Aufgaben:Exercise 1.14: Bhattacharyya Bound for BEC

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Possible received vectors for the  $(5, 2)$ code and BEC

In this exercise, we consider the systematic  $(5, 2)$ code

• with the  $2×5$ generator matrix

$${ \boldsymbol{\rm G}}_{(5, 2)} = \begin{pmatrix} 1 &0 &1 &1 &0\\ 0 &1 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$

• the  $3 × 5$ parity-check matrix

$${ \boldsymbol{\rm H}}_{(5, 2)} = \begin{pmatrix} 1 &0 &1 &0 &0\\ 1 &1 &0 &1 &0\\ 0 &1 &0 &0 &1 \end{pmatrix} \hspace{0.05cm},$$

• and the  $2^k = 4$  code words

$$\underline{x}_0 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} (0, 0, 0, 0, 0) \hspace{0.05cm},\hspace{0.2cm}\underline{x}_1 = (0, 1, 0, 1, 1)\hspace{0.05cm},\hspace{0.2cm}\underline{x}_2 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} (1, 0, 1, 1, 0) \hspace{0.05cm},\hspace{0.2cm}\underline{x}_3 = (1, 1, 1, 0, 1)\hspace{0.05cm}.$$

At the output of the digital channel defined by the  "BEC model"  (Binary Erasure Channel) with the erasure probability  $\lambda = 0.001$ the received vector

$$\underline{y} = (y_1, \hspace{0.05cm}y_2, \hspace{0.05cm}y_3, \hspace{0.05cm}y_4, \hspace{0.05cm}y_5)$$

occurs, where for  $i = 1, \ \text{...} \ , 5$  holds:   $y_{i} \in \{0, 1, \rm E\}$.

The BEC channel is characterized by the fact that.

• corruptions  $(0 → 1, 1 → 0)$  are excluded,
• but cancellations  $(0 → \rm E, 1 → E)$  may occur.

The graph explicitly shows all possible received vectors  $\underline{y}$  with three or more erasures $\rm E$  assuming that the all-zero vector  $(0, 0, 0, 0, 0)$ was sent.

• For less than three extinctions, for the considered  $(5, 2)$ code, the codeword finder always returns the correct decision:   $\underline{z} = \underline{x}$.

• On the other hand, if there are three or more erasures, wrong decisions may occur. In this case, the following applies to the block error probability:

$$ {\rm Pr(block\:error)}= {\rm Pr} (\underline{z} \ne \underline{x}) = {\rm Pr}\left \{ \hspace{0.1cm} [\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}1}] \hspace{0.05cm}\cup\hspace{0.05cm}[\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}2}] \hspace{0.05cm}\cup \hspace{0.05cm}[\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}3}] \hspace{0.1cm}\right \} \hspace{0.05cm}.$$

Please note:

• The event  $[\underline{x}_{0} → \underline{x}_{1}]$  does not necessarily say that at the received vector under consideration  $\underline{y}$  is actually decided for the codeword  $\underline{x}_{1}$  is decided, but only that the decision for  $x_{1}$  would be more reasonable than the decision for  $\underline{x}_{0}$ due to statistics.
• But it could also be decided for  $\underline{x}_{2}$  or  $\underline{x}_{3}$  if the  "maximum-likelihood criterion"  is in favor.

Computing the block error probability is difficult because the events  $[\underline{x}_{0} → \underline{x}_{1}]$ ,  $[\underline{x}_{0} → \underline{x}_{2}]$  and  $[\underline{x}_{0} → \underline{x}_{3}]$  are not necessarily  "disjoint" . An upper bound is provided by  "Union Bound":

$${\rm Pr(Union \hspace{0.15cm}Bound)} = {\rm Pr}[\underline{x}_{\hspace{0.02cm}0} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}1}] +{\rm Pr}[\underline{x}_{\hspace{0.02cm}0} \hspace{-0.02cm}\mapsto \hspace{-0.02cm} \underline{x}_{\hspace{0.02cm}2}] +{\rm Pr}[\underline{x}_{\hspace{0.02cm}0} \hspace{-0.02cm}\mapsto \hspace{-0.02cm} \underline{x}_{\hspace{0.02cm}3}] \ge {\rm Pr(Blockfehler)} \hspace{0.05cm}.$$

Another bound was given by Bhattacharyya:

$${\rm Pr(Bhattacharyya)} = W(\beta)-1 \ge {\rm Pr(Union \hspace{0.15cm}Bound)} \ge {\rm Pr(block\:error)} \hspace{0.05cm},$$

where, for the Binary Erasure Channel, Bhattacharyya parameter  $\beta = \lambda$  and  $W(X)$  the  "weight enumerator function"  where the pseudo-variable  $X$  is to be replaced here by the Bhattacharyya parameter  $\lambda$ .

• The Bhattacharyya bound is more or less far above the  Union Bound. depending on the channel.
• Its importance lies in the fact that the bound can be specified in the same way for different channels.

Hints:

• The exercise belongs to the chapter  "Bounds on block error probability".

Questions

1

What is the pairwise error probability between the codewords  $\underline{x}_{0} = (0, 0, 0, 0, 0)$  and  $\underline{x}_{1} = (0, 1, 0, 1, 1)$?

${\rm Pr}[\underline{x}_{0} → \underline{x}_{1}] \ = \ $ $\ \cdot 10^{-3} $

2

Which statements are true regarding  ${\rm Pr}[\underline{x}_{0} → \underline{x}_{i}]$  with index  $i = 1, \ \text{...} \ , 3$?  
$d_{{\rm H},\hspace{0.05cm}i}$  denotes here the Hamming distance between  $x_{0}$  and  $x_{i}$.

	It holds  ${\rm Pr}[\underline{x}_{0} → \underline{x}_{i}] \ = \ \lambda ^{d_{{\rm H},\hspace{0.05cm}i}} \ · \ (1 – \lambda)^{n \hspace{0.05cm}– \hspace{0.05cm}d_{{\rm H},\hspace{0.05cm}i}}$.
	It holds  ${\rm Pr}[\underline{x}_{0} → \underline{x}_{i}] \ = \ 1/2 · \lambda ^{d_{{\rm H},\hspace{0.05cm}i}}.$
	${\rm Pr}[\underline{x}_{0} → \underline{x}_{i}]$ is the corruption probability of $x_{0}$ nach $x_{i}$.

3

What are the following probabilities?

$\ {\rm Pr}[\underline{x}_{0} → \underline{x}_{2}] \ = \ $ $\ \cdot 10^{-3} $

$\ {\rm Pr}[\underline{x}_{0} → \underline{x}_{3}] \ = \ $ $\ \cdot 10^{-3} $

4

Specify the Union Bound  for the block error probability.

$\ {\rm Pr(Union\ Bound)} \ = \ $ $\ \cdot 10^{-3} $

5

What is the Bhattacharyya bound in the present case?

$\ {\rm Pr(Bhattacharyya)} \ = \ $ $\ \cdot 10^{-3} $

Solution

Solution

(1)  The code words $\underline{x}_{0}$ and $\underline{x}_{1}$ differ in bit $2, \ 4$ and $5$. If only one of these three binary values is transmitted correctly, the entire codeword is thus uniquely determined. No information about the codeword is obtained for the following received vectors (see table on the information page):

• $\underline{y} = (0, {\rm E}, 0, {\rm E}, {\rm E})$ The code words $\underline{x}_{0}$ and $\underline{x}_{1}$ differ in bit $2, \ 4$ and $5$. If only one of these three binary values is transmitted correctly, the entire codeword is thus uniquely determined. No information about the codeword is obtained for the following received vectors (see table on the information page): $\lambda^3 \ · \ (1 – \lambda)^2$,
• $\underline{y} = (0, {\rm E}, {\rm E}, {\rm E}, {\rm E})$ with probability $\lambda^4 \ · \ (1 – \lambda)$,
• $\underline{y} = ({\rm E}, {\rm E}, 0, {\rm E}, {\rm E})$ with probability $\lambda^4 \ · \ (1 – \lambda)$,
• $\underline{y} = ({\rm E}, {\rm E}, {\rm E}, {\rm E}, {\rm E})$ with probability $\lambda^5$.

The probability that, due to the specific received vector $\underline{y}$, the codeword $\underline{x}_{1}$ is just as likely as $\underline{x}_{0}$, is given by

$$\ {\rm Pr}\ [\underline{x}_0 \hspace{0.12cm}{\rm and}\hspace{0.12cm} \underline{x}_1 \hspace{0.15cm}{\rm are \hspace{0.15cm}equally \hspace{0.15cm}probable}] = \lambda^3 \cdot (1- \lambda)^2 + 2 \cdot \lambda^4 \cdot (1- \lambda) + \lambda^5 =\lambda^3 \cdot \left [ (1- \lambda)^2 + 2 \cdot \lambda \cdot (1- \lambda) + \lambda^2 \right ] = \lambda^3 \hspace{0.05cm}.$$

In this case, one decides at random for $\underline{x}_{0}$ (would be correct) or for $\underline{x}_{1}$ (unfortunately wrong), with equal probability. From this follows:

$${\rm Pr} [\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}1}] = 1/2 \cdot \lambda^3 \hspace{0.15cm} \underline{= 0.5 \cdot 10^{-3}} \hspace{0.05cm}.$$

(2)  According to subtask (1), answer 2 is correct and not answer 1. Statement 3 is also incorrect:

• ${\rm Pr}[\underline{x}_{0} → \underline{x}_{1}]$ does not state that with this probability the codeword $\underline{x}_{0}$ actually transitions to the incorrect codeword $\underline{x}_{1}$, but only that it could transition to $\underline{x}_{1}$ with this probability.
• ${\rm Pr}[\underline{x}_{0} → \underline{x}_{1}]$ also includes constellations where the decision is actually for $\underline{x}_{2}$ or $\underline{x}_{3}$.

(3)  Because of  $d_{\rm H}(\underline{x}_{0}, \underline{x}_{2}) = 3$  and  $d_{\rm H}(\underline{x}_{0}, \underline{x}_{3}) = 4$  it follows that

$${\rm Pr} [\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}2}] = 1/2 \cdot \lambda^3 \hspace{0.15cm} \underline{= 0.5 \cdot 10^{-3}} \hspace{0.05cm},\hspace{0.2cm} {\rm Pr} [\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}3}] = 1/2 \cdot \lambda^4 \hspace{0.15cm} \underline{= 0.05 \cdot 10^{-3}} \hspace{0.05cm}.$$

(4)  The block error probability is never larger (with a certain probability rather smaller) than the so-called Union Bound:

$${\rm Pr(Union \hspace{0.15cm}Bound)} \hspace{-0.15cm}\ = \ \hspace{-0.15cm} {\rm Pr}[\underline{x}_{\hspace{0.02cm}0} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}1}] +{\rm Pr}[\underline{x}_{\hspace{0.02cm}0} \hspace{-0.02cm}\mapsto \hspace{-0.02cm} \underline{x}_{\hspace{0.02cm}2}] +{\rm Pr}[\underline{x}_{\hspace{0.02cm}0} \hspace{-0.02cm}\mapsto \hspace{-0.02cm} \underline{x}_{\hspace{0.02cm}3}] = 2 \cdot \lambda^3/2 + \lambda^4/2 = 0.001 + 0.00005 \hspace{0.15cm} \underline{= 1.05 \cdot 10^{-3}} \hspace{0.05cm}.$$

(5)  Generally speaking:

$${\rm Pr(block\:error) ≤ {\rm Pr(Union \hspace{0.15cm}Bound)} \le Pr(Bhattacharyya)} = W(\beta) - 1.$$

• For the distance spectrum or the weight function, we obtain in the present case:

$$W_0 = 1 \hspace{0.05cm}, \hspace{0.2cm} W_3 = 2 \hspace{0.05cm}, \hspace{0.2cm}W_4 = 1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} W(X) = 1+ 2 \cdot X^{3} +X^{4} \hspace{0.05cm}.$$

• For the BEC channel, $\beta = \lambda$ also applies. From this follows as a final result for $\lambda = 0.001$:

$${\rm Pr(Bhattacharyya)} = 2 \cdot \lambda^3 + \lambda^4 \hspace{0.15cm} \underline{= 2.1 \cdot 10^{-3}} \hspace{0.05cm}.$$

Note that in the BEC model, the Bhattacharyya bound is always twice as large as the Union Bound, which is itself an upper bound on the block error probability.