Information Theory/AWGN Channel Capacity for Continuous-Valued Input - Revision history

Guenter at 14:22, 28 February 2023

2023-02-28T14:22:04Z

Guenter: Guenter moved page Information Theory/AWGN Channel Capacity for Continuous Input to Information Theory/AWGN Channel Capacity for Continuous-Valued Input

2023-02-25T15:14:38Z

Guenter moved page Information Theory/AWGN Channel Capacity for Continuous Input to Information Theory/AWGN Channel Capacity for Continuous-Valued Input

Guenter at 17:03, 24 February 2023

2023-02-24T17:03:57Z

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Hwang at 10:55, 22 February 2023

2023-02-22T10:55:13Z

Hwang at 10:53, 22 February 2023

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Hwang at 10:51, 22 February 2023

2023-02-22T10:51:46Z

Hwang at 10:50, 22 February 2023

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Hwang at 10:49, 22 February 2023

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Hwang at 10:47, 22 February 2023

2023-02-22T10:47:35Z

Hwang at 13:06, 23 January 2023

2023-01-23T13:06:46Z

@@ Line 168: / Line 168: @@
 ==Parallel Gaussian channels ==
 <br>
-[[File:EN_Inf_T_4_2_S4c_v3.png|frame|Parallel AWGN channels '''KORREKTUR: transmission power''']]
+[[File:EN_Inf_T_4_2_S4c.png|frame|Parallel AWGN channels]]
 We now consider   according to the graph&nbsp; $K$&nbsp; parallel Gaussian channels&nbsp; $X_1 → Y_1$,&nbsp; ... ,&nbsp;  $X_k → Y_k$,&nbsp; ... , $X_K → Y_K$.

@@ Line 161: / Line 161: @@
 ==Parallel Gaussian channels ==
 <br>
-[[File:EN_Inf_T_4_2_S4c_v3.png|frame|Parallel AWGN channels]]
+[[File:EN_Inf_T_4_2_S4c_v3.png|frame|Parallel AWGN channels '''KORREKTUR: transmission power''']]
 We now consider, according to the graph&nbsp; $K$&nbsp; parallel Gaussian channels&nbsp; $X_1 → Y_1$,&nbsp; ... ,&nbsp;  $X_k → Y_k$,&nbsp; ... , $X_K → Y_K$.

@@ Line 133: / Line 133: @@
 - \hspace{0.05cm}{n^2}/(2 \sigma_N^2) } \hspace{0.05cm}, $$
-we obtain the model sketched on the right for calculating the channel capacity of the so-called&nbsp; [[Modulation_Methods/Qualitätskriterien#Einige_Anmerkungen_zum_AWGN.E2.80.93Kanalmodell|$\text{AWGN channel}$]]&nbsp; ("Additive White Gaussian Noise").&nbsp;
+we obtain the model sketched on the right for calculating the channel capacity of the so-called&nbsp; [[Modulation_Methods/Quality_Criteria#Some_remarks_on_the_AWGN_channel_model|$\text{AWGN channel}$]]&nbsp; ("Additive White Gaussian Noise").&nbsp;
 In the following, we usually replace the variance&nbsp; $\sigma_N^2$&nbsp; by the power&nbsp; $P_N$.

@@ Line 97: / Line 97: @@
 *The additive noise&nbsp; $N$&nbsp; is given by the&nbsp; (mean-free)&nbsp;  PDF&nbsp; $f_N(n)$&nbsp; and the noise power&nbsp; $P_N = σ_N^2$&nbsp;.
 *If&nbsp; $X$&nbsp; and&nbsp; $N$&nbsp; are assumed to be statistically independent &nbsp; &rArr; &nbsp;  signal-independent noise, then&nbsp; $\text{E}\big[X · N \big] = \text{E}\big[X \big] · \text{E}\big[N\big] = 0$ .
-*The received signal is &nbsp;$Y = X + N$.&nbsp; The output PDF&nbsp; $f_Y(y)$&nbsp; can be calculated with the [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_Time_Domain|$\text{convolution operation}$]]&nbsp; &nbsp; ⇒ &nbsp;  $f_Y(y) = f_X(x) ∗ f_N(n)$.
+*The received signal is &nbsp;$Y = X + N$.&nbsp; The output PDF&nbsp; $f_Y(y)$&nbsp; can be calculated with the [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_the_time_domain|$\text{convolution operation}$]]&nbsp; &nbsp; ⇒ &nbsp;  $f_Y(y) = f_X(x) ∗ f_N(n)$.
 [[File:Inf_T_4_2_S3neu.png|right|frame|Message transmission system with additive noise]]

@@ Line 94: / Line 94: @@
 <br>
 We now consider a very simple model of message transmission:
-*The random variable&nbsp; $X$&nbsp; stands for the (zero mean) transmitted signal and is characterised by the PDF&nbsp; $f_X(x)$&nbsp; and the variance&nbsp; $σ_X^2$&nbsp; .&nbsp; The transmission power is $P_X = σ_X^2$.
+*The random variable&nbsp; $X$&nbsp; stands for the (zero mean) transmitted signal and is characterized by the PDF&nbsp; $f_X(x)$&nbsp; and the variance&nbsp; $σ_X^2$&nbsp; .&nbsp; The transmission power is $P_X = σ_X^2$.
 *The additive noise&nbsp; $N$&nbsp; is given by the&nbsp; (mean-free)&nbsp;  PDF&nbsp; $f_N(n)$&nbsp; and the noise power&nbsp; $P_N = σ_N^2$&nbsp;.
 *If&nbsp; $X$&nbsp; and&nbsp; $N$&nbsp; are assumed to be statistically independent &nbsp; &rArr; &nbsp;  signal-independent noise, then&nbsp; $\text{E}\big[X · N \big] = \text{E}\big[X \big] · \text{E}\big[N\big] = 0$ .
@@ Line 138: / Line 138: @@
 We know from previous sections:
-*The&nbsp; [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Definition_and_meaning_of_channel_capacity|$\text{channel capacity}$]]&nbsp; $C_{\rm AWGN}$&nbsp; specifies the maximum mutual information&nbsp; $I(X; Y)$&nbsp; between the input quantity&nbsp;  $X$&nbsp;  and the output quantity&nbsp;  $Y$&nbsp;  of the AWGN channel.&nbsp;  The maximisation refers to the best possible input PDF.&nbsp;  Thus, under the&nbsp;  [[Information_Theory/Differentielle_Entropie#Differential_entropy_of_some_power-constrained_random_variables|$\text{power constraint}$]] the following applies:
+*The&nbsp; [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Definition_and_meaning_of_channel_capacity|$\text{channel capacity}$]]&nbsp; $C_{\rm AWGN}$&nbsp; specifies the maximum mutual information&nbsp; $I(X; Y)$&nbsp; between the input quantity&nbsp;  $X$&nbsp;  and the output quantity&nbsp;  $Y$&nbsp;  of the AWGN channel.&nbsp;  The maximization refers to the best possible input PDF.&nbsp;  Thus, under the&nbsp;  [[Information_Theory/Differentielle_Entropie#Differential_entropy_of_some_power-constrained_random_variables|$\text{power constraint}$]] the following applies:
 :$$C_{\rm AWGN} = \max_{f_X:\hspace{0.1cm} {\rm E}[X^2 ] \le P_X} \hspace{-0.35cm}  I(X;Y)
@@ Line 144: / Line 144: @@
 \hspace{0.05cm}.$$
-:It is already taken into account that the maximisation relates solely to the differential entropy &nbsp;$h(Y)$ &nbsp; ⇒ &nbsp; probability density function &nbsp;$f_Y(y)$.&nbsp;  Indeed, for a given noise power&nbsp;  $P_N$&nbsp;, &nbsp;$h(N) = 1/2 · \log_2 (2π{\rm e} · P_N)$&nbsp; is a constant.
+:It is already taken into account that the maximization relates solely to the differential entropy &nbsp;$h(Y)$ &nbsp; ⇒ &nbsp; probability density function &nbsp;$f_Y(y)$.&nbsp;  Indeed, for a given noise power&nbsp;  $P_N$&nbsp;, &nbsp;$h(N) = 1/2 · \log_2 (2π{\rm e} · P_N)$&nbsp; is a constant.
 *The maximum for &nbsp;$h(Y)$&nbsp;  is obtained for a Gaussian PDF &nbsp;$f_Y(y)$&nbsp; with &nbsp;$P_Y = P_X + P_N$&nbsp;, see section&nbsp; [[Information_Theory/Differentielle_Entropie#Proof:_Maximum_differential_entropy_with_power_constraint|"Maximum differential entropy under power constraint"]]:
 :$${\rm max}\big[h(Y)\big] = 1/2 · \log_2 \big[2πe · (P_X + P_N)\big].$$
@@ Line 231: / Line 231: @@
-The two identical and independent channels can be realised in different ways, for example by multiplexing in time, frequency or space.
+The two identical and independent channels can be realized in different ways, for example by multiplexing in time, frequency or space.
-However, the case&nbsp; $K = 2$&nbsp; can also be realised by using orthogonal basis functions such as&nbsp; "cosine"&nbsp; and&nbsp; "sine"&nbsp; as for example with
+However, the case&nbsp; $K = 2$&nbsp; can also be realized by using orthogonal basis functions such as&nbsp; "cosine"&nbsp; and&nbsp; "sine"&nbsp; as for example with
 *&nbsp; [[Modulation_Methods/Quadratur–Amplitudenmodulation|$\text{quadrature amplitude modulation}$]]&nbsp; $\rm (QAM)$&nbsp; or

@@ Line 20: / Line 20: @@
 In order to derive the mutual information&nbsp; $I(X; Y)$&nbsp; between two continuous random variables&nbsp; $X$&nbsp; and&nbsp; $Y$,&nbsp; one proceeds as follows, whereby inverted commas indicate a quantized variable:
 *One quantizes the random variables&nbsp; $X$&nbsp; and&nbsp; $Y$&nbsp; $($with the quantization intervals&nbsp; ${\it Δ}x$&nbsp; and&nbsp; ${\it Δ}y)$&nbsp; and thus obtains the probability functions&nbsp; $P_{X\hspace{0.01cm}′}$&nbsp; and&nbsp; $P_{Y\hspace{0.01cm}′}$.
-*The „vectors”&nbsp; $P_{X\hspace{0.01cm}′}$&nbsp; and&nbsp; $P_{Y\hspace{0.01cm}′}$&nbsp; become infinitely long after the boundary transitions&nbsp; ${\it Δ}x → 0,\hspace{0.1cm} {\it Δ}y → 0$&nbsp;, and the joint PMF&nbsp; $P_{X\hspace{0.01cm}′\hspace{0.08cm}Y\hspace{0.01cm}′}$&nbsp; is also infinitely extended in area.
+*The "vectors"&nbsp; $P_{X\hspace{0.01cm}′}$&nbsp; and&nbsp; $P_{Y\hspace{0.01cm}′}$&nbsp; become infinitely long after the boundary transitions&nbsp; ${\it Δ}x → 0,\hspace{0.1cm} {\it Δ}y → 0$&nbsp;, and the joint PMF&nbsp; $P_{X\hspace{0.01cm}′\hspace{0.08cm}Y\hspace{0.01cm}′}$&nbsp; is also infinitely extended in area.
 *These boundary transitions give rise to the probability density functions of the continuous random variables according to the following equations:
@@ Line 202: / Line 202: @@
-[[File:EN_Inf_T_4_2_S4d_v2.png|right|frame|Best possible power allocation for&nbsp; $K = 4$&nbsp; („Water–Filling”)]]
+[[File:EN_Inf_T_4_2_S4d_v2.png|right|frame|Best possible power allocation for&nbsp; $K = 4$&nbsp; ("Water–Filling")]]
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp; We consider&nbsp; $K = 4$&nbsp; parallel Gaussian channels with four different noise powers&nbsp; $σ_1^2$,&nbsp; ... ,&nbsp; $σ_4^2$&nbsp; according to the adjacent figure (faint green background).

@@ Line 9: / Line 9: @@
 ==Mutual information between continuous random variables ==
 <br>
-In the chapter &nbsp;[[Information_Theory/Application_to_Digital_Signal_Transmission#Information-theoretical_model_of_digital_signal_transmission|Information-theoretical model of digital signal transmission]]&nbsp; the&nbsp; "mutual information" between the two discrete random variables&nbsp; $X$&nbsp; and&nbsp; $Y$&nbsp; was given, among other things, in the following form:
+In the chapter &nbsp;[[Information_Theory/Application_to_Digital_Signal_Transmission#Information-theoretical_model_of_digital_signal_transmission|"Information-theoretical model of digital signal transmission"]]&nbsp; the&nbsp; "mutual information" between the two discrete random variables&nbsp; $X$&nbsp; and&nbsp; $Y$&nbsp; was given, among other things, in the following form:
 :$$I(X;Y) = \hspace{0.5cm} \sum_{\hspace{-0.9cm}y \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp}\hspace{0.05cm} (P_{Y}\hspace{-0.08cm})} \hspace{-1.1cm}\sum_{\hspace{1.3cm} x \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp}\hspace{0.05cm} (P_{X}\hspace{-0.08cm})}
   \hspace{-0.9cm} P_{XY}(x, y) \cdot {\rm log} \hspace{0.1cm} \frac{ P_{XY}(x, y)}{P_{X}(x) \cdot P_{Y}(y)} \hspace{0.05cm}.$$
-This equation simultaneously corresponds to the &nbsp;[[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Informational_divergence_-_Kullback-Leibler_distance|Kullback&ndash;Leibler distance]]&nbsp; between the joint probability function&nbsp; $P_{XY}$&nbsp;  and the product of the two individual probability functions&nbsp; $P_X$&nbsp; and&nbsp; $P_Y$ :
+This equation simultaneously corresponds to the &nbsp;[[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Informational_divergence_-_Kullback-Leibler_distance|$\text{Kullback&ndash;Leibler distance}$]]&nbsp; between the joint probability function&nbsp; $P_{XY}$&nbsp;  and the product of the two individual probability functions&nbsp; $P_X$&nbsp; and&nbsp; $P_Y$ :
 :$$I(X;Y) = D(P_{XY} \hspace{0.05cm} ||  \hspace{0.05cm}P_{X} \cdot P_{Y}) \hspace{0.05cm}.$$
@@ Line 67: / Line 67: @@
 The outflowing or inflowing differential entropy characterises
-*the&nbsp; '''equivocation''':
+*the&nbsp; &raquo;'''equivocation'''&laquo;:
 :$$h(X \hspace{-0.05cm}\mid \hspace{-0.05cm} Y)   = - \hspace{-0.3cm}\int\limits_{\hspace{-0.9cm}y \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp}\hspace{0.05cm} (P_{Y}\hspace{-0.08cm})} \hspace{-1.1cm}\int\limits_{\hspace{1.3cm} x \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp}\hspace{0.05cm} (P_{X}\hspace{-0.08cm})}
@@ Line 73: / Line 73: @@
   \hspace{0.15cm}{\rm d}x\hspace{0.15cm}{\rm d}y,$$
-*the&nbsp; '''irrelevance''':
+*the&nbsp; &raquo;'''irrelevance'''&laquo;:
 :$$h(Y \hspace{-0.05cm}\mid \hspace{-0.05cm} X)   = - \hspace{-0.3cm}\int\limits_{\hspace{-0.9cm}y \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp}\hspace{0.05cm} (P_{Y}\hspace{-0.08cm})} \hspace{-1.1cm}\int\limits_{\hspace{1.3cm} x \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp}\hspace{0.05cm} (P_{X}\hspace{-0.08cm})}
@@ Line 79: / Line 79: @@
   \hspace{0.15cm}{\rm d}x\hspace{0.15cm}{\rm d}y.$$
-The significance of these two information-theoretic quantities will be discussed in more detail in&nbsp; [[Aufgaben:Exercise_4.5Z:_Again_Mutual_Information|Exercise 4.5Z]]&nbsp;.
+The significance of these two information-theoretic quantities will be discussed in more detail in&nbsp; [[Aufgaben:Exercise_4.5Z:_Again_Mutual_Information|"Exercise 4.5Z"]]&nbsp;.
 If one compares the graphical representations of the mutual information for
-*discrete  random variables in the section &nbsp;[[Information_Theory/Application_to_Digital_Signal_Transmission#Information-theoretical_model_of_digital_signal_transmission|Information-theoretical model of digital signal transmission]],&nbsp;
+*discrete  random variables in the section &nbsp;[[Information_Theory/Application_to_Digital_Signal_Transmission#Information-theoretical_model_of_digital_signal_transmission|"Information-theoretical model of digital signal transmission"]],&nbsp;
 *continuous  random variables according to the above diagram,
@@ Line 97: / Line 97: @@
 *The additive noise&nbsp; $N$&nbsp; is given by the&nbsp; (mean-free)&nbsp;  PDF&nbsp; $f_N(n)$&nbsp; and the noise power&nbsp; $P_N = σ_N^2$&nbsp;.
 *If&nbsp; $X$&nbsp; and&nbsp; $N$&nbsp; are assumed to be statistically independent &nbsp; &rArr; &nbsp;  signal-independent noise, then&nbsp; $\text{E}\big[X · N \big] = \text{E}\big[X \big] · \text{E}\big[N\big] = 0$ .
-*The received signal is &nbsp;$Y = X + N$.&nbsp; The output PDF&nbsp; $f_Y(y)$&nbsp; can be calculated with the [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_Time_Domain|convolution operation]]&nbsp; &nbsp; ⇒ &nbsp;  $f_Y(y) = f_X(x) ∗ f_N(n)$.
+*The received signal is &nbsp;$Y = X + N$.&nbsp; The output PDF&nbsp; $f_Y(y)$&nbsp; can be calculated with the [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_Time_Domain|$\text{convolution operation}$]]&nbsp; &nbsp; ⇒ &nbsp;  $f_Y(y) = f_X(x) ∗ f_N(n)$.
 [[File:Inf_T_4_2_S3neu.png|right|frame|Message transmission system with additive noise]]
@@ Line 108: / Line 108: @@
 The sketched probability density functions&nbsp; (rectangular or trapezoidal)&nbsp; are only intended to clarify the calculation process and have no practical relevance.
 <br clear=all>
-To calculate the mutual information between input&nbsp; $X$&nbsp; and output&nbsp; $Y$&nbsp; there are three possibilities according to the&nbsp; [[Information_Theory/AWGN–Kanalkapazität_bei_wertkontinuierlichem_Eingang#On_equivocation_and_irrelevance|graphic on the previous subchapter]]:
+To calculate the mutual information between input&nbsp; $X$&nbsp; and output&nbsp; $Y$&nbsp; there are three possibilities according to the&nbsp; [[Information_Theory/AWGN–Kanalkapazität_bei_wertkontinuierlichem_Eingang#On_equivocation_and_irrelevance|"graphic in the previous subchapter"]]:
 * Calculation according to &nbsp;$I(X, Y) = h(X) + h(Y) - h(XY)$:
 :The first two terms can be calculated in a simple way from &nbsp;$f_X(x)$&nbsp; and &nbsp;$f_Y(y)$&nbsp; respectively.&nbsp; The&nbsp; "joint differential entropy" &nbsp;$h(XY)$ is problematic.&nbsp; For this, one needs the 2D joint PDF &nbsp;$f_{XY}(x, y)$, which is usually not given directly.
@@ Line 129: / Line 129: @@
 [[File:P_ID2884__Inf_T_4_2_S4_neu.png|right|frame|Derivation of the AWGN channel capacity]]
-If one specifies the probability density function of the noise in the previous&nbsp;   [[Information_Theory/AWGN–Kanalkapazität_bei_wertkontinuierlichem_Eingang#Calculation_of_mutual_information_with_additive_noise|general system model]]&nbsp; as Gaussian corresponding to
+If one specifies the probability density function of the noise in the previous&nbsp;   [[Information_Theory/AWGN–Kanalkapazität_bei_wertkontinuierlichem_Eingang#Calculation_of_mutual_information_with_additive_noise|$\text{general system model}$]]&nbsp; as Gaussian corresponding to
 :$$f_N(n) = \frac{1}{\sqrt{2\pi  \sigma_N^2}} \cdot {\rm e}^{
 - \hspace{0.05cm}{n^2}/(2 \sigma_N^2) } \hspace{0.05cm}, $$
-we obtain the model sketched on the right for calculating the channel capacity of the so-called&nbsp; [[Modulation_Methods/Qualitätskriterien#Einige_Anmerkungen_zum_AWGN.E2.80.93Kanalmodell|AWGN channel]]&nbsp; ("Additive White Gaussian Noise").&nbsp;
+we obtain the model sketched on the right for calculating the channel capacity of the so-called&nbsp; [[Modulation_Methods/Qualitätskriterien#Einige_Anmerkungen_zum_AWGN.E2.80.93Kanalmodell|$\text{AWGN channel}$]]&nbsp; ("Additive White Gaussian Noise").&nbsp;
 In the following, we usually replace the variance&nbsp; $\sigma_N^2$&nbsp; by the power&nbsp; $P_N$.
 We know from previous sections:
-*The&nbsp; [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Definition_and_meaning_of_channel_capacity|channel capacity]]&nbsp; $C_{\rm AWGN}$&nbsp; specifies the maximum mutual information&nbsp; $I(X; Y)$&nbsp; between the input quantity&nbsp;  $X$&nbsp;  and the output quantity&nbsp;  $Y$&nbsp;  of the AWGN channel.&nbsp;  The maximisation refers to the best possible input PDF.&nbsp;  Thus, under the&nbsp;  [[Information_Theory/Differentielle_Entropie#Differential_entropy_of_some_power-constrained_random_variables|power constraint]] the following applies:
+*The&nbsp; [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Definition_and_meaning_of_channel_capacity|$\text{channel capacity}$]]&nbsp; $C_{\rm AWGN}$&nbsp; specifies the maximum mutual information&nbsp; $I(X; Y)$&nbsp; between the input quantity&nbsp;  $X$&nbsp;  and the output quantity&nbsp;  $Y$&nbsp;  of the AWGN channel.&nbsp;  The maximisation refers to the best possible input PDF.&nbsp;  Thus, under the&nbsp;  [[Information_Theory/Differentielle_Entropie#Differential_entropy_of_some_power-constrained_random_variables|$\text{power constraint}$]] the following applies:
 :$$C_{\rm AWGN} = \max_{f_X:\hspace{0.1cm} {\rm E}[X^2 ] \le P_X} \hspace{-0.35cm}  I(X;Y)
@@ Line 175: / Line 175: @@
-which we call the&nbsp; '''total channel capacity'''&nbsp; of this AWGN configuration.
+which we call the&nbsp; &raquo;'''total channel capacity'''&laquo;&nbsp; of this AWGN configuration.
 {{BlaueBox|TEXT=
@@ Line 199: / Line 199: @@
 *The equal sign&nbsp; (identity)&nbsp; is valid for mean-free Gaussian input variables&nbsp; $X_k$&nbsp; as well as for statistically independent disturbances&nbsp; $N_k$.
 *One arrives from this equation at the&nbsp; "maximum mutual information" &nbsp;  ⇒ &nbsp;  "channel capacity",&nbsp;  if the total transmission power&nbsp; $P_X$&nbsp; is divided as best as possible, taking into account the different interferences in the individual channels &nbsp;$(σ_k^2)$&nbsp;.
-*This optimization problem can again be elegantly solved with the method of&nbsp; [https://en.wikipedia.org/wiki/Lagrange_multiplier Lagrange multipliers].&nbsp;  The following example only explains the result.
+*This optimization problem can again be elegantly solved with the method of&nbsp; [https://en.wikipedia.org/wiki/Lagrange_multiplier $\text{Lagrange multipliers}$].&nbsp;  The following example only explains the result.
@@ Line 215: / Line 215: @@
-This allocation procedure is called a '''Water–Filling algorithm'''.}}
+This allocation procedure is called a &raquo;'''Water–Filling algorithm'''&laquo;.}}
@@ Line 235: / Line 235: @@
 However, the case&nbsp; $K = 2$&nbsp; can also be realised by using orthogonal basis functions such as&nbsp; "cosine"&nbsp; and&nbsp; "sine"&nbsp; as for example with
-*&nbsp; [[Modulation_Methods/Quadratur–Amplitudenmodulation|quadrature amplitude modulation]]&nbsp; $\rm (QAM)$&nbsp; or
+*&nbsp; [[Modulation_Methods/Quadratur–Amplitudenmodulation|$\text{quadrature amplitude modulation}$]]&nbsp; $\rm (QAM)$&nbsp; or
-*&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Other_signal_space_constellations|multi-level phase modulation]]&nbsp; such as&nbsp; $\rm QPSK$&nbsp; or&nbsp; $\rm  8–PSK$.}}
+*&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Other_signal_space_constellations|$\text{multi-level phase modulation}$]]&nbsp; such as&nbsp; $\rm QPSK$&nbsp; or&nbsp; $\rm  8–PSK$.}}
 ==Exercises for the chapter ==