Exercise 4.5Z: Tangent Hyperbolic and Inverse

$y = \tanh {(x)}$ represented on a table

In "Theory Part" it was shown, using the example of single parity–check code that the extrinsic $L$ value with respect to the $i$th symbol is defined as follows:

$$L_{\rm E}(i) = {\rm ln} \hspace{0.2cm}\frac{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} even} \hspace{0.05cm} | \hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]}{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd} \hspace{0.05cm} | \hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]} \hspace{0.05cm}.$$

This equation is also applicable to many other channel codes. The code word $\underline{x}^{(-i)}$ in this definition includes all symbols except $x_i$ and thus has length $n-1$ only.

In the "Exercise 4.4" it was shown that the extrinsic $L$ value can also be written as follows:

$$L_{\rm E}(i) = {\rm ln} \hspace{0.2cm} \frac{1 + \pi}{1 - \pi}\hspace{0.05cm}, \hspace{0.3cm} {\rm mit} \hspace{0.3cm} \pi = \prod\limits_{j \ne i}^{n} \hspace{0.15cm}{\rm tanh}(L_j/2) \hspace{0.05cm}.$$

In this exercise, we will now look for another calculation possibility.

Hints:

This exercise belongs to the chapter "Soft–in Soft–out Decoder".
Reference is made in particular to the "Calculations of extrinsic LLRs" page.
Above you can see a table with the numerical values of the function $y = \tanh(x)$ ⇒ hyperbolic tangent.
With the rows highlighted in red you can read the values of the inverse function $x = \tanh^{-1}(y)$ needed for the subtask (5).

Questions

Solution

(1) According to the specification applies:

$$L_{\rm E}(i) = {\rm ln} \hspace{0.2cm} \frac{1 + \pi}{1 - \pi}\hspace{0.05cm}, \hspace{0.3cm} {\rm with} \hspace{0.3cm} \pi = \prod\limits_{j \ne i}^{3} \hspace{0.15cm}{\rm tanh}(L_j/2) \hspace{0.05cm}.$$

From the table on the specification page can be read:

$$\tanh {(L_1/2)} = \tanh {(0.5)} = 0.4621,$$

$$\tanh {(L_2/2)} = \tanh {(0.2)} = 0.1974.$$

Since the hyperbolic tangent is an odd function, the following applies further

$$\tanh {(L_3/2)} = -\tanh {(0.5)} = -0.4621.$$

Calculation of $L_{\rm E}(1)$:

$$\pi = {\rm tanh}(L_2/2) \cdot {\rm tanh}(L_3/2) = (+0.1974) \cdot (-0.4621) = - 0.0912\hspace{0.3cm} \Rightarrow \hspace{0.3cm} L_{\rm E}(1) = {\rm ln} \hspace{0.2cm} \frac{1 -0.0912}{1 +0.0912}\hspace{0.15cm}\underline{=-0.1829} \hspace{0.05cm}.$$

Calculation of $L_{\rm E}(2)$:

$$\pi = {\rm tanh}(L_1/2) \cdot {\rm tanh}(L_3/2) = (+0.4621) \cdot (-0.4621) = - 0.2135\hspace{0.3cm} \Rightarrow \hspace{0.3cm} L_{\rm E}(2) = {\rm ln} \hspace{0.2cm} \frac{1 -0.2135}{1 +0.2135}\hspace{0.15cm}\underline{=-0.4337} \hspace{0.05cm}.$$

Calculation of $L_{\rm E}(3)$:

$$\pi = {\rm tanh}(L_1/2) \cdot {\rm tanh}(L_2/2) = (+0.4621) \cdot (+0.1974) = + 0.0912\hspace{0.3cm} \Rightarrow \hspace{0.3cm} L_{\rm E}(3) = {\rm ln} \hspace{0.2cm} \frac{1 +0.0912}{1 -0.0912}\hspace{0.15cm}\underline{=+0.1829}= - L_{\rm E}(1) \hspace{0.05cm}.$$

(2) The correct solutions are 1, 2, 3, and 5:

The function

$$y ={\rm tanh}(x) = \frac{{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm e}^{-x}} = \frac{1-{\rm e}^{-2x}}{1+{\rm e}^{-2x}}$$

is computable for all $x$ values and $\tanh(-x) = -\tanh(x)$ holds.

For large values of $x$, ${\rm e}^{-2x}$ becomes very small, so that in the limiting case $x → ∞$ the limit $y = 1$ is obtained.

(3) Since the hyperbolic tangent only yields values between $±1$, the inverse function $x = \tanh^{-1}(y)$ can also only be evaluated for $|y| ≤ 1$.

By rearranging the given equation

$$x ={\rm tanh}^{-1}(y) = 1/2 \cdot {\rm ln} \hspace{0.2cm} \frac{1+y}{1-y}$$

one obtains:

$${\rm e}^{2x} = \frac{1+y}{1-y} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} {\rm e}^{-2x} = \frac{1-y}{1+y} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} (1+y) \cdot {\rm e}^{-2x} = 1-y \hspace{0.3cm} \Rightarrow \hspace{0.3cm}y = \frac{1-{\rm e}^{-2x}}{1+{\rm e}^{-2x}} = {\rm tanh}(x) \hspace{0.05cm}.$$

This means:

The equation given in the proposed solution 2 is correct.
In the limiting case $y → 1$, $x = \tanh^{-1}(y) → ∞$ holds.
Also the inverse function is odd ⇒ in the limiting case $y → -1$ goes $x → -∞$.

Accordingly, the proposed solutions 2 and 4 are correct.

(4) Starting from the equation.

$$L_{\rm E}(i) = {\rm ln} \hspace{0.2cm} \frac{1 + \pi}{1 - \pi}$$

one arrives with the result of (3) at the equivalent equation corresponding to proposed solution 2:

$$L_{\rm E}(i) = 2 \cdot {\rm tanh}^{-1}(\pi)\hspace{0.05cm}.$$

(5) With the result of the subtask (1) we get.

for the first extrinsic $L$ value, since $\pi_1 = -0.0912$:

$$L_{\rm E}(1) = 2 \cdot {\rm tanh}^{-1}(-0.0912)= -2 \cdot {\rm tanh}^{-1}(0.0912) = -2 \cdot 0.0915\hspace{0.15cm}\underline{=-0.1830} \hspace{0.05cm}.$$

for the second extrinsic $L$ value, since $\pi_2 = -0.2135$:

$$L_{\rm E}(2) = -2 \cdot {\rm tanh}^{-1}(0.2135) = -2 \cdot 0.2168\hspace{0.15cm}\underline{=-0.4336} \hspace{0.05cm}.$$

for the third extrinsic $L$ value, since $\pi_3 = +0.0912 = -\pi_1$:

$$L_{\rm E}(3) = -L_{\rm E}(1) \hspace{0.15cm}\underline{=+0.1830} \hspace{0.05cm}.$$

The result was determined using the red table entries on the information page and, except for rounding errors (multiplication/division by $2$), agrees with the results of subtask (1).

	$\tanh\hspace{-0.05cm} {(x)} = ({\rm e}^x - {\rm e}^{-x}) \ / \ ({\rm e}^x + {\rm e}^{-x})$ is valid.
	$\tanh\hspace{-0.05cm} {(x)} = (1 - {\rm e}^{-2x}) \ / \ (1 + {\rm e}^{-2x})$ is valid.
	The function $y = \tanh\hspace{-0.05cm} {(x)}$ is defined for all $x$ values.
	$y_{\rm min} = 0$ and $y_{\rm max} → ∞$ is valid.
	$y_{\rm min} = -1$ and $y_{\rm max} = +1$ is valid.

	The function $x = \tanh^{-1}\hspace{-0.05cm} (y)$ is defined for all $y$ values.
	$x = \tanh^{-1}\hspace{-0.08cm} {(y)} = 1/2 \cdot \ln {[(1 + y) \ / \ (1 - y)]}$ is valid.
	$x_{\rm min} = -1$ and $x_{\rm max} = +1$.
	$x_{\rm min} → -∞$ and $x_{\rm max} → +∞$ is valid.

	$L_{\rm E}(i) = \tanh^{-1}\hspace{-0.08cm} {(\pi)}$ is valid.
	$L_{\rm E}(i) = 2 \cdot \tanh^{-1}\hspace{-0.08cm} {(\pi)}$ is valid.
	$L_{\rm E}(i) = 2 \cdot \tanh^{-1}\hspace{-0.05cm}\big [ {\ln {[(1 + \pi) \ / \ (1 - \pi)]}}\big ]$ is valid.