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

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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^]]
	[[de:~~Zusatzaufgaben~~:4.~~12_Weißes_Rauschen~~]]		[[de:Aufgaben:Aufgabe 4.12Z: Weißes Rauschen]]

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	'''(2)'''  The power-spectral density is a rectangle of width  $2B$  and height  $N_0/2$.		'''(2)'''  The power-spectral density is a rectangle of width  $2B$  and height  $N_0/2$.
	*The inverse Fourier transform yields an sinc–function:		*The inverse Fourier transform yields an sinc–function:
	:$$\varphi_b(\tau) = N_0 \cdot B \cdot {\rm si} (2 \pi B \tau)\hspace{0.3cm}		:$$\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.$$
	\Rightarrow \hspace{0.3cm}\varphi_b(\tau = 0) = N_0 \cdot B \hspace{0.15cm}\underline {=4}\cdot 10^{-6} \ \rm V^2.$$

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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^]]
			[[de:Zusatzaufgaben:4.12_Weißes_Rauschen]]

Guenter at 14:27, 25 March 2022

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	The bottom graph shows the two power-spectral densities  ${\it \Phi}_{b}(f)$  and  ${\it \Phi}_{b+}(f)$  of bandlimited white noise signal  $b(t)$.  It holds with the one-sided bandwidth $B$:		The bottom graph shows the two power-spectral densities  ${\it \Phi}_{b}(f)$  and  ${\it \Phi}_{b+}(f)$  of bandlimited white noise signal  $b(t)$.  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,  band-limited noise must always be assumed,  since only discrete-time processes can be handled.  For this,  the  [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem\|Sampling Theorem]]  must be obeyed.  This states that the bandwidth  $B$  must be set according to the interpolation distance  $T_{\rm A}$  of the simulation.		For computer simulation of noise processes,  band-limited noise must always be assumed,  since only discrete-time processes can be handled.  For this,  the  [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem\|Sampling Theorem]]  must be obeyed.  This states that the bandwidth  $B$  must be set according to the interpolation distance  $T_{\rm A}$  of the simulation.

Guenter at 14:11, 25 March 2022

2022-03-25T14:11:15Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Correct are <u>solutions 2, 3, and 4</u>:		'''(1)'''  Correct are the  <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  $\rm (ACF)$  is the Fourier transform of the power-spectral density  $\rm (PSD)$.  Here:
	:$${\it \Phi}_n (f) = {N_0}/{2} \hspace{0.3cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.3cm} \varphi_n (\tau)={N_0}/{2} \cdot {\rm \delta} ( \tau).$$		:$${\it \Phi}_n (f) = {N_0}/{2} \hspace{0.3cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.3cm} \varphi_n (\tau)={N_0}/{2} \cdot {\rm \delta} ( \tau).$$
	*However, there is no "real" white noise in physics, since such a noise would have to have an infinitely large signal power  $($the integral over the PSD as well as the ACF value at  $\tau = 0$  are both infinitely large$)$.		*However,  there is no  "real"  white noise in physics,  since such a noise would have to have an infinitely large signal power  $($the integral over the PSD as well as the ACF value at  $\tau = 0$  are both infinitely large$)$.
	*Thermal noise has a constant PSD up to frequencies of about  $\text{6000 GHz}$ . 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  $\text{6000 GHz}$.  Since all  (current)  transmission systems operate in a much lower frequency range,  thermal noise can be said to be  "white"  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  "white"  says nothing about the amplitude distribution,  which is determined by the probability density function  $\rm (PDF)$  alone.
	*When considering the phase of a bandpass signal as the stochastic variable, it is often modeled as uniformly distributed between  $0$  and  $2\pi$~~ ~~ .		*When considering the phase of a bandpass signal as the stochastic variable,  it is often modeled as uniformly distributed between  $0$  and  $2\pi$.
	*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,  this random process is also  "white".






	[[File:P_ID410__Sto_Z_4_12_b.png\|right\|frame\|ACF of band-limited noise]]		[[File:P_ID410__Sto_Z_4_12_b.png\|right\|frame\|ACF of band-limited noise]]
	'''(2)'''  The ~~Power~~ spectral density ~~spectrum~~ is a rectangle of width  $2B$  and height  $N_0/2$.		'''(2)'''  The power-spectral density is a rectangle of width  $2B$  and height  $N_0/2$.
	*The inverse Fourier ~~transformation~~ yields an si function:		*The inverse Fourier transform yields an sinc–function:
	:$$\varphi_b(\tau) = N_0 \cdot B \cdot {\rm si} (2 \pi B \tau)\hspace{0.3cm}		:$$\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.$$		\Rightarrow \hspace{0.3cm}\varphi_b(\tau = 0) = N_0 \cdot B \hspace{0.15cm}\underline {=4}\cdot 10^{-6} \ \rm V^2.$$
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	'''(3)'''  The ACF value at the point  $\tau = 0$  gives the power.		'''(3)'''  The ACF value at the point  $\tau = 0$  gives the power.
	*The root of this is called the rms value:		*The root of this is called the  "rms value":
	:$$\sigma_b = \sqrt{\varphi_b(\tau = 0)} \hspace{0.15cm}\underline {=2 \hspace{0.05cm}\rm V}.$$		:$$\sigma_b = \sqrt{\varphi_b(\tau = 0)} \hspace{0.15cm}\underline {=2 \hspace{0.05cm}\rm V}.$$


	'''(4)'''  The ACF computed in  '''(3)'''  has zeros at equidistant distance from  $T_{\rm A}= 1/(2B)\hspace{0.15cm}\underline {=5\hspace{0.05cm} \rm ns}$:		'''(4)'''  The ACF computed in  '''(3)'''  has zeros at equidistant distance from  $T_{\rm A}= 1/(2B)\hspace{0.15cm}\underline {=5\hspace{0.05cm} \rm ns}$:
	*There are no statistical bindings between the two signal values  $b(t)$  and  $b(t + \nu \cdot T_{\rm A})$,		*There are no statistical bindings between the two signal values  $b(t)$  and  $b(t + \nu \cdot T_{\rm A})$,
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	'''(5)'''  The correct solution is <u>suggested solution 2</u>.		'''(5)'''  The correct solution is the  <u>suggested solution 2</u>.
	*The ACF value at  $\tau = T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  ~~amounts to.~~		*The ACF value at  $\tau = T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  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:   Two signal values separated by  $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  are positively correlated:		*This result says:   Two signal values separated by  $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  are positively correlated:
	*If  $b(t)$  is positive and large;, then with high probability  $b(t+1 \hspace{0.05cm}\rm ns)$  is also positive and large;.		:*If  $b(t)$  is positive and large,  then with high probability  $b(t+1 \hspace{0.05cm}\rm ns)$  is also positive and large.
	*In contrast, there is a negative correlation between  $b(t)$  and  $b(t+7 \hspace{0.05cm}\rm ns)$  If  $b(t)$  is positive, then  $b(t+7 \hspace{0.05cm}\rm ns)$  is probably negative.		:*In contrast,  there is a negative correlation between  $b(t)$  and  $b(t+7 \hspace{0.05cm}\rm ns)$.  If  $b(t)$  is positive,  then  $b(t+7 \hspace{0.05cm}\rm ns)$  is probably negative.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 13:50, 25 March 2022

2022-03-25T13:50:08Z

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

Guenter at 13:48, 25 March 2022

2022-03-25T13:48:56Z

Noah at 18:59, 7 March 2022

2022-03-07T18:59:16Z

Show changes

Guenter at 09:44, 28 February 2022

2022-02-28T09:44:46Z

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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^]]

Guenter at 09:42, 28 February 2022

2022-02-28T09:42:27Z

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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^]]

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	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Power-Spectral_Density		{{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 $n(t)$  is called ~~<i>~~white;~~</i>~~ if it contains all spectral components without preference of any frequencies.		A noise signal $n(t)$  is called  "white"  if it contains all spectral components without preference of any frequencies.
	* The physical ~~Power~~ spectral density defined only for positive frequencies ~~$f$~~  ${\it \Phi}_{n+}(f)$  is constant  $($equal  $N_0)$  and extends frequency-wise to infinity.		* The physical power-spectral density defined only for positive frequencies   ⇒   ${\it \Phi}_{n+}(f)$  is constant  $($equal  $N_0)$  and extends frequency-wise to infinity.
	* ${\it \Phi}_{n+}(f)$  is shown in green in the upper graph.  The plus sign in the index is to indicate that the function is valid only for positive values of $f$~~ ~~ .		* ${\it \Phi}_{n+}(f)$  is shown in green in the upper graph.  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~~  ${\it \Phi}_{n}(f)$.  Here applies for;all frequencies from  $-\infty$  to  $+\infty$  (blue curve in the upper ~~picture~~):		* For mathematical description one usually uses the two-sided power-spectral density  ${\it \Phi}_{n}(f)$.  Here applies for all frequencies from  $-\infty$  to  $+\infty$  (blue curve in the upper graph):
	:$${\it \Phi}_n (f) ={N_0}/{2}.$$		:$${\it \Phi}_n (f) ={N_0}/{2}.$$


	The bottom graph shows the two ~~Power~~ spectral densities  ${\it \Phi}_{b}(f)$  and  ${\it \Phi}_{b+}(f)$  of a bandlimited white noise signal  $b(t)$  It holds with the one-sided bandwidth~~ ~~ $B$:		The bottom graph shows the two power-spectral densities  ${\it \Phi}_{b}(f)$  and  ${\it \Phi}_{b+}(f)$  of bandlimited white noise signal  $b(t)$.  It holds with the one-sided bandwidth $B$:
	:$${\it \Phi}_b(f)=\left\{ {N_0/2\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad \|f\|\le B \atop {\rm else}}\right.,$$		:$${\it \Phi}_b(f)=\left\{ {N_0/2\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad \|f\|\le B \atop {\rm else}}\right.,$$
	:$${\it \Phi}_{b+}(f)=\left\{ {N_0\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad 0 \le f\le B \atop {\rm else}}\right.$$		:$${\it \Phi}_{b+}(f)=\left\{ {N_0\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad 0 \le f\le B \atop {\rm else}}\right.$$

	For computer simulation of noise processes, band-limited noise must always be assumed, since only discrete-time processes can be handled. For this, the  [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem\|~~SamplingTheorem~~]]  must be obeyed.  This states that the bandwidth  $B$  must be set according to the ~~pitch~~  $T_{\rm A}$  of the simulation.		For computer simulation of noise processes,  band-limited noise must always be assumed,  since only discrete-time processes can be handled.  For this,  the  [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem\|Sampling Theorem]]  must be obeyed.  This states that the bandwidth  $B$  must be set according to the interpolation distance  $T_{\rm A}$  of the simulation.

	Assume the following numerical values throughout ~~the~~ exercise:		Assume the following numerical values throughout this exercise:
	* The noise power density –  with respect to the resistor  $1 \hspace{0.05cm}\rm \Omega$  –  ~~beträgt~~  $N_0 = 4 \cdot 10^{-14}\hspace{0.05cm}\rm V^2/Hz$.		* The noise power density –  with respect to the resistor  $1 \hspace{0.05cm}\rm \Omega$  –  is  $N_0 = 4 \cdot 10^{-14}\hspace{0.05cm}\rm V^2/Hz$.
	* The (one-sided) bandwidth of the band-limited white noise is $B = 100 \hspace{0.08cm}\rm MHz$.		* The  (one-sided)  bandwidth of the band-limited white noise is  $B = 100 \hspace{0.08cm}\rm MHz$.


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	*This exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Power-Spectral_Density\|Power-Spectral Density]].		*This exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Power-Spectral_Density\|Power-Spectral Density]].
	*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function\|Auto-correlation function]].		*Reference is also made to the chapter  [[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  [[Der_AWGN-Kanal_(Lernvideo)\|The AWGN channel ~~(German)~~]]~~ ~~.		*The properties of white noise are summarized in the second part of the  (German language)  learning video  [[Der_AWGN-Kanal_(Lernvideo)\|The AWGN channel]].


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	<quiz display=simple>		<quiz display=simple>
	{Which statements are always true for a white noise signal  $n(t)$  Give reasons for your answers.		{Which statements are always true for a white noise signal  $n(t)$.  Give reasons for your answers.
	\|type="[]"}		\|type="[]"}
	- The ACF  $\varphi_n(t)$  has a si-shaped progression.		- The ACF  $\varphi_n(t)$  has a sinc-shaped progression.
	+ The ACF  $\varphi_n(\tau)$  is a Dirac at  $\tau = 0$  with weight  $N_0/2$.		+ The ACF  $\varphi_n(\tau)$  is a Dirac delta function at  $\tau = 0$  with weight  $N_0/2$.
	+ In practice, there is no (exact) white noise.		+ In practice,  there is no  (exact)  white noise.
	+ Thermal noise can always be approximated as white.		+ Thermal noise can always be approximated as white.
	- White noise is always Gaussian distributed.		- White noise is always Gaussian distributed.


	{Calculate the ACF  $\varphi_b(\tau)$  of the random signal $b(t)$ bandlimited to  $B = 100 \hspace{0.08cm}\rm MHz$  What value results for  $\tau = 0$?		{Calculate the ACF  $\varphi_b(\tau)$  of the random signal  $b(t)$  bandlimited to  $B = 100 \hspace{0.08cm}\rm MHz$.  What value results for  $\tau = 0$?
	\|type="{}"}		\|type="{}"}
	$\varphi_b(\tau = 0) \ = \ $ { 4 3% } $\ \cdot 10^{-6} \ \rm V^2$		$\varphi_b(\tau = 0) \ = \ $ { 4 3% } $\ \cdot 10^{-6} \ \rm V^2$
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	{What sampling distance  $T_{\rm A}$  should be ~~chosen~~ (at most) if the band-limited signal  $b(t)$  is used for discrete-time simulation of white noise?		{What sampling distance  $T_{\rm A}$  should be  (at most)  chosen if the band-limited signal  $b(t)$  is used for discrete-time simulation of white noise?
	\|type="{}"}		\|type="{}"}
	$T_{\rm A} \ = \ $ { 5 3% } $\ \rm ns$		$T_{\rm A} \ = \ $ { 5 3% } $\ \rm ns$


	{Assume sampling distance  $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  Then, which of the statements are true for two consecutive samples of the signal  $b(t)$ ?		{Assume sampling distance  $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$.  Then,  which of the statements are true for two consecutive samples of the signal  $b(t)$ ?
	\|type="[]"}		\|type="()"}
	- The samples are uncorrelated.		- The samples are uncorrelated.
	+ The samples are positively correlated.		+ The samples are positively correlated.