Exercise 3.10: Metric Calculation

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Only partially evaluated trellis

In the  $\text{theory section}$  of this chapter,  the calculation of the branch metrics  ${\it \Gamma}_i(S_{\mu})$  has been discussed in detail,  based on the Hamming distance   $d_{\rm H}(\underline{x}\hspace{0.05cm}', \ \underline{y}_i)$   between

• the possible code words   $\underline{x}\hspace{0.05cm}' ∈ \{00, \, 01, \, 10, \, 11\}$ 

• and the 2–bit–words  $\underline{y}_i$  received at time  $i$.

The exercise deals exactly with this topic.  In the adjacent graph

• the considered trellis is shown  – valid for the code with rate  $R = 1/2$,   memory  $m = 2$   and 

$$\mathbf{G}(D) = (1 + D + D^2, \ 1 + D^2),$$

• the received words  $\underline{y}_1 = (01), \hspace{0.05cm}\text{ ...} \hspace{0.05cm} , \ \underline{y}_7 = (11)$  are indicated in the rectangles,

• all branch metrics  ${\it \Gamma}_0(S_{\mu}), \hspace{0.05cm}\text{ ...} \hspace{0.05cm} , \ {\it \Gamma}_4(S_{\mu})$  are already entered.

For example,  the branch metric  ${\it \Gamma}_4(S_0)$  with  $\underline{y}_4 = (01)$  as the minimum of the two comparison values

• ${\it \Gamma}_3(S_0) + d_{\rm H}((00), \ (01)) = 3 + 1 = 4$,  and

• ${\it \Gamma}_3(S_2) + d_{\rm H}((11), \ (01)) = 2 + 1 = 3$.

The surviving branch  – here from   ${\it \Gamma}_3(S_2)$   to   ${\it \Gamma}_4(S_0)$   – is drawn solid,  the eliminated branch from   ${\it \Gamma}_3(S_0)$   to   ${\it \Gamma}_4(S_0)$   dotted.  Red arrows represent the information bit  $u_i = 0$,  blue arrows  $u_i = 1$.

In subtask  (4)  shall be worked out the relationship between

• the  ${\it \Gamma}_i(S_{\mu})$  minimization and
• the  ${\it \Lambda}_i(S_{\mu})$  maximization.

Here,  we refer to the nodes  ${\it \Lambda}_i(S_{\mu})$  as  "correlation metrics",  where the metric increment over the predecessor nodes results from the correlation value  $〈\underline{x}_i\hspace{0.05cm}', \, \underline{y}_i 〉$.  For more details on this topic,  see the following theory sections:

1. "Relationship between Hamming distance and correlation"
2. "Viterbi algorithm based on correlation and metrics"
3. "Viterbi decision for non–terminated convolutional codes".

Hints:

• The exercise refers to the chapter  "Decoding Convolutional Codes".

• For the time being,  the search of surviving paths is not considered. 

• This will be dealt with for the same example in the later  $\text{Exercise 3.11}$.

Questions

1

What are the branch metrics for time  $i = 5$?

${\it \Gamma}_5(S_0) \ = \ $

${\it \Gamma}_5(S_1) \ = \ $

${\it \Gamma}_5(S_2) \ = \ $

${\it \Gamma}_5(S_3) \ = \ $

2

What are the branch metrics for time  $i = 6$?

${\it \Gamma}_6(S_0) \ = \ $

${\it \Gamma}_6(S_2) \ = \ $

3

What is the final value of this trellis based on  ${\it \Gamma}_i(S_{\mu})$?

	It holds  ${\it \Gamma}_7(S_0) = 3$.
	This final value suggests one error-free transmission.
	This final value suggests three transmission errors.

4

Which statements are true for the  ${\it \Lambda}_i(S_{\mu})$  evaluation?

	The correlation metrics  ${\it \Lambda}_i(S_{\mu})$  provide the same information as  ${\it \Gamma}_i(S_{\mu})$.
	For all nodes,  ${\it \Lambda}_i(S_{\mu}) = 2 \cdot \big [i \, –{\it \Gamma}_i(S_{\mu})\big ]$.
	For the metric increments,  $〈 \underline{x}_i', \, \underline{y}_i 〉 ∈ \{0, \, 1, \, 2\}$.

Solution

Solution

(1)  At all nodes  $S_{\mu}$  a decision must be made between the two incoming branches.  The branch that led to the (minimum) error metric  ${\it \Gamma}_5(S_{\mu})$  is then selected in each case.  With  $\underline{y}_5 = (01)$  one obtains:

$${\it \Gamma}_5(S_0) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}{\rm min} \left [{\it \Gamma}_{4}(S_0) + d_{\rm H} \big ((00)\hspace{0.05cm},\hspace{0.05cm} (01) \big )\hspace{0.05cm},\hspace{0.2cm}{\it \Gamma}_{4}(S_2) + d_{\rm H} \big ((11)\hspace{0.05cm},\hspace{0.05cm} (01) \big ) \right ] ={\rm min} \left [ 3+1\hspace{0.05cm},\hspace{0.05cm} 2+1 \right ] \hspace{0.15cm}\underline{= 3}\hspace{0.05cm},$$

$${\it \Gamma}_5(S_1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}{\rm min} \left [{\it \Gamma}_{4}(S_0) + d_{\rm H} \big ((11)\hspace{0.05cm},\hspace{0.05cm} (01) \big )\hspace{0.05cm},\hspace{0.2cm}{\it \Gamma}_{4}(S_2) + d_{\rm H} \big ((00)\hspace{0.05cm},\hspace{0.05cm} (01) \big ) \right ] ={\rm min} \left [ 3+1\hspace{0.05cm},\hspace{0.05cm} 2+1 \right ] \hspace{0.15cm}\underline{= 3}\hspace{0.05cm},$$

$${\it \Gamma}_5(S_2) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}{\rm min} \left [{\it \Gamma}_{4}(S_1) + d_{\rm H} \big ((10)\hspace{0.05cm},\hspace{0.05cm} (01) \big )\hspace{0.05cm},\hspace{0.2cm}{\it \Gamma}_{4}(S_3) + d_{\rm H} \big ((01)\hspace{0.05cm},\hspace{0.05cm} (01) \big ) \right ] = {\rm min} \left [ 3+2\hspace{0.05cm},\hspace{0.05cm} 2+0 \right ] \hspace{0.15cm}\underline{= 2}\hspace{0.05cm},$$

$${\it \Gamma}_5(S_3) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}{\rm min} \left [{\it \Gamma}_{4}(S_1) + d_{\rm H} \big ((01)\hspace{0.05cm},\hspace{0.05cm} (01) \big )\hspace{0.05cm},\hspace{0.2cm}{\it \Gamma}_{4}(S_3) + d_{\rm H} \big ((10)\hspace{0.05cm},\hspace{0.05cm} (01) \big ) \right ] ={\rm min} \left [ 3+0\hspace{0.05cm},\hspace{0.05cm} 2+2 \right ] \hspace{0.15cm}\underline{= 3}\hspace{0.05cm}.$$

Evaluated trellis diagrams

The left sketch in the graph shows the final evaluated ${\it \Gamma}_i(S_{\mu})$ trellis.
(2)  At time  $i = 6$  the termination is already effective and there are only two branch metrics left.  For these one obtains with  $\underline{y}_6 = (01)$:

$${\it \Gamma}_6(S_0) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}{\rm min} \left [{\it \Gamma}_{5}(S_0) + d_{\rm H} \big ((00)\hspace{0.05cm},\hspace{0.05cm} (01) \big )\hspace{0.05cm},\hspace{0.2cm}{\it \Gamma}_{5}(S_2) + d_{\rm H} \big ((11)\hspace{0.05cm},\hspace{0.05cm} (01) \big ) \right ] ={\rm min} \left [ 3+1\hspace{0.05cm},\hspace{0.05cm} 2+1 \right ] \hspace{0.15cm}\underline{= 3}\hspace{0.05cm},$$

$${\it \Gamma}_6(S_2) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}{\rm min} \left [{\it \Gamma}_{5}(S_1) + d_{\rm H} \big ((10)\hspace{0.05cm},\hspace{0.05cm} (01) \big )\hspace{0.05cm},\hspace{0.2cm}{\it \Gamma}_{5}(S_3) + d_{\rm H} \big ((01)\hspace{0.05cm},\hspace{0.05cm} (01) \big ) \right ] ={\rm min} \left [ 3+2\hspace{0.05cm},\hspace{0.05cm} 3+0 \right ] \hspace{0.15cm}\underline{= 3}\hspace{0.05cm}.$$

(3)  The final value results to

$${\it \Gamma}_7(S_0) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}{\rm min} \left [{\it \Gamma}_{6}(S_0) + d_{\rm H} \big ((00)\hspace{0.05cm},\hspace{0.05cm} (11) \big )\hspace{0.05cm},\hspace{0.2cm}{\it \Gamma}_{6}(S_2) + d_{\rm H} \big ((11)\hspace{0.05cm},\hspace{0.05cm} (11) \big ) \right ] ={\rm min} \left [ 3+2\hspace{0.05cm},\hspace{0.05cm} 3+0 \right ] \hspace{0.15cm}\underline{= 3}\hspace{0.05cm}.$$

• With the BSC model,  one can infer from  ${\it \Gamma}_7(S_{\mu}) = 3$  that three transmission errors occurred   ⇒   solutions 1 and 3.

(4)  Correct are the   statements 1 and 2:

• Maximizing the correlation branch metrics  ${\it \Lambda}_i(S_{\mu})$  according to the right sketch in the above graph gives the same result as minimizing the Hamming branch metrics ${\it \Gamma}_i(S_{\mu})$ shown on the left.

• Also,  the surviving and deleted branches are identical in both graphs.

• The given equation is also correct,  which is shown here only on the example  $i = 7$:

$${\it \Lambda}_7(S_0)) = 2 \cdot \big [i - {\it \Gamma}_7(S_0) \big ] = 2 \cdot \big [7 - 3 \big ] \hspace{0.15cm}\underline{= 8}\hspace{0.05cm}.$$

• The last statement is false.  Rather applies  $〈x_i', \, y_i〉 ∈ \{–2, \, 0, \, +2\}$.

• In  $\text{Exercise 3.11}$,  the path finding will be demonstrated for the same example,  assuming ${\it \Lambda}_i(S_{\mu})$  metrics as shown in the right graph.