Digital Signal Transmission/Optimization of Baseband Transmission Systems - Revision history

Hwang at 13:51, 23 January 2023

2023-01-23T13:51:00Z

Guenter at 07:21, 20 May 2022

2022-05-20T07:21:24Z

Guenter at 07:20, 20 May 2022

2022-05-20T07:20:10Z

Hwang at 12:36, 16 May 2022

2022-05-16T12:36:53Z

Hwang at 15:57, 10 May 2022

2022-05-10T15:57:55Z

Hwang at 14:58, 10 May 2022

2022-05-10T14:58:11Z

Hwang at 13:15, 10 May 2022

2022-05-10T13:15:50Z

Hwang at 13:05, 10 May 2022

2022-05-10T13:05:17Z

Hwang at 12:52, 10 May 2022

2022-05-10T12:52:07Z

Guenter at 14:41, 9 May 2022

2022-05-09T14:41:46Z

@@ Line 124: / Line 124: @@
 *The left diagram compares the systems at the same average power &nbsp;$(P_{\rm S})$&nbsp; and at constant energy per bit &nbsp;$(E_{\rm B})$,&nbsp; resp.&nbsp; Since the abscissa value is additionally related to &nbsp;$N_0$,&nbsp; the equation &nbsp;$p_{\rm B}(E_{\rm B}/N_0)$&nbsp; and its graphical representation correctly reflects the situation even for different noise power densities &nbsp;$N_0$.&nbsp;<br>
-*When power is limited,&nbsp; configurations &nbsp;$\rm A$&nbsp; and &nbsp;$\rm C$&nbsp; are equivalent and represent the optimum in each case.&nbsp; As will be shown on the next pages,&nbsp; an optimal system with power limitation always exists if &nbsp;$g_s(t)$&nbsp; and &nbsp;$h_{\rm E}(t)$&nbsp; have the same shape&nbsp; $($"matched filter"$)$.&nbsp; The smaller power of system &nbsp;$\rm C$&nbsp; is compensated by the abscissa chosen here.<br>
+*When power is limited,&nbsp; configurations &nbsp;$\rm A$&nbsp; and &nbsp;$\rm C$&nbsp; are equivalent and represent the optimum in each case.&nbsp; As will be shown in the next sections,&nbsp; an optimal system with power limitation always exists if &nbsp;$g_s(t)$&nbsp; and &nbsp;$h_{\rm E}(t)$&nbsp; have the same shape&nbsp; $($"matched filter"$)$.&nbsp; The smaller power of system &nbsp;$\rm C$&nbsp; is compensated by the abscissa chosen here.<br>
 *In contrast,&nbsp; for system &nbsp;$\rm B$&nbsp; the matched filter condition is not met &nbsp;$(T_{\rm E} \ne T_{\rm S})$&nbsp; and the error probability curve is now &nbsp;$\text{3 dB}$&nbsp; to the right of the boundary curve given by systems &nbsp;$\rm A$&nbsp; and &nbsp;$\rm C$.&nbsp;<br>

@@ Line 141: / Line 141: @@
 \hspace{0.3cm}{\rm Maximum}\hspace{0.05cm}.$$
 Here, &nbsp;$g_0 = g_d(t=0)$&nbsp; indicates the amplitude of the considered Nyquist pulse and &nbsp;$\sigma_d^2$&nbsp; denotes the detection noise power for the given receiver filter.&nbsp; At the same time it must be ensured that<br>
-*the basic detection pulse '''KORREKTUR: basic transmitter pulse''' &nbsp;$g_d(t) = g_s(t) \star h_{\rm E}(t)$&nbsp;  satisfies the first Nyquist criterion, and<br>
+*the basic detection pulse &nbsp;$g_d(t) = g_s(t) \star h_{\rm E}(t)$&nbsp;  satisfies the first Nyquist criterion, and<br>
 *the energy of the basic transmission pulse &nbsp;$g_s(t)$&nbsp;  does not exceed a predetermined value &nbsp;$E_{\rm B}$.&nbsp;
@@ Line 200: / Line 200: @@
 <br>
 The main result of the calculations on the last sections was that for the optimal binary system under the constraint of &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation|"power limitation"]]
-*the basic detection pulse '''KORREKTUR: basic transmitter pulse''' &nbsp;$g_d(t) = g_s(t) \star h_{\rm E}(t)$&nbsp; must satisfy the first Nyquist condition,&nbsp; and
+*the basic detection pulse &nbsp;$g_d(t) = g_s(t) \star h_{\rm E}(t)$&nbsp; must satisfy the first Nyquist condition,&nbsp; and
 *the impulse response &nbsp;$h_{\rm E}(t)$&nbsp; of the receiver filter must be chosen to be equal in shape to the basic transmission pulse &nbsp;$g_s(t)$;&nbsp; &nbsp;
 *the same applies to the spectral functions &nbsp;$H_{\rm E}(f)$&nbsp; and &nbsp;$G_s(f)$.
@@ Line 281: / Line 281: @@
   E}(f)= \sqrt{H_{\rm CRO}(f)}
 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} H_{\rm S}(f) \cdot H_{\rm E}(f) = {H_{\rm CRO}(f)}\hspace{0.05cm}.$$
-Here,&nbsp; the frequency responses&nbsp; $H_{\rm S}(f)$&nbsp; and&nbsp; $H_{\rm CRO}(f)$&nbsp; give the normalized spectral functions of the basic transmission pulse and the basic detection pulse '''KORREKTUR: basic transmitter pulse'''.&nbsp; It holds:
+Here,&nbsp; the frequency responses&nbsp; $H_{\rm S}(f)$&nbsp; and&nbsp; $H_{\rm CRO}(f)$&nbsp; give the normalized spectral functions of the basic transmission pulse and the basic detection pulse.&nbsp; It holds:
 :$$G_s(f) = g_0 \cdot T \cdot \sqrt{H_{\rm S}(f)},\hspace{0.5cm} G_d(f) = g_0 \cdot T \cdot {H_{\rm CRO}(f)} = G_{\rm Nyq}(f)\hspace{0.05cm}.$$
 The graphic shows the eye diagrams at the transmitter&nbsp; (top)&nbsp; resp. at the receiver&nbsp; (bottom)&nbsp; for the rolloff&ndash;factors &nbsp;$r = 0.25$, &nbsp;$r = 0.5$&nbsp; and &nbsp;$r = 0. 1$.&nbsp; It should be recalled,&nbsp; that such a configuration is optimal under the constraint of power limitation independent of the rolloff&ndash;factor &nbsp;$r$.&nbsp; In front of the decision device there is always a fully open eye&nbsp; (see lower row of figures)&nbsp; and it holds for the system efficiency: &nbsp; $\eta_{\rm L} = 1$.

@@ Line 72: / Line 72: @@
 *NRZ basic transmission pulse,
 *Matched filter,&nbsp; since &nbsp;$T_{\rm E} = T_{\rm S}$,
-*Basic detection pulse '''KORREKTUR: basic transmitter pulse''': &nbsp; triangle,
+*Basic detection pulse: &nbsp; triangle,
 *Energy per bit:  &nbsp; $E_{\rm B} = s_0^2 \cdot T$,
 *Noise power: &nbsp; $\sigma_d^2 = N_0/(2T)$,
@@ Line 86: / Line 86: @@
 *NRZ basic transmission pulse,
 *No matched filter,&nbsp; since &nbsp;$T_{\rm E} \ne T_{\rm S}$,
-*Basic detection pulse '''KORREKTUR: basic transmitter pulse''': &nbsp; trapezoid,
+*Basic detection pulse: &nbsp; trapezoid,
 *Energy per bit:  &nbsp; $E_{\rm B} = s_0^2 \cdot T$,
 *Noise power: &nbsp; $\sigma_d^2 = N_0/T$,
@@ Line 101: / Line 101: @@
 *RZ basic transmission pulse,
 *Matched filter,&nbsp; since &nbsp;$T_{\rm E} = T_{\rm S}$,
-*Basic detection pulse '''KORREKTUR: basic transmitter pulse''': &nbsp;  smaller triangle,
+*Basic detection pulse: &nbsp;  smaller triangle,
 *Energy per bit:  &nbsp; $E_{\rm B} = 1/2 \cdot s_0^2 \cdot T$,
 *Noise power: &nbsp; $\sigma_d^2 = N_0/T$,

@@ Line 233: / Line 233: @@
 :$$p_{\rm B, \hspace{0.08cm}min} = {\rm Q} \left( \sqrt{\rho_{d, \hspace{0.05cm}{\rm max \hspace{0.05cm} | \hspace{0.05cm} L}}}\right)\hspace{0.5cm}{\rm with}\hspace{0.5cm}
   \rho_{d, \hspace{0.05cm}{\rm max \hspace{0.05cm} | \hspace{0.05cm} L}}={2 \cdot E_{\rm B}}/{N_0}\hspace{0.05cm}.$$
-*In the case of &nbsp;"peak limitation"&nbsp; (German:&nbsp; "Spitzenwertbegrenzung"&nbsp; or&nbsp; ""Amplitudenbegrenzung" &nbsp; &rArr; &nbsp; subscript:&nbsp; "'''A'''"),&nbsp; on the other hand,&nbsp; the modulation range of the transmitter device is limited &nbsp; &rArr; &nbsp; $\vert s(t) \vert \le s_0\hspace{0.4cm}{\rm{for} }\hspace{0.15cm}{\rm
+*In the case of &nbsp;"peak limitation"&nbsp; (German:&nbsp; "Spitzenwertbegrenzung"&nbsp; or&nbsp; "Amplitudenbegrenzung" &nbsp; &rArr; &nbsp; subscript:&nbsp; "'''A'''"),&nbsp; on the other hand,&nbsp; the modulation range of the transmitter device is limited &nbsp; &rArr; &nbsp; $\vert s(t) \vert \le s_0\hspace{0.4cm}{\rm{for} }\hspace{0.15cm}{\rm
 all}\hspace{0.15cm}t$. Here, the following applies to the corresponding quantities:
 :$$p_{\rm B, \hspace{0.08cm}min} = {\rm Q} \left( \sqrt{\rho_{d, \hspace{0.05cm}{\rm max \hspace{0.05cm} | \hspace{0.05cm}A}}}\right)\hspace{0.5cm}{\rm with}\hspace{0.5cm}

@@ Line 225: / Line 225: @@
 $\rm Notes$:
 *For baseband transmission,&nbsp; the bandwidth requirement plays only a minor role.
-*However,&nbsp; the diagram also applies to &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation#Gemeinsames_Blockschaltbild_f.C3.BCr_ASK_und_BPSK|"carrier frequency systems"]]&nbsp; when displayed in the equivalent low-pass range.
+*However,&nbsp; the diagram also applies to &nbsp;[[Digital_Signal_Transmission/Linear_Digital_Modulation_-_Coherent_Demodulation#Common_block_diagram_for_ASK_and_BPSK|"carrier frequency systems"]]&nbsp; when displayed in the equivalent low-pass range.
 *In these systems,&nbsp; bandwidth plays a very important role.&nbsp; Because:&nbsp; &nbsp; Every additional Hertz of bandwidth can be very expensive. <br clear=all>
 == System optimization with peak limitation==

@@ Line 199: / Line 199: @@
 ==Root Nyquist systems==
 <br>
-The main result of the calculations on the last sections was that for the optimal binary system under the constraint of &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation|power limitation]]
+The main result of the calculations on the last sections was that for the optimal binary system under the constraint of &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation|"power limitation"]]
 *the basic detection pulse '''KORREKTUR: basic transmitter pulse''' &nbsp;$g_d(t) = g_s(t) \star h_{\rm E}(t)$&nbsp; must satisfy the first Nyquist condition,&nbsp; and
 *the impulse response &nbsp;$h_{\rm E}(t)$&nbsp; of the receiver filter must be chosen to be equal in shape to the basic transmission pulse &nbsp;$g_s(t)$;&nbsp; &nbsp;
@@ Line 225: / Line 225: @@
 $\rm Notes$:
 *For baseband transmission,&nbsp; the bandwidth requirement plays only a minor role.
-*However,&nbsp; the diagram also applies to &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation#Gemeinsames_Blockschaltbild_f.C3.BCr_ASK_und_BPSK|carrier frequency systems]]&nbsp; when displayed in the equivalent low-pass range.
+*However,&nbsp; the diagram also applies to &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation#Gemeinsames_Blockschaltbild_f.C3.BCr_ASK_und_BPSK|"carrier frequency systems"]]&nbsp; when displayed in the equivalent low-pass range.
 *In these systems,&nbsp; bandwidth plays a very important role.&nbsp; Because:&nbsp; &nbsp; Every additional Hertz of bandwidth can be very expensive. <br clear=all>
 == System optimization with peak limitation==
@@ Line 248: / Line 248: @@
-A comparison with the &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation|system efficiency under power limitation]] &nbsp; &rArr; &nbsp; $\eta_{\rm L}$&nbsp; shows:
+A comparison with the &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation|"system efficiency under power limitation"]] &nbsp; &rArr; &nbsp; $\eta_{\rm L}$&nbsp; shows:
 * $\eta_{\rm A}$&nbsp; differs from &nbsp;$\eta_{\rm L}$&nbsp; in that now the denominator contains &nbsp;$s_0^2 \cdot T$&nbsp; instead of &nbsp;$E_{\rm B}$.&nbsp; The following relationship holds:
 :$$\eta_{\rm A} =  \frac{E_{\rm B}}{s_0^2 \cdot T} \cdot \eta_{\rm L}= \frac{\eta_{\rm L}}{C_{\rm S}^2}\hspace{0.05cm}.$$
-*Here, the &nbsp;[https://en.wikipedia.org/wiki/Crest_factor crest factor]&nbsp; $C_{\rm S}$&nbsp; denotes the ratio of the maximum value &nbsp;$s_0$&nbsp; and the rms value &nbsp;$s_{\rm eff}$&nbsp; of the transmitted signal:
+*Here, the &nbsp;[https://en.wikipedia.org/wiki/Crest_factor "crest factor"]&nbsp; $C_{\rm S}$&nbsp; denotes the ratio of the maximum value &nbsp;$s_0$&nbsp; and the rms value &nbsp;$s_{\rm eff}$&nbsp; of the transmitted signal:
 :$$C_{\rm S} =  \frac{s_0}{\sqrt{E_{\rm B}/T}} = \frac{{\rm Max}[s(t)]}{\sqrt{{\rm E}[s^2(t)]}}=  \frac{s_0}{s_{\rm eff}}
 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} s_{\rm eff} = \sqrt {E_{\rm B}/T}.$$
@@ Line 257: / Line 257: @@
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; As in &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation| $\text{Example 1}$]]&nbsp; we consider three different configurations,&nbsp; each with rectangular functions &nbsp;$g_s(t)$&nbsp; and &nbsp;$h_{\rm E}(t)$.&nbsp; The system efficiencies are:
+$\text{Example 3:}$&nbsp; As in &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation| "$\text{Example 1}$"]]&nbsp; we consider three different configurations,&nbsp; each with rectangular functions &nbsp;$g_s(t)$&nbsp; and &nbsp;$h_{\rm E}(t)$.&nbsp; The system efficiencies are:
 * $\text{System A:}$ &nbsp; &nbsp;$\rho_d =  {2 \cdot E_{\rm B} }/{N_0} = { 2 \cdot s_0^2 \cdot
   T}/{N_0}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\eta_{\rm L} = 1.0,\hspace{0.3cm}\eta_{\rm A} =
@@ Line 277: / Line 277: @@
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp;
-We now consider a &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems |root&ndash;Nyquist&ndash;configuration]]&nbsp; with cosine&ndash;rolloff total frequency response:
+We now consider a &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems |"root&ndash;Nyquist&ndash;configuration"]]&nbsp; with cosine&ndash;rolloff total frequency response:
 :$$H_{\rm S}(f)= \sqrt{H_{\rm CRO}(f)}, \hspace{0.5cm}H_{\rm
   E}(f)= \sqrt{H_{\rm CRO}(f)}
@@ Line 324: / Line 324: @@
 *The absolute optimum at peak limitation &nbsp; &#8658; &nbsp; $\eta_{\rm A}= 1$&nbsp; results only with a rectangular basic transmission pulse &nbsp;$g_s(t)$&nbsp; and a likewise rectangular receiver filter pulse response &nbsp;$h_{\rm E}(t)$&nbsp; of the same width &nbsp;$T$.
 *The best cosine rolloff Nyquist spectrum with &nbsp;$r = 0.8$&nbsp; (blue curve)&nbsp; is about &nbsp;$0.5 \ \rm  dB$&nbsp; worse compared to the matched filter&nbsp; (violet-dashed curve),&nbsp; because the area under the blue curve is about &nbsp;$12\%$&nbsp; larger than the area under the violet curve.
-*The so-called &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|root&ndash;root configuration]] &nbsp; &rArr; &nbsp; $H_{\rm S}(f) = H_{\rm E}(f) =\sqrt{H_{\rm CRO}(f)}$&nbsp; thus only makes sense if one assumes power limitation.}}
+*The so-called &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|"root&ndash;root configuration"]] &nbsp; &rArr; &nbsp; $H_{\rm S}(f) = H_{\rm E}(f) =\sqrt{H_{\rm CRO}(f)}$&nbsp; thus only makes sense if one assumes power limitation.}}

@@ Line 156: / Line 156: @@
 In the following it is proved that
 *the quantity thus defined actually satisfies the condition &nbsp;$0 \le \eta_{\rm L} \le 1$&nbsp; and thus can be interpreted as "efficiency",
-*the value   &nbsp;$\eta_{\rm L} = 1$&nbsp; is obtained when the receiver filter impulse response &nbsp;$h_{\rm E}(t)$&nbsp; is equal in shape to the basic transmission pulse &nbsp;$g_s(t)$&.}}
+*the value   &nbsp;$\eta_{\rm L} = 1$&nbsp; is obtained when the receiver filter impulse response &nbsp;$h_{\rm E}(t)$&nbsp; is equal in shape to the basic transmission pulse &nbsp;$g_s(t)$.}}

@@ Line 115: / Line 115: @@
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp; The same conditions as in $\text{Example 1}$ apply.&nbsp; Graphically shown are the results from&nbsp; $\text{Example 1}$.&nbsp; The diagram shows the bit error probability &nbsp;$p_{\rm B}$&nbsp; as
-[[File:P_ID1288__Dig_T_1_4_S2_v1.png|right|frame|System comparison for power and peak limitation '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_1_4_S2.png|right|frame|System comparison for power and peak limitation|class=fit]]
 * a function of the ratio &nbsp;$E_{\rm B}/N_0$&nbsp; (left diagram)&nbsp; and