Exercise 4.06Z: Signal Space Constellations

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Three signal space constellations

The  (mean)  symbol error probability of an optimal binary system is:

$$p_{\rm S} = {\rm Pr}({ \cal E} ) = {\rm Q} \left ( \frac{d/2}{\sigma_n} \right )\hspace{0.05cm}.$$

It should be noted here:

• ${\rm Q}(x)$  denotes the complementary Gaussian error function  (definition and approximation):

$${\rm Q}(x) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} {\rm e}^{-u^2/2} \,{\rm d} u \approx \frac{1}{\sqrt{2\pi} \cdot x} \cdot {\rm e}^{-x^2/2} \hspace{0.05cm}.$$

• The parameter  $d$  specifies the distance between the two transmitted signal points  $s_0$  and  $s_1$  in vector space:

$$d = \sqrt{ || \boldsymbol{ s }_1 - \boldsymbol{ s }_0||^2} \hspace{0.05cm}.$$

• $\sigma_n^2$  is the variance of the AWGN noise after the detector, which e.g. can be implemented as a matched filter.
  It is assumed that  $\sigma_n^2 = N_0/2$.

The graphic shows three different signal space constellations,  namely

• Variant $\rm A$:   $s_0 = (+1, \, +5), \hspace{0.4cm} s_1 = (+4, \, +1)$,
• Variant $\rm B$:   $s_0 = (-1.5, \, +2), \, s_1 = (+1.5, \, -2)$,
• Variant $\rm C$:   $s_0 = (-2.5, \, 0), \hspace{0.45cm} s_1 = (+2.5, \, 0)$.

The mean energy per symbol  $(E_{\rm S})$  can be calculated as follows:

$$E_{\rm S} = {\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_0) \cdot || \boldsymbol{ s }_0||^2 + {\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_1) \cdot || \boldsymbol{ s }_1||^2\hspace{0.05cm}.$$

Notes:

• The chapter belongs to the chapter  "Approximation of the Error Probability".

• For numeric calculations, the energy  $E = 1$  can be set for simplification.

• Unless otherwise specified, equally probable symbols can be assumed:

$${\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_0) = {\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_1) = 0.5\hspace{0.05cm}.$$

Questions

1

Which prerequisites must absolutely  (in any case)  be fulfilled so that the given error probability equation is valid?

	Additive white Gaussian noise with variance  $\sigma_n^2$.
	Optimal binary receiver.
	Decision boundary in the middle between the symbols.
	Equally likely symbols  $s_0$  and  $s_1$.

2

Which statement applies to the error probability with  $\sigma_n^2 = E$?

	Variant  $\rm A$  has the lowest error probability.
	Variant  $\rm B$  has the lowest error probability.
	Variant  $\rm C$  has the lowest error probability.
	All variants show the same error behavior.

3

Give the error probability for variant  $\rm A$  with  $\sigma_n^2 = E$.   You can calculate  ${\rm Q}(x)$  according to the approximation.

$p_{\rm S} \ = \ $ $\ \%$

4

It is assumed that  $N_0 = 2 \cdot 10^{\rm –6} \ {\rm W/Hz}$,   $E_{\rm S} = 6.25 \cdot 10^{\rm –6} \ \rm Ws$.   What is the error probability for variant $\rm C$  with equally probable symbols?

$p_{\rm S} \ = \ $ $\ \%$

5

What is the error probability for variant   $\rm B$  under the same conditions?

$p_{\rm S} \ = \ $ $\ \%$

6

How large should the average energy per symbol  $(E_{\rm S})$  be chosen for variant $\rm A$  in order to obtain the same error probability as for variant  $\rm C$? 

$E_{\rm S} \ = \ $ $\ \cdot 10^{\rm –6} \ \rm Ws$

Solution

Solution

(1)  The  first three prerequisites  must be met in any case:

• The equation then applies independently of the occurrence probabilities.
• In the case of  ${\rm Pr}(\boldsymbol{s} = \boldsymbol{s}_0) ≠ {\rm Pr}(\boldsymbol{s} = \boldsymbol{s}_1)$,  a lower error probability can be achieved by shifting the decision threshold.

(2)  The noise rms value  $\sigma_n$  and thus also the signal energy  $E = \sigma_n^2$  are the same for all three considered variants. 

• The same applies to the distance of the signal space points.  For variant  $\rm A$,  for example,  the following applies:

$$d = \sqrt{ || \boldsymbol{ s }_1 - \boldsymbol{ s }_0||^2} = \sqrt{ E \cdot (4-1)^2 + E \cdot (1-5)^2} = 5 \cdot \sqrt{E}\hspace{0.05cm}.$$

• Due to the shifting of the coordinate system,  the distance between  $\boldsymbol{s}_0$  and  $\boldsymbol{s}_1$  does not change (variant  $\rm B$),  the same distance results in variant  $\rm C$  (after rotation).

• Solution 4 is correct:

1. By rotating the coordinate system,  one can always get by with one basis function  $(N = 1)$  for a binary system  $(M = 2)$.
2. Since the two-dimensional noise is circularly symmetric   ⇒   equal standard deviation  $\sigma_n$  in all directions, 
   the noise term can also be described one-dimensionally as in the chapter  "Error Probability for Baseband Transmission".

(3)  For all variants considered here,  i.e.,  also for variant  $\rm A$,  the following holds:

$$p_{\rm S} = {\rm Pr}({ \cal E} ) = {\rm Q} \left ( \frac{d/2}{\sigma_n} \right )= {\rm Q} \left ( \frac{5/2 \cdot \sqrt{E}}{\sigma_n} \right ) = {\rm Q}(2.5)\hspace{0.05cm}.$$

• With the given approximation we obtain

$$p_{\rm S} = \frac{1}{\sqrt{2\pi} \cdot 2.5} \cdot {\rm e}^{-2.5^2/2} \hspace{0.1cm} \hspace{0.15cm}\underline {\approx 0.7 \%}\hspace{0.05cm}.$$

(4)  For variant  $\rm C$,  the average energy per symbol is given by:

$$E_{\rm S} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_0) \cdot (-2.5 \cdot \sqrt{E})^2 + {\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_1) \cdot (+ 2.5 \cdot \sqrt{E})^2 = \left [ {\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_0) + {\rm Pr}(\boldsymbol{ s } = \boldsymbol{ s }_0) \right ] \cdot 6.25 \cdot E = 6.25 \cdot E$$

$$\Rightarrow \hspace{0.3cm} E = \frac {E_{\rm S}}{6.25} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \sqrt{E}= \frac {\sqrt{E_{\rm S}}}{2.5} \hspace{0.05cm}.$$

• Substituting this result into the equation found in  (3),  we obtain with  $\sigma_n^2 = N_0/2$:

$$p_{\rm S} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm Q} \left ( \frac{2.5 \cdot \sqrt{E}}{\sigma_n} \right )= {\rm Q} \left ( \frac{ \sqrt{E_{\rm S}}}{\sigma_n} \right ) = {\rm Q} \left ( \frac{ \sqrt{2 \cdot E_{\rm S}}}{N_0} \right ) ={\rm Q} \left ( \sqrt{\frac{ 2 \cdot 6.25 \cdot 10^{-6}\,{\rm Ws}}{2 \cdot 10^{-6}\,{\rm W/Hz}}} \right ) ={\rm Q}(2.5) \hspace{0.1cm} \hspace{0.15cm}\underline {\approx 0.7 \%}\hspace{0.05cm}. $$

(5)  Rotating the coordinate system does not change the energy ratios.

• Therefore,  $p_{\rm S} \ \underline {\approx 0.7\%}$  is obtained again.

(6)  In variant  $\rm A$,  the average energy per symbol is

$$E_{\rm S} = {1}/{2} \cdot \left [ (1^2 + 5^2) \cdot E + (4^2 + 1^2) \cdot E \right ] = 21.5 \cdot E \hspace{0.05cm}. $$

• The distance from the threshold,  which should be midway between  $\boldsymbol{s}_0$  and  $\boldsymbol{s}_1$  for equally probable symbols,  is  $d/2 = 2.5 \cdot E^{\rm 1/2}$,  as in the other variants.

• Thus,  with  $\sigma_n^2 = N_0/2$,  we obtain the governing equation:

$$p_{\rm S} = {\rm Q} \left ( \frac{ 2.5 \cdot \sqrt{E}}{\sqrt{N_0/2}} \right ) ={\rm Q}(2.5)\approx 0.7 \cdot 10^{-2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \sqrt{\frac {2E}{N_0}} = 1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \frac {E}{N_0} = 0.5 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\frac {E_{\rm S}}{21.5 \cdot N_0} = 0.5$$

$$\Rightarrow \hspace{0.3cm} {E_{\rm S}} = 0.5 \cdot {21.5 \cdot N_0} \hspace{0.1cm} \hspace{0.15cm}\underline { = 21.5 \cdot 10^{-6}\,{\rm Ws}}\hspace{0.05cm}.$$

• This means:  For variant  $\rm A$,  compared to the other two variants,  a mean symbol energy  $E_{\rm S}$  larger by a factor of  $3.44$  is required to achieve the same  $p_{\rm S} = 0.7\%$.  Because:

1. This signal space constellation is very unfavorable.  It results in a very large  $E_{\rm S}$  without increasing the distance  $d$  at the same time.
2. With $E_{\rm S} = 6.25 \cdot 10^{\rm –6} \ \rm Ws$,  on the other hand,  $p_{\rm S} = {\rm Q}(2.5/3.44^{\rm 1/2}) \approx {\rm Q}(1.35) \approx 9\%$  would result.
3. That means:   The error probability would be larger by more than one power of ten.