Aufgaben:Exercise 1.17Z: BPSK Channel Capacity - Revision history

Hwang at 16:04, 23 January 2023

2023-01-23T16:04:40Z

Guenter at 15:12, 28 September 2022

2022-09-28T15:12:55Z

Guenter at 13:27, 28 September 2022

2022-09-28T13:27:53Z

Guenter at 14:38, 17 August 2022

2022-08-17T14:38:33Z

Noah at 15:17, 16 August 2022

2022-08-16T15:17:51Z

Noah at 15:17, 16 August 2022

2022-08-16T15:17:29Z

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Noah at 21:21, 14 August 2022

2022-08-14T21:21:53Z

Guenter at 11:21, 11 May 2022

2022-05-11T11:21:04Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.17Z: BPSK–Kanalkapazität to Aufgaben:Exercise 1.17Z: BPSK Channel Capacity

2022-05-11T10:16:32Z

Guenter moved page Aufgabe 1.17Z: BPSK–Kanalkapazität to Exercise 1.17Z: BPSK Channel Capacity

Javier: Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

2021-03-23T12:51:14Z

Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

@@ Line 99: / Line 99: @@
-'''(2)'''&nbsp; The tangent increase at the zero point can be estimated from the graph on the data page:
+'''(2)'''&nbsp; The tangent increase at the zero point can be estimated from the graph in the data section:
 :$$\frac{{\rm d}C_2}{{\rm d}x} (x=0) = \frac{1.6 + 1.5}{1.25} =2.48 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} a = \frac{1}{2.48} \hspace{0.15cm} \underline{\approx 0.4}\hspace{0.05cm}.$$
@@ Line 120: / Line 120: @@
 The values&nbsp; $C_{2}\hspace{0.01cm}'$&nbsp; of the channel capacity for binary input approximated in this way are somewhat too small.
-From the graph on the information page the exact values&nbsp; $C_{2}$&nbsp; can be estimated:
+From the graph in the information section the exact values&nbsp; $C_{2}$&nbsp; can be estimated:
 :$$10 \cdot {\rm lg} \hspace{0.1cm} E_{\rm B}/N_0 =\text{2 dB:} \hspace{0.3cm} C_2 \hspace{-0.15cm}\ \approx \ \hspace{-0.15cm} {0.78}\hspace{0.05cm},$$

@@ Line 86: / Line 86: @@
 {{ML-Kopf}}
-'''(1)'''&nbsp; In the lower&nbsp; $E_{\rm B} /N_{0}$ range, the capacity curves
+'''(1)'''&nbsp; In the lower&nbsp; $E_{\rm B} /N_{0}$&nbsp; range,&nbsp; the capacity curves
-*$C_{2}$&nbsp; (valid for binary input, for example BPSK) and
+*$C_{2}$&nbsp; $($valid for binary input,&nbsp; for example BPSK$)$&nbsp; and
-*$C$&nbsp; (valid for analog real-valued input).
+*$C$&nbsp; $($valid for analog real-valued input$)$
-converge. For a given rate&nbsp; $R$&nbsp; $E_{\rm B}/N_{0}$&nbsp; must be greater than&nbsp; $(2^{2R} - 1)/2R.$
+converge.&nbsp; For a given rate&nbsp; $R$ &nbsp; &rArr; &nbsp; $E_{\rm B}/N_{0}$&nbsp; must be greater than&nbsp; $(2^{2R} - 1)/2R.$
 The limit transition for&nbsp; $R → 0$&nbsp; provides the absolute Shannon limit above which error-free transmission is no longer possible:
@@ Line 103: / Line 103: @@
 :$$\frac{{\rm d}C_2}{{\rm d}x} (x=0) = \frac{1.6 + 1.5}{1.25} =2.48 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} a = \frac{1}{2.48} \hspace{0.15cm} \underline{\approx 0.4}\hspace{0.05cm}.$$
-Thus, the approximation for the BPSK channel capacity as a function of the abscissa value&nbsp; $x$ is:
+Thus, the approximation for the BPSK channel capacity as a function of the abscissa value&nbsp; $x$&nbsp; is:
 :$$C_2' = \hspace{0.15cm} \left\{ \begin{array}{c} 1 - {\rm e}^{- 0.4 \hspace{0.05cm}\cdot \hspace{0.05cm}x} \\ 0 \end{array} \right.\quad \begin{array}{*{1}c} {\rm for\hspace{0.15cm}} x > 0, \\{\rm for\hspace{0.15cm}} x < 0. \end{array}$$
-'''(3)'''&nbsp; From&nbsp; $E_{\rm B} = N_{0}$&nbsp; follows&nbsp; $\ 10  ·  \lg {(E_{\rm B} = N_0)} = 0 \ {\rm dB}$ and $x = 1.6$:
+'''(3)'''&nbsp; From&nbsp; $E_{\rm B} = N_{0}$&nbsp; follows&nbsp; $\ 10  ·  \lg {(E_{\rm B} = N_0)} = 0 \ {\rm dB}$&nbsp; and&nbsp; $x = 1.6$:
 :$$C_2\hspace{0.01cm}' = 1 - {\rm e}^{- 0.4 \hspace{0.05cm}\cdot \hspace{0.05cm}1.6}\hspace{0.15cm}\underline{\approx 0.47}\hspace{0.05cm}.$$
@@ Line 118: / Line 118: @@
 :$$10 \cdot {\rm lg} \hspace{0.1cm} E_{\rm B}/N_0 = \text{6 dB:} \hspace{0.3cm} C_2\hspace{0.01cm}' \hspace{-0.15cm}\ = \ \hspace{-0.15cm} 1 - {\rm e}^{- 0.4 \hspace{0.05cm}\cdot \hspace{0.05cm}7.6}\hspace{0.15cm}\underline{\approx 0.95}.$$
-The values $C_{2}\hspace{0.01cm}'$ of the channel capacity for binary input approximated in this way are somewhat too small.
+The values&nbsp; $C_{2}\hspace{0.01cm}'$&nbsp; of the channel capacity for binary input approximated in this way are somewhat too small.
-From the graph on the information page the exact values $C_{2}$ can be estimated:
+From the graph on the information page the exact values&nbsp; $C_{2}$&nbsp; can be estimated:
 :$$10 \cdot {\rm lg} \hspace{0.1cm} E_{\rm B}/N_0 =\text{2 dB:} \hspace{0.3cm} C_2 \hspace{-0.15cm}\ \approx \ \hspace{-0.15cm} {0.78}\hspace{0.05cm},$$

@@ Line 3: / Line 3: @@
 [[File:EN_KC_Z_1_17.png|right|frame|For clarification of the BPSK channel capacity]]
-According to the&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#Channel_coding_theorem_and_channel_capacity|"channel coding theorem"]]&nbsp; binary signals can be transmitted error-free over the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|"AWGN channel"]]&nbsp; if and only if
+Binary signals can be transmitted error-free over the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|"AWGN channel"]]&nbsp; according to the&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#Channel_coding_theorem_and_channel_capacity|"channel coding theorem"]]&nbsp;  if and only if
 *the block length&nbsp; $n$&nbsp; of this code is chosen very large &nbsp; &rArr; &nbsp; $n → ∞$,
 *the rate&nbsp; $R$&nbsp; is smaller than the channel capacity valid for binary input&nbsp; $C_{2}$,
 *where the BPSK channel capacity&nbsp; $C_{2}$&nbsp; depends on the AWGN quotient&nbsp; $E_{\rm B}/N_{0}$&nbsp;.
-The permissible range for the code rate&nbsp; $R$&nbsp; is highlighted in green in the graph. The limit curve&nbsp; $C_{2}$, valid for binary input signals $($therefore the index &nbsp;$2)$ and sometimes also referred to as BPSK channel capacity (stands for ''Binary Phase Shift Keying''), however, cannot be specified in mathematically closed form, but is the result of an integral that can only be evaluated numerically.
+<u>Note:</u>
-As blue curve the channel capacity&nbsp; $C$&nbsp; is entered, if arbitrary real input signals are allowed.
+Conversely,&nbsp; the rate&nbsp; $R$&nbsp; for the given AWGN quotient&nbsp; $E_{\rm B}/N_{0}$&nbsp; is upper bounded:
-*For multilevel signals the rate may well assume values&nbsp; $R > 1$&nbsp;.
+*The just allowable code rate&nbsp; $R_{\rm max}$&nbsp; for a given channel&nbsp; $(E_{\rm B}/N_{0} = \rm const.)$&nbsp; we refer to as the channel capacity&nbsp; $C$.
-:$$\left (E_{\rm B}/N_0 \right)_{\rm min} = \frac{2^{2R}-1}{2R}\hspace{0.05cm}.$$
+*For&nbsp; $E_{\rm B}/N_{0} = 1 ⇒ 10 &nbsp; \ \lg {E_{\rm B}/N_0} = 0\ {\rm dB}$&nbsp; we obtain,&nbsp; for example&nbsp; $C = 0.5$.
-−
-Conversely, the rate&nbsp; $R$&nbsp; for the given AWGN quotient&nbsp; $E_{\rm B}/N_{0}$&nbsp; is upper bounded:
+*This means: &nbsp; Even with the best possible amplitude distribution of the real input signal, the code rate must not exceed&nbsp; $R = 0.5$.
-*The just allowable code rate&nbsp; $R_{\rm max}$&nbsp; for a given channel&nbsp; $(E_{\rm B}/N_{0} = \rm const.)$&nbsp; we refer to as the channel capacity&nbsp; $C$.
-*For&nbsp; $E_{\rm B}/N_{0} = 1 ⇒ 10 - \ \lg {E_{\rm B}/N_0} = 0 {\rm dB}$&nbsp; we obtain, for example&nbsp; $C = 0.5$.
+*For binary input,&nbsp; a slightly smaller value results according to&nbsp; $C_{2}.$
-In this exercise, try to approximate the graphically given channel capacitance&nbsp; $C_{2}$&nbsp; by an exponential function:
+In this exercise,&nbsp; try to approximate the graphically given channel capacitance&nbsp; $C_{2}$&nbsp; by an exponential function:
-*For the abscissa, use the auxiliary variable (see graph)
+*For the abscissa,&nbsp; use the auxiliary variable&nbsp; (see graph)
 :$$x = \frac {x_0 + 10 \cdot {\rm lg} \hspace{0.1cm} E_{\rm B}/N_0 }{1\,{\rm dB}}\hspace{0.05cm}.$$
-:That is, &nbsp; $x$&nbsp; is without unit; the pseudo unit "$\rm dB$" is omitted.
+:That is, &nbsp; $x$&nbsp; is without unit;&nbsp; the pseudo unit&nbsp; "$\rm dB$"&nbsp; is omitted.
-*Consider that for a small&nbsp; $E_{\rm B}/N_{0}$&nbsp; the approximation&nbsp; $C_{2} \approx C$&nbsp; is valid (see graph), from which the parameter&nbsp; $x_{0}$&nbsp; can be determined.
+*Consider that for a small&nbsp; $E_{\rm B}/N_{0}$&nbsp; the approximation&nbsp; $C_{2} \approx C$&nbsp; is valid&nbsp; (see graph),&nbsp; from which the parameter&nbsp; $x_{0}$&nbsp; can be determined.
-*Set for&nbsp; $C_{2}\hspace{0.01cm}' = 1 - {\rm e}^{-a\hspace{0.05cm} · \hspace{0.05cm}x}$&nbsp; and determine the parameter&nbsp; $a$&nbsp; from the dashed tangent such that&nbsp; $C_{2}\hspace{0.01cm} ' \approx C$&nbsp; holds.
+*Set for&nbsp; $C_{2}\hspace{0.01cm}' = 1 - {\rm e}^{-a\hspace{0.05cm} · \hspace{0.05cm}x}$&nbsp; and determine the parameter&nbsp; $a$&nbsp; from the red dashed tangent such that&nbsp; $C_{2}\hspace{0.01cm} ' \approx C$&nbsp; holds.
@@ Line 45: / Line 53: @@
 Hints:
 * The exercise belongs to the chapter&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding|"Information Theoretical Limits of Channel Coding"]]. It complements the&nbsp; [[Aufgaben:Exercise_1.17:_About_the_Channel_Coding_Theorem|"Exercise 1.17"]].
 * The "bit/channel use" pseudo-unit of channel capacity is omitted in these exercises.
@@ Line 58: / Line 67: @@
 $x_{0} \ = \ $ { 1.6 3% }$\ \rm dB$
-{Approximate&nbsp; $C_{2}(x)$&nbsp; by&nbsp; $C_{2}\hspace{0.01cm}'(x) = 1 -{\rm e}^{-a\hspace{0.05cm} · \hspace{0.05cm}x}$. What is the value of&nbsp; $a$?
+{Approximate&nbsp; $C_{2}(x)$&nbsp; by&nbsp; $C_{2}\hspace{0.01cm}'(x) = 1 -{\rm e}^{-a\hspace{0.05cm} · \hspace{0.05cm}x}$.&nbsp; What is the value of&nbsp; $a$?
 |type="{}"}
 $a \ = \ $ { 0.4 3% }
@@ Line 117: / Line 126: @@
 :$$10 \cdot {\rm lg} \hspace{0.1cm} E_{\rm B}/N_0 = \text{6 dB:} \hspace{0.3cm} C_2 \hspace{-0.15cm}\ \approx \ \hspace{-0.15cm} {0.99}\hspace{0.05cm}.$$
-Ab etwa&nbsp; $\ 10  ·  \lg {(E_{\rm B} / N_0)} = 8 \ {\rm dB}$&nbsp;  gilt innerhalb der Zeichengenauigkeit&nbsp; $C_{2}\hspace{0.01cm}'= C_{2} = 1$&nbsp; (bit/channel use).
+From about&nbsp; $\ 10 \cdot \lg {(E_{\rm B} / N_0)} = 8 \ {\rm dB}$&nbsp; applies within character accuracy:&nbsp; $C_{2}\hspace{0.01cm}'= C_{2} = 1$&nbsp; $($bit/channel use$)$.
 {{ML-Fuß}}

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding}}
-[[File:P_ID2413__KC_Z_1_16.png|right|frame|For clarification of the BPSK channel capacity]]
+[[File:EN_KC_Z_1_17.png|right|frame|For clarification of the BPSK channel capacity]]
 According to the&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#Channel_coding_theorem_and_channel_capacity|"channel coding theorem"]]&nbsp; binary signals can be transmitted error-free over the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|"AWGN channel"]]&nbsp; if and only if

@@ Line 46: / Line 46: @@
 Hints:
 * The exercise belongs to the chapter&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding|"Information Theoretical Limits of Channel Coding"]]. It complements the&nbsp; [[Aufgaben:Exercise_1.17:_About_the_Channel_Coding_Theorem|"Exercise 1.17"]].
-* The "bit/channel access" pseudo-unit of channel capacity is omitted in these exercises.
+* The "bit/channel use" pseudo-unit of channel capacity is omitted in these exercises.

← Older revision		Revision as of 21:21, 14 August 2022
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−	{{quiz-Header\|Buchseite=~~Kanalcodierung~~/~~Informationstheoretische Grenzen der Kanalcodierung~~}}	+	{{quiz-Header\|Buchseite=Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding}}

	[[File:P_ID2413__KC_Z_1_16.png\|right\|frame\|Zur Verdeutlichung der BPSK–Kanalkapazität]]		[[File:P_ID2413__KC_Z_1_16.png\|right\|frame\|Zur Verdeutlichung der BPSK–Kanalkapazität]]

← Older revision		Revision as of 11:21, 11 May 2022
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−	[[Category:Channel Coding: Exercises\|^1.7 ~~Informationstheoretische Grenzen~~^]]	+	[[Category:Channel Coding: Exercises\|^1.7 Channel Capacity^]]

← Older revision		Revision as of 12:51, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^1.7 Informationstheoretische Grenzen^]]	+	[[Category:Channel Coding: Exercises\|^1.7 Informationstheoretische Grenzen^]]