Aufgaben:Exercise 1.3Z: Thermal Noise - Revision history

Guenter at 16:27, 10 April 2022

2022-04-10T16:27:27Z

Guenter at 16:26, 10 April 2022

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Guenter at 16:25, 10 April 2022

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Guenter at 16:24, 10 April 2022

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Reed at 16:59, 23 March 2022

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Guenter at 15:16, 17 February 2022

2022-02-17T15:16:59Z

Bene: Text replacement - "rms value" to "standard deviation"

2022-02-17T11:11:29Z

Text replacement - "rms value" to "standard deviation"

Reed at 10:11, 24 November 2021

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Reed at 09:56, 24 November 2021

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Guenter at 14:15, 16 November 2021

2021-11-16T14:15:51Z

@@ Line 25: / Line 25: @@
 All equations apply regardless of whether the noise is low-pass or band-pass.  The diagram shows two noise signals &nbsp;$n_1(t)$&nbsp; and &nbsp;$n_2(t)$&nbsp; of equal bandwidth.&nbsp; Question &nbsp; '''(4)'''&nbsp; asks which of these signals will appear at the output of a low-pass and a band-pass, respectively.
-The two-sided noise power density of band-limited low-pass noise &nbsp;$n_{\rm TP}(t)$&nbsp; is:
+The two-sided noise power-spectral density of band-limited low-pass noise &nbsp;$n_{\rm TP}(t)$&nbsp; is:
 :$$ {\it \Phi}_{n, {\hspace{0.05cm}\rm LP}}(f) = \left\{ \begin{array}{c} N_0/2 \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.005cm} f\hspace{0.05cm} \right| < B,} \\ {\rm otherwise.} \\ \end{array}$$
 In contrast,&nbsp; for band-pass noise &nbsp;$n_{\rm BP}(t)$&nbsp; with center frequency &nbsp;$f_{\rm M}$, it holds that:

@@ Line 65: / Line 65: @@
 - The noise signal &nbsp;$n_1(t)$&nbsp; is characteristically band-pass.
-{What is the value of the noise power density of the low-pass&nbsp; $\rm (LP)$&nbsp; noise at frequency &nbsp;$f = 20\text{ kHz}$?&nbsp; Let &nbsp;$B = 30\text{ kHz}$.
+{What is the value of the noise power-spectral density of the low-pass&nbsp; $\rm (LP)$&nbsp; noise at frequency &nbsp;$f = 20\text{ kHz}$?&nbsp; Let &nbsp;$B = 30\text{ kHz}$.
 |type="{}"}
 ${\it Φ}_{n, \hspace{0.05cm}\rm LP}(f = 20 \ \rm kHz) \ = \ $ { 2 3% } $\ \cdot 10^{ -12 }\ \text{W/Hz}$
-{What is the value of the noise power density of the band-pass&nbsp; $\rm (BP)$&nbsp; noise at  &nbsp;$f = 120\text{ kHz}$?&nbsp; Let &nbsp;$f_{\rm M} = 100\text{ kHz}$&nbsp; and &nbsp;$B = 30\text{ kHz}$.
+{What is the value of the noise power-spectral density of the band-pass&nbsp; $\rm (BP)$&nbsp; noise at  &nbsp;$f = 120\text{ kHz}$?&nbsp; Let &nbsp;$f_{\rm M} = 100\text{ kHz}$&nbsp; and &nbsp;$B = 30\text{ kHz}$.
 |type="{}"}
 ${\it Φ}_{n, \hspace{0.05cm}\rm BP}(f = 120 \ \rm kHz) \ = \ $ { 0. } $\ \cdot 10^{ -12 }\ \text{W/Hz}$

@@ Line 12: / Line 12: @@
 However,&nbsp; this noise is limited to &nbsp;$6\text{ THz}$&nbsp; for physical reasons.&nbsp; Furthermore,&nbsp; it can be observed that this minimum value can only be achieved with exact impedance matching.
-In the realization of a circuit unit - for example, an amplifier - the effective noise power density is usually significantly greater,&nbsp; since several noise sources add up,&nbsp; and mismatches also play a role.&nbsp;  This effect is captured by the noise factor &nbsp;$F \ge 1$&nbsp; .&nbsp; It holds that:
+In the realization of a circuit unit - for example, an amplifier - the effective noise power-spectral density is usually significantly greater,&nbsp; since several noise sources add up,&nbsp; and mismatches also play a role.&nbsp;  This effect is captured by the noise factor &nbsp;$F \ge 1$&nbsp; .&nbsp; It holds that:
 :$$N_0 = F \cdot {N_{\rm 0, \hspace{0.05cm}min}}= F \cdot k_{\rm B} \cdot \theta \hspace{0.05cm}.$$
 With a bandwidth &nbsp;$B$,&nbsp; the effective noise power is characterized by:
@@ Line 46: / Line 46: @@
 <quiz display=simple>
-{Calculate the noise power density &nbsp;$N_0$&nbsp; with a noise factor of  &nbsp;$F = 10$&nbsp; and &nbsp;$θ = 290^\circ$&nbsp; Kelvin.
+{Calculate the noise power-spectral density &nbsp;$N_0$&nbsp; with a noise factor of  &nbsp;$F = 10$&nbsp; and &nbsp;$θ = 290^\circ$&nbsp; Kelvin.
 |type="{}"}
 $N_0 \ = \ $ { 4 3% }  $\ \cdot 10^{ -20 }\ \text{W/Hz}$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID953__Mod_Z_1_3.png|right|frame|Example signals for low-pass and band-pass noise]]
+[[File:P_ID953__Mod_Z_1_3.png|right|frame|Examplary signals for low-pass and band-pass noise]]
-A fundamental disturbance and one that occurs in any communication system is&nbsp; "thermal noise",&nbsp; since any resistance &nbsp;$R$&nbsp; with an absolute temperature of &nbsp;$θ$&nbsp; (in "degrees Kelvin")&nbsp; produces a noise signal &nbsp;$n(t)$&nbsp; with a&nbsp; (one-sided)&nbsp; noise power density
+A fundamental disturbance and one that occurs in any communication system is&nbsp; "thermal noise",&nbsp; since any resistance &nbsp;$R$&nbsp; with an absolute temperature of &nbsp;$θ$&nbsp; (in "degrees Kelvin")&nbsp; produces a noise signal &nbsp;$n(t)$&nbsp; with a&nbsp; (one-sided)&nbsp; noise power-spectral density
 :$${N_{\rm 0, \hspace{0.05cm}min}}= k_{\rm B} \cdot \theta
 \hspace{0.3cm}\left(k_{\rm B} = 1.38 \cdot 10^{-23}
@@ Line 102: / Line 102: @@
 W}\hspace{0.05cm} \cdot 50 \; {\rm \Omega}= 6\hspace{0.05cm}\cdot
 ^{-14}\;{\rm V^2}\hspace{0.05cm}.$$
-[[File:EN_Mod_Z_1_3_e.png|rechts|frame|Power density spectra with band-limited noise]]
+[[File:EN_Mod_Z_1_3_e.png|rechts|frame|Power-spectral densities with band-limited noise]]
 *The noise standard deviation $σ_n$&nbsp; is the square root of this:
 :$$\sigma_n= \sqrt{6\hspace{0.05cm}\cdot 10^{-14}\;{\rm V^2}} \hspace{0.15cm}\underline {= 0.245 \hspace{0.05cm}\cdot 10^{-6}\;{\rm V}}\hspace{0.05cm}.$$

@@ Line 18: / Line 18: @@
 :$$N = N_0 \cdot B\cdot R = \sigma_n^2 \hspace{0.01cm}.$$
 *According to the first equation,&nbsp; the result is the actual,&nbsp; physical power in "watts"&nbsp; $\rm (W)$.
-*According to the second equation,&nbsp; the result has the unit &nbsp; „$\rm V^{ 2 }$”.
+*According to the second equation,&nbsp; the result has the unit &nbsp; "$\rm V^{ 2 }$".
 *This means that the power is here converted to the reference resistance &nbsp;$R = 1\ Ω$&nbsp; – as is often the case in Communications Engineering.
 *This equation must also be used to calculate the standard deviation&nbsp; $σ_n$&nbsp; of the noise signal &nbsp;$n(t)$&nbsp;.

@@ Line 20: / Line 20: @@
 *According to the second equation,&nbsp; the result has the unit &nbsp; „$\rm V^{ 2 }$”.
 *This means that the power is here converted to the reference resistance &nbsp;$R = 1\ Ω$&nbsp; – as is often the case in Communications Engineering.
-*This equation must also be used to calculate the rms value&nbsp; ("standard deviation")&nbsp; $σ_n$&nbsp; of the noise signal &nbsp;$n(t)$&nbsp;.
+*This equation must also be used to calculate the standard deviation&nbsp; ("standard deviation")&nbsp; $σ_n$&nbsp; of the noise signal &nbsp;$n(t)$&nbsp;.
@@ Line 55: / Line 55: @@
-{What is the noise power &nbsp;$N$&nbsp;  with bandwidth &nbsp;$B = 30\text{ kHz}$?&nbsp; What is the rms value $σ_n$?
+{What is the noise power &nbsp;$N$&nbsp;  with bandwidth &nbsp;$B = 30\text{ kHz}$?&nbsp; What is the standard deviation $σ_n$?
 |type="{}"}
 $N \ = \ $ { 12 3% }  $\ \cdot 10^{ -16 }\ \text{W/Hz}$
@@ Line 103: / Line 103: @@
 ^{-14}\;{\rm V^2}\hspace{0.05cm}.$$
 [[File:EN_Mod_Z_1_3_e.png|rechts|frame|Power density spectra with band-limited noise]]
-*The noise rms value $σ_n$&nbsp; is the square root of this:
+*The noise standard deviation $σ_n$&nbsp; is the square root of this:
 :$$\sigma_n= \sqrt{6\hspace{0.05cm}\cdot 10^{-14}\;{\rm V^2}} \hspace{0.15cm}\underline {= 0.245 \hspace{0.05cm}\cdot 10^{-6}\;{\rm V}}\hspace{0.05cm}.$$

@@ Line 36: / Line 36: @@
 Hints:
-*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Qualitätskriterien|Quality Criteria]].
+*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quality_Criteria|Quality Criteria]].
 *Particular reference is made to the page&nbsp;  [[Modulation_Methods/Quality_Criteria#Some_remarks_on_the_AWGN_channel_model|Some remarks on the AWGN channel model]].
 *By specifying the powers in &nbsp;$\rm W$atts&nbsp;, they are independent of the reference resistance &nbsp;$R$,&nbsp; while power with the unit  &nbsp;$\rm V^2$&nbsp; can only be evaluated directly for &nbsp;$R = 1\ \Omega$.

@@ Line 26: / Line 26: @@
 The two-sided noise power density of band-limited low-pass noise &nbsp;$n_{\rm TP}(t)$&nbsp; is:
-:$$ {\it \Phi}_{n, {\hspace{0.05cm}\rm LP}}(f) = \left\{ \begin{array}{c} N_0/2 \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{f\ddot{u}r}} \\ \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.005cm} f\hspace{0.05cm} \right| < B,} \\ {\rm otherwise.} \\ \end{array}$$
+:$$ {\it \Phi}_{n, {\hspace{0.05cm}\rm LP}}(f) = \left\{ \begin{array}{c} N_0/2 \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.005cm} f\hspace{0.05cm} \right| < B,} \\ {\rm otherwise.} \\ \end{array}$$
 In contrast,&nbsp; for band-pass noise &nbsp;$n_{\rm BP}(t)$&nbsp; with center frequency &nbsp;$f_{\rm M}$, it holds that:
-:$${\it \Phi}_{n, {\hspace{0.05cm}\rm BP}}(f) = \left\{ \begin{array}{c} N_0/2 \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{f\ddot{u}r}} \\ \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.005cm} f - f_{\rm M}\hspace{0.05cm} \right| < B/2,} \\ {\rm otherwise.} \\ \end{array}.$$
+:$${\it \Phi}_{n, {\hspace{0.05cm}\rm BP}}(f) = \left\{ \begin{array}{c} N_0/2 \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.005cm} f - f_{\rm M}\hspace{0.05cm} \right| < B/2,} \\ {\rm otherwise.} \\ \end{array}.$$
 For all subsequent numerical calculations it is assumed:
 :$$ F = 10, \hspace{0.2cm}\theta = 290\,{\rm K},\hspace{0.2cm}R = 50\,{\rm \Omega},\hspace{0.2cm}B = 30\,{\rm kHz},\hspace{0.2cm}f_{\rm M} = 0 \hspace{0.5cm}{\rm or}\hspace{0.5cm}f_{\rm M} =100\,{\rm kHz}\hspace{0.05cm}.$$
@@ Line 88: / Line 84: @@
 Ws}{\rm K}\cdot 290\,{\rm K} \hspace{0.15cm}\underline {\approx 4\hspace{0.05cm}\cdot
 ^{-20} \hspace{0.05cm}{\rm W}/{\rm Hz}}\hspace{0.05cm}.$$
@@ Line 96: / Line 91: @@
 \hspace{0.08cm}{\rm Hz}\hspace{0.15cm}\underline {= 0.24\hspace{0.08cm}\cdot 10^{-6}\;{\rm
 W}}\hspace{0.05cm}.$$
@@ Line 114: / Line 108: @@
+'''(4)'''&nbsp; <u>Answer 1</u>&nbsp; is correct:
-'''(4)'''&nbsp; <u>Answer 1</u> is correct:
+*In the random signal&nbsp; $n_2(t)$&nbsp;one can recognize certain regularities similar to a harmonic oscillation &nbsp; &rArr; &nbsp;  it is bandpass noise.
 *In contrast, the signal&nbsp; $n_1(t)$&nbsp; is low-pass noise.
 '''(5)'''&nbsp; The noise power density of the random signal&nbsp; $n_1(t)$&nbsp; is constant in the frequency range&nbsp; $|f| < 30$&nbsp; kHz:
-:$${\it \Phi}_{n,\hspace{0.05cm}{   \rm TP} }(f) \hspace{-0.05cm}=\hspace{-0.05cm} \frac{N_0}{2}  \hspace{0.15cm}\underline {=2\hspace{0.05cm}\hspace{-0.05cm}\cdot \hspace{-0.05cm}
+:$${\it \Phi}_{n,\hspace{0.05cm}{   \rm LP} }(f) \hspace{-0.05cm}=\hspace{-0.05cm} \frac{N_0}{2}  \hspace{0.15cm}\underline {=2\hspace{0.05cm}\hspace{-0.05cm}\cdot \hspace{-0.05cm}
 ^{-12} \hspace{0.05cm}{\rm W}/{\rm Hz}}\hspace{0.05cm}.$$
-*Thus, this value is also valid for the frequency&nbsp; $f = 20$&nbsp; kHz.
+*Thus,&nbsp; this value is also valid for the frequency&nbsp; $f = 20$&nbsp; kHz.
-'''(6)'''&nbsp; As can be seen from the diagram, &nbsp; ${\it Φ}_{n, \hspace{0.05cm}\rm BP}(f)$&nbsp; is non-zero only in the range between&nbsp; $85$&nbsp; kHz and&nbsp; $115$&nbsp; kHz, when the bandwidth is&nbsp; $B = 30$&nbsp; kHz.
+'''(6)'''&nbsp; As can be seen from the diagram, &nbsp; ${\it Φ}_{n, \hspace{0.05cm}\rm BP}(f)$&nbsp; is non-zero only in the range between&nbsp; $85$&nbsp; kHz and&nbsp; $115$&nbsp; kHz,&nbsp; when the bandwidth is&nbsp; $B = 30$&nbsp; kHz.
-*Thus, at a frequency of&nbsp; $f = 120$&nbsp; kHz, the noise power density is zero:
+*Thus,&nbsp; at the frequency &nbsp; $f = 120$&nbsp; kHz, the noise power density is zero:
 :$${\it Φ}_{n, \hspace{0.05cm}\rm BP}(f = 120 \ \rm kHz)\hspace{0.15cm}\underline{=0}.$$