Decoding error probability of convolutional code

The technique of a decoding noise immunity estimation by convolutional codes is not differed from a technique stated in section 8.1 for a case of block codes. Here the code rate R _code, code distance properties (in a case of the convolutional codes – the free distance d _f), and decoding algorithm is played the main role. By using of probability decoding algorithm (Viterbi algorithm) the approximately expression for bit error probability looks like:

, (11.1)

where P_k – error probability of the way choice on a code trellis;

w_k – spectrum of weights of erroneous ways;

At transmission of code symbols through a channel with BPSK with white noise power spectral density N ₀/2 is defined so:

. (11.2)

Evaluations under formulas (11.1) and (11.2) show, that in the sum (11.1) by a big ratio signal/noise the first member (by k = d _f) has the greatest value, and remaining members of the sum with growth k fast decrease. Therefore in practice they are limited to use of the simplified formula:

. (11.4)

As well as by the block coding, comparison of a decoding noise immunity can be made with a noise immunity of coherent receiving of signals with binary phase modulation BPSK. Thus the calculation formula for bit error probability can be received from expression (11.2) having supposed k = 1, R _code = 1:

, (11.5)

where – the ratio of the signal energy expended on transmission of bit E _b to a power spectral density of a noise N ₀ on an input of the demodulator.

Exercise 11.1 The analysis of a decoding noise immunity

Let's make a calculations of a bit error probability on exits of the demodulator of signals BPSK and Viterbi decoder included after it, using formulas (11.5) and (11.4) for next codes:

1. Code (5, 7), R _code = 1/2, d _f = 5, n = 2;

2. Code (133, 171), R _code = 1/2, d _f = 10, n = 6.

The calculation results are given in table 11.1 and presented on figure 11.1. In the table given values of argument z are specified function Q (z), used in formulas.