Digital Signal Transmission/Structure of the Optimal Receiver - Revision history

Hwang at 14:18, 23 January 2023

2023-01-23T14:18:47Z

Hwang at 10:06, 17 November 2022

2022-11-17T10:06:11Z

Guenter at 11:55, 20 August 2022

2022-08-20T11:55:20Z

Hwang at 08:56, 12 August 2022

2022-08-12T08:56:35Z

Hwang at 08:53, 12 August 2022

2022-08-12T08:53:30Z

Guenter at 18:18, 18 July 2022

2022-07-18T18:18:03Z

Guenter at 13:42, 18 July 2022

2022-07-18T13:42:59Z

Guenter at 14:08, 15 July 2022

2022-07-15T14:08:04Z

Guenter at 14:02, 15 July 2022

2022-07-15T14:02:03Z

Guenter at 13:53, 15 July 2022

2022-07-15T13:53:10Z

@@ Line 81: / Line 81: @@
 :$$\hat{m} = {\rm arg}\hspace{0.05cm} \max_i \hspace{0.1cm} P_{\hspace{0.02cm}m\hspace{0.05cm} \vert \hspace{0.05cm}\boldsymbol{ r} } (  m_i \hspace{0.05cm} \vert \hspace{0.05cm} \boldsymbol{\rho})$$
-are &nbsp;'''a posteriori probabilities'''. &nbsp; [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Inference_probability|"Bayes' theorem"]]&nbsp; can be used to write for this:
+are &nbsp;'''a-posteriori probabilities'''. &nbsp; [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Inference_probability|"Bayes' theorem"]]&nbsp; can be used to write for this:
 :$$P_{\hspace{0.02cm}m\hspace{0.05cm} \vert \hspace{0.05cm} \boldsymbol{ r} } (  m_i \hspace{0.05cm} \vert \hspace{0.05cm}\boldsymbol{\rho}) =
 \frac{ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r}\hspace{0.05cm} \vert \hspace{0.05cm}m } (\boldsymbol{\rho}\hspace{0.05cm} \vert \hspace{0.05cm}m_i )}{p_{\boldsymbol{ r} } (\boldsymbol{\rho})}

@@ Line 43: / Line 43: @@
 == Fundamental approach to optimal receiver design==
 <br>
-Compared to the&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Block_diagram_and_prerequisites|"block diagram"]]&nbsp; shown on the previous page, we now perform some essential generalizations:
+Compared to the&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Block_diagram_and_prerequisites|"block diagram"]]&nbsp; shown in the previous section, we now perform some essential generalizations:
 [[File:EN_Dig_T_4_2_S2b.png|right|frame|Model for deriving the optimal receiver|class=fit]]

@@ Line 196: / Line 196: @@
 \hspace{0.05cm}.$$
-*According to the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|"Wiener-Chintchine theorem" '''KORREKTUR: Wiener-Khintchine Theorem''']],&nbsp; the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is obtained as the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral| "Fourier retransform"]]&nbsp; of&nbsp; ${\it \Phi_n(f)}$:
+*According to the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|"Wiener-Khintchine theorem"]],&nbsp; the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is obtained as the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral| "Fourier retransform"]]&nbsp; of&nbsp; ${\it \Phi_n(f)}$:
 :$${\varphi_n(\tau)} = {\rm E}\big [n(t) \cdot n(t+\tau)\big  ] = {N_0}/{2} \cdot \delta(t)\hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} {\rm E}\big [n(t) \cdot n(t+\tau)\big  ]  =

@@ Line 196: / Line 196: @@
 \hspace{0.05cm}.$$
-*According to the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|"Wiener-Chintchine theorem"]],&nbsp; the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is obtained as the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral| "Fourier retransform"]]&nbsp; of&nbsp; ${\it \Phi_n(f)}$:
+*According to the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|"Wiener-Chintchine theorem" '''KORREKTUR: Wiener-Khintchine Theorem''']],&nbsp; the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is obtained as the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral| "Fourier retransform"]]&nbsp; of&nbsp; ${\it \Phi_n(f)}$:
 :$${\varphi_n(\tau)} = {\rm E}\big [n(t) \cdot n(t+\tau)\big  ] = {N_0}/{2} \cdot \delta(t)\hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} {\rm E}\big [n(t) \cdot n(t+\tau)\big  ]  =

← Older revision		Revision as of 08:53, 12 August 2022
Line 166:		Line 166:
	*In the diagram all vectorial quantities are red inscribed.		*In the diagram all vectorial quantities are red inscribed.

−	*Moreover,red inscribed the quantities  $\boldsymbol{s}$,  $\boldsymbol{ n}_1$  and  $\boldsymbol{ n}_2$  are independent of each other.<br>	+	*Moreover, red inscribed the quantities  $\boldsymbol{s}$,  $\boldsymbol{ n}_1$  and  $\boldsymbol{ n}_2$  are independent of each other.<br>

@@ Line 320: / Line 320: @@
 *with the two signal space points&nbsp; $s_i \in \{s_0, \hspace{0.05cm}s_1\}$,&nbsp; because of&nbsp; $M = 2$.
+<br clear=all>
+For the signal&nbsp; $r(t) = s(t) + n(t)$&nbsp; at the AWGN channel output,&nbsp; the noise-free case &nbsp; &#8658; &nbsp;  $r(t) = s(t)$&nbsp; yields exactly the same constellation;&nbsp; The signal space points are at
 :$$r_0 = s_0 = \sqrt{E}\hspace{0.05cm},\hspace{0.2cm}r_1 = s_1 = -\sqrt{E}\hspace{0.05cm}.$$
@@ Line 374: / Line 373: @@
 *$||\boldsymbol{ \eta}||$&nbsp; is called the&nbsp; "norm"&nbsp; (length)&nbsp; of the vector &nbsp;$\boldsymbol{ \eta}$.<br>
-==Dummy==
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp;
-Shown on the right is the two-dimensional Gaussian probability density function&nbsp; $p_{\boldsymbol{ n} } (\boldsymbol{ \eta})$&nbsp; of the two-dimensional random variable&nbsp; $\boldsymbol{ n} = (n_1,\hspace{0.05cm}n_2)$.&nbsp;  Arbitrary realizations of the random variable&nbsp; $\boldsymbol{ n}$&nbsp; are denoted by&nbsp; $\boldsymbol{ \eta} = (\eta_1,\hspace{0.05cm}\eta_2)$.&nbsp;
+Shown on the right is the two-dimensional Gaussian probability density function&nbsp; $p_{\boldsymbol{ n} } (\boldsymbol{ \eta})$&nbsp; of the two-dimensional random variable&nbsp; $\boldsymbol{ n} = (n_1,\hspace{0.05cm}n_2)$.&nbsp;  Arbitrary realizations of the random variable&nbsp; $\boldsymbol{ n}$&nbsp; are denoted by&nbsp; $\boldsymbol{ \eta} = (\eta_1,\hspace{0.05cm}\eta_2)$.&nbsp; The equation of the represented two-dimensional&nbsp; "Gaussian bell curve"&nbsp; is:
 [[File:EN_Dig_T_4_2_S8.png|right|frame|Two-dimensional Gaussian PDF]]
 :$$p_{n_1, n_2}(\eta_1, \eta_2) = \frac{1}{\left( \sqrt{2\pi}  \cdot \sigma_n \right)^2 }  \cdot
 {\rm exp} \left [ - \frac{ \eta_1^2 + \eta_2^2}{2 \sigma_n^2}\right ]\hspace{0.05cm}. $$

@@ Line 303: / Line 303: @@
 = \int_{-\infty}^{+\infty}r(\tau) \cdot \varphi_j(T-t+\tau) \,{\rm d} \tau \hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} r_j (t = \tau) = \int_{-\infty}^{+\infty}r(\tau) \cdot \varphi_j(\tau) \,{\rm d} \tau = r_j
-\hspace{0.05cm}.$$}}
+\hspace{0.05cm}.$$
 == Probability density function of the received values ==

@@ Line 387: / Line 387: @@
 :$$p_{\boldsymbol{ n} }(\boldsymbol{ \eta}) = \frac{1}{\pi \cdot N_0 }  \cdot
 {\rm exp} \left [ - \frac{\vert \vert \boldsymbol{ \eta} \vert \vert ^2}{N_0}\right ]\hspace{0.05cm}.$$
-*This rotationally symmetric PDF is suitable e.g. for describing/investigating a&nbsp; "two-dimensional modulation process"&nbsp; such as&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"M&ndash;QAM"]],&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Multi-level_phase.E2.80.93shift_keying_.28M.E2.80.93PSK.29|"M&ndash;PSK"]]&nbsp; or&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29|"2&ndash;FSK"]].<br>
+*This rotationally symmetric PDF is suitable e.g. for describing/investigating a&nbsp; "two-dimensional modulation process"&nbsp; such as&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"M&ndash;QAM"]],&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Multi-level_phase.E2.80.93shift_keying_.28M.E2.80.93PSK.29|"M&ndash;PSK"]]&nbsp; or&nbsp; [[Modulation_Methods/Non-Linear_Digital_Modulation#FSK_.E2.80.93_Frequency_Shift_Keying|"2&ndash;FSK"]].<br>
 *However,&nbsp; two-dimensional real random variables are often represented in a one-dimensional complex way,&nbsp; usually in the form&nbsp; $n(t) = n_{\rm I}(t) + {\rm j} \cdot n_{\rm Q}(t)$.&nbsp; The two components are then called the&nbsp; "in-phase component"&nbsp; $n_{\rm I}(t)$&nbsp; and the&nbsp; "quadrature component"&nbsp; $n_{\rm Q}(t)$&nbsp; of the noise.<br>

@@ Line 46: / Line 46: @@
 [[File:EN_Dig_T_4_2_S2b.png|right|frame|Model for deriving the optimal receiver|class=fit]]
-*The transmission channel is described by the&nbsp; [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_Probability|"conditional probability density function"]]&nbsp;  $p_{\hspace{0.02cm}r(t)\hspace{0.02cm} \vert \hspace{0.02cm}s(t)}$&nbsp; which determines the dependence of the received signal&nbsp; $r(t)$&nbsp; on the transmitted signal&nbsp; $s(t)$.&nbsp; <br>
+*The transmission channel is described by the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|"conditional probability density function"]]&nbsp;  $p_{\hspace{0.02cm}r(t)\hspace{0.02cm} \vert \hspace{0.02cm}s(t)}$&nbsp; which determines the dependence of the received signal&nbsp; $r(t)$&nbsp; on the transmitted signal&nbsp; $s(t)$.&nbsp; <br>
 *If a certain signal&nbsp; $r(t) = \rho(t)$&nbsp; has been received,&nbsp; the receiver has the task to determine the probability density functions based on this&nbsp; "signal realization" &nbsp; $\rho(t)$&nbsp; and the&nbsp; $M$&nbsp; conditional probability density functions