be obeyed.&nbsp; This states that the bandwidth&nbsp; $B$&nbsp; must be set according to the interpolation distance&nbsp; $T_{\rm A}$&nbsp; of the simulation.

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 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>solutions 2, 3, and 4</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <u>solutions 2, 3, and 4</u>:
-*The auto-correlation function (ACF) is the Fourier transform of the Power spectral density (PSD). Here:
+*The auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is the Fourier transform of the power-spectral density&nbsp; $\rm (PSD)$.&nbsp; Here:
 :$${\it \Phi}_n (f) = {N_0}/{2} \hspace{0.3cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.3cm} \varphi_n (\tau)={N_0}/{2}  \cdot {\rm \delta} ( \tau).$$
-*However, there is no "real" white noise in physics, since such a noise would have to have an infinitely large signal power&nbsp; $($the integral over the PSD as well as the ACF value at&nbsp; $\tau = 0$&nbsp; are both infinitely large$)$.
+*However,&nbsp; there is no&nbsp; "real"&nbsp; white noise in physics,&nbsp; since such a noise would have to have an infinitely large signal power&nbsp; $($the integral over the PSD as well as the ACF value at&nbsp; $\tau = 0$&nbsp; are both infinitely large$)$.
-*Thermal noise has a constant PSD up to frequencies of about&nbsp; $\text{6000 GHz}$&nbsp;. Since all (current) &uuml;b transmission systems operate in a much lower frequency range, thermal noise can be said to be "white" to a good approximation.
+*Thermal noise has a constant PSD up to frequencies of about&nbsp; $\text{6000 GHz}$.&nbsp; Since all&nbsp; (current)&nbsp; transmission systems operate in a much lower frequency range,&nbsp; thermal noise can be said to be&nbsp; "white"&nbsp; to a good approximation.
-*The statistical property "white" says nothing about the amplitude distribution, which is determined by the probability density function (PDF) alone.
+*The statistical property&nbsp; "white"&nbsp; says nothing about the amplitude distribution,&nbsp; which is determined by the probability density function&nbsp; $\rm (PDF)$&nbsp; alone.
-*When considering the phase of a bandpass signal as the stochastic variable, it is often modeled as uniformly distributed between&nbsp; $0$&nbsp; and&nbsp; $2\pi$&nbsp; .
+*When considering the phase of a bandpass signal as the stochastic variable,&nbsp; it is often modeled as uniformly distributed between&nbsp; $0$&nbsp; and&nbsp; $2\pi$.
-*If there are no statistical bindings between the respective phase angles at different times, this random process is also "white".
+*If there are no statistical bindings between the respective phase angles at different times,&nbsp; this random process is also&nbsp; "white".
 [[File:P_ID410__Sto_Z_4_12_b.png|right|frame|ACF of band-limited noise]]
-'''(2)'''&nbsp; The Power spectral density spectrum is a rectangle of width&nbsp; $2B$&nbsp; and height&nbsp; $N_0/2$.
+'''(2)'''&nbsp; The power-spectral density is a rectangle of width&nbsp; $2B$&nbsp; and height&nbsp; $N_0/2$.
-*The inverse Fourier transformation yields an si function:
+*The inverse Fourier transform yields an sinc&ndash;function:
 :$$\varphi_b(\tau) = N_0 \cdot B \cdot {\rm si} (2 \pi B \tau)\hspace{0.3cm}
 \Rightarrow \hspace{0.3cm}\varphi_b(\tau = 0) = N_0 \cdot B \hspace{0.15cm}\underline {=4}\cdot 10^{-6} \ \rm V^2.$$
@@ Line 97: / Line 93: @@
 '''(3)'''&nbsp; The ACF value at the point&nbsp; $\tau = 0$&nbsp; gives the power.&nbsp;
-*The root of this is called the rms value:
+*The root of this is called the&nbsp; "rms value":
 :$$\sigma_b = \sqrt{\varphi_b(\tau = 0)} \hspace{0.15cm}\underline {=2 \hspace{0.05cm}\rm V}.$$
 '''(4)'''&nbsp; The ACF computed in&nbsp; '''(3)'''&nbsp; has zeros at equidistant distance from&nbsp; $T_{\rm A}= 1/(2B)\hspace{0.15cm}\underline {=5\hspace{0.05cm}  \rm ns}$:&nbsp;
 *There are no statistical bindings between the two signal values&nbsp; $b(t)$&nbsp; and&nbsp; $b(t + \nu \cdot T_{\rm A})$,
@@ Line 105: / Line 103: @@
-'''(5)'''&nbsp; The correct solution is <u>suggested solution 2</u>.
+'''(5)'''&nbsp; The correct solution is the&nbsp; <u>suggested solution 2</u>.
-*The ACF value at&nbsp; $\tau = T_{\rm A} = 1 \hspace{0.05cm}\rm ns$&nbsp; amounts to.
+*The ACF value at&nbsp; $\tau = T_{\rm A} = 1 \hspace{0.05cm}\rm ns$&nbsp; is
-:$$\varphi_b(\tau = T_{\rm A}) = {\rm 4 \cdot 10^{-6} \hspace{0.1cm}V^2 \cdot si (\pi/5) \approx 3.742 \cdot 10^{-6} \hspace{0.1cm}V^2} > 0.$$
+:$$\varphi_b(\tau = T_{\rm A}) = {\rm 4 \cdot 10^{-6} \hspace{0.1cm}V^2 \cdot sinc (1/5) \approx 3.742 \cdot 10^{-6} \hspace{0.1cm}V^2} > 0.$$
 *This result says: &nbsp; Two signal values separated by&nbsp; $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$&nbsp; are positively correlated:
-*If&nbsp; $b(t)$&nbsp; is positive and large;, then with high probability&nbsp; $b(t+1 \hspace{0.05cm}\rm ns)$&nbsp; is also positive and large;.
+:*If&nbsp; $b(t)$&nbsp; is positive and large,&nbsp; then with high probability&nbsp; $b(t+1 \hspace{0.05cm}\rm ns)$&nbsp; is also positive and large.
-*In contrast, there is a negative correlation between&nbsp; $b(t)$&nbsp; and&nbsp; $b(t+7 \hspace{0.05cm}\rm ns)$&nbsp; If&nbsp; $b(t)$&nbsp; is positive, then&nbsp; $b(t+7 \hspace{0.05cm}\rm ns)$&nbsp; is probably negative.
+:*In contrast,&nbsp; there is a negative correlation between&nbsp; $b(t)$&nbsp; and&nbsp; $b(t+7 \hspace{0.05cm}\rm ns)$.&nbsp; If&nbsp; $b(t)$&nbsp; is positive,&nbsp; then&nbsp; $b(t+7 \hspace{0.05cm}\rm ns)$&nbsp; is probably negative.
 {{ML-Fuß}}

@@ Line 29: / Line 29: @@
 Hints:
 *This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|Power-Spectral Density]].
-*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|Auto-correlation function]].
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|Auto-Correlation Function]].
 *The properties of white noise are summarized in the second part of the&nbsp; (German language)&nbsp; learning video&nbsp; [[Der_AWGN-Kanal_(Lernvideo)|The AWGN channel]].

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Power-Spectral_Density
 }}
-[[File:P_ID409__Sto_Z_4_12.png|right|frame|Power spectral densities <br> of white noise]]
+[[File:P_ID409__Sto_Z_4_12.png|right|frame|PSD of white noise]]
-A noise signal $n(t)$&nbsp; is called <i>white;</i> if it contains all spectral components without preference of any frequencies.
+A noise signal $n(t)$&nbsp; is called&nbsp; "white"&nbsp; if it contains all spectral components without preference of any frequencies.
-* The physical Power spectral density defined only for positive frequencies $f$&nbsp; ${\it \Phi}_{n+}(f)$&nbsp; is constant&nbsp; $($equal&nbsp; $N_0)$&nbsp; and extends frequency-wise to infinity.
+* The physical power-spectral density defined only for positive frequencies &nbsp; &rArr; &nbsp; ${\it \Phi}_{n+}(f)$&nbsp; is constant&nbsp; $($equal&nbsp; $N_0)$&nbsp; and extends frequency-wise to infinity.
-* ${\it \Phi}_{n+}(f)$&nbsp; is shown in green in the upper graph.&nbsp; The plus sign in the index is to indicate that the function is valid only for positive values of $f$&nbsp; .
+* ${\it \Phi}_{n+}(f)$&nbsp; is shown in green in the upper graph.&nbsp; The plus sign in the index is to indicate that the function is valid only for positive values of $f$.
-* For mathematical description one usually uses the two-sided Power spectral density spectrum&nbsp; ${\it \Phi}_{n}(f)$.&nbsp; Here applies for;all frequencies from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$&nbsp; (blue curve in the upper picture):
+* For mathematical description one usually uses the two-sided power-spectral density&nbsp; ${\it \Phi}_{n}(f)$.&nbsp; Here applies for all frequencies from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$&nbsp; (blue curve in the upper graph):
 :$${\it \Phi}_n (f) ={N_0}/{2}.$$
-The bottom graph shows the two Power spectral densities&nbsp; ${\it \Phi}_{b}(f)$&nbsp; and&nbsp; ${\it \Phi}_{b+}(f)$&nbsp; of a bandlimited white noise signal&nbsp; $b(t)$&nbsp; It holds with the one-sided bandwidth&nbsp; $B$:
+The bottom graph shows the two power-spectral densities&nbsp; ${\it \Phi}_{b}(f)$&nbsp; and&nbsp; ${\it \Phi}_{b+}(f)$&nbsp; of bandlimited white noise signal&nbsp; $b(t)$.&nbsp; It holds with the one-sided bandwidth $B$:
 :$${\it \Phi}_b(f)=\left\{ {N_0/2\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad |f|\le B \atop {\rm else}}\right.,$$
 :$${\it \Phi}_{b+}(f)=\left\{ {N_0\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad 0 \le f\le B \atop {\rm else}}\right.$$
-For computer simulation of noise processes, band-limited noise must always be assumed, since only discrete-time processes can be handled. For this, the&nbsp; [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|SamplingTheorem]]&nbsp; must be obeyed.&nbsp; This states that the bandwidth&nbsp; $B$&nbsp; must be set according to the pitch&nbsp; $T_{\rm A}$&nbsp; of the simulation.
+For computer simulation of noise processes,&nbsp; band-limited noise must always be assumed,&nbsp; since only discrete-time processes can be handled.&nbsp; For this,&nbsp; the&nbsp; [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|Sampling Theorem]]&nbsp; must be obeyed.&nbsp; This states that the bandwidth&nbsp; $B$&nbsp; must be set according to the interpolation distance&nbsp; $T_{\rm A}$&nbsp; of the simulation.
-Assume the following numerical values throughout the exercise:
+Assume the following numerical values throughout this exercise:
-* The noise power density &ndash;&nbsp; with respect to the resistor&nbsp; $1 \hspace{0.05cm}\rm \Omega$&nbsp; &ndash;&nbsp; betr&auml;gt&nbsp; $N_0 = 4 \cdot 10^{-14}\hspace{0.05cm}\rm V^2/Hz$.
+* The noise power density &ndash;&nbsp; with respect to the resistor&nbsp; $1 \hspace{0.05cm}\rm \Omega$&nbsp; &ndash;&nbsp; is&nbsp; $N_0 = 4 \cdot 10^{-14}\hspace{0.05cm}\rm V^2/Hz$.
-* The (one-sided) bandwidth of the band-limited white noise is $B = 100 \hspace{0.08cm}\rm MHz$.
+* The&nbsp; (one-sided)&nbsp; bandwidth of the band-limited white noise is&nbsp; $B = 100 \hspace{0.08cm}\rm MHz$.
@@ Line 30: / Line 30: @@
 *This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|Power-Spectral Density]].
 *Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|Auto-correlation function]].
-*The properties of white noise are summarized in the second part of the tutorial video&nbsp; [[Der_AWGN-Kanal_(Lernvideo)|The AWGN channel (German)]]&nbsp;.
+*The properties of white noise are summarized in the second part of the&nbsp; (German language)&nbsp; learning video&nbsp; [[Der_AWGN-Kanal_(Lernvideo)|The AWGN channel]].
@@ Line 38: / Line 38: @@
 <quiz display=simple>
-{Which statements are always true for a white noise signal&nbsp; $n(t)$&nbsp; Give reasons for your answers.
+{Which statements are always true for a white noise signal&nbsp; $n(t)$.&nbsp; Give reasons for your answers.
 |type="[]"}
-- The ACF&nbsp; $\varphi_n(t)$&nbsp; has a si-shaped progression.
+- The ACF&nbsp; $\varphi_n(t)$&nbsp; has a sinc-shaped progression.
-+ The ACF&nbsp; $\varphi_n(\tau)$&nbsp; is a Dirac at&nbsp; $\tau = 0$&nbsp; with weight&nbsp; $N_0/2$.
++ The ACF&nbsp; $\varphi_n(\tau)$&nbsp; is a Dirac delta function  at&nbsp; $\tau = 0$&nbsp; with weight&nbsp; $N_0/2$.
-+ In practice, there is no (exact) white noise.
++ In practice,&nbsp; there is no&nbsp; (exact)&nbsp; white noise.
 + Thermal noise can always be approximated as white.
 - White noise is always Gaussian distributed.
-{Calculate the ACF&nbsp; $\varphi_b(\tau)$&nbsp; of the random signal $b(t)$ bandlimited to&nbsp; $B = 100 \hspace{0.08cm}\rm MHz$&nbsp; What value results for&nbsp; $\tau = 0$?
+{Calculate the ACF&nbsp; $\varphi_b(\tau)$&nbsp; of the random signal&nbsp; $b(t)$&nbsp; bandlimited to&nbsp; $B = 100 \hspace{0.08cm}\rm MHz$.&nbsp; What value results for&nbsp; $\tau = 0$?
 |type="{}"}
 $\varphi_b(\tau = 0) \ = \ $ { 4 3% } $\ \cdot 10^{-6} \ \rm V^2$
@@ Line 57: / Line 57: @@
-{What sampling distance&nbsp; $T_{\rm A}$&nbsp; should be chosen (at most) if the band-limited signal&nbsp; $b(t)$&nbsp; is used for discrete-time simulation of white noise?
+{What sampling distance&nbsp; $T_{\rm A}$&nbsp; should be&nbsp; (at most)&nbsp; chosen if the band-limited signal&nbsp; $b(t)$&nbsp; is used for discrete-time simulation of white noise?
 |type="{}"}
 $T_{\rm A} \ = \ $ { 5 3% } $\ \rm ns$
-{Assume sampling distance&nbsp; $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$&nbsp; Then, which of the statements are true for two consecutive samples of the signal&nbsp; $b(t)$&nbsp;?
+{Assume sampling distance&nbsp; $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$.&nbsp; Then,&nbsp; which of the statements are true for two consecutive samples of the signal&nbsp; $b(t)$&nbsp;?
-|type="[]"}
+|type="()"}
 - The samples are uncorrelated.
 + The samples are positively correlated.

← Older revision		Revision as of 09:44, 28 February 2022
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−	[[Category:Theory of Stochastic Signals: Exercises\|^4.5 Power-~~spectral~~ Density^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.5 Power-Spectral Density^]]

← Older revision		Revision as of 09:42, 28 February 2022
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−	[[Category:Theory of Stochastic Signals: Exercises\|^4.5 ~~Leistungsdichtespektrum (LDS)~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.5 Power-spectral Density^]]

← Older revision	Revision as of 09:33, 28 February 2022
(No difference)

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 *Die Autokorrelationsfunktion (AKF) ist die Fouriertransformierte des Leistungsdichtespektrums (LDS). Dabei gilt:
 :$${\it \Phi}_n (f) =  {N_0}/{2} \hspace{0.3cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.3cm} \varphi_n (\tau)={N_0}/{2}  \cdot {\rm \delta} ( \tau).$$
-*Allerdings gibt es in der Physik kein "echt&rdquo; wei&szlig;es Rauschen, da ein solches eine unendlich gro&szlig;e Signalleistung aufweisen m&uuml;sste&nbsp; $($das Integral &uuml;ber das LDS sowie der AKF-Wert bei&nbsp; $\tau = 0$&nbsp; sind jeweils unendlich groß$)$.
+*Allerdings gibt es in der Physik kein "echt" wei&szlig;es Rauschen, da ein solches eine unendlich gro&szlig;e Signalleistung aufweisen m&uuml;sste&nbsp; $($das Integral &uuml;ber das LDS sowie der AKF-Wert bei&nbsp; $\tau = 0$&nbsp; sind jeweils unendlich groß$)$.
-*Thermisches Rauschen hat bis zu Frequenzen von etwa&nbsp; $\text{6000 GHz}$&nbsp; ein konstantes LDS. Da alle (derzeitigen) &Uuml;bertragungssysteme in einem sehr viel niedrigeren Frequenzbereich arbeiten, kann man thermisches Rauschen mit guter N&auml;herung als "wei&szlig;&rdquo; bezeichnen.
+*Thermisches Rauschen hat bis zu Frequenzen von etwa&nbsp; $\text{6000 GHz}$&nbsp; ein konstantes LDS. Da alle (derzeitigen) &Uuml;bertragungssysteme in einem sehr viel niedrigeren Frequenzbereich arbeiten, kann man thermisches Rauschen mit guter N&auml;herung als "wei&szlig;" bezeichnen.
-*Die statistische Eigenschaft "wei&szlig;&rdquo; sagt nichts &uuml;ber die Amplitudenverteilung aus, die allein durch die Wahrscheinlichkeitsdichtefunktion (WDF) festgelegt ist.
+*Die statistische Eigenschaft "wei&szlig;" sagt nichts &uuml;ber die Amplitudenverteilung aus, die allein durch die Wahrscheinlichkeitsdichtefunktion (WDF) festgelegt ist.
 *Betrachtet man die Phase eines bandpassf&ouml;rmigen Signals als die stochastische Gr&ouml;&szlig;e, so wird diese oft als gleichverteilt zwischen&nbsp; $0$&nbsp; und&nbsp; $2\pi$&nbsp; modelliert.
-*Bestehen zwischen den jeweiligen Phasenwinkeln zu unterschiedlichen Zeiten keine statistischen Bindungen, so ist auch dieser Zufallsprozess "wei&szlig;&rdquo;.
+*Bestehen zwischen den jeweiligen Phasenwinkeln zu unterschiedlichen Zeiten keine statistischen Bindungen, so ist auch dieser Zufallsprozess "wei&szlig;".