Key parameters and classification of convolutional codes

code rate is defined as

, (9.3)

where k – an amount of information symbols simultaneously arriving on k encoder inputs, n – an amount of code symbols corresponding to them on n encoder outputs.

It is used several parameters for definition of memory length by coding. The length of encoder register K is equal to an amount of delay elements containing in encoder scheme. length of encoder registeroften apply to memory definition by coding with rate , when encoder contains one register. The encoder represented on figure 9.1 has register length K = 3. If encoder contains some inputs (k > 1) so lengths of registers connected to each input, can be various. In this case it is defined a code constrained length.

The code constrained length on each input is defined by the higher degree of corresponding generator polynomials

n _i = max [deg ].

The resultant code constrained length is defined by the sum:

. (9.4)

For codes with one register (k = 1) the values n and K are connected by a simple relation

n = K. (9.5)

For comparison of a decoding algorithm complexity it is used complexity performance. as it was marked earlier, development of trellis diagram consists in a repetition of the same step (see figure 9.3). diagram complexity is accepted to define an amount of branches on a step of trellis diagram. The number of states of a trellis is defined by number of variables K = n on inputs of register elements. As a result complexity of one trellis step can be defined an amount of branches on this step

C = m ^{(n + k)} (9.6)

The decoding noise immunity depends on distance properties of code sequences on encoder input. Thus for binary codes the distance between sequences is often estimated in Hamming metric.

Free distance of a convolution code d _f – is the minimum distance between two arbitrary semi-infinite sequences on the encoder output which differing from the first branch. For short codes free distance can be defined under the state diagram. If the binary code diagram is set free distance is equally to minimum Hamming weight of a way under the diagram from a state 00 in the same state (excepting a loop at this state). On the diagram figure 9.2 it is visible, that free distance d _f= 5. On the value of free distance judge about control properties of convolution codes. In particular, if two ways on encoder output, going out from one state on the trellis diagram, differ in Hamming metric on the value d _f, that by decoding on a minimum distance (with analogy to a case of block codes decoding (see Section 3.1)) the multiplicity of corrected errors is defined by expression

, (d _f is odd). (9.7)

The free distance is used for an estimation of a noise immunity of convolution codes decoding with decoding algorithms by a maximum a posterior probability or close to them (Viterbi algorithm etc.). In a systematic code on k (from n possible) encoder outputs there are information sequences of transmitted symbols, and on remaining (n – k) exits – the sequences of the additional symbols formed as linear combination of information symbols. By rate R _code =1/2 generator polynomials of a systematic code look like

g ⁽¹⁾(D) = 1 and g ⁽²⁾(D) = g ₀⁽²⁾ + g ₁⁽²⁾ D + g ₂⁽²⁾ D ² +... + g _n⁽²⁾ D ⁿ.

Systematic codes allow to receive on a receiving site an estimation of information symbols, without decoding or any other processing of received symbols. Nonsystematic codes do not possess such property. As well as in case of a block codes the using of convolution coding with rate leads to expansion of a signal frequency band in the channel. Thus the of band expansion factor is defined by expression:

. (9.8)

By small code rates the considerable band expansion becomes unacceptable, therefore try to apply encoding with a high code rate. Practically, a choice of code parameters is made on the basis of the compromise, proceeding from demanded level energy coding gain and admissible value of frequency band expansion factor.

Exercise 9.1 The analysis of code parameters connections.

Using consecutive modification of the structure of initial encoder (7, 5) and corresponding to it state diagram and a trellis (figures 9.1, 9.2 and 9.3) we will establish connections of the encoder parameters k, n, R _code, S and generator polynomial with code free distance d _f. We will consider some variants of the codes:

1. Initial code (7, 5). Its scheme is resulted in figure 9.1. The diagram of states is constructed in figure 9.2.

Parameters of the code (7, 5): k = 1, n = 2, k = 2, R _code = 1/2, since code is binary (m =2) then S = 2 ^k = 4, free distance d _f = 5, code is nonsystematic.

2. Forming of a systematic code (1, 5).

Let modify the first polynomial of an initial code, having left one connection, as shown in figure 9.1. The state diagram will partially vary. The number of a states remains former as the structure of encoder register has not varied. Nonzero branches vary: according to a modification of the first generator polynomial on a place of the first branch numeral it is necessary to write down the first numeral of a state to which this branch is directed (figure 9.4). The code rate also has not varied.

Parameters of the code (1, 5): k = 1, n = 2, k = 2, R _code = 1/2, since code is binary (m =2) then S = 2 ^k = 4, free distance has decreased d _f = 3, code is systematic.

This example illustrates the general conclusion of a coding theory: on the free distance the systematic code appear worse of a nonsystematic codesfrom which they are organised. Therefore in practice it is preferred to use the nonsystematic codes.

3. Forming of a nonredundancy code (1,0).

This, apparently, the "exotic" example allows to reveal a role of the nonzero generator polynomials forming additional symbols (figure 9.5).

Parameters of the code (1, 0): k = 1, n = 1, k = 2, R _code = 1, since code is binary (m =2) then S = 2 ^k = 4, free distance has considerably decreased d _f = 1.

Encoder is systematic without a additional symbols.

Actually, nonredundancy coding is present (memory of the encoder is not used). Therefore the code free distance is equally d _f = 1, also corresponds to a rate of the nonredundancy code R _code = 1. All increment of free distance in the a code is considered in variant 2 spoke presence of nonzero additional symbols.

In Attachment А.3 performances of binary convolutional codes with maximum free Hamming distance for various code rates are given.