View source for Digital Signal Transmission/Error Probability with Intersymbol Interference

 
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== Gaussian receiver filter==
<br>
We start from the sketched block diagram. The following configuration is assumed for quantitative consideration of the &nbsp;[[Digital_Signal_Transmission/Ursachen_und_Auswirkungen_von_Impulsinterferenzen#Definition_des_Begriffs_.E2.80.9EImpulsinterferenz.E2.80.9D|intersymbol interference]]:&nbsp; 
*Rectangular NRZ basic transmission pulse &nbsp;$g_s(t)$&nbsp; with height &nbsp;$s_0$&nbsp; and duration &nbsp;$T$,<br>
*Gaussian-shaped receiver filter with cutoff frequency &nbsp;$f_{\rm G}$ &nbsp;(''Note:'' &nbsp; In this section, we often also denote the exponential function by &nbsp;$\rm exp [ . ]$):
:$$H_{\rm E}(f) = H_{\rm G}(f) = {\rm exp}\left [-   \frac{\pi  \cdot f^2}{(2f_{\rm G})^2} \right ] \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ
 \hspace{0.2cm}h_{\rm E}(t) = h_{\rm G}(t) = {\rm exp}\left [- \pi  \cdot (2
 f_{\rm G} t)^2\right ]
  \hspace{0.05cm}.$$
*AWGN channel, that is, $H_{\rm K}(f) = 1 $ and ${\it \Phi}_n(f) = N_0/2$.


[[File:Dig_T_3_2_S1_version2.png|center|frame|Block diagram for the chapter "Error probability for intersymbol interference"]]

Based on the assumptions made here, the following holds for the basic detection pulse:

:$$g_d(t) = g_s(t) \star h_{\rm G}(t) = 2 f_{\rm G} \cdot s_0 \cdot \int_{t-T/2}^{t+T/2}
{\rm e}^{- \pi  \hspace{0.05cm}\cdot\hspace{0.05cm} (2 \hspace{0.05cm}\cdot\hspace{0.05cm}
 f_{\rm G}\hspace{0.05cm}\cdot\hspace{0.05cm} \tau )^2} \,{\rm d} \tau \hspace{0.05cm}.$$

The integration leads to the following equivalent results:

:$$g_d(t) =  s_0 \cdot \big [ {\rm Q} \left (  2 \cdot \sqrt {2 \pi}
\cdot f_{\rm G}\cdot  ( t - {T}/{2})\right )-  {\rm Q} \left (
2 \cdot \sqrt {2 \pi} \cdot f_{\rm G}\cdot ( t + {T}/{2}
)\right ) \big ],$$
:$$g_d(t) =  s_0 \cdot\big [ {\rm erfc} \left (  2 \cdot
\sqrt {\pi} \cdot f_{\rm G}\cdot  ( t - {T}/{2})\right )- {\rm
erfc} \left (  2 \cdot \sqrt {\pi} \cdot f_{\rm G}\cdot ( t +
{T}/{2} )\right ) \big ]\hspace{0.05cm}.$$

Here, two variants of the complementary Gaussian error function are used, viz.

:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it
x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d {\it u}
\hspace{0.05cm},\hspace{0.5cm}
 {\rm erfc} (\it x) = \frac{\rm 2}{\sqrt{\rm
\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}}\,d \it u
\hspace{0.05cm}.$$

The module &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Complementary Gaussian Error Functions]]&nbsp; provides the numerical values of &nbsp;${\rm Q} (x)$&nbsp; and &nbsp;$0.5 \cdot {\rm erfc} (x)$.<br>


The noise power at the output of the Gaussian receiver filter &nbsp;$H_{\rm G}(f)$&nbsp; is equal to
:$$\sigma_d^2 = \frac{N_0}{2} \cdot \int_{-\infty}^{+\infty}
|H_{\rm G}(f)|^2 \,{\rm d} f = \frac{N_0\cdot f_{\rm
G}}{\sqrt{2}}\hspace{0.05cm}.$$

From these two equations one can already see:
*The smaller the cutoff frequency &nbsp;$f_{\rm G}$&nbsp; of the Gaussian low-pass filter, the smaller the noise rms value &nbsp;$\sigma_d$&nbsp; and consequently the better the noise performance.<br>

*However, a small cutoff frequency leads to a strong deviation of the basic detection pulse &nbsp;$g_d(t)$&nbsp; from the square wave form and thus to intersymbol interference.<br><br>

{{GraueBox|TEXT=  
$\text{Example 1:}$&nbsp; The left graph shows the basic detection pulse &nbsp;$g_d(t)$&nbsp; at the output of a Gaussian low-pass filter &nbsp;$H_{\rm G}(f)$&nbsp; with the cutoff frequency &nbsp;$f_{\rm G}$ when an NRZ rectangular pulse (blue curve) is applied at the input.<br>

[[File:P_ID1373__Dig_T_3_2_S1b_version1.png|center|frame|Basic pulse and noise power density with Gaussian receiver filter]]
One can see from this plot:
*The Gaussian low-pass filter &nbsp;$H_{\rm G}(f)$&nbsp; causes the detection pulse &nbsp;$g_d(t)$&nbsp; to be reduced and broadened compared to the transmitted pulse &nbsp;$g_s(t)$&nbsp; &rArr; &nbsp; ''time dispersion''.<br>

*This pulse deformation is the stronger, the smaller the cutoff frequency &nbsp;$f_{\rm G}$&nbsp; is. For example, with &nbsp;$f_{\rm G} \cdot T = 0.4$&nbsp; (red curve) the pulse maximum is already reduced to about &nbsp;$68\%$.&nbsp;<br>

*In the limiting case &nbsp;$f_{\rm G} \to \infty$&nbsp; the Gaussian low-pass has no effect &nbsp; &#8658; &nbsp; $g_d(t) = g_s(t)$. However, in this case, no noise limitation is effective at all, as can be seen from the right figure.}}


{{GraueBox|TEXT=  
$\text{Example 2:}$&nbsp; The same preconditions apply as for the last example. The graph shows the detection signal &nbsp;$d(t)$&nbsp; after the Gaussian low-pass (before the decision) for two different cutoff frequencies, namely &nbsp;$f_{\rm G} \cdot T = 0.8$&nbsp; and &nbsp;$f_{\rm G} \cdot T = 0.4$

[[File:P ID1384 Dig T 3 2 S1c version2.png|center|frame|Detection signal with Gaussian receiver filter]] 
Shown equally in both diagrams (but admittedly difficult to see as a screen capture) are:
*the component &nbsp;$d_{\rm S}(\nu \cdot T)$&nbsp; without considering the noise (blue circles at the detection times),<br>
*the total detection signal &nbsp;$d(t)$&nbsp; including the noise component (yellow),<br>
*the transmitted signal &nbsp;$s(t)$&nbsp; as reference signal (green dotted in the upper graph; equally valid for the lower graph).<br><br>

By comparing these images, the following statements can be verified:
*With the cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.8$&nbsp; (upper graph), only minorintersymbol interferences result at the detection times $($at multiples of &nbsp;$T)$.&nbsp; Due to the Gaussian low-pass here primarily the corners of the transmitted signal &nbsp;$s(t)$&nbsp; are rounded.<br>
*In contrast, in the lower image &nbsp;$(f_{\rm G} \cdot T = 0.4)$&nbsp; the effects of the intersymbol interferences are clearly visible. At the detection times &nbsp;$(\nu \cdot T)$,&nbsp; the signal component of the detection signal &nbsp;$d_{\rm S}(\nu \cdot T)$&nbsp; shown in blue can assume six different values (grid lines drawn).<br>
*The noise component &nbsp;$d_{\rm N}(t)$ &ndash; recognizable as the difference between the yellow curve and the blue circles &ndash; is statistically larger on average with &nbsp;$f_{\rm G} \cdot T = 0.8$&nbsp; than with &nbsp;$f_{\rm G} \cdot T = 0.4$.<br>
*This result can be explained by the right graph of &nbsp;$\text{example 1}$,&nbsp; which shows the power-spectral density of the noise component &nbsp;$d_{\rm N}(t)$:&nbsp;
:$${\it \Phi}_{d{\rm N} }(f) = {N_0}/{2} \cdot \vert H_{\rm G}(f) \vert^2 =
{N_0}/{2} \cdot {\rm exp}\left [- 
\frac{2\pi   f^2}{(2f_{\rm G})^2} \right ] .$$

*The integral over &nbsp;${\it \Phi}_{d{\rm N} }(f)$&nbsp; &ndash; i.e. the noise power &nbsp;$\sigma_d^2$&nbsp; &ndash; is twice as large for &nbsp;$f_{\rm G} \cdot T = 0.8$&nbsp; (purple curve) than with &nbsp;$f_{\rm G} \cdot T = 0.4$&nbsp; (red curve).}}


== Definition and statements of the eye diagram==
<br>
The above mentioned facts can also be explained by the eye diagram. 

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; The &nbsp;'''eye diagram'''&nbsp; (or <i>eye pattern</i>)&nbsp; is the sum of all superimposed sections of the detection signal, whose duration is an integer multiple of the symbol duration &nbsp;$T$.&nbsp;}}


{{GraueBox|TEXT=  
$\text{Example 3:}$&nbsp; We assume a redundancy-free binary bipolar NRZ rectangular signal &nbsp;$s(t)$&nbsp; and the Gaussian low-pass filter with cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.4$.&nbsp; 
[[File:P ID1375 Dig T 3 2 S2 version1.png|right|frame|Eye diagrams with and without noise|class=fit]]
<br>Shown are the eye diagrams after the Gaussian low-pass,
*left with noise &nbsp; &#8658; &nbsp; signal &nbsp;$d(t)$, 
*on the right without taking noise into account &nbsp; &#8658; &nbsp; signal &nbsp;$d_{\rm S}(t)$.


This diagram has a certain resemblance to an eye, which led to its naming.
<br clear=all>
This representation allows important statements about the quality of a digital transmission system:

*Only the eye diagram of the signal &nbsp;$d(t)$&nbsp; can be displayed metrologically on an oscilloscope, which is triggered with the clock signal. From this eye diagram (left graph), for example, the noise rms value &nbsp;$\sigma_d$&nbsp; can be read &ndash; or rather: &nbsp;estimated &ndash; werden.<br>

*The eye diagram without noise (right graph) refers to the signal component of the detection signal &nbsp;$d_{\rm S}(t)$&nbsp; and can only be determined by means of a computer simulation. For an implemented system, this eye diagram cannot be displayed, since the noise component &nbsp;$d_{\rm N}(t)$&nbsp; cannot be eliminated.<br>

*In both diagrams, &nbsp;$2048$&nbsp; eye lines were drawn in each case. In the right graph, however, only &nbsp;$2^5 = 32$&nbsp; eye lines are distinguishable because the present basic transmitter pulse &nbsp;$g_d(t)$&nbsp; is limited to the time range &nbsp;$\vert t\vert \le  2T$&nbsp; (see &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Gaussian_receiver_filter|graph in example 1]]&nbsp; with &nbsp;$f_{\rm G} \cdot T = 0.4$, red curve).<br>

*The inner eye lines determine the '''vertical eye opening''' &nbsp;$\ddot{o}(T_{\rm D})$. The smaller this is, the greater is the influence of intersymbol interference. For a (intersymbol interference free) Nyquist system the vertical eye opening is maximum. Normalized to the transmitted amplitude, &nbsp;$\ddot{o}(T_{\rm D})/s_0 = 2$ is then valid. <br>

*With symmetrical basic pulse, the detection time &nbsp;$T_{\rm D} = 0$&nbsp; is optimal. With a different value $($for example  &nbsp;$T_{\rm D} = T/10) $,&nbsp; &nbsp;$\ddot{o}(T_{\rm D})$&nbsp; would be somewhat smaller and thus the error probability would be significantly larger. This case is indicated by the purple&ndash;dashed vertical line in the right graph.}}

== Mean error probability==
<br>
As with the previous graphs in this chapter, we assume the following:
*NRZ rectangles with amplitude &nbsp;$s_0$, &nbsp;AWGN noise with &nbsp;$N_0$, where
:$$10 \cdot {\rm lg}\hspace{0.1cm} \frac{s_0^2 \cdot T}{N_0}\approx
13\,{\rm dB}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
\frac{N_0}{s_0^2 \cdot T} = 0.05\hspace{0.05cm}.$$
*Gaussian receiver filter with cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.4$:
:$$\sigma_d^2 = \frac{(N_0 /T)\cdot (f_{\rm G}\cdot T)}{\sqrt{2}}= \frac{0.05 \cdot
s_0^2\cdot0.4}{\sqrt{2}} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \sigma_d = \sqrt{0.0141}\cdot s_0
\approx 0.119 \cdot s_0 \hspace{0.05cm}.$$
* Let &nbsp;$g_d(\nu \cdot T) \approx 0$&nbsp; be valid for &nbsp;$|\nu| \ge 2$. The other basic detection pulse values are given as follows:
:$$g_0  =  g_d(t=0) \approx 0.68 \cdot s_0,  \hspace{0.5cm}g_1  =  g_d(t=T) \approx 0.16 \cdot s_0,  \hspace{0.2cm} g_{-1} = g_d(t=-T) \approx
0.16 \cdot s_0\hspace{0.05cm}.$$

[[File:P ID1377 Dig T 3 2 S3 version1.png|center|frame|Eye diagram and PDF of the signal component|class=fit]]

Let us now analyze the possible values for the signal component of the detection signal &nbsp;$d_{\rm S}(t)$&nbsp; at the detection times:
*Of the total &nbsp;$32$&nbsp; eye lines, four lines intersect the ordinate &nbsp;$(t = 0)$&nbsp; at &nbsp;$g_0 + 2 \cdot g_1 = s_0$. These lines belong to the amplitude coefficients "$\text{...}\hspace{0.05cm} +\hspace{-0.1cm}1,\hspace{0.05cm} {\it +\hspace{-0.05cm}1},\hspace{0.05cm} +\hspace{-0.05cm}1\hspace{0.05cm} \text{...}$". &nbsp; Here, the "middle" coefficient &nbsp;$a_{\nu = 0}$ is highlighted in italics.<br>
*The four eye lines, each representing the coefficients &nbsp;"$\text{...}\hspace{0.05cm} -\hspace{-0.1cm}1,\hspace{0.05cm} {\it +\hspace{-0.05cm}1},\hspace{0.05cm} -\hspace{-0.05cm}1,\hspace{0.05cm} \text{...}$"&nbsp; result in the useful sample &nbsp;$d_{\rm S}(T_{\rm D} = 0) =g_0 - 2 \cdot g_1 = 0.36 \cdot s_0$.<br>
*In contrast, the useful sample &nbsp;$d_{\rm S}(T_{\rm D} = 0) =g_0 = 0.68 \cdot s_0$&nbsp; occurs twice as often. This goes back either to the coefficients &nbsp;"$\text{...}\hspace{0.05cm} +\hspace{-0.1cm}1,\hspace{0.05cm} {\it +\hspace{-0.05cm}1},\hspace{0.05cm} -\hspace{-0.05cm}1\hspace{0.05cm} \text{...}$"&nbsp; or to  &nbsp;"$\text{...}\hspace{0.05cm} -\hspace{-0.1cm}1,\hspace{0.05cm} {\it +\hspace{-0.05cm}1},\hspace{0.05cm} +\hspace{-0.05cm}1\hspace{0.05cm} \text{...}$".&nbsp; <br>
*For the &nbsp;$16$&nbsp; eye lines which intersect the ordinate &nbsp;$T_{\rm D} = 0$&nbsp; below the decision threshold &nbsp;$E = 0$,&nbsp; exactly mirror-image relations result.<br><br>

The possible values &nbsp;$d_{\rm S}(T_{\rm D})$&nbsp; and their occurrence probabilities can be found in the above graph on the left side in the &nbsp;[[Theory_of_Stochastic_Signals/Probability_Density_Function#PDF_definition_for_discrete_random_variables|probability density function]]&nbsp; (PDF) of the detection useful samples:
:$$f_{d{\rm S}}(d_{\rm S})  =  {1}/{8} \cdot \delta (d_{\rm S}
- s_0)+ {1}/{4} \cdot \delta (d_{\rm S} - 0.68 \cdot s_0)+
{1}/{8} \cdot \delta (d_{\rm S} - 0.36 \cdot s_0)+ $$
$$\hspace{2.15cm} + \hspace{0.2cm} {1}/{8} \cdot \delta (d_{\rm S} + s_0)+{1}/{4}
\cdot \delta (d_{\rm S} + 0.68 \cdot s_0)+{1}/{8} \cdot
\delta (d_{\rm S} + 0.36 \cdot s_0)\hspace{0.05cm}.$$

Thus, the (average) symbol error probability of the intersymbol interference free system can be given. Taking advantage of the symmetry, one obtains with &nbsp;$\sigma_d/s_0 = 0.119$:
:$$p_{\rm S}  =  {1}/{4} \cdot {\rm Q} \left( \frac{s_0}{ \sigma_d}
  \right)+ {1}/{2} \cdot {\rm Q} \left( \frac{0.68 \cdot s_0}{ \sigma_d}
  \right)+{1}/{4} \cdot {\rm Q} \left( \frac{0.36 \cdot s_0}{ \sigma_d}
  \right)$$
:$$\Rightarrow \hspace{0.3cm}p_{\rm S}   \approx  {1}/{4} \cdot {\rm Q}(8.40) +{1}/{2}  \cdot {\rm Q}(5.71)+ {1}/{4}  \cdot {\rm
  Q}(3.02)\approx
 {1}/{4}  \cdot 2.20 \cdot 10^{-17}+ {1}/{2}  \cdot 1.65 \cdot
  10^{-9}+ {1}/{4}  \cdot 1.26 \cdot 10^{-3} \approx 3.14 \cdot 10^{-4}
 \hspace{0.05cm}.$$

{{BlaueBox|TEXT=  
$\text{Conclusion:}$&nbsp; On the basis of this numerical example one recognizes:
*In the presence of intersymbol interference, the (average) symbol error probability &nbsp;$p_{\rm S}$&nbsp; is essentially determined by the inner eye lines.<br>

*The computational cost of determining the error probability &nbsp;$p_{\rm S}$&nbsp; can become very large, especially if the intersymbol interference comes from very many basic pulse values &nbsp;$g_\nu$.&nbsp; }}


{{GraueBox|TEXT=  
$\text{Example 4:}$&nbsp; 
*If the basic pulse values &nbsp;$g_{-5}, \text{...} \ , g_{+5}$&nbsp; are different from zero and &nbsp;$E \ne  0$, an averaging over &nbsp;$p_{\rm S}$&nbsp; eye lines is necessary to determine the error probability &nbsp;$2^{11} = 2048$.&nbsp; 
*If, on the other hand, only the basic pulse values &nbsp;$g_{-1}, \ g_0, \ g_{+1}$&nbsp; are different from zero and, in addition, the symmetry with respect to the threshold &nbsp;$E = 0$&nbsp; is taken into account, the effort is reduced to averaging over four terms.
* If, in addition, the symmetry &nbsp;$g_{-1} = g_{+1}$&nbsp; applies as with the above numerical values, then the symmetry with respect to &nbsp;$T_{\rm D}$&nbsp; can also be exploited and averaging over three terms is sufficient. }} <br>

== Worst-case error probability==
<br>
In the past, a variety of approximations for the average error probability have been given, among others:

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; As a very simple approximation for the actual error probability &nbsp;$p_{\rm S}$,&nbsp;
the &nbsp;'''worst-case error probability'''&nbsp; 
[[File:P ID1379 Dig T 3 2 S4 version1.png|right|frame|Relationship between mean and worst-case error probability|class=fit]]
:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}
  \right) \hspace{0.05cm}$$ is often used.

For their calculation, the worst-case symbol sequences are always assumed. This means:

*The actual PDF of the useful samples (left graph: &nbsp;six red Diracs) is replaced by a simplified PDF with only the inner Dirac delta functions (right graph: &nbsp;two green Diracs).<br>

*For the half vertical eye opening, with the basic pulse values &nbsp;$g_\nu = g_d( T_{\rm D}+ \nu \cdot T)$&nbsp; generally holds:
:$$\ddot{o}(T_{\rm D})/{ 2}= g_0 - \sum_{\nu = 1}^{n} \vert g_{\nu} \vert- \sum_{\nu = 1}^{v} \vert g_{-\nu} \vert   \hspace{0.05cm}.$$}}


This equation can be interpreted as follows:
*$g_0 = g_d( T_{\rm D})$&nbsp; is the so-called ''main value''&nbsp; of the basic pulse. For Nyquist systems &nbsp;$\ddot{o}(T_{\rm D})/{ 2}= g_0$ is always valid. In the following (mostly) &nbsp;$T_{\rm D}= 0$&nbsp; is set.<br>
*The first sum describes the intersymbol interferences of the &nbsp;$n$&nbsp; ''trailing pulses''&nbsp; of preceding pulses. Tacitly assumed is &nbsp;$g_\nu = 0$&nbsp; for &nbsp;$\nu  \gt n$.<br>
*The second sum considers the influence of the &nbsp;$v$&nbsp; ''leading pulses''&nbsp; of following pulses under the condition &nbsp;$g_{-\nu} = 0$&nbsp; for &nbsp;$\nu  \gt v$.<br>
*If all leading and trailing pulses are positive, the two worst-case symbol sequences are &nbsp;"$\text{...}\hspace{0.05cm} -\hspace{-0.1cm}1,\hspace{0.05cm} -\hspace{-0.05cm}1,\hspace{0.05cm} {\it +\hspace{-0.05cm}1},\hspace{0.05cm} -\hspace{-0.05cm}1,\hspace{0.05cm} -\hspace{-0.05cm}1\hspace{0.05cm} \text{...}$"&nbsp; and &nbsp;"$\text{...}\hspace{0.05cm} +\hspace{-0.1cm}1,\hspace{0.05cm} +\hspace{-0.05cm}1,\hspace{0.05cm} {\it -\hspace{-0.05cm}1},\hspace{0.05cm} +\hspace{-0.05cm}1,\hspace{0.05cm} +\hspace{-0.05cm}1\hspace{0.05cm} \text{...}$"&nbsp; (the coefficient &nbsp;$a_{\nu = 0}$&nbsp; is in italics in each case). These specifications apply, for example, to the Gaussian receiver filter considered here.<br>
*If some basic pulse values are negative, this is taken into account in the above equation by the magnitude formation. This results in other "worst&ndash;case" sequences than those just mentioned.<br>


{{GraueBox|TEXT=  
$\text{Example 5:}$&nbsp; The graph shows the error probabilities of the AWGN channel as a function of the (logarithmized) quotient &nbsp;$E_{\rm B}/N_0$, namely
*the average error probability &nbsp;$p_{\rm S}$&nbsp; with a Gaussian receive filter (blue circles),<br>
*the worst-case error probability &nbsp;$p_{\rm U}$&nbsp; with Gaussian receiver filter (blue rectangles),<br>
*the smallest possible error probability according to the section &nbsp;[[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Optimal_binary_receiver_.E2.80.93_.22Matched_Filter.22_realization| Optimal binary receiver]]&nbsp; (red curve).<br><br>

Here, the energy per bit is equal to &nbsp;$E_{\rm B} = s_0^2 \cdot T$&nbsp; (NRZ rectangular transmitted pulses).<br>

[[File:P ID1385 Dig T 3 2 S4b version2.png|center|frame|Mean and worst-case error probability vs. &nbsp;$E_{\rm B}/N-0$|class=fit]]<br>

The left graph is for the (normalized) cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.4$, the right one for a broader band receiver filter with &nbsp;$f_{\rm G} \cdot T = 0.8$. The results can be interpreted as follows:
*The worst-case error probability &nbsp;$p_{\rm U}$&nbsp; is always an upper bound for the actual symbol error probability &nbsp;$p_{\rm S}$. The smaller the influence of the intersymbol interference (large cutoff frequency), the closer &nbsp;$p_{\rm S}$&nbsp; and &nbsp;$p_{\rm U}$&nbsp; are to each other. For the optimal receiver &nbsp;$p_{\rm S} = p_{\rm U}.$
*For Gaussian receiver filter with &nbsp;$f_{\rm G} \cdot T \ge 0.3$,&nbsp; the intersymbol interferences are caused by the neighboring pulses alone &nbsp;$(g_2 = g_3 = \text{...} \approx 0)$, so that a lower bound can also be given for &nbsp;$p_{\rm S}$:&nbsp; 
:$${p_{\rm U} }/{ 4} \le p_{\rm S} \le p_{\rm U}
   \hspace{0.05cm}.$$
*The strong intersymbol interference of a Gaussian receiver filter with &nbsp;$f_{\rm G} \cdot T = 0.4$&nbsp; leads to the fact that compared to the optimal receiver a &nbsp;$6 \ \rm dB$ larger $E_{\rm B}/N_0$&nbsp; must be applied (four times the power), so that the error probability does not exceed the value &nbsp;$10^{-8}$.&nbsp; <br>
*However, the horizontal distance between the blue &nbsp;$p_{\rm S}$ curve (marked by circles) and the red comparison curve is not constant. At &nbsp;$p_{\rm S} = 10^{-2}$&nbsp; the distance is only &nbsp;$4 \ \rm dB$.<br>
*The right graph shows that with &nbsp;$f_{\rm G} \cdot T = 0.8$&nbsp; the distance to the comparison system is less than &nbsp;$1 \ \rm dB$.&nbsp; In the next section it is shown that with a Gaussian receiver filter the (normalized) cutoff frequency &nbsp;$f_{\rm G} \cdot T \approx 0.8$&nbsp; is the optimum.}}<br>

== Optimization of the cutoff frequency==
<br>
[[File:P ID1380 Dig T 3 2 S5 version1.png|right|frame|SNR as a function of the cutoff frequency of a Gaussian low-pass filter|class=fit]]
For system optimization and system comparison, it turns out to be convenient,
*instead of ''worst-case error probability'' &nbsp;$p_{\rm U}$ 
*to use the ''worst&ndash;case signal&ndash;to&ndash;noise power ratio''&nbsp;  (S/N ratio):
:$$\rho_{\rm U} = [\ddot{o}(T_{\rm D})]^2/ \sigma_d^2.$$ 

In the case of Gaussian perturbation, the following relationship exists:
:$$p_{\rm U} = {\rm Q} \left( \sqrt{\rho_{\rm U}}
  \right) \hspace{0.05cm}.$$

The mean symbol error probability &nbsp;$p_{\rm S}$&nbsp; can also be formally expressed by a S/N ratio via the Q function:

:$$\rho_d = \left[{\rm Q}^{-1} \left( p_{\rm S}
  \right)\right]^2 \hspace{0.05cm}.$$


The graph shows the two quantities &nbsp;$\rho_d$&nbsp; and &nbsp;$\rho_{\rm U}$&nbsp; in logarithmic form depending on the normalized cutoff frequency &nbsp;$f_{\rm G} \cdot T$&nbsp; of a Gaussian receiver filter, where &nbsp;$10 \cdot {\rm lg}\hspace{0.1cm} E_{\rm B}/N_0 = 13 \ \rm dB$&nbsp; is the basis.
*The circles outlined in blue are for &nbsp;$10 \cdot {\rm lg}\hspace{0.1cm} \rho_d$ &nbsp; &#8658; &nbsp; "mean" detection SNR,<br>
*The blue outlined squares mark &nbsp;$10 \cdot {\rm lg}\hspace{0.1cm} \rho_{\rm U}$ &nbsp; &#8658; &nbsp; "worst-case" SNR.


For comparison, the result for the &nbsp;[[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Optimal_binary_receiver_.E2.80.93_.22Matched_Filter.22_realization| optimal binary receiver]]&nbsp; is also plotted as a red horizontal line. For this one holds:
:$$\rho_d = \rho_{\rm U} = {2 \cdot E_{\rm B}}/{ N_0}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
  10 \cdot {\rm lg}\hspace{0.1cm} \rho_d = 10 \cdot {\rm lg}\hspace{0.1cm} \rho_{\rm U} \approx 16\,{\rm dB}
   \hspace{0.05cm}.$$

One can see from the plot:
*The optimization criterion &nbsp;$\rho_d$&nbsp; leads to the optimal cutoff frequency &nbsp;$f_\text{G, opt} \cdot T = 0.8$. A smaller cutoff frequency results in stronger intersymbol interference (smaller eye opening), a larger cutoff frequency results in a larger noise rms value &nbsp;$\sigma_d$.<br>
*Such a Gaussian receiver filter with &nbsp;$f_\text{G, opt} \cdot T \approx 0.8$&nbsp; leads to the signal-to-noise ratio &nbsp;$10 \cdot {\rm lg}\hspace{0.1cm} \rho_d  \approx 15 \ \rm dB$&nbsp; and thus to the error probability &nbsp;$p_{\rm S} \approx 10^{-8}$. For comparison: &nbsp; For the optimal receiver (impulse response matched to the transmitter), the results are &nbsp;$10 \cdot {\rm lg}\hspace{0.1cm} \rho_d  \approx 16 \ \rm dB$&nbsp; and &nbsp;$p_{\rm S} \approx 10^{-10}$.<br>
*However, the graph also shows that the much simpler optimization criterion &nbsp;$ \rho_{\rm U}$&nbsp;  &nbsp;$($or &nbsp;$ p_{\rm U})$&nbsp; leads approximately to the same optimal cutoff frequency &nbsp;$f_\text{G, opt} \cdot T = 0.8$.&nbsp; For this cutoff frequency, we obtain &nbsp;$10 \cdot {\rm lg}\hspace{0.1cm} \rho_{\rm U}  \approx 14.7 \ \rm dB$&nbsp; and the worst-case error probability &nbsp;$p_{\rm U} \approx 3 \cdot 10^{-8}$.<br>
*If the cutoff frequency &nbsp;$f_\text{G} \cdot T < 0.27$, the vertical eye opening will always be &nbsp;$\ddot{o}(T_{\rm D}) = 0$. This is called a <i>closed eye</i>. As a consequence, some worst-case pulse sequences would always be wrongly decided even without noise. A systematic error occurs.<br>
*Further investigations have shown that the optimization criterion &nbsp;$ \rho_{\rm U}$&nbsp; is sufficient even with smaller &nbsp;$E_{\rm B}/N_0$.&nbsp; Thus, for a distortion-free channel &nbsp; &rArr; &nbsp;  $H_{\rm K}(f) = 1$, the optimal cutoff frequency of the Gaussian low-pass always results in &nbsp;$f_\text{G, opt} \cdot T \approx 0.8$, at least in a realistic approach.<br><br>

All statements of this chapter can be reproduced with the interactive applet [[Applets:Augendiagramm|Eye diagram and eye opening]].

==Exercises for the chapter ==
<br>

[[Aufgaben:3.2_Augendiagramm_nach_Gaußtiefpass|Aufgabe 3.2: Augendiagramm nach Gaußtiefpass]]

[[Aufgaben:3.2Z_Optimale_Grenzfrequenz_bei_Gauß-Tiefpass|Aufgabe 3.2Z: Optimale Grenzfrequenz bei Gauß-Tiefpass]]


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