Exercise 4.4Z: Supplement to Exercise 4.4

Hamming weights $w_{\rm H}(\underline{x})$,
sequence probabilities ${\rm Pr}(\underline{x})$

The information theorist $\text{Robert G. Gallager}$ dealt already in 1963 with the following problem:

Given a random vector $\underline{x} = (x_1, \, x_2, \hspace{-0.04cm} \text{ ...} \hspace{0.08cm} , x_n)$ with $n$ binary elements $x_i ∈ \{0, \, 1\}$.

Known are all probabilities $p_i = {\rm Pr}(x_i = 1)$ and $q_i = {\rm Pr}(x_i = 0) = 1 - p_i$ with index $i = 1, \hspace{-0.04cm}\text{ ...} \hspace{0.08cm} ,\ n$.

What we are looking for is the probability that the number of "ones" in this vector is even.

Or expressed using the "Hamming weight": What is the probability ${\rm Pr}[w_{\rm H}(\underline{x}) {\rm \ is \ even}]$?

The graph illustrates the task for the example $n = 4$ and $p_1 = 0.2$, $p_2 = 0.9$, $p_3 = 0.3$ and $p_4 = 0.6$.

For the row highlighted in green ⇒ $\underline{x} = (1, \, 0, \, 0, \, 1)$ holds $w_{\rm H}(\underline{x}) = 2$ and

$${\rm Pr}(\underline{x}) = p_1 \cdot q_2 \cdot q_3 \cdot p_4 = 0.0084.$$

Blue font means "$w_{\rm H}(\underline{x})$ is even". Red font stands for "$w_{\rm H}(\underline{x})$ is odd."

The probability ${\rm Pr}[w_{\rm H}(\underline{x}) {\rm \ is \ even}]$ is the sum of the blue numbers in the last column.

The sum of the red numbers gives ${\rm Pr}[w_{\rm H}(\underline{x}) {\rm \ is \ odd}] = 1 - {\rm Pr}[w_{\rm H}(\underline{x}) {\rm \ is \ even}]$.

Gallager solved the problem in an analytical way:

$$ {\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even} \right ] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 1/2 \cdot [1 + \pi]\hspace{0.05cm},$$

$${\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd} \right ] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 1/2 \cdot [1 - \pi]\hspace{0.05cm}.$$

Here the following auxiliary variable is used:

$$\pi = \prod\limits_{i =1}^{n} \hspace{0.25cm}(1-2p_i) \hspace{0.05cm}.$$

One applies the equation, for example, to calculate the extrinsic L–values of a "single parity–check code".

Indeed, as already pointed out in $\text{Exercise 4.4}$ the extrinsic log likelihood ratio with Hamming–weight $w_{\rm H}$ is the truncated sequence $\underline{x}^{(-i)}$:

$$L_{\rm E}(i) = {\rm ln} \hspace{0.15cm}\frac{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} even} \hspace{0.05cm} | \hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]}{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd} \hspace{0.05cm} | \hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]} \hspace{0.05cm}.$$

Here it is taken into account that for $L_{\rm E}(i)$ one may refer only to the other symbols $(j ≠ i)$ :

$$\underline{x}^{(-i)} = \big ( \hspace{0.03cm}x_1, \hspace{-0.04cm} \text{ ...} \hspace{0.08cm} , \hspace{0.03cm} x_{i-1}, \hspace{0.43cm} x_{i+1}, \hspace{-0.04cm} \text{ ...} \hspace{0.08cm} , x_{n} \hspace{0.03cm} \big )\hspace{0.03cm}. $$

Hints:

This exercise belongs to the chapter "Soft–in Soft–out Decoder".

Reference is made in particular to the section "For calculating the extrinsic log likelihood ratios".

The exercise is intended as a supplement to $\text{Exercise 4.4}$ .

Questions

${\rm Pr}\big [w_{\rm H}(\underline{x}) \ {\rm is \ even}\big ] \ = \ $

${\rm Pr(blue) = Pr}\big [w_{\rm H}(\underline{x}) \ {\rm is \ even}\big ] \hspace{0.33cm} = \ $

${\rm Pr(red) = Pr}\big [w_{\rm H}(\underline{x}) \ {\rm is \ odd}\big ] \hspace{0.1cm} = \ $

$\text{Quotient }Q = {\rm Pr(blau)/Pr(rot)} \hspace{0.4cm} = \ $

$L_{\rm E}(i = 5) \ = \ $

	$L_{\rm E}(i = 5)$ becomes larger.
	$L_{\rm E}(i = 5)$ becomes smaller.
	$L_{\rm E}(i = 5)$ is not changed compared to subtask (4).

Solution

Derivation "$w_{\rm H}$ is even" for code length $n = 2$.

(1) According to the adjacent table applies:

$${\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.10cm}{\rm is \hspace{0.10cm} even}\right ] = {\rm Pr} \left [w_{\rm H} = 0 \right] + {\rm Pr} \left [w_{\rm H} = 2 \right] \hspace{0.05cm}. $$

With the probabilities

$$p_1 = {\rm Pr} (x_1 = 1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.2\hspace{0.05cm},\hspace{0.3cm}q_1 = {\rm Pr} (x_1 = 0) = 0.8\hspace{0.05cm},$$

$$p_2 = {\rm Pr} (x_2 = 1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.9\hspace{0.05cm},\hspace{0.3cm}q_2 = {\rm Pr} (x_2 = 0) = 0.1$$

one obtains:

$${\rm Pr} \left [w_{\rm H}(\underline{x}) = 0\right] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Pr} \left [(x_1 = 0)\cap (x_2 = 0) \right] = q_1 \cdot q_2 = 0.8 \cdot 0.1 = 0.08 \hspace{0.05cm},$$

$${\rm Pr} \left [w_{\rm H}(\underline{x}) = 2\right] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Pr} \left [(x_1 = 1)\cap (x_2 = 1) \right] = p_1 \cdot p_2 = 0.2 \cdot 0.9 = 0.18$$

$$\Rightarrow \hspace{0.3cm} {\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even}\right] = 0.8 + 0.18 \hspace{0.15cm} \underline{= 0.26} \hspace{0.05cm}.$$

The Gallager's equation provides for the same set of parameters:

$${\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even}\right] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.5 + 0.5 \cdot \prod\limits_{i =1}^{2} \hspace{0.25cm}(1-2\cdot p_i) = 0.5 + 0.5 \cdot (1 - 2 \cdot 0.2)\cdot (1 - 2 \cdot 0.9) = 0.26 \hspace{0.05cm}.$$

The equation given by Gallager 1963 was hereby verified for $n = 2$.

(2) In the second table, the four combinations with an even number of "ones" are marked in blue.

Derivation "$w_{\rm H}$ is even" for code length $n = 3$.

The occurrence probabilities of each combination are given in the last column. Thus, the result is:

$$ {\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even}\right] = 0.056 + 0.216 + 0.006 + 0.126 \hspace{0.15cm} \underline{= 0.404} \hspace{0.05cm}.$$

The red rows provide the complementary event:

$$ {\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd}\right] = 0.024 + 0.504 + 0.014 + 0.054= 0.596 \hspace{0.05cm}.$$

Gallager's equation again gives the exact same result, although it should be noted that this equation is valid for all $n$ and all arbitrary probabilities:

$${\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even}\right] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.5 + 0.5 \cdot \prod\limits_{i =1}^{3} \hspace{0.25cm}(1-2\cdot p_i) $$

$$\Rightarrow\hspace{0.3cm}{\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even}\right] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.5 + 0.5 \cdot (+0.6) \cdot (-0.8) \cdot (+0.4) = 0.404 \hspace{0.05cm}.$$

(3) According to the specification page applies:

$$\pi = \prod\limits_{i =1}^{4} \hspace{0.25cm}(1-2\cdot p_i) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (1 - 2 \cdot 0.2) \cdot (1 - 2 \cdot 0.9) \cdot (1 - 2 \cdot 0.3) \cdot (1 - 2 \cdot 0.6) $$

$$\Rightarrow\hspace{0.3cm}\pi = \prod\limits_{i =1}^{4} \hspace{0.25cm}(1-2\cdot p_i) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}(+0.6) \cdot (-0.8) \cdot (+0.4) \cdot (-0.2) = 0.0384 \hspace{0.05cm}.$$

From this can be calculated:

$${\rm Pr}({\rm blue}) = {\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even}\right] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.5 + 0.5 \cdot \pi = 0.5 + 0.5 \cdot 0.0384\hspace{0.15cm} \underline{= 0.5192}\hspace{0.05cm},$$

$${\rm Pr}({\rm red}) = {\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd}\right] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.5 - 0.5 \cdot \pi = 0.5 - 0.5 \cdot 0.0384\hspace{0.15cm} \underline{= 0.4808}\hspace{0.05cm}. $$

If you add up the blue and red probabilities on the information page,V you get exactly the values calculated here. For the quotient we get:

$$Q = \frac{{\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} even}\right]} { {\rm Pr} \left [w_{\rm H}(\underline{x})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd}\right]} = \frac{0.5192}{0.4808}\hspace{0.15cm} \underline{= 1.0799} \hspace{0.05cm}. $$

(4) For the "single parity–check code", the extrinsic log likelihood ratio with respect to the $i^{th}$ bit was specified as follows:

$$L_{\rm E}(i) = {\rm ln} \hspace{0.15cm}\frac{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} even} \hspace{0.05cm} | \hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]}{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd} \hspace{0.05cm} | \hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]} \hspace{0.05cm},$$

Or:

$$L_{\rm E}(i) = {\rm ln} \hspace{0.15cm}\frac{1+\prod_{j \ne i} \hspace{0.25cm}(1-2\cdot p_j)}{1-\prod_{j \ne i} \hspace{0.25cm}(1-2\cdot p_j)} \hspace{0.05cm}.$$

At $\text{SPC (5, 4, 2}$) ⇒ $n = 5$, this product for $i = 5$ results from the following four factors:

$$\pi = \prod\limits_{j = 1, \hspace{0.05cm}2, \hspace{0.05cm}3, \hspace{0.05cm}4} \hspace{0.05cm}(1-2\cdot p_j) = (1-2\cdot p_1) \cdot (1-2\cdot p_2) \cdot (1-2\cdot p_3) \cdot (1-2\cdot p_4) \hspace{0.05cm}.$$

The comparison with subtask (3) shows that $L_{\rm E}(i = 5) = \ln {Q} = \ln {(1.0799)} \ \underline{\approx 0.077}$.

(5) Correct is the proposed solution 3 because the result for $L_{\rm E}(i = 5)$ is independent of $p_5$.