Synthesis of a cyclic code combination

The process of the cyclic code combination generation is considered in item 1.2.1 of this teaching manual. To get the separable code combination of CC we use formula (1.4).

For convertion of binary information sequence in polynomial form each bit (1 or 0) is multiplied by х in the degree corresponding to a site of this bit.

Let's transform the sequence received in item 2.1.1 in polynomial form.

S

I

х ^15

х ^14

х ^13

х ^12

х ^11

х ^10

х ^9

х ^8

х ^7

х ^6

х ^5

х ^4

х ^3

х ^2

х ^1

х ^0

The received code combination can be written down as:

G (x) = х ¹⁴ + х ⁹ + х ⁸ + х ⁵ + х ⁴ + х ³ +1.

Let's increase G (x) by a monomial х^r. As the amount of check bits, calculated in item 2.1.2 is seven, it is multiplied by х ⁷

G (x) х ⁷ = х ²¹ + х ¹⁶ + х ¹⁵ + х ¹² + х ¹¹ + х ¹⁰ + х ⁷.

For reception of the resolved combination of a cyclic code we will divide the received sequence into generator polynomial chosen in item 2.1.2.

Å	x ^2^1+ х ^16+ х ^15+ х ^12+ х ^11+ х ^10+ х ^7	x ^7+ x ^4+ x ^3+1
x ^2^1+ x ^18+ x ^17+ x ^14		x ^14+ x ^11+ x ^10+ x ^9++ x ^7+ x ^4+ x ^3+ x ^2+ x
Å	x ^18+ x ^17+ х ^16+ х ^15+ x ^14 +х ^12+ х ^11+ х ^10+ х ^7
x ^18+ x ^15+ x ^14+ x ^11
Å	x ^17+ х ^16+ х ^12+ х ^10+ х ^7
x ^17+ x ^1^4+ x ^1^3+ x ^1^0
Å	х ^16+ x ^1^4+ x ^1^3+ х ^12+ х ^7
x ^16+ x ^1^3+ x ^1^2+ x ^9
Å	x ^1^4+ x ^9+ х ^7
x ^14+ x ^1^1+ x ^1^0+ x ^7
Å	x ^1^1+ x ^1^0+ x ^9
x ^11+ x ^8+ x ^7+ x ^4
Å	x ^1^0+ x ^9+ x ^8+ x ^7+ x ^4
x ^10+ x ^7+ x ^6+ x ^3
Å	x ^9+ x ^8+ x ^6+ x ^4+ x ^3
x ^9+ x ^6+ x ^5+ x ^2
Å	x ^8+ x ^5+ x ^4+ x ^3+ x ^2
x ^8+ x ^5+ x ^4+ x
	x ^3+ x ^2+ x = R (x)

So, the resolved cyclic code combination, according to the formula (1.4) looks like:

F (x)= х ²¹ + х ¹⁶ + х ¹⁵ + х ¹² + х ¹¹ + х ¹⁰ + х ⁷+ x ³+ x ²+ x.

Let's convert it to a binary form:

0100 0011 0011 1001 0001110

Check of correctness of reception resolved CCC

Check of correctness of cyclic code combination we will execute in the binary form. For this purpose it is necessary sequence F (x) in the binary form to add modulo 2 with generator polynomial Р (x), also taken in the binary form (Р (x) ® 1001 1001). In case of correctness of construction, we will receive zero. Let’s check up it on the example resulted above.

Å
Å
Å
Å
Å
Å
Å
Å
Å

As the remainder of division is zero, composition of the resolved cyclic code combination was correct.

Coding and decoding of concatenated codes

Objective: To learn the encoding process by the concatenated code. To code the received in item 2.1.3 sequence by inner convolutional code (7, 5).

Task to the practical seminar:

1. To learn items 1.3.1, 1.3.3 – 1.3.4, 1.4.1 – 1.4.2 of this teaching manual.

2. To rearrange the sequence, got in item 2.1.3, by the block interleaver.

3. To encode received combination by the convolutional (7, 5) code.

4. To calculate the redundancy of the concatenated code and the amount of corrected errors.

5. To check up correction of double errors by entering the errors into received sequence (numbers of erroneous elements correspond to two last figures of the credit book).

Дата добавления: 2015-11-14