View source for Digital Signal Transmission/Properties of Nyquist Systems

 
{{Header
|Untermenü=Digital Signal Transmission under Idealized Conditions
|Vorherige Seite=Fehlerwahrscheinlichkeit bei Basisbandübertragung
|Nächste Seite=Optimierung der Basisbandübertragungssysteme
}}


== First Nyquist criterion in the time domain ==
<br>
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;$g_d(t)$
*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; 


For the sake of simplicity,&nbsp; the noise component of the detection signal is assumed to be negligibly small in the following &nbsp; &rArr; &nbsp; $d_{\rm N}(t) =0$.

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; One denotes a basic detection pulse '''KORREKTUR: basic transmitter pulse''' with the properties
:$$g_d ( t = \nu  T)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
\nu = \pm 1, \pm 2,\pm 3,\hspace{0.05cm}\text{...}$$
as&nbsp; '''Nyquist pulse''' &nbsp;$g_{\hspace{0.05cm}\rm Nyq}(t)$,&nbsp; named after the physicist &nbsp;[https://en.wikipedia.org/wiki/Harry_Nyquist "Harry Nyquist"].}} 


{{GraueBox|TEXT=
[[File:P_ID1272__Dig_T_1_3_S1_v1.png|right|frame|Detection signal with Nyquist pulse shaping]]  
$\text{Example 1:}$&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'''
*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.
'''Ende Check 1'''


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;$r = 0.5$:
:$$g_{\hspace{0.05cm}\rm Nyq} ( t )= g_0 \cdot {\rm sinc} \left ( \frac{t}{T}\right)\cdot {\rm sinc} \left ( \frac{ t}{2 \cdot
T}\right),$$
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".}}


== First Nyquist criterion in the frequency domain ==
<br>
[https://en.wikipedia.org/wiki/Harry_Nyquist "Harry Nyquist"]&nbsp; formulated the condition for detection without intersymbol interfering not only for the time domain,&nbsp; but in 1928 he also gave the corresponding criterion in the frequency domain.

{{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
:$$\sum \limits_{\it k = -\infty}^{+\infty} G_d \left ( f - {k}/{T} \right)=
g_0 \cdot T = {\rm const.} \hspace{0.05cm}, $$
then &nbsp;$g_d(t)$&nbsp; is a Nyquist pulse
*with equidistant zero crossings at the times &nbsp;$\nu \cdot T$&nbsp; for &nbsp;$\nu \ne 0$&nbsp; and 
*the amplitude &nbsp;$g_d(t = 0) = g_0$.  


$\text{Note}$:&nbsp; The proof follows on the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#Proof_of_the_first_Nyquist_criterion|"next page"]].}}


{{GraueBox|TEXT=  
$\text{Example 2:}$&nbsp; Sketched are two Nyquist spectra &nbsp;$G_1(f)$&nbsp; and &nbsp;$G_2(f)$, which are composed of rectangular and triangular segments:

The purely real spectrum sketched&nbsp; $G_1(f)$&nbsp; on the left satisfies the condition formulated above with the smallest possible bandwidth:
[[File:P_ID1273__Dig_T_1_3_S2_v1.png|right|frame|Illustration of the first Nyquist criterion|class=fit]]
:$$G_1(f)  \hspace{-0.05cm}=\hspace{-0.05cm}   \left\{ \begin{array}{c} g_0 \cdot T  \\
 0 \\  \end{array} \right.
\begin{array}{*{1}c} \text{for}
\\ \text{for}  \\ \end{array}\begin{array}{*{20}c}
\vert f \vert \hspace{-0.08cm}<\hspace{-0.08cm} {1}/(2T), \\
\vert f \vert \hspace{-0.08cm}>\hspace{-0.08cm} {1}/(2T),   \\
\end{array}$$
But the associated Nyquist pulse &nbsp;$g_1(t) = g_0 \cdot {\rm sinc}(t/T)$&nbsp;  decays very slowly, asymptotically with &nbsp;$1/t$.


The real part of the spectrum &nbsp;$G_2(f)$&nbsp; shown on the upper right was constructed from the rectangular spectrum by&nbsp;$G_1(f)$&nbsp; by shifting parts of them  by &nbsp;$1/T$&nbsp; to the right or to the left.&nbsp; 
*$G_2(f)$&nbsp; is also a Nyquist spectrum because of
:$$\sum \limits_{\it k = -\infty}^{+\infty} {\rm Re}\big[G_2 \left ( f -
{k}/{T} \right)\big]= g_0 \cdot T \hspace{0.05cm},$$
:$$\sum \limits_{\it k = -\infty}^{+\infty} {\rm Im}\left[G_2 \big ( f -
{k}/{T} \right)\big]= 0.$$
*In the imaginary part,&nbsp; the respective equally shaded parts,&nbsp; each &nbsp;$2/T$&nbsp; apart,&nbsp; cancels out.
* However,&nbsp; the specification of the corresponding Nyquist pulse &nbsp;$g_2(t)$&nbsp; is very complicated.}}


== Proof of the first Nyquist criterion ==
<br>
{{BlaueBox|TEXT=   
'''(1)''' &nbsp; We start from the first Nyquist condition in the time domain:
:$$g_{\hspace{0.05cm}\rm Nyq}(\nu
T)  =   \left\{ \begin{array}{c} g_0  \\
 0 \\  \end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for} }
\\  {\rm{for} }  \\ \end{array}\begin{array}{*{20}c}
\nu = 0 \hspace{0.05cm}, \\
\nu \ne 0  \hspace{0.1cm}.  \\
\end{array}$$
'''(2)''' &nbsp; Thus,&nbsp; from the second Fourier integral,&nbsp; we obtain for &nbsp;$\nu \ne 0$:
:$$g_{\hspace{0.05cm}\rm Nyq}(\nu
T)  =   \int_{-\infty}^{+\infty}G_{\rm Nyq}(f) \cdot {\rm
e}^{ \hspace{0.05cm}{\rm j} \hspace{0.05cm}2 \pi f \hspace{0.05cm}\nu
\hspace{0.05cm}T}\,{\rm d} f = 0 \hspace{0.05cm}.$$
'''(3)''' &nbsp; Decomposing the Fourier integral into partial integrals of width &nbsp;$1/T$,&nbsp; the conditional equations are:
:$$\sum_{k = -\infty}^{+\infty}  \hspace{0.2cm} \int_{(k-1/2)/T}^{(k+1/2)/T}G_{\rm Nyq}(f) \cdot {\rm
e}^{ \hspace{0.05cm}{\rm j} \hspace{0.05cm}2 \pi f \hspace{0.05cm}\nu
\hspace{0.05cm}T}\,{\rm d} f = 0 \hspace{0.05cm}.$$
'''(4)''' &nbsp; With the substitution &nbsp;$f\hspace{0.08cm}' = f + k/T$&nbsp; it follows:
:$$\sum_{k = -\infty}^{+\infty}  \hspace{0.2cm} \int_{-1/(2T)}^{1/(2T)}G_{\rm Nyq}(f\hspace{0.08cm}' -
\frac{k}{T} ) \cdot {\rm e}^{\hspace{0.05cm}{\rm j}
\hspace{0.05cm}2 \pi \hspace{0.05cm} \cdot \hspace{0.05cm} (f\hspace{0.08cm}'-
k/T) \hspace{0.05cm} \cdot \hspace{0.05cm}\nu
\hspace{0.05cm}T}\,{\rm d} f \hspace{0.08cm}' = 0 \hspace{0.05cm}.$$
'''(5)''' &nbsp; For all integer values of &nbsp;$k$&nbsp; and &nbsp;$\nu$&nbsp; holds:
:$${\rm e}^{ - {\rm j} \hspace{0.05cm}2 \pi
\hspace{0.05cm} k \hspace{0.05cm} \nu } = 1
\hspace{0.4cm} \Rightarrow \hspace{0.4cm}\sum_{k = - \infty}^{+\infty}  \hspace{0.2cm} \int_{- 1/(2T)}^{1/(2T)}G_{\rm Nyq}(f\hspace{0.08cm}'  - 
\frac{k}{T} ) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}2 \pi
\hspace{0.02cm}f\hspace{0.08cm}' \hspace{0.02cm} \nu \hspace{0.05cm}T}\,{\rm d} f
\hspace{0.08cm}' = 0 \hspace{0.05cm}.$$
'''(6)''' &nbsp; Swapping summation and integration and renaming &nbsp;$f\hspace{0.08cm}'$&nbsp;  to  &nbsp;$f$,&nbsp; it further follows:
:$$\int_{-1/(2T)}^{1/(2T)}\hspace{0.2cm} \sum_{k = - \infty}^{+\infty} G_{\rm Nyq}(f -
\frac{k}{T} ) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}2 \pi
\hspace{0.02cm}f \hspace{0.02cm} \nu \hspace{0.05cm}T}\,{\rm d} f
 = 0 \hspace{0.05cm}.$$
'''(7)''' &nbsp; This requirement can be satisfied for all &nbsp;$\nu \ne 0$&nbsp; only if the infinite sum is independent of &nbsp;$f$,&nbsp; i.e.,&nbsp; has a constant value:
:$$\sum_{k = -\infty}^{+\infty} G_{\rm Nyq}(f -
\frac{k}{T} ) =  K_{\rm Nyq} \hspace{0.05cm}.$$
'''(8)''' &nbsp; From the penultimate equation,&nbsp; we obtain simultaneously for &nbsp;$\nu = 0$:
:$$\int_{-1/(2T)}^{1/(2T)}\hspace{0.2cm} \sum_{k = -\infty}^{+\infty} G_{\rm Nyq}(f -
\frac{k}{T} ) \,{\rm d} f
 = K_{\rm Nyq} \cdot \frac{1}{T} = g_0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}K_{\rm Nyq}  = g_0 \cdot T \hspace{0.05cm}.$$
<div align="right">'''q.e.d.'''</div>}}


== 1/T Nyquist spectra==
<br>
Nyquist spectra which are limited to the frequency range &nbsp;$-1/T \le  f \le +1/T$&nbsp; and are coherent,&nbsp; have a particular importance for digital signal transmission.

The diagram shows the&nbsp; "trapezoidal characteristic"&nbsp; and the&nbsp; "cosine rolloff characteristic",&nbsp; which are two variants in this respect. 
The same applies to both Nyquist spectra:
*The rolloff occurs between the two corner frequencies &nbsp;$f_1$&nbsp; and &nbsp;$f_2$&nbsp; point-symmetrically about the&nbsp; '''Nyquist frequency''' &nbsp;$f_{\rm Nyq} = (f_1+f_2)/2$.&nbsp; <br>That is,&nbsp; for &nbsp;$0 \le f \le f_{\rm Nyq}$:&nbsp;
:$$G_{\rm Nyq}(f_{\rm Nyq}+f) + G_{\rm Nyq}(f_{\rm Nyq}-f) = g_0 \cdot T \hspace{0.05cm}.$$
*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_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:
:$$\frac{G_{\rm Nyq}(f)}{g_0 \cdot T }  =   \left\{ \begin{array}{c} \frac{f_2 - |f|}{f_2 -f_1 }
 \\ \\
 \cos^2( \frac{\pi}{2}\cdot \frac{f_2 - |f|}{f_2 -f_1 }) \\  \end{array} \right.\quad
\begin{array}{*{1}c} {\rm{at \hspace{0.15cm}trapezoidal}\hspace{0.05cm},}
\\ \\ {\rm{\rm{at \hspace{0.15cm}cosine \hspace{0.15cm}rolloff}}\hspace{0.05cm}.}  \\ \end{array}$$
*To parameterize the slope,&nbsp;  we use the&nbsp; '''rolloff factor''' &nbsp;$r$,&nbsp; which can take values between &nbsp;$0$&nbsp; and &nbsp;$1$&nbsp; (including these limits):
:$$r = \frac{f_2 -f_1 }
{f_2 +f_1 } \hspace{0.05cm}.$$
:In the literature,&nbsp; the rolloff factor is sometimes referred to as &nbsp;$\alpha$&nbsp; ("alpha").

*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 
#resp.&nbsp; the cosine-square spectrum,&nbsp; 

:depending on which of the two basic structures shown above one assumes.&nbsp; These frequency curves are shown in red dashed lines.




==Time domain description of the  1/T Nyquist spectra ==
<br>
Let us now consider the Nyquist pulses.&nbsp; For the &nbsp;'''trapezoidal spectrum'''&nbsp; with rolloff factor &nbsp;$r$,&nbsp; we obtain:
:$$g_{_{\rm trapezoid}} ( t )= g_0 \cdot {\rm sinc} \left ( \frac{t}{T}\right)\cdot {\rm sinc} \left ( \frac{r \cdot
t}{T}\right) \hspace{0.5cm}{\rm with }\hspace{0.5cm}{\rm sinc}(x) = {\rm sin}(\pi x)/(\pi x)  .$$
In contrast,&nbsp; the Fourier inverse transform of the&nbsp; '''cosine rolloff spectrum'''&nbsp; (short:&nbsp; "CRO spectrum") yields:
:$$g_{_{\rm CRO}} ( t )= g_0 \cdot {\rm sinc} \left ( \frac{
t}{T}\right)\cdot \frac{\cos(\pi \cdot r \cdot t/T)}{1 - (2 \cdot
r \cdot t/T)^2 } \hspace{0.3cm}{\rm with }\hspace{0.3cm}{\rm sinc}(x) = {\rm sin}(\pi x)/(\pi x).$$
These 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:
[[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).
#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 &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.


== Second Nyquist criterion==
<br>
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/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]]


$\text{Interpretation:}$
*The diagram on the left shows the eye diagram of a Nyquist system with cosine rolloff characteristic,&nbsp; where the rolloff factor &nbsp;$r= 0.5$&nbsp; was chosen.&nbsp; Since the first Nyquist criterion is fulfilled here&nbsp; (there is point symmetry around the Nyquist frequency &nbsp;$f_{\rm Nyq}$,&nbsp; the  vertical eye opening at time &nbsp;$t = 0$&nbsp; is &nbsp;$2 \cdot g_d(0)$.&nbsp; All eye lines pass  at time &nbsp;$t = 0$&nbsp; through one of the two points marked in red &nbsp; &rArr;  &nbsp; the eye is vertically maximally open.


*The middle spectrum does not show any symmetry with respect to the rolloff,&nbsp; so that the first Nyquist criterion is not fulfilled here &ndash; in contrast to the second one.&nbsp; All eyelines here intersect the time axis at the same times&nbsp; (marked by the green dots),&nbsp; which facilitates,&nbsp; for example,&nbsp; clock recovery by means of a&nbsp; $\rm PLL$&nbsp; ("Phase-Locked Loop").&nbsp; When the second Nyquist criterion is met,&nbsp; the horizontal eye opening is maximally equal to the symbol duration $T$ &nbsp; &rArr;  &nbsp; the eye is maximally open horizontally.


*The right eye diagram illustrates that for the CRO spectrum with &nbsp;$r = 1$,&nbsp; both the first and second Nyquist criteria are satisfied.&nbsp; Here,&nbsp; the Nyquist pulse
:$$g_d ( t )= g_0 \cdot \frac{\pi }{4}\cdot {\rm sinc} \left (
\frac{ t}{T}\right)\cdot \left [ {\rm sinc}
(\frac{t}{T} + \frac{1}{2}) + {\rm sinc} (\frac{t}{T} -
\frac{1}{2})\right]$$
: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'''.



{{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
::$$g_d ( t= 0) \ne  0, \hspace{1cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
t = \pm T, \pm 2T,\pm 3T,\hspace{0.05cm}\text{...}$$
: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:
::$$g_d ( t= 0.5) \ne  0, \hspace{0.8cm} g_d ( t)= 0 \hspace{0.3cm}{\rm{for} }\hspace{0.3cm}
t = \pm 1.5T, \ \pm 2.5T,\ \pm 3.5T,\hspace{0.05cm}\text{...}$$
: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; 
::$$g_{\rm Nyq-2} ( t )= g_{\rm Nyq-1} ( t +T/2)+g_{\rm Nyq-1} ( t -T/2)\hspace{0.05cm}.$$
'''(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:
::$$\sum_{k = -\infty}^{+\infty} \frac {G_d \left ( f -k/T
\right)}{\cos(\pi \cdot f \cdot T - k \cdot \pi)}= {\rm const.}$$}}


{{GraueBox|TEXT=  
$\text{Example 3:}$&nbsp; Starting from the Nyquist&ndash;1 pulse &nbsp;$g_{\rm Nyq-1}( t )= g_0 \cdot {\rm sinc}(t/T)$&nbsp; the corresponding Nyquist&ndash;2 pulse is:
:$$g_{\rm Nyq-2}( t ) = g_0 \cdot  \left [ {\rm sinc}( \frac{t + T/2}{T})  + {\rm sinc}( \frac{t- T/2}{T}) \right] =\frac{2 \cdot g_0}{\pi} \cdot \frac{\cos(\pi \cdot t/T)}{1 - (2 \cdot t/T)^2}\hspace{0.05cm}.$$
*Due to the limitation of the spectrum &nbsp;$G_{\rm Nyq-1}( f)$&nbsp; to the range &nbsp;$\vert f \vert \le f_{\rm Nyq} = 1/(2T)$,&nbsp; the sum in equation&nbsp; '''(4)'''&nbsp; is limited to the term with &nbsp;$k = 0$,&nbsp; and we obtain:
:$$G_{\rm Nyq-2}(f)  =   \left\{ \begin{array}{c} g_0 \cdot T  \cdot \cos(\pi/2 \cdot f/f_{\rm Nyq}) \\
  \\ 0 \\\end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for} }\hspace{0.15cm} \vert f \vert < f_{\rm
Nyq}\hspace{0.05cm},
\\    \\ {\rm{otherwise} }\hspace{0.05cm}. \\
\end{array}$$
*This frequency response and the corresponding eye diagram is sketched in the middle column of the above diagram.
*From the bottom diagram, one can clearly see the fulfillment of the second Nyquist criterion.}}



== Exercises for the chapter ==
<br>
[[Aufgaben:Exercise_1.4:_Nyquist_Criteria|Exercise 1.4: Nyquist Criteria]]

[[Aufgaben:Exercise_1.4Z:_Complex_Nyquist_Spectrum|Exercise 1.4Z: Complex Nyquist Spectrum]]

[[Aufgaben:Exercise_1.5:_Cosine-Square_Spectrum|Exercise 1.5: Cosine-Square Spectrum]]

==References==

<references/>


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