Digital Signal Transmission/Properties of Nyquist Systems - Revision history

Hwang at 23:20, 12 November 2022

2022-11-12T23:20:58Z

Guenter at 11:37, 30 May 2022

2022-05-30T11:37:42Z

Guenter at 07:15, 20 May 2022

2022-05-20T07:15:40Z

Guenter at 07:13, 20 May 2022

2022-05-20T07:13:41Z

Hwang at 15:51, 10 May 2022

2022-05-10T15:51:04Z

Hwang at 14:50, 10 May 2022

2022-05-10T14:50:17Z

Hwang at 15:30, 9 May 2022

2022-05-09T15:30:11Z

Hwang at 15:24, 9 May 2022

2022-05-09T15:24:27Z

Guenter at 12:58, 1 May 2022

2022-05-01T12:58:34Z

Guenter at 14:20, 28 April 2022

2022-04-28T14:20:14Z

@@ Line 37: / Line 37: @@
 :$$g_{\hspace{0.05cm}\rm Nyq} ( t )= g_0 \cdot {\rm sinc} \left ( \frac{t}{T}\right)\cdot {\rm sinc} \left ( \frac{ t}{2 \cdot
 T}\right),$$
-which has already been discussed on the page &nbsp;[[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filter|"Trapezoidal low-pass filter"]]&nbsp; of the book&nbsp; "Linear Time Invariant Systems".}}
+which has already been discussed in the section &nbsp;[[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filter|"Trapezoidal low-pass filter"]]&nbsp; of the book&nbsp; "Linear Time Invariant Systems".}}
@@ Line 53: / Line 53: @@
-$\text{Note}$:&nbsp; The proof follows on the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#Proof_of_the_first_Nyquist_criterion|"next page"]].}}
+$\text{Note}$:&nbsp; The proof follows in the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#Proof_of_the_first_Nyquist_criterion|"next section"]].}}
@@ Line 177: / Line 177: @@
 #In the limiting case &nbsp;$r \to 0$,&nbsp; both cases yield the rectangular Nyquist spectrum and the &nbsp;$\rm sinc$&ndash;shaped Nyquist pulse, &nbsp; which decays asymptotically with &nbsp;$1/t$&nbsp; (thin green curves).
 #For an average rolloff &nbsp;$(r \approx 0.5)$,&nbsp; the first overshoots are smaller for the trapezoidal spectrum than for the CRO spectrum,&nbsp; because here the Nyquist slope is flatter for a given &nbsp;$r$&nbsp; (blue curves).
-#With the rolloff factor &nbsp;$r = 1$,&nbsp; the trapezoid becomes a triangle in the frequency domain and the CRO spectrum becomes the&nbsp; "cosine&ndash;square spectrum".&nbsp; In the diagrams on the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#1.2FT_Nyquist_spectra|"last page"]]&nbsp; these spectral functions are drawn in red.
+#With the rolloff factor &nbsp;$r = 1$,&nbsp; the trapezoid becomes a triangle in the frequency domain and the CRO spectrum becomes the&nbsp; "cosine&ndash;square spectrum".&nbsp; In the diagrams in the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#1.2FT_Nyquist_spectra|"last section"]]&nbsp; these spectral functions are drawn in red.
 #With &nbsp;$r = 1$&nbsp; the asymptotically decay of the upper time function&nbsp; (according to the trapezoidal spectrum)&nbsp; occurs with &nbsp;$1/t^2$,&nbsp; the decay of the lower function&nbsp; (according to the CRO spectrum)&nbsp; with &nbsp;$1/t^3$.
 #This means: &nbsp; After a longer time,&nbsp; the CRO Nyquist pulse has settled better than the trapezoidal Nyquist pulse.

@@ Line 29: / Line 29: @@
 $\text{Example 1:}$&nbsp; The diagram shows the detection signal &nbsp;$d(t)$&nbsp; of such a Nyquist system.&nbsp; Dotted in red are the&nbsp; (weighted and shifted)&nbsp; Nyquist pulses &nbsp;$a_\nu \cdot g_{\hspace{0.05cm}\rm Nyq}(t -\nu \cdot T)$.&nbsp;
-Please note: '''Bitte checken'''
+Please note:
 *At the detection times hold &nbsp; $d(\nu \cdot T) = a_\nu \cdot g_{\hspace{0.05cm}\rm Nyq}(0)$,&nbsp; as shown by the blue circles and the green grid.
 *The trailing edge &nbsp; &rArr; &nbsp; "trailers"&nbsp; of preceding pulses &nbsp;$(\nu < 0)$&nbsp; as well as the rising edge &nbsp; &rArr; &nbsp; "precursors"&nbsp; of following pulses &nbsp;$(\nu > 0)$&nbsp; do not affect the detection of the symbol &nbsp;$a_0$&nbsp; in the Nyquist system.

@@ Line 19: / Line 19: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; One denotes a basic detection pulse '''KORREKTUR: basic transmitter pulse''' with the properties
+$\text{Definition:}$&nbsp; One denotes a basic detection pulse  with the properties
 :$$g_d ( t = \nu  T)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
 \nu = \pm 1, \pm 2,\pm 3,\hspace{0.05cm}\text{...}$$

@@ Line 11: / Line 11: @@
 For the entire first main chapter it was assumed that the detection of a symbol should not be affected by neighboring pulses.&nbsp; This is achieved by the detection of the signal
 :$$d(t) =  d_{\rm S}(t) = \sum \limits_{\it (\nu)} a_\nu \cdot g_d ( t - \nu T)$$
-at the detection times &nbsp;$(\nu \cdot T)$&nbsp; whenever the basic detection pulse '''KORREKTUR: basic transmitter pulse'''&nbsp;$g_d(t)$
+at the detection times &nbsp;$(\nu \cdot T)$&nbsp; whenever the basic detection pulse &nbsp;$g_d(t)$
 *is restricted to the range &nbsp;$|t| < T$,&nbsp; which was assumed for the last chapter &nbsp;[[Digital_Signal_Transmission/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|"Error Probability for Baseband Transmission"]], &nbsp; or
 *has equidistant zero crossings at the detection times &nbsp;$\nu \cdot T$.&nbsp;
@@ Line 35: / Line 35: @@
-It should be mentioned that this graph is valid for the basic detection pulse '''KORREKTUR: basic transmitter pulse''' with trapezoidal spectrum and rolloff factor &nbsp;$r = 0.5$:
+It should be mentioned that this graph is valid for the basic detection pulse  with trapezoidal spectrum and rolloff factor &nbsp;$r = 0.5$:
 :$$g_{\hspace{0.05cm}\rm Nyq} ( t )= g_0 \cdot {\rm sinc} \left ( \frac{t}{T}\right)\cdot {\rm sinc} \left ( \frac{ t}{2 \cdot
 T}\right),$$
@@ Line 46: / Line 46: @@
 {{BlaueBox|TEXT=
-$\text{First Nyquist criterion:}$&nbsp; If the spectrum &nbsp;$G_d(f)$&nbsp;  of the basic detection pulse '''KORREKTUR: basic transmitter pulse''' &nbsp;$g_d(t)$&nbsp; fulfills the condition
+$\text{First Nyquist criterion:}$&nbsp; If the spectrum &nbsp;$G_d(f)$&nbsp;  of the basic detection pulse  &nbsp;$g_d(t)$&nbsp; fulfills the condition
 :$$\sum \limits_{\it k = -\infty}^{+\infty} G_d \left ( f - {k}/{T} \right)=
 g_0 \cdot T = {\rm const.} \hspace{0.05cm}, $$
@@ Line 210: / Line 210: @@
 {{BlaueBox|TEXT=
 $\text{Summary of Nyquist criteria:}$<br><br>
-'''(1)''' &nbsp; In memory of the physicist &nbsp;[https://en.wikipedia.org/wiki/Harry_Nyquist "Harry Nyquist"]&nbsp;  we denote a basic detection pulse '''KORREKTUR: basic transmitter pulse''' &nbsp;$g_d( t)$&nbsp; with the properties
+'''(1)''' &nbsp; In memory of the physicist &nbsp;[https://en.wikipedia.org/wiki/Harry_Nyquist "Harry Nyquist"]&nbsp;  we denote a basic detection pulse  &nbsp;$g_d( t)$&nbsp; with the properties
 ::$$g_d ( t= 0) \ne  0, \hspace{1cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
 t = \pm T, \pm 2T,\pm 3T,\hspace{0.05cm}\text{...}$$
 :as the&nbsp; '''Nyquist&ndash;1 pulse''' &nbsp;$g_{\hspace{0.05cm}\rm Nyq-1}(t)$.&nbsp; This satisfies the first Nyquist criterion and leads to the maximum vertical eye opening.
-'''(2)''' &nbsp; A basic detection pulse '''KORREKTUR: basic transmitter pulse''' &nbsp;$g_d( t)$&nbsp; satisfying the second Nyquist criterion must have zero crossings at &nbsp;$t = \pm 1.5T$, &nbsp;$t = \pm 2.5T$, ...&nbsp; have:
+'''(2)''' &nbsp; A basic detection pulse  &nbsp;$g_d( t)$&nbsp; satisfying the second Nyquist criterion must have zero crossings at &nbsp;$t = \pm 1.5T$, &nbsp;$t = \pm 2.5T$, ...&nbsp; have:
 ::$$g_d ( t= 0.5) \ne  0, \hspace{0.8cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
 t = \pm 1.5T, \ \pm 2.5T,\ \pm 3.5T,\hspace{0.05cm}\text{...}$$

@@ Line 187: / Line 187: @@
 Before the exact mathematical definition,&nbsp; the significance of the&nbsp; '''second Nyquist criterion'''&nbsp; for the evaluation of a digital system is illustrated by means of diagrams. The graphic shows three examples of Nyquist systems,&nbsp; in each case:
 *at the top,&nbsp; the Nyquist spectrum &nbsp;$G_d(f)$,
-*below the corresponding&nbsp; "eye diagram"&nbsp; referring to the &nbsp;[[Digital_Signal_Transmission/Fehlerwahrscheinlichkeit_unter_Berücksichtigung_von_Impulsinterferenzen#Definition_und_Aussagen_des_Augendiagramms|"third main chapter"]].
+*below the corresponding&nbsp; "eye diagram"&nbsp; referring to the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram|"third main chapter"]].
 [[File:EN_Dig_T_1_3_S4_v3.png|right|frame|Clarification of first and second Nyquist criterion]]

@@ Line 154: / Line 154: @@
 *For &nbsp;$r = 0$ &nbsp; &rArr; &nbsp; $f_1 = f_2 = f_{\rm Nyq}$&nbsp; we obtain in both cases the&nbsp; (green&ndash;dotted)&nbsp; rectangular Nyquist spectrum.
 * The rolloff factor  &nbsp;$r = 1$ &nbsp; &rArr; &nbsp;  $f_1 = 0, \ f_2 = 2 f_{\rm Nyq}$&nbsp; stands
-#for the triangular spectrum #
+#for the triangular spectrum
 #resp.&nbsp; the cosine-square spectrum,&nbsp;

@@ Line 171: / Line 171: @@
 t}{T}\right)\cdot \frac{\cos(\pi \cdot r \cdot t/T)}{1 - (2 \cdot
 r \cdot t/T)^2 } \hspace{0.3cm}{\rm with }\hspace{0.3cm}{\rm sinc}(x) = {\rm sin}(\pi x)/(\pi x).$$
-These Nyquist pulses can be viewed using the HTML5/JS applet &nbsp;[[Applets:Frequency_%26_Impulse_Responses|"Frequency & Impulse Responses"]]&nbsp; $($with setting &nbsp;$ \Delta \cdot f = 1)$,&nbsp; illustrating the influence of the rolloff factor &nbsp;$r$.&nbsp;
+These pulses can be viewed using the HTML5/JS applet &nbsp;[[Applets:Frequency_%26_Impulse_Responses|"Frequency & Impulse Responses"]]&nbsp; $($with setting &nbsp;$ \Delta \cdot f = 1)$,&nbsp; illustrating the influence of the rolloff factor &nbsp;$r$.&nbsp;
 The following upper diagram shows the Nyquist pulse with trapezoidal spectrum for different rolloff factors. Below is the corresponding time course for the cosine rolloff spectrum. One can see:
@@ Line 204: / Line 204: @@
 (\frac{t}{T} + \frac{1}{2}) + {\rm sinc} (\frac{t}{T} -
 \frac{1}{2})\right]$$
-:exhibits the required zero crossings at &nbsp;$t = \pm T$, &nbsp;$t = \pm 1.5T$, &nbsp;$t = \pm 2T$, &nbsp;$t = \pm 2.5T$,  ... but not at &nbsp;$t = \pm 0.5T$. The pulse amplitude is &nbsp;$g_d(t = 0) = g_0$. &nbsp; &nbsp;<br><u>Note</u>: &nbsp; No other pulse satisfies both the first and second Nyquist criteria simultaneously.
+:exhibits the required zero crossings at &nbsp;$t = \pm T$, &nbsp;$t = \pm 1.5T$, &nbsp;$t = \pm 2T$, &nbsp;$t = \pm 2.5T$,  ... but not at &nbsp;$t = \pm 0.5T$. The pulse amplitude is &nbsp;$g_d(t = 0) = g_0$. &nbsp; &nbsp;<br><u>Note</u>: &nbsp; '''No other pulse satisfies both the first and second Nyquist criteria simultaneously'''.
@@ Line 211: / Line 211: @@
 $\text{Summary of Nyquist criteria:}$<br><br>
 '''(1)''' &nbsp; In memory of the physicist &nbsp;[https://en.wikipedia.org/wiki/Harry_Nyquist Harry Nyquist]&nbsp;  we denote a basic detection pulse &nbsp;$g_d( t)$&nbsp; with the properties
-:$$g_d ( t= 0) \ne  0, \hspace{1cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
+::$$g_d ( t= 0) \ne  0, \hspace{1cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
 t = \pm T, \pm 2T,\pm 3T,\hspace{0.05cm}\text{...}$$
 :as the&nbsp; '''Nyquist&ndash;1 pulse''' &nbsp;$g_{\hspace{0.05cm}\rm Nyq-1}(t)$.&nbsp; This satisfies the first Nyquist criterion and leads to the maximum vertical eye opening.
 '''(2)''' &nbsp; A basic detection pulse &nbsp;$g_d( t)$&nbsp; satisfying the second Nyquist criterion must have zero crossings at &nbsp;$t = \pm 1.5T$, &nbsp;$t = \pm 2.5T$, ...&nbsp; have:
-:$$g_d ( t= 0.5) \ne  0, \hspace{0.8cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
+::$$g_d ( t= 0.5) \ne  0, \hspace{0.8cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
 t = \pm 1.5T, \ \pm 2.5T,\ \pm 3.5T,\hspace{0.05cm}\text{...}$$
 :Such a&nbsp; '''Nyquist&ndash;2 pulse'''&nbsp; leads to the maximum horizontal eye opening.
 '''(3)''' &nbsp; A Nyquist&ndash;2 pulse can always be represented as the sum of two Nyquist&ndash;1 pulses shifted by &nbsp;$t = \pm T/2$:&nbsp;
-:$$g_{\rm Nyq-2} ( t )= g_{\rm Nyq-1} ( t +T/2)+g_{\rm Nyq-1} ( t -T/2)\hspace{0.05cm}.$$
+::$$g_{\rm Nyq-2} ( t )= g_{\rm Nyq-1} ( t +T/2)+g_{\rm Nyq-1} ( t -T/2)\hspace{0.05cm}.$$
 '''(4)''' &nbsp; In the frequency domain,&nbsp; the second Nyquist criterion&nbsp; (see [TS87] <ref>Tröndle, K.; Söder, G.:&nbsp; Optimization of Digital Transmission Systems.&nbsp; Boston – London:&nbsp; Artech House, 1987.</ref>) is:
-:$$\sum_{k = -\infty}^{+\infty} \frac {G_d \left ( f -k/T
+::$$\sum_{k = -\infty}^{+\infty} \frac {G_d \left ( f -k/T
 \right)}{\cos(\pi \cdot f \cdot T - k \cdot \pi)}= {\rm const.}$$}}

@@ Line 140: / Line 140: @@
 :$$G_{\rm Nyq}(f_{\rm Nyq}+f) + G_{\rm Nyq}(f_{\rm Nyq}-f) = g_0 \cdot T \hspace{0.05cm}.$$
 *For all frequencies &nbsp;$|f| \le f_1$,&nbsp; the function&nbsp; $G_{\rm Nyq}(f)$&nbsp; is constantly equal to &nbsp;$g_0 \cdot T$&nbsp; and for &nbsp;$|f| \ge f_2$&nbsp; it is identically zero.
-[[File:EN_Dig_T_1_3_S3a.png|right|frame|$1/T$&nbsp; Nyquist spectra '''Korrektur''' cosine|class=fit]]
+[[File:EN_Dig_T_1_3_S3a_vers2.png|right|frame|$1/T$&nbsp; Nyquist spectra|class=fit]]
 *In the range between &nbsp;$f_1$&nbsp; and &nbsp;$f_2$&nbsp; holds:
 :$$\frac{G_{\rm Nyq}(f)}{g_0 \cdot T }  =   \left\{ \begin{array}{c} \frac{f_2 - |f|}{f_2 -f_1 }
@@ Line 174: / Line 174: @@
 The following upper diagram shows the Nyquist pulse with trapezoidal spectrum for different rolloff factors. Below is the corresponding time course for the cosine rolloff spectrum. One can see:
-[[File:EN_Dig_T_1_3_S3b.png|right|frame|Nyquist pulses with trapezoidal and cosine rolloff spectrum  '''Korrektur''' cosine|class=fit]]
+[[File:EN_Dig_T_1_3_S3b_v2.png|right|frame|Nyquist pulses with trapezoidal and cosine rolloff spectrum|class=fit]]
 #The smaller the rolloff factor &nbsp;$r$,&nbsp; the slower the decay of the Nyquist pulse. This statement is true for both the trapezoidal and the cosine rolloff spectra.
 #In the limiting case &nbsp;$r \to 0$,&nbsp; both cases yield the rectangular Nyquist spectrum and the &nbsp;$\rm sinc$&ndash;shaped Nyquist pulse, &nbsp; which decays asymptotically with &nbsp;$1/t$&nbsp; (thin green curves).
@@ Line 189: / Line 189: @@
 *below the corresponding&nbsp; "eye diagram"&nbsp; referring to the &nbsp;[[Digital_Signal_Transmission/Fehlerwahrscheinlichkeit_unter_Berücksichtigung_von_Impulsinterferenzen#Definition_und_Aussagen_des_Augendiagramms|third main chapter]].
-[[File:EN_Dig_T_1_3_S4.png|right|frame|Clarification of first and second Nyquist criterion]]
+[[File:EN_Dig_T_1_3_S4_v3.png|right|frame|Clarification of first and second Nyquist criterion]]