Digital Signal Transmission/Linear Nyquist Equalization - Revision history

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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Lineare_Nyquistentzerrung]]

Guenter at 10:21, 13 July 2022

2022-07-13T10:21:50Z

← Older revision		Revision as of 12:21, 13 July 2022
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	72.25\,{\rm dB} \hspace{0.05cm}.$$		72.25\,{\rm dB} \hspace{0.05cm}.$$

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

Guenter at 10:20, 13 July 2022

2022-07-13T10:20:50Z

← Older revision		Revision as of 12:20, 13 July 2022
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	<br clear=all>		<br clear=all>
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$  The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient  $k_0$ when the real, symmetric, and periodic ~~transverse '''KORREKTUR:~~ transversal~~'''~~ filter frequency response  $H_{\rm TF}(f)$  is represented as a Fourier series.		$\text{Conclusion:}$  The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient  $k_0$ when the real, symmetric, and periodic transversal filter frequency response  $H_{\rm TF}(f)$  is represented as a Fourier series.

	[[File:P ID1430 Dig T 3 5 S5b version3.png\|right\|frame\|Coefficients of the optimal Nyquist equalizer  $\rm (ONE)$\|class=fit]]		[[File:P ID1430 Dig T 3 5 S5b version3.png\|right\|frame\|Coefficients of the optimal Nyquist equalizer  $\rm (ONE)$\|class=fit]]

Hwang at 08:59, 11 July 2022

2022-07-11T08:59:46Z

← Older revision		Revision as of 10:59, 11 July 2022
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	For a system comparison, the  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation\|"system efficiency"]]  is suitable,  which relates the achievable detection SNR  $\rho_d$  to the maximum SNR  $\rho_{d, \ {\rm max}}$,  which,  however,  is only achievable for ideal channel  $H_{\rm K}(f) \equiv 1$.		For a system comparison, the  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation\|"system efficiency"]]  is suitable,  which relates the achievable detection SNR  $\rho_d$  to the maximum SNR  $\rho_{d, \ {\rm max}}$,  which,  however,  is only achievable for ideal channel  $H_{\rm K}(f) \equiv 1$.

	[[File:EN_Dig_T_3_5_S6_neu.png\|right\|frame\|Comparison of binary and ~~multilevel~~ transmission systems according to  $\text{GLP}$  and  $\text{ONE}$\|class=fit]]		[[File:EN_Dig_T_3_5_S6_neu.png\|right\|frame\|Comparison of binary and multi-level transmission systems according to  $\text{GLP}$  and  $\text{ONE}$\|class=fit]]
	For the system efficiency, with  $M$–level transmission and optimal Nyquist equalization:		For the system efficiency, with  $M$–level transmission and optimal Nyquist equalization:
	:$$\eta = \frac{\rho_d}{s_0^2 \cdot T / N_0}=\frac{{\rm log_2}\hspace{0.1cm}M}{(M-1)^2 \cdot k_0}.$$		:$$\eta = \frac{\rho_d}{s_0^2 \cdot T / N_0}=\frac{{\rm log_2}\hspace{0.1cm}M}{(M-1)^2 \cdot k_0}.$$
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	$\text{Conclusion:}$  The results of this system comparison can be summarized as follows:		$\text{Conclusion:}$  The results of this system comparison can be summarized as follows:
	#In the binary case  $(M = 2)$,  the intersymbol interference-free system  $\text{(ONE)}$  outperforms the intersymbol interference system  $\text{(GLP)}$ by about  $6 \ \rm dB$.  <br>		#In the binary case  $(M = 2)$,  the intersymbol interference-free system  $\text{(ONE)}$  outperforms the intersymbol interference system  $\text{(GLP)}$ by about  $6 \ \rm dB$.  <br>
	#If the optimal Nyquist equalization is applied to ~~multilevel~~ systems, a further, significant gain in signal-to-noise ratio is possible compared to  $\text{GLP}$.		#If the optimal Nyquist equalization is applied to multi-level systems, a further, significant gain in signal-to-noise ratio is possible compared to  $\text{GLP}$.
	#For  $M =4$,  this gain is about  $18.2 \ \rm dB$.<br>		#For  $M =4$,  this gain is about  $18.2 \ \rm dB$.<br>
	#However, the narrowband  $\text{GLP}$ system can be significantly improved by using a receiver with decision feedback. This will be discussed in the next chapter.}}<br>		#However, the narrowband  $\text{GLP}$ system can be significantly improved by using a receiver with decision feedback. This will be discussed in the next chapter.}}<br>

Hwang at 08:58, 11 July 2022

2022-07-11T08:58:46Z

← Older revision		Revision as of 10:58, 11 July 2022
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	*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  $a_\star = 80 \ \rm dB$.

Hwang at 08:54, 11 July 2022

2022-07-11T08:54:08Z

← Older revision		Revision as of 10:54, 11 July 2022
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	72.25\,{\rm dB} \hspace{0.05cm}.$$		72.25\,{\rm dB} \hspace{0.05cm}.$$

	*It can be shown that the normalized noise power can be calculated using the transverse filter frequency response  $H_{\rm TF}(f)$  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  $H_{\rm TF}(f)$  alone, as shown in the right graph:
	:$$\sigma_{d,\hspace{0.05cm} {\rm norm}}^2 = T \cdot		:$$\sigma_{d,\hspace{0.05cm} {\rm norm}}^2 = T \cdot
	\int_{-1/(2T)}^{+1/(2T)} H_{\rm TF}(f) \,{\rm d} f		\int_{-1/(2T)}^{+1/(2T)} H_{\rm TF}(f) \,{\rm d} f
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	<br clear=all>		<br clear=all>
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$  The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient  $k_0$ when the real, symmetric, and periodic transverse filter frequency response  $H_{\rm TF}(f)$  is represented as a Fourier series.		$\text{Conclusion:}$  The normalized noise power of the optimal Nyquist equalizer is equal to the Fourier coefficient  $k_0$ when the real, symmetric, and periodic transverse '''KORREKTUR: transversal''' filter frequency response  $H_{\rm TF}(f)$  is represented as a Fourier series.

	[[File:P ID1430 Dig T 3 5 S5b version3.png\|right\|frame\|Coefficients of the optimal Nyquist equalizer  $\rm (ONE)$\|class=fit]]		[[File:P ID1430 Dig T 3 5 S5b version3.png\|right\|frame\|Coefficients of the optimal Nyquist equalizer  $\rm (ONE)$\|class=fit]]

Hwang at 08:20, 11 July 2022

2022-07-11T08:20:22Z

← Older revision		Revision as of 10:20, 11 July 2022
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	The graph shows the following properties of the  '''optimal Nyquist equalizer'''  $\rm (ONE)$:		The graph shows the following properties of the  '''optimal Nyquist equalizer'''  $\rm (ONE)$:
	*If the cable attenuation is sufficiently large  $(a_\star \ge 10 \ \rm dB)$,  the overall frequency response can be described with good approximation by the  [[Digital_Signal_Transmission/~~Eigenschaften_von_Nyquistsystemen~~#1.~~2FT.E2.80.93Nyquistspektren~~\| "cosine rolloff low-pass"]].  <br>		*If the cable attenuation is sufficiently large  $(a_\star \ge 10 \ \rm dB)$,  the overall frequency response can be described with good approximation by the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#1.2FT_Nyquist_spectra\| "cosine rolloff low-pass"]].  <br>

	*The larger  $a_\star$  is,  the smaller is the rolloff factor  $r$  and the steeper is the edge drop.  For the characteristic cable attenuation  $a_\star = 40 \ \rm dB$  (blue curve)  we get  $r \approx 0.4$, for  $a_\star = 80 \ \rm dB$  (red curve) $r \approx 0.18$.<br>		*The larger  $a_\star$  is,  the smaller is the rolloff factor  $r$  and the steeper is the edge drop.  For the characteristic cable attenuation  $a_\star = 40 \ \rm dB$  (blue curve)  we get  $r \approx 0.4$, for  $a_\star = 80 \ \rm dB$  (red curve) $r \approx 0.18$.<br>

Hwang at 08:01, 11 July 2022

2022-07-11T08:01:56Z

← Older revision		Revision as of 10:01, 11 July 2022
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	*In some derivations,  transmitter and channel are combined by the  "common frequency response"  $H_{\rm SK}(f) = H_{\rm S}(f) \cdot H_{\rm K}(f)$.  <br>		*In some derivations,  transmitter and channel are combined by the  "common frequency response"  $H_{\rm SK}(f) = H_{\rm S}(f) \cdot H_{\rm K}(f)$.  <br>

	*The receiver filter  $H_{\rm E}(f)$  is multiplicatively composed of the  [[Theory_of_Stochastic_Signals/Matched_Filter\|matched filter]]  $H_{\rm MF}(f) = H_{\rm SK}^\star(f)$  and the  [[Digital_Signal_Transmission/Linear_Nyquist_Equalization#~~Mode_of_action_of_the_transversal_filter~~\|transversal filter]]  $H_{\rm TF}(f)$,  at least it can be split up mentally in this way.		*The receiver filter  $H_{\rm E}(f)$  is multiplicatively composed of the  [[Theory_of_Stochastic_Signals/Matched_Filter\|matched filter]]  $H_{\rm MF}(f) = H_{\rm SK}^\star(f)$  and the  [[Digital_Signal_Transmission/Linear_Nyquist_Equalization#Operating_principle_of_the_transversal_filter\|transversal filter]]  $H_{\rm TF}(f)$,  at least it can be split up mentally in this way.

	*The overall frequency response between Dirac source and threshold decision should satisfy the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain\| "first Nyquist condition"]].  Thus, it must hold:		*The overall frequency response between Dirac source and threshold decision should satisfy the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain\| "first Nyquist condition"]].  Thus, it must hold:

Guenter at 14:50, 28 June 2022

2022-06-28T14:50:08Z

← Older revision		Revision as of 16:50, 28 June 2022
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	For a system comparison, the  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation\|"system efficiency"]]  is suitable,  which relates the achievable detection SNR  $\rho_d$  to the maximum SNR  $\rho_{d, \ {\rm max}}$,  which,  however,  is only achievable for ideal channel  $H_{\rm K}(f) \equiv 1$.		For a system comparison, the  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation\|"system efficiency"]]  is suitable,  which relates the achievable detection SNR  $\rho_d$  to the maximum SNR  $\rho_{d, \ {\rm max}}$,  which,  however,  is only achievable for ideal channel  $H_{\rm K}(f) \equiv 1$.

	[[File:~~P ID1431 Dig T 3 5 S6 version1~~.png\|right\|frame\|Comparison of binary and multilevel transmission systems according to  $\text{GLP}$  and  $\text{ONE}$\|class=fit]]		[[File:EN_Dig_T_3_5_S6_neu.png\|right\|frame\|Comparison of binary and multilevel transmission systems according to  $\text{GLP}$  and  $\text{ONE}$\|class=fit]]
	For the system efficiency, with  $M$–level transmission and optimal Nyquist equalization:		For the system efficiency, with  $M$–level transmission and optimal Nyquist equalization:
	:$$\eta = \frac{\rho_d}{s_0^2 \cdot T / N_0}=\frac{{\rm log_2}\hspace{0.1cm}M}{(M-1)^2 \cdot k_0}.$$		:$$\eta = \frac{\rho_d}{s_0^2 \cdot T / N_0}=\frac{{\rm log_2}\hspace{0.1cm}M}{(M-1)^2 \cdot k_0}.$$