Example of calculations and code optimisation procedure

Input data:

1 The rate of digital signal R = 64 kbit/s.

2 S/N ratio = 4,5 dB.

3 A modulation method is QPSK.

4 Mode of reception is coherent.

5 Pass band of transmission channel F_ch = 100 kHz.

6 Acceptable bit error probability p_acc = 10^–5.

7 Admissible code trellis complexity no more C_perm = 150.

Solution

1 Calculation of a necessary channel pass band for transmission with the method QPSK is made under formula F_QPSK = [R(1 + α)]/2, where α is roll-off factor of spectrum . Being set by value α = 0,4, we receive

F_QPSK =[R(1 + α)]/2 = [64 (1+0,4)]/2 = 44,8 kHz.

2 According to the formula (B.1) it is defined limiting value of code rate

.

3 Under code tables we select the codes, satisfying to the requirement on a rate. Data about these codes are shown in table B.1. From the table it is visible, that for the solving given task can be used codes with the rate R_code = 1/2 which ensure enough big A-gain. in the table data the code with generator polynomials (133, 171) which at rate R_code = 0,5 ensures A-gain =6,99 dB is chosen for the project. Data of bit error probability calculation is given on figure 8.1 (signals BPSK and QPSK have the same noise immunity). It is visible, that the using of a such code ensures such performance: by the ratio signal/noise = 4,5 dB the bit error probability is less than 10^–5. Comparison with curves for uncoded QPSK shows that by p = 10^–5 this code ensures coding gain 6 dB.

table B.1 – Performances for a code choice

Code rate Rcode	Generator polynomials	Code length 	Trellis complexity C	A-gain, dB
1/8	25,27,33,35, 37,25,33,37	4	32	6,02
1/8	115,127,131,135, 157,173,175,123	6	128	6,99
1/4	25,27,33,37	4	32	6,02
1/4	463,535,733,745	8	512	8,29
1/3	47,53,75	5	64	6,42
1/3	557,663,711	8	512	7,78
1/2	53,75	5	64	6,02
1/2	61,73	5	64	6,02
1/2	71,73	5	64	6,02
1/2	133,171	6	128	6,99
1/2	247,371	7	256	6,99

Table B.2 – Input data for the course work

variant number for an elaborating of course work should correspond to the number of student surname in the academic group register
variant number	S/N ratio , dB	modulation method	Rate of signal R, kbit/s	Bandwidth of channel Fch, kHz	Bit error probability pacc	Trellis complexity Cperm
1	4,0	QPSK	64	80	10-6	150
2	5,0	QPSK	16	25	10-4	160
3	6,0	BPSK	256	800	10-5	170
4	6,5	BPSK	64	200	10-6	180
5	4,0	QPSK	16	25	10-4	250
6	7,0	QPSK	128	200	10-5	350
7	5,0	BPSK	2400	7000	10-8	560
8	6,0	QPSK	32	50	10-6	200
9	5,0	BPSK	24	70	10-4	300
10	4,5	QPSK	256	400	10-5	250
11	5,5	BPSK	300	1200	10-8	550
12	4,0	QPSK	48	70	10-6	150
13	4,0	QPSK	32	50	10-4	250
14	5,0	BPSK	256	800	10-5	300
15	4,0	QPSK	450	1300	10-9	550
16	7,0	QPSK	56	90	10-6	150
17	5,0	BPSK	24	70	10-4	160
18	4,5	QPSK	256	400	10-5	200
19	5,5	QPSK	500	1400	10-9	550
20	6,0	BPSK	64	200	10-6	150
21	7,5	QPSK	32	400	10-4	250
23	6,5	QPSK	16	50	10-5	150
24	6,0	QPSK	64	150	10-6	150
25	4,5	BPSK	16	25	10-4	200
26	5,0	BPSK	6000	16000	10-9	550
27	6,0	QPSK	384	600	10-5	250
28	4,5	QPSK	56	100	10-6	150
29	5,0	BPSK	16	50	10-5	250
30	5,5	BPSK	5500	32000	10-9	560
31	4,5	QPSK	64	200	10-5	150
32	5,0	QPSK	64	300	10-5	250

Attachment C. Education manual for laboratory works

LW 4.1 Studying of block error-control Hamming code codecs structure

1 Objectives

1.1 Studying of Hamming systematic code (7, 4) codec structure.

1.2 Research of the code (7, 4) control ability.

2 Main principles

The systematic code is error-control code, which code word contain k information bits and r = n – k checking symbols (checking symbols are linear combination of information bits). Systematic codes are denoted as (n, k) or (n, k, d_min). In this work code (7, 4) or (7, 4, 3) is studied.

Error-control codes with code distance d_min = 3, allowing to correct first order errors at decoding, name as Hamming codes [5, p. 149]. We will determine connection between error-control code parameters n and k. It is known that for any natural number r the Hamming code of lengths n = 2^r – 1 or k + r = 2^r – 1 exists [5, p. 149]. These equalities can be used and as inequalities k  2^r ‑ r ‑ 1. The last expression allows choosing n and r at given k.

The matrix method of linear block codes coding and decoding processes description is most useful (see Sections 5, 6). So, coding by systematic code (n, k) consist in addition to code words checking symbols and can be described by matrix equality

AG = B, (1)

where A = (b₁ b₂ … b_k) is row matrix size k, correspond to information code word;

B = (b₁ b₂ … b_k b_k+1 … b_n) is row matrix size n, correspond to error-control code word.

G – generator matrix kn, the elements of which g_ij take on values 1 or 0.

The G matrix rows must satisfy next conditions [6, p. 86…88].

1 Distance between any two rows must not be less d_min.

2 Every row must contain no less then d_min units.

3 All rows must be linearly independent, i.e. none of rows can be got by adding (XOR) of some other rows.

For example, for a code (7, 4) generator matrix looks like formulas (4.4), (4.7).

The coder operation algorithm:

1 k information bits in a parallel code or in a series code (in last case a shift register is needed) on the coder input.

2 The checking symbols r = n – k by adders with r calculates.

3 k information bits and r checking symbols in a parallel or series code (in last case the converter of parallel in series code is needed) on the coder output.

On the figure 5.1 the code (7, 4) with generator matrix (4.4) encoder functional diagram is resulted. Input and output code words are represented in a parallel code.

The process of decoding includes the syndrome calculation. In matrix form is written down as:

S= H , (2)

where H is check matrix rn.

is matrix-column, size n, correspond to the code word on the decoder input;

S is syndrome, matrix-column, size r.

The decoder algorithm is following:

1 n symbols of the code word come on the decoder input.

2 Using (6) a syndrome calculates.

3 Syndromes analyzer, built on the syndromes table basis, forms signals for error symbols correction.

4 Error symbols correction consist in their inverting: XOR for error symbol and unit (lets is some symbol, then ).

5 After correction of error the information code word of k symbols come on the decoder output.

On the figure 5.2 the code (7, 4) decoder functional diagram is resulted. Input and output code words are represented in a parallel code.

A code, encoder and decoder of which, is built on multiinput adders (with XOR operation), name as Hamming code or even parity check code. Using the indicated principles it is possible to build encoder and decoder for different values of n and k.