Digital Signal Transmission/Linear Nyquist Equalization - Revision history

Guenter at 10:21, 13 July 2022

2022-07-13T10:21:50Z

Guenter at 10:20, 13 July 2022

2022-07-13T10:20:50Z

Hwang at 08:59, 11 July 2022

2022-07-11T08:59:46Z

Hwang at 08:58, 11 July 2022

2022-07-11T08:58:46Z

Hwang at 08:54, 11 July 2022

2022-07-11T08:54:08Z

Hwang at 08:20, 11 July 2022

2022-07-11T08:20:22Z

Hwang at 08:01, 11 July 2022

2022-07-11T08:01:56Z

Guenter at 14:50, 28 June 2022

2022-06-28T14:50:08Z

Guenter at 14:48, 28 June 2022

2022-06-28T14:48:30Z

Guenter at 14:21, 22 June 2022

2022-06-22T14:21:47Z

@@ Line 217: / Line 217: @@
 .25\,{\rm dB} \hspace{0.05cm}.$$
-*It can be shown that the normalized noise power can be calculated using the transverse '''KORREKTUR: transversal''' filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; alone, as shown in the right graph:
+*It can be shown that the normalized noise power can be calculated using the transversal filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; alone, as shown in the right graph:
 :$$\sigma_{d,\hspace{0.05cm} {\rm norm}}^2 = T \cdot
 \int_{-1/(2T)}^{+1/(2T)} H_{\rm TF}(f) \,{\rm d} f

@@ Line 225: / Line 225: @@
 <br clear=all>
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient &nbsp;$k_0$ when the real, symmetric, and periodic transverse '''KORREKTUR: transversal''' filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; is represented as a Fourier series.
+$\text{Conclusion:}$&nbsp; The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient &nbsp;$k_0$ when the real, symmetric, and periodic transversal filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; is represented as a Fourier series.
 [[File:P ID1430 Dig T 3 5 S5b version3.png|right|frame|Coefficients of the optimal Nyquist equalizer&nbsp; $\rm (ONE)$|class=fit]]

@@ Line 240: / Line 240: @@
 For a system comparison, the &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation|"system efficiency"]]&nbsp; is suitable,&nbsp; which relates the achievable detection SNR &nbsp;$\rho_d$&nbsp; to the maximum SNR &nbsp;$\rho_{d, \ {\rm max}}$,&nbsp; which,&nbsp; however,&nbsp; is only achievable for ideal channel &nbsp;$H_{\rm K}(f) \equiv 1$.&nbsp;
-[[File:EN_Dig_T_3_5_S6_neu.png|right|frame|Comparison of binary and multilevel transmission systems according to &nbsp;$\text{GLP}$&nbsp; and &nbsp;$\text{ONE}$|class=fit]]
+[[File:EN_Dig_T_3_5_S6_neu.png|right|frame|Comparison of binary and multi-level transmission systems according to &nbsp;$\text{GLP}$&nbsp; and &nbsp;$\text{ONE}$|class=fit]]
 For the system efficiency, with &nbsp;$M$&ndash;level transmission and optimal Nyquist equalization:
 :$$\eta = \frac{\rho_d}{s_0^2 \cdot T / N_0}=\frac{{\rm log_2}\hspace{0.1cm}M}{(M-1)^2 \cdot k_0}.$$
@@ Line 260: / Line 260: @@
 $\text{Conclusion:}$&nbsp; The results of this system comparison can be summarized as follows:
 #In the binary case &nbsp;$(M = 2)$,&nbsp; the intersymbol interference-free system &nbsp;$\text{(ONE)}$&nbsp; outperforms the intersymbol interference system &nbsp;$\text{(GLP)}$ by about &nbsp;$6 \ \rm dB$.&nbsp; <br>
-#If the optimal Nyquist equalization is applied to multilevel systems, a further, significant gain in signal-to-noise ratio is possible compared to &nbsp;$\text{GLP}$.&nbsp;
+#If the optimal Nyquist equalization is applied to multi-level systems, a further, significant gain in signal-to-noise ratio is possible compared to &nbsp;$\text{GLP}$.&nbsp;
 #For &nbsp;$M =4$,&nbsp; this gain is about &nbsp;$18.2 \ \rm dB$.<br>
 #However, the narrowband &nbsp;$\text{GLP}$ system can be significantly improved by using a receiver with decision feedback. This will be discussed in the next chapter.}}<br>

← Older revision		Revision as of 08:58, 11 July 2022
Line 248:		Line 248:
	*Note the normalization of the characteristic cable attenuation  $a_\star$  in the first column.		*Note the normalization of the characteristic cable attenuation  $a_\star$  in the first column.

−	*The table un the right from  [TS87]<ref name='TS87'/>  allows a system comparison for the characteristic cable attenuation  $a_\star = 80 \ \rm dB$.	+	*The table on the right from  [TS87]<ref name='TS87'/>  allows a system comparison for the characteristic cable attenuation  $a_\star = 80 \ \rm dB$.

@@ Line 217: / Line 217: @@
 .25\,{\rm dB} \hspace{0.05cm}.$$
-*It can be shown that the normalized noise power can be calculated using the transverse filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; alone, as shown in the right graph:
+*It can be shown that the normalized noise power can be calculated using the transverse '''KORREKTUR: transversal''' filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; alone, as shown in the right graph:
 :$$\sigma_{d,\hspace{0.05cm} {\rm norm}}^2 = T \cdot
 \int_{-1/(2T)}^{+1/(2T)} H_{\rm TF}(f) \,{\rm d} f
@@ Line 225: / Line 225: @@
 <br clear=all>
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient &nbsp;$k_0$ when the real, symmetric, and periodic transverse filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; is represented as a Fourier series.
+$\text{Conclusion:}$&nbsp; The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient &nbsp;$k_0$ when the real, symmetric, and periodic transverse '''KORREKTUR: transversal''' filter frequency response &nbsp;$H_{\rm TF}(f)$&nbsp; is represented as a Fourier series.
 [[File:P ID1430 Dig T 3 5 S5b version3.png|right|frame|Coefficients of the optimal Nyquist equalizer&nbsp; $\rm (ONE)$|class=fit]]

@@ Line 190: / Line 190: @@
 The graph shows the following properties of the&nbsp; '''optimal Nyquist equalizer'''&nbsp; $\rm (ONE)$:
-*If the cable attenuation is sufficiently large &nbsp;$(a_\star \ge 10 \ \rm dB)$,&nbsp; the overall frequency response can be described with good approximation by the &nbsp;[[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#1.2FT.E2.80.93Nyquistspektren| "cosine rolloff low-pass"]].&nbsp; <br>
+*If the cable attenuation is sufficiently large &nbsp;$(a_\star \ge 10 \ \rm dB)$,&nbsp; the overall frequency response can be described with good approximation by the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#1.2FT_Nyquist_spectra| "cosine rolloff low-pass"]].&nbsp; <br>
 *The larger &nbsp;$a_\star$&nbsp; is,&nbsp; the smaller is the rolloff factor &nbsp;$r$&nbsp; and the steeper is the edge drop.&nbsp; For the characteristic cable attenuation &nbsp;$a_\star = 40 \ \rm dB$&nbsp; (blue curve)&nbsp; we get &nbsp;$r \approx 0.4$, for &nbsp;$a_\star = 80 \ \rm dB$&nbsp; (red curve)  $r \approx 0.18$.<br>

@@ Line 17: / Line 17: @@
 *In some derivations,&nbsp; transmitter and channel are combined by the&nbsp; "common frequency response" &nbsp;$H_{\rm SK}(f) = H_{\rm S}(f) \cdot H_{\rm K}(f)$.&nbsp; <br>
-*The receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; is multiplicatively composed of the &nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter|matched filter]]&nbsp; $H_{\rm MF}(f) = H_{\rm SK}^\star(f)$&nbsp; and the &nbsp;[[Digital_Signal_Transmission/Linear_Nyquist_Equalization#Mode_of_action_of_the_transversal_filter|transversal filter]]&nbsp; $H_{\rm TF}(f)$,&nbsp; at least it can be split up mentally in this way.
+*The receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; is multiplicatively composed of the &nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter|matched filter]]&nbsp; $H_{\rm MF}(f) = H_{\rm SK}^\star(f)$&nbsp; and the &nbsp;[[Digital_Signal_Transmission/Linear_Nyquist_Equalization#Operating_principle_of_the_transversal_filter|transversal filter]]&nbsp; $H_{\rm TF}(f)$,&nbsp; at least it can be split up mentally in this way.
 *The overall frequency response between Dirac source and threshold decision should satisfy the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain| "first Nyquist condition"]].&nbsp; Thus, it must hold:

@@ Line 9: / Line 9: @@
 <br>
 In this section we assume the following block diagram of a binary system.&nbsp; In this regard,&nbsp; it should be noted:
-[[File:P ID1423 Dig T 3 5 S1 version1.png|right|frame|Block diagram of the optimal Nyquist equalizer|class=fit]]
+[[File:EN_Dig_T_3_5_S1.png|right|frame|Block diagram of the optimal Nyquist equalizer|class=fit]]
 *The&nbsp; "Dirac source"&nbsp; provides the message to be transmitted in binary bipolar form  &nbsp; &rArr; &nbsp; amplitude coefficients &nbsp;$a_\nu \in \{ -1, \hspace{0.05cm}+1\}$.&nbsp; The source is assumed to be redundancy-free.

@@ Line 135: / Line 135: @@
 follows from the first Nyquist criterion.&nbsp; By applying the &nbsp;[https://en.wikipedia.org/wiki/Calculus_of_variations&nbsp; "Calculus of Variations"],&nbsp; the frequency response of the transversal filter is obtained &nbsp; &ndash; see [TS87]<ref name='TS87'>Tröndle, K.; Söder, G.:&nbsp; Optimization of Digital Transmission Systems.&nbsp; Boston – London: Artech House, 1987,&nbsp; ISBN:&nbsp; 0-89006-225-0.</ref>:
 [[File:Dig_T_3_5_S3b_version2.png|right|frame|Magnitude frequency response of the transversal filter&nbsp; (left) and the entire optimal Nyquist equalizer&nbsp; (right)|class=fit]]
-:$$H_{\rm TF}(f) = \frac{1}{\sum\limits_{\kappa = -\infty}^{+\infty}  |H_{\rm SK}(f -
+$$H_{\rm TF}(f) = \frac{1}{\sum\limits_{\kappa = -\infty}^{+\infty}  |H_{\rm SK}(f -
   \frac{\kappa}{T})
   |^2},$$
-:$$\text{where }H_{\rm SK}(f) = H_{\rm S}(f)\cdot H_{\rm K}(f).$$
+$$\text{where }H_{\rm SK}(f) = H_{\rm S}(f)\cdot H_{\rm K}(f).$$
 The left graph shows &nbsp;$20 \cdot \lg \ H_{\rm TF}(f)$&nbsp; in the range &nbsp;$| f | \le 1/T$. This assumes rectangular NRZ transmission pulses and a coaxial cable with the characteristic cable attenuation &nbsp;$a_\star$.