1.3.2 Decoding of the systematic convolutional codes

The sufficient condition of systematic CvC with threshold decoding construction with R = 1/2 is implementation of two positions:

, (1.27)

where is perfect difference set.

Subset consisting of j integers is difference set (DS) with parameters (v, j, λ), if all differences on the modulo v at i ≠ k will be equal to 1, 2, 3, ..., v – 1 exactly λ times.

DS with λ = 1 are perfect, or simple. Binary signals built on the basis of perfect DS possess a two-digit autocorrelation function and find application not only in a theory and technique of the noiseproof encoding but also in the different systems of telemetry and radio-locations.

Perfect DS, i.e. with parameters (v, j, 1) have property "no more than one coincidence". It means that at any time shift of impulsive sequence consistent with the elements of DS, will be no more than one coincidence of impulses of initial and moved sequence. This property allows realize the threshold decoding of systematic convolutional codes.

For example, the set Q = {0, 2, 5, 6} is perfect difference set with parameters j = 4, v = 13. The choice of these four numbers can be explained by the fact that all possible differences on v modulo form sequence of positive natural numbers N = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. This can be explained as follows. Completing all possible difference subtractions of Q, given that 13 ≡ 0 mod 13, we obtain:

0 – 0 = 0,

0 – 2  13 – 2 = 11, 2 – 0 = 2,

0 – 5  13 – 5 = 8, 5 – 0 = 5,

0 – 6  13 – 6 = 7, 6 – 0 = 6,

2 – 5  2 + 13 – 5 = 10, 5 – 2 = 3,

2 – 6  2 + 13 – 6 = 9, 6 – 2 = 4,

5 – 6  5 + 13 – 6 = 12, 6 – 5 = 1.

Perfect DS Q = {0, 2, 5, 6} determines systematic CvC with generator polynomials

G₁(D) = 1, G₂(D) = 1 + D² + D⁵ + D⁶. (1.28)

Here the first polynomial corresponds with the formation of information symbols a_i, and the second with the formation of check symbols b_i.

The flow diagram of this coder is shown in the fig. 1.5. The shift register Rg1, consists of six memory cells D1, ..., D6, which save received binary symbols. Multi-input adder modulo 2 A1 intends for the formation of check symbols. The multiplexer MP realizes time multiplexing of information symbols a_i, a_i–1, a_i–2, …, a_i–n and check bits b_i, b_i–1, b_i–2, …, b_i–n, where i is the number of symbol at the entrance of Rg1. The frequency of reference generator f₀ must be selected according to

Figure 1.5 – Flow diagram of the coder CvC with polynomials G₁(D) = 1, G₂(D) = 1 + D² + D⁵ + D⁶.

transmission rate of the communication channel. Divisor D:2 with factor of division 2 determines the frequency of timed pulses of the coder.

To generate the check symbols b_i inputs of A1 connect to the outputs of the register Rg1 as follows

b_i = a_i  a_i–2  a_i–5  a_{i–6 .}

To determine free code distance of CvC, which is shown in the fig. 1.5, we have to encode the single influence of kind e_i = 1 0 0 0 0 0 0 . . . 0 by the formula (1.22):

v = 11 00 01 00 00 01 01 00 00 …

W(v) = 5, so d_f = 5.

Constraint length of such code and the amount of informative bits in the limits of CL are

N_a = n₀ (K + 1) = 2 (6 + 1) = 14, K_a = K + 1 = 7.

So, this code can be denoted as (14, 7).

As the amount of inputs of the coder (14,7) k₀ = 1, and the amount of the outputs n₀ = 2, relative information transfer rate and redundancy are

R = k₀/n₀ = 1/2, ρ = 1 – R = 1/2.

Flow diagram of the convolutional decoder with threshold decoding method is shown in fig. 1.6. The algorithm of threshold decoding method consists in the following. After digital demultiplexing in DMP of the information and check bits of the received sequence

= a_i  e_i and = b_i  e_i ,

where e_i – noise symbols, informative enter register Rg2. As a result, as forming of check symbols of the coder b_i, the sequence of local check symbols appears in a decoder

=    .

After adding of the check symbols, formed in the decoder and received from the channel , on the output of A2 the syndrome sequence S_i appears, that

Figure 1.6 – Flow diagram of the convolutional decoder (14,7)

enters the register Rg3. Note that units in this sequence will appear only in the case of mismatch of and . Connecting of this register to the threshold device TD corresponds to the reverse order of connections in Rg2. Unit on the output of threshold device is formed for the correction of error bit, which is on the output of Rg2, only at the presence on its inputs of three in any order or four units.

As a result of analysis of S_i by TD a decoder makes decision about possible validity of the received informative sequence . If during some time error in a channel does not appear, then in a syndrome register Rg3 will follow only zero symbols, because = 0. At the presence of errors on the output of decoder in the sequence S_i, except zeros, units appear.

A threshold device (fig. 1.7) with connecting to the syndrome register, determines character of the following units and zeros. If necessary, TD gives out the signal of correction of error on the second entrance of adder A3 (i.e. unit) that synchronizes with the moment of read-out of the error information bit from the output of Rg2.

Figure 1.7 – Flow diagram of threshold device

In the given flow diagram of decoder (see a fig. 1.6) there is a feed-back with memory cells of syndrome register for the correction of Rg3 for preparation of favourable conditions at the correction of the next error information bits. Note that at intensive interferences in a communication channel the feed-back of decoder can give a result in additional formation of errors.