Exercise 3.13: Path Weighting Function again

For the reduction of the state transition diagram

In the "rules for manipulating the state transition diagram" section, for the example of our rate 1/2 standard code with memory $m = 2$ and the transfer function matrix

$${\boldsymbol{\rm G}}(D) = \big ( 1 + D + D^2\hspace{0.05cm},\hspace{0.1cm} 1 + D^2 \hspace{0.05cm}\big )$$

the calculation of path weighting enumerator functions was described in great detail. The results given were:

$$T_{\rm enh}(X, U) \hspace{-0.2cm} \ = \ \hspace{-0.2cm} \frac{U\hspace{-0.05cm} X^5}{1- 2U\hspace{-0.05cm}X} =U\hspace{-0.05cm}X^5 \cdot \big [ 1 + (2U\hspace{-0.08cm}X) + (2U\hspace{-0.08cm}X)^2 + \text{...} \hspace{0.05cm} \big ] \hspace{0.01cm},$$

$$T(X) \hspace{-0.2cm} \ = \ \hspace{-0.2cm} \frac{X^5}{1- 2X} = X^5 \cdot \big [ 1 + (2X) + (2X)^2 + \text{...} \hspace{0.05cm} \big ] \hspace{0.05cm}.$$

Now the same calculations shall be done for the "equivalent systematic code" with the transfer function matrix

$${\boldsymbol{\rm G}}(D) = \big ( 1 \hspace{0.05cm},\hspace{0.1cm} (1 + D^2)/(1 + D + D^2) \hspace{0.05cm}\big )$$

can be performed.

The diagram shows the state transition diagram $\rm (A)$ and the structure of the reduced diagram $\rm (B)$, where the transitions are denoted by $A(X, \, U), \ \text{...}\ , \ G(X, \, U)$ in general.
In the subtask (1) these abbreviations are to be adapted to the state transition diagram $\rm (A)$ .

Hints:

This exercise belongs to the chapter "Distance Characteristics and Error Probability Barriers".
For the solution of the subtasks (2) and (3)' we refer here again to the section "Rules for manipulating the state transition diagram" in the theory section.

Solution

(1) All proposed solutions are correct. In the adapted diagram $\rm (B)$ all transitions are drawn:

Reduction of the state transition diagram

The transition from $S_0$ to $S_1$ is denoted by "$1 \ | \ 11$".
The output sequence $\underline{x}_i = (11)$ is expressed by $X^2$, the input bit $u_i = 1$ by $U$.
The same result is obtained for $G(X, \, U)$:

$$A(X, U) = G(X, U)= UX^2 \hspace{0.05cm}.$$

The output sequences $\underline{x}_i = (01)$ as well as $\underline{x}_i = (10)$ are both marked with $X$.
Taking into account the input bits, we thus obtain:

$$u_i = 1\hspace{-0.1cm}:\hspace{0.15cm} B(X, U) = D(X, U)= UX\hspace{0.05cm},$$

$$u_i = 0\hspace{-0.1cm}:\hspace{0.15cm} C(X, U) = E(X, U)= X \hspace{0.05cm}.$$

The transition "$0 \ | \ 00$" from $S_2$ to $S_1$ is expressed by $F(X, \, U) = 1$.

(2) According to the procedure in section "Rules for manipulating the state transition diagram" in the theory section, first the transition from $S_1$ to $S_2$ via $S_3$ is summarized by an ring.

One obtains for the red background in the diagram $\rm (B)$:

$$T_1(X, U) = \frac{A(X, U) \cdot B(X, U)}{1- C(X, U)} = \frac{X \cdot X}{1- U \cdot X} \hspace{0.05cm}.$$

The two parallel transitions corresponding to the blue background in the diagram $\rm (C)$ can be combined as follows:

$$T_2(X, U) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} T_1(X, U) + B(X, U) =\frac{X^2}{1- U X}+ U X = \frac{X^2 + U- U^2X^2}{1- U X} \hspace{0.05cm}.$$

The extended path weighting enumerator function results according to diagram $\rm (D)$ as feedback:

$$T_{\rm enh}(X, U) = \frac{A(X, U) \cdot G(X, U)\cdot T_2(X, U)}{1- F(X, U) \cdot T_2(X, U)} = \frac{UX^2 \cdot UX^2\cdot \frac{X^2 + UX- U^2X^2}{1- U X}}{1- 1 \cdot \frac{X^2 + UX- U^2X^2}{1- U X}}\hspace{0.05cm}.$$

Even after several attempts, the author has not succeeded in further simplifying this expression in a target-oriented way. He tends to the proposed solution 3 with the addition "without guarantee".

However, this result would imply that the extended path weighting enumerator function of the equivalent systematic code is different from that of the nonsystematic code.
We will still clarify this issue with an expert.

(3) Correct are the proposed solutions 1 and 2:

If one sets the formal parameter $U = 1$ in the extended function $T_{\rm enh}(X, \, U)$, one obtains the solution proposition 1:

$$T(X) = \frac{X^4 \cdot \frac{X^2 + X- X^2}{1- X}}{1- \frac{X^2 + X- X^2}{1- X}}= \frac{X^5 }{1- X - X} = \frac{X^5 }{1- 2X} \hspace{0.05cm}.$$

Using the series expansion $1/(1 \, –x) = 1 + x + x^2 + \ \text{...}\ $, we arrive at the proposed solution 2.
That is, the simple path weighting enumerator function $T(X)$ matches for both codes.

	$A(X, \, U) = UX^2$,
	$B(X, \, U) = UX$,
	$C(X, \, U) = X$,
	$D(X, \, U) = UX$,
	$E(X, \, U) = X$,
	$F(X, \, U) = 1$,
	$G(X, \, U) = UX^2$.

	$T_{\rm enh}(X, \, U) = UX^5 \ / \ (1 \, –2UX)$.
	$T_{\rm enh}(X, \, U) = UX^5 + 2 U^2X^6 + 4U^3X^7 + 8U^4X^8 + \ \text{...}$,
	None of the proposals are correct.

	$T(X) = X^5 \ / \ (1 \, –2X)$.
	$T(X) = X^5 + 2X^6 + 4X^7 + 8X^8 + \ \text{...} $
	None of the proposals are correct.