Aufgaben:Exercise 2.4Z: Error Probabilities for the Octal System

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Octal  "random coding"  and  Gray coding

A digital system with  $M = 8$  amplitude levels  ("octal system")  is considered,  whose  $M – 1 = 7$  decision thresholds lie exactly at the respective interval centers.

Each of the equally probable amplitude coefficients  $a_{\mu}$  with  $1 ≤ \mu ≤ 8$  can be falsified only into the immediate neighbor coefficients  $a_{\mu–1}$  and  $a_{\mu+1}$,  respectively,  and in both directions with the same probability  $p = 0.01$.  Here are some examples:

• $a_5$  passes into coefficient $a_4$ with probability  $p = 0.01$  and into coefficient  $a_6$ with the same probability  $p = 0.01$. 

• $a_8$  is falsified with probability  $p = 0.01$  into coefficient  $a_7$.  No falsification is possible in the other direction.

The mapping of each three binary source symbols into an octal amplitude coefficient happens alternatively according to

• the second column in the given table,  which was generated  "randomly"  - without strategy,

• the Gray coding,  which is only incompletely indicated in column 3 and is still to be supplemented.

The Gray code is given for  $M = 4$.  For  $M = 8$  the last two binary characters are to be mirrored at the dashed line.  For the first four amplitude coefficients a  $\rm L$  is to be added at the first place,  for  $a_{5}, ..., a_{8}$  the binary symbol  $\rm H$.

For the two mappings  "Random"  and  "Gray"  are to be calculated:

• the  "symbol error probability"  $p_{\rm S}$,  which is the same in both cases;  $p_{\rm S}$  indicates the average falsifcation probability of an amplitude coefficient  $a_{\mu}$; 

• the  "bit error probability"  $p_{\rm B}$  related to the (decoded) binary symbols.

Notes:

• The exercise belongs to the chapter  "Basics of Coded Transmission".

• Reference is also made to the chapter  "Redundancy-Free Coding" .

Questions

1

To which amplitude coefficient  $a_{ \mu}$  do the binary sequences  $\rm {LHH}$  and  $\rm {HLL}$ correspond in the Gray code?
Please enter index  $ \mu$   $(1 < \mu < 8)$.

$ \rm {LHH}\text{:}\hspace{0.4cm} \mu \ = \ $

$ \rm {HLL}\text{:}\hspace{0.45cm} \mu \ = \ $

2

Calculate the symbol error probability  $p_{\rm S}$.

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

3

Calculate the bit error probability  $p_{\rm B}$  for the  Gray code.

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

4

Calculate the bit error probability  $p_{\rm B}$  for the  random code.

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

Solution

Solution

(1)  According to the description on the specification page

• $\rm LHH$  for the amplitude coefficient  $a_{3}$   ⇒   $\underline{\mu =3}$.
• $\rm HLL$  for the amplitude coefficient  $a_{8}$   ⇒   $\underline{\mu =8}$.

(2)  The outer coefficients  $(a_{1}$  and  $a_{8})$  are each falsified with probability  $p = 1 \%$,  the  $M – 2 = 6$  inner ones with twice the probability  $(2p= 2 \%)$.  By averaging,  we obtain:

$$p_{\rm S} = \frac{2 \cdot 1 + 6 \cdot 2} { 8} \cdot p\hspace{0.15cm}\underline { = 1.75 \,\%} \hspace{0.05cm}.$$

(3)  Each transmission error  (symbol error)  results in exactly one bit error in Gray code.  However,  since each octal symbol contains three binary characters,  the following applies

$$p_{\rm B} ={p_{\rm S}}/ { 3}\hspace{0.15cm}\underline { = 0.583 \,\%} \hspace{0.05cm}.$$

(4)  Of the total of seven possible transitions  (each in both directions)  lead to

• one error:     $\rm HLH \ \Leftrightarrow \ LLH$,
• two errors:      $\rm HLL \ \Leftrightarrow \ HHH$,     $\rm LLL \ \Leftrightarrow \ LHH$,     $\rm HHL \ \Leftrightarrow \ HLH$,     $\rm LLH \ \Leftrightarrow \ LHL$,
• three errors:      $\rm HHH \ \Leftrightarrow \ LLL$,     $\rm LHH \ \Leftrightarrow \ HHL$.

It follows that:

$$p_{\rm B} = \frac{p} { 3} \cdot \frac{1 + 4 \cdot 2 + 2 \cdot 3} { 7} = \frac{15} { 21} \cdot p \hspace{0.15cm}\underline { = 0.714 \,\%} \hspace{0.05cm}.$$