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LDPC encodingINFORMATION THEORY AND CODING (ECE 307)

Group Members:

1. Lokesh Jindal (11BEC1043)2. Bhagwat Singh (11BEC1070)3. Devanshu (11BEC1100)4. Gurpartap Singh (11BEC1124)

LDPC(Low Density Parity Codes)

LDPC Codes are characterized by the sparseness of ones in the parity-check matrix.

This low number of ones allows for a large minimum distance of the code, resulting in improved

performance.

PARITY-CHECK MATRIX

LDPC codes are classified into two different classes of codes: regular and irregular codes.

Regular codes are the set of codes in which there is a constant number of ๐ค๐ถ 1โs distributed throughout

each column and a constant number of ๐ค๐ 1โs per row.

For a determined column weight (๐ค๐ถ), we can determine the row weight as๐โ๐ค๐ถ

๐โ๐, (N is the block-length

of the code and k is the message length).

Irregular codes are those of which do not belong to this set (do not maintain a consistent row weight)

MINIMUM DISTANCE OF LDPC CODES

The minimum distance is a property of any coding scheme.

Ideally this minimum distance should be as large as possible, but there is a practical limit on how large

this minimum distance can be.

LDPC posses a large problem when calculating this minimum distance efficiently as an effective LDPC

code requires rather large block-lengths.

Using random generation it is very difficult to specify the minimum distance as a parameter, rather

minimum distance will become a property of the code.

CYCLE LENGTH OF LDPC CODES

Using a Tanner Graph it is possible to view the definition of the minimum cycle length of a code.

It is the minimum number of edges travelled from one check node to return to the same check node.

Length 4 and Length 6 cycles with the corresponding parity-check matrix configurations are shown in

Figures 5 and 6 respectively.

Contdโฆ

It has been shown that the existence of these cycles degrade the performance during iterative decoding

process.

Therefore when generating the parity-check matrix, the minimum cycle length permitted must be

determined.

It is possible control the minimum cycle length when generating the matrix, however computational

complexity and time increases exponentially with each increase in minimum cycle length.

LINEAR INDEPENDENCE

The generator matrix G, is defined such that:

๐ = ๐ฎ๐ป๐

Where,

๐ = [๐1, ๐2, ๐3 , โฆโฆโฆ๐๐]๐ โ Code-word

๐ = [๐1, ๐2, ๐3 , โฆโฆโฆ๐๐]๐ โ Message Word

๐บ = ๐ by ๐ Generator matrix

In order to guarantee the existence of such a matrix G, the linear independence of all rows of the parity-

check matrix must be assured.

LDPC SYSTEM OVERVIEW

Where:

โข m- Message

โข c - Code-word

โข x- Modulated signal

โข n- AWGN noise

โข y- Received signal

โข cห- Estimated code-word

โข mห- Estimated message

Encoding

๐ = ๐ฎ๐ป๐

we define a complete set of successful parity-checks as:

๐ป๐ = 0

Where:

๐ = [๐1, ๐2, ๐3 , โฆโฆโฆ๐๐]๐

๐ป(๐โ๐)โ๐ = ๐ โ ๐ ๐๐ฆ ๐ Parity-Check Matrix

Contdโฆ The location of the parity-bits in the code-word is arbitrary, therefore we will form our code-word such

that:

๐ = [๐:๐]๐

Where:

๐ = [๐1, ๐2, ๐3 , โฆโฆโฆ๐๐]๐ โ Message Word

๐ = [๐1, ๐2, ๐3 , โฆโฆโฆ๐๐โ๐]๐โ Parity Bits

Therefore:

๐ป[๐:๐]๐= 0

H can be partitioned as:

๐ป = [๐: ๐]

Where:

X = N-k by N-k Sub-matrix

Y = N-k by k Sub-matrix

Contdโฆ

From this we can find:

๐๐ + ๐๐ = 0

Using modulo-2 arithmetic we can solve for p as:

๐ = ๐โ1๐๐

Then we solve for c as:

๐ = [ ๐โ1๐ ๐: ๐ผ]๐๐

Where I is the k by k identity matrix and we define G as:

๐บ = [ ๐โ1๐ ๐: ๐ผ]

G Matrix for this project

In our project we directly chose the G matrix from โCommunication Systemsโ by Simon Haykin.

The G matrix is:

๐บ =

1 0 0 1 1 0 1 0 0 00 0 0 1 1 1 0 1 0 00 0 1 1 1 0 0 0 1 00 1 0 1 1 0 0 0 0 1

Hamming Code

Linear block code is said to be Hamming code if following parameters are obeyed:

1. No. of bits in the code-word (n) = 2๐ โ 1

2. No. of message bits (k) = 2๐ โ๐ โ 1

3. No. of parity bits (n-k) = ๐ and ๐ โฅ 3

Encoding

p= k by (n-k) Parity Matrix

Generator Matrix:

๐บ = [๐: ๐ผ๐]

Parity Check Matrix:

๐ป = [๐ผ๐โ๐: ๐๐]

Encoded message:

๐ฅ = ๐:๐

Where,

๐ = ๐๐

Syndrome:

๐ = ๐ฆ๐ป๐

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