Questions

7.1 What are the key parameters of Hamming codes?

7.2 What are the advantages of cyclic codes?

7.3 Is it possible to use Hamming codes and cyclical codes for correction of single errors? What will parameters of these codes be?

Tasks

7.1 The generator matrix of a code (7, 4) is set:

.

Define the allowed code word of this code b if word of simple code on encoder input a = (1110) is set.

7.2 The parity check matrix of a code (7, 4) is set:

.

Give a functional chart of the decoder of this code.

7.4 Consider an example of formation of a allowed code word if a word of a simple code is a = (10010).

7.5 By analogy to an example of section 7.1 make the table of Hamming codes parameters for values r = 2, 3, 4. As these codes have identical minimum distance, compare them on suitability for realization in practical systems. Formulate the recommendation and a substantiation of application of the best (in your opinion) code from this list.

7.6 For Hamming code recommended in the previous section, form generator and parity check matrixes.

7.7 By the rules stated in Exercise 4.1 define value of minimum distance by a generator matrix of code from the Task 7.5.

8 Decoding noise immunity of block codes

8.1 Decoding noise immunity of block codes

Let's define of an error probability by decoding of block codes in the binary symmetric channel. We will consider code (n, k) with minimal distance d_min. In such channel errors in sequentially transmitted code symbols (signals) occur independently with probability p (decoding in discrete channel without memory). Then the probability of that on length of the block n will occur a error multiplicity q, will be equal

.

Here – number of combinations from n elements on q. If the code corrects all errors of multiplicity q_corr = (d_min – 1)/2 (d_min – odd) and less then the probability of reception on decoder output the word with not corrected errors will be equal

.

Hence, the probability of erroneous decoding of block will satisfy to an inequality:

. (8.1)

In this expression equality takes place, if the perfect code is used. Parities between parameters n, k and q_corr are defined by the concrete chosen code.

Expression (8.1) allows to define the upper estimation of error probability of a code words by decoding of block codes in binary symmetric channel without memory. For calculation of probability of an error in concrete information (or additional) symbols it is necessary to know used decoding algorithm and structure of an error-control code(in particular, a set of distances from a transmitted code word to all allowed words). Such data for block codes are not published in a code tables and for calculations of error probability decoding of code symbols (information or additional) use the approximated formula [1]:

_. (8.2)

For channels with coherent receiving of signals BPSK the probability of an signal error reception is defined by the formula:

, (8.3)

where – the ratio of the energy spent on transfer of one binary symbol E_s to power spectral density of noise N₀ on a demodulator input;

– gaussian Q-function (probability integral) which tables contain the handbooks on probability theory and statistical calculations. For practical calculations it is convenient to use enough exact approximation:

Q(z) = 0,65 exp[–0,44(z + 0,75)²]. (8.4)

The introduction of redundancy by using of error-control coding leads to expansion of a frequency band that occupied with coded signal. if the frequency band in system without coding is F_s (Hz), the using of the code with a rate R_code= demands expansions of a frequency band

. (8.5)

i.e. there is an expansion of a frequency band in time. For codes with low code rate ( ) such expansion can appear appreciable. Therefore the problem of a code choice by designing of transmission system consists of search of a compromise between desirable degree of a noise immunity and expansion of a frequency band of the coded signal. Under formulas (8.2) and (8.3) taking into account expansion of a frequency band of coded signal according to the formula (8.5) following conclusions allow to draw on efficiency of application error-control coding:

1 With growth of a code word length n the error probability of an decoding p_d goes down.

2 Codes with the big redundancy (small code rate R_code) provide considerable decreasing of a decoding probability error.

3 By using of error-control codes in transmission systems as a payment for noise immunity increasing is expansion of frequency band of transmitted signal, caused by the redundancy entered by coding on size:

. (8.6)