Aufgaben:Exercise 1.6: Non-Binary Markov Sources

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Markov sources with
$M = 3$  and  $M = 4$

The graph shows two ergodic Markov sources   $($German:  "Markovquellen"   ⇒   "$\rm MQ$"$)$:

• The source  $\rm MQ3$  is denoted by  $M = 3$  states  (symbols)  $\rm N$,  $\rm M$,  $\rm P$.  Due to stationarity, the probabilities have the following values:

$$p_{\rm N} = 1/2\hspace{0.05cm},\hspace{0.2cm}p_{\rm M} = p_{\rm P} = 1/4\hspace{0.05cm}.$$

• For the source  $\rm MQ4$  the state  $\rm O$  is additionally possible   ⇒   $M = 4$.  Due to the symmetric transitions, the stationary probabilities are all equal:

$$p_{\rm N} = p_{\rm M} = p_{\rm O} = p_{\rm P} = 1/4\hspace{0.05cm}.$$

In terms of information theory, Markov sources are of particular importance, since with these – and only with these – by

• $H_1$  (first entropy approximation, based only on the symbol probabilities), and
• $H_2$  (second entropy approximation, computable with the composite probabilities for all two-tuples)

are also determined at the same time:

• the further entropy approximations  $H_k$  with  $k = 3, \ 4$,  ..., and
• the actual (final) source entropy  $H$.

The following equations of determination apply:

$$H= H_{k \to \infty} = 2 \cdot H_2 - H_1\hspace{0.05cm},$$

$$H_k = {1}/{k} \cdot \big [ H_{\rm 1} + (k-1) \cdot H \big ] \hspace{0.05cm}.$$

Hint:

• The exercise belongs to the chapter  Discrete Sources with Memory.
• Reference is made in particular to the page  Non-binary Markov Sources.
• For all entropies, add the pseudo-unit  "bit/symbol".

Fragebogen

1

Calculate the entropy approximation  $H_1$  of the Markov source  $\rm MQ3$.

$H_1 \ = \ $ $\ \rm bit/symbol$

2

Calculate the entropy approximation  $H_2$  of the Markov source  $\rm MQ3$.

$H_2 \ = \ $ $\ \rm bit/symbol$

3

What are the actual source entropy  $H= H_{k \to \infty}$  and the approximations  $H_3$  and  $H_4$ for  $\rm MQ3$ ?

$H \ = \ $ $\ \rm bit/symbol$

$H_3 \ = \ $ $\ \rm bit/symbol$

$H_4 \ = \ $ $\ \rm bit/symbol$

4

Calculate the entropy approximation  $H_1$  of the Markov source  $\rm MQ4$.

$H_1 \ = \ $ $\ \rm bit/symbol$

5

Calculate the entropy approximation  $H_2$  of the Markov source  $\rm MQ4$

$H_2 \ = \ $ $\ \rm bit/symbol$

6

What are the actual source entropy  $H= H_{k \to \infty}$  and the approximations  $H_3$  and  $H_4$ for  $\rm MQ4$ ?

$H \ = \ $ $\ \rm bit/symbol$

$H_3 \ = \ $ $\ \rm bit/symbol$

$H_4 \ = \ $ $\ \rm bit/symbol$

Musterlösung

Solution

(1)  The symbol probabilities of the ternary Markov source are given.

• From this, the entropy approximation  $H_1$  can be calculated:

$$H_{\rm 1} = 1/2 \cdot {\rm log}_2\hspace{0.1cm} (2) + 2 \cdot 1/4 \cdot {\rm log}_2\hspace{0.1cm}(4 ) \hspace{0.15cm} \underline {= 1.5 \,{\rm bit/symbol}} \hspace{0.05cm}.$$

(2)  The composite probability is  $p_{\rm XY} = p_{\rm X} \cdot p_{\rm Y|X}$,  where  $p_{\rm X}$  gives the symbol probability of  $\rm X$  and  $p_{\rm Y|X}$  gives the conditional probability of  $\rm Y$, given that previously  $\rm X$  occurred.

• $\rm X$  and  $\rm Y$  are here placeholders for the symbols  $\rm N$,  $\rm P$  and  $\rm M$.  Then holds:

$$p_{\rm NN} = 1/2 \cdot 1/2 = 1/4\hspace{0.05cm},\hspace{0.2cm}p_{\rm PP} = 1/4 \cdot 0 = 0\hspace{0.05cm},\hspace{0.2cm}p_{\rm MM} = 1/4 \cdot 0 = 0 \hspace{0.05cm},$$

$$p_{\rm NP} = 1/2 \cdot 1/4 = 1/8\hspace{0.05cm},\hspace{0.2cm} p_{\rm PM} = 1/4 \cdot 1/2 = 1/8\hspace{0.05cm},\hspace{0.2cm}p_{\rm MN} = 1/4 \cdot 1/2 = 1/8 \hspace{0.05cm},$$

$$p_{\rm NM} = 1/2 \cdot 1/4 = 1/8\hspace{0.05cm},\hspace{0.2cm} p_{\rm MP} = 1/4 \cdot 1/2 = 1/8\hspace{0.05cm},\hspace{0.2cm}p_{\rm PN} = 1/4 \cdot 1/2 = 1/8$$

$$\Rightarrow \hspace{0.3cm} H_{\rm 2} = {1}/{2} \cdot \big [ 1/4 \cdot {\rm log}_2\hspace{0.1cm}( 4) + 6 \cdot 1/8 \cdot {\rm log}_2\hspace{0.1cm} (8) \big ] \hspace{0.15cm} \underline {= 1.375 \,{\rm bit/Symbol}} \hspace{0.05cm}.$$

(3)  $\rm MQ3$  has Markov properties.

• Therefore, from  $H_1$  and  $H_2$  all approximations  $H_3$,  $H_4$,  ... and also the limit  $H =H_\infty$  for  $k \to \infty$  are given:

$$H = 2 \cdot H_2 - H_1 = 2\cdot 1.375 - 1.5 \hspace{0.15cm} \underline {= 1.250 \,{\rm bit/symbol}}\hspace{0.05cm},$$

$$H_3 \hspace{0.1cm} = \hspace{0.1cm}(H_1 + 2 \cdot H)/3 = (1.5 + 2 \cdot 1.25)/3 \hspace{0.15cm} \underline {= 1.333 \,{\rm bit/symbol}}\hspace{0.05cm},$$

$$H_4 = (H_1 + 3 \cdot H)/4 = (1.5 + 3 \cdot 1.25)/4 \hspace{0.15cm} \underline {= 1.3125 \,{\rm bit/symbol}}\hspace{0.05cm}.$$

• The tenth entropy approximation still differs from the final value  $H = 1.25 \,\rm bit/symbol$,  albeit only slightly  $($by $2\%)$ :

$$H_{10} = (H_1 + 9 \cdot H)/10 = (1.5 + 9 \cdot 1.25)/10 {= 1.275 \,{\rm bit/symbol}}\hspace{0.05cm}.$$

(4)  According to the specification, for  $\rm MQ4$  the  $M = 4$  symbols are equally probable.

• From this follows:

$$H_{\rm 1} = H_{\rm 0} = {\rm log}_2\hspace{0.1cm} (4) \hspace{0.15cm} \underline {= 2 \,{\rm bit/symbol}} \hspace{0.05cm}.$$

(5)  Of the  $M^2 = 16$  possible two-tuples, eight combinations are now not possible:

$$\rm NP, NO, PP, PO, OM, ON, MM, MN.$$

• The eight other combinations (two-tuples) each yield the composite probability value  $1/8$, as shown in two examples:

$$p_{\rm NN} = p_{\rm N} \cdot p_{\rm N\hspace{0.01cm}|\hspace{0.01cm}N} = 1/4 \cdot 1/2 = 1/8 \hspace{0.05cm},\hspace{0.5cm} p_{\rm MP} = p_{\rm M} \cdot p_{\rm P\hspace{0.01cm}|\hspace{0.01cm}M} = 1/4 \cdot 1/2 = 1/8 \hspace{0.05cm}.$$

$$\Rightarrow \hspace{0.3cm} H_2 = {1}/{2} \cdot \big [ 8 \cdot 1/8 \cdot {\rm log}_2\hspace{0.1cm} (8) \big ] \hspace{0.15cm} \underline {= 1.5 \,{\rm bit/symbol}} \hspace{0.05cm}.$$

(6)  Because of the Markove property, here:

$$H = 2 \cdot H_2 - H_1 = 2\cdot 1.5 - 2 \hspace{0.15cm} \underline {= 1 \,{\rm bit/symbol}}\hspace{0.05cm},$$

$$H_3 = (H_1 + 2 \cdot H)/3 = (2 + 2 \cdot 1)/3 \hspace{0.15cm} \underline {= 1.333 \,{\rm bit/symbol}}\hspace{0.05cm},$$

$$H_4 = (H_1 + 3 \cdot H)/4 = (2 + 3 \cdot 1)/4 \hspace{0.15cm} \underline {= 1.250 \,{\rm bit/symbol}}\hspace{0.05cm}.$$

• Also here, the tenth approximation still differs significantly from the final value, namely by  $10\%$:

$$H_{10} = (H_1 + 9 \cdot H)/10 = (2 + 9 \cdot 1)/10 {= 1.1 \,{\rm bit/symbol}}\hspace{0.05cm}.$$

• A deviation of only  $2\%$  results here only for  $k = 50$.  For comparison:   For the Markov source  $\rm MQ3$  this approximation was already achieved with  $k = 10$ .