5.3 Coding of sequence by convolutional code

Coding is multiplication of a polynomial information sequence и(D) on generating polynomials. For example, for codes with a speed R = 1/п sequence of symbols on i-th coder output

(10)

Let's consider coding process on an example of the sequence¹ 11000001 and generating polynomials G ⁽¹⁾ = D² + D + 1 and G ⁽²⁾ = D² + 1.

u (D) = D⁷ + D⁶ + 1

v ⁽¹⁾ (D) = (D⁷ + D⁶ + 1) (D² + D + 1) = D⁹ + D⁸ + D² + D⁸ + D⁷ + D + D⁷ + D⁶ + 1 =

= D⁹ + D⁶ + D² + D +1 1001000111.

v ⁽²⁾ (D) = (D⁷ + D⁶ + 1) (D² + 1) = D⁹ + D⁸ + D² + D⁷ + D⁶ + 1 =

= D⁹ + D⁸ +D⁷ + D⁶ +D² +1 1111000101.

Let's write down now the general sequence on a coder output, reading out in pairs v ⁽¹⁾ v ⁽²⁾:

v = … 11.01.01.11.00.00.00.11.10.11 …

5.4 Decoding of sequence on Viterbi's algorithm

Decoding is tracking on a code lattice of a way with maximum a posteriori probability. At decoding choose such sequence of links , which provides a sum minimum

WM = , (11)

where - input sequence of the decoder; - set of code links estimation.

The expression (11) is named the decoded way metrics (WM). The way metrics is the sum of links metric: WM = , where links metrics is . In the discrete channel for an estimation of distances use Heming's metrics.

The periodic structure of the trellis diagram essentially simplifies comparison and a choice of ways according to rules (11). The number of states on a lattice is limited, and two at random enough long ways have, as a rule, the general states. Pieces of the ways entering into these states, it is necessary to compare and choose a way with the least metrics. Such way name survived. According to Viterbi's algorithm comparison and rejection of pieces of ways is made periodically on each step of decoding. We will consider decoding (7, 5) which symbols are transferred on the discrete channel. In this case the links metrics LM is equal to Heming's distance between a character set z ⁽¹⁾ z ⁽²⁾ on an decoder input and a character set v ⁽¹⁾ v ⁽²⁾, corresponding to the given link on the trellis diagram. If z ⁽¹⁾ z ⁽²⁾ = 01, possible values LM for a code (7, 5) with a lattice represented on fig. 5, will be: LM (00) = 1, LM (01) = 0, LM (11) = 1 and LM (10) = 2. The way metrics is the sum of metrics of the links forming some way on the trellis diagram. The state metrics SM is equal to WM which comes to an end in the given state.

On fig. 6 development of decoding process for code (7, 5) is shown. On a decoder input pairs of symbols from the channel z ⁽¹⁾ z ⁽²⁾ = 11 10 00 11 01 arrived... Figures about links designate links metrics; figures in circles designate state metrics. In the initial moment of time we believe, that the decoder is in a state 00 and initial metrics SM (00) = 0. If from the channel symbols 11, metrics of two links leaving this state, LM (00) = 2 and LM (11) = 0 have arrived. As other links from a state 00 in states 00 and 11 are not present, metrics of these states accept equal to metrics of entering links: SM (00) = 2 and SM (10) = 0. Similarly and on a following step when from the channel symbols 10 arrive. Here LM (00) = 1, LM (11) = 1 and LM (10) = 0, LM (01) = 2. States metrics on this step are defined now as the sums of metrics of entering links with metrics of the previous states: SM (00) = 2 + 1 = 3, SM (10) = 2 + 1 = 3, SM (01) = 0 + 0 = 0, SM (11) = 0 + 2 = 2. On it development of the trellis diagram comes to an end.

Links metrics

States

Survived

ways

Figure 6

To each new state conducts two ways. For example, to a condition 00 conduct ways from states 00 and 01. The decoder calculates metrics of ways as the sums of metrics of the previous states and metrics of entering links.

Further make paired comparison of metrics of the ways entering into each state. As a result of comparison the decoder choose the smaller metrics and it consider as the metrics of the given state for the subsequent step of decoding. The way entering into the given state with the smaller metrics, consider survived. The ways entering into states with the big metrics, consider torn off. They are shown on the trellis diagram by shaped lines.

If metrics of compared ways are identical, a choice of one of two ways make in a random way. On each step as a result of comparison half of possible ways is rejected and further is not used. Other half forms continuations of ways for a following step of decoding. From each condition on a following step again there are two variants of continuation of ways. It provides a constancy of the calculations made on each step. The decoder traces a way having the minimum distance from a way which generates the coder on a code lattice.

1 Coding process is necessary to exercise for the sequence received in item 5.1