Low Density Parity Check Codes Ppt

LOW DENSITY PARITY CHECK CODES

BLOCK DIAGRAM OF A GENERAL COMMUNICATION SYSTEM

WHAT IS CODING?

Coding is the conversion of information to another form for some purpose.

Source Coding : The purpose is lowering the redundancy in the information. (e.g. ZIP, JPEG, MPEG2)

Channel Coding : The purpose is to defeat channel noise.

CHANNEL CODING

Channel encoding : The application of redundant symbols to correct data errors.

Modulation : Conversion of symbols to a waveform for transmission.

Demodulation : Conversion of the waveform back to symbols, usually one at a time.

Decoding: Using the redundant symbols to correct errors.

WHY CHANNEL CODING?

Trade-off between Bandwidth, Energy and Complexity.

Coding provides the means of patterning signals so as to reduce their energy or bandwidth consumption for a given error performance.

CHANNELS The Binary Symmetric Channel(BSC) The Binary Erasure Channel (BEC)

HOW TO EVALUATE CODE PERFORMANCE? Need to consider Code Rate (R), SNR (Eb/No),

and Bit Error Rate (BER). Coding Gain is the saving in Eb/No required

to achieve a given BER when coding is used vs. that with no coding.

Generally the lower the code rate, the higher the coding gain.

Better Codes provides better coding gains. Better Codes usually are more complicated

and have higher complexity.

SHANNON’S CODING THEOREMS

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• If C is a code with rate R>C, then the probability of error in decoding this code is bounded away from 0. (In other words, at any rate R>C, reliable communication is not possible.)

• It Tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise.

(x))H – (H(x)Max C y

SHANNON’S CODING THEOREMS

STATEMENT OF THE THEOREM

Where, C is the channel capacity in bits per second. B is the bandwidth of the channel in hertz.  S is the average received signal power over the

bandwidth N is the average noise or interference power over

the bandwidth S/N is the signal-to-noise ratio (SNR) or the 

carrier-to-noise ratio (CNR) of the communication signal.

COMMON FORWARD ERROR CORRECTION CODES

Convolutional Codes Block Codes (e.g. Reed-Solomon Code) Trellis-Coded-Modulation (TCM) Concatenated Codes

LINEAR BLOCK CODESThe parity bits of linear block codes are

linear combination of the message. Therefore, we can represent the encoder by a linear system described by matrices.

BASIC DEFINITIONS Linearity:

where m is a k-bit information sequence c is an n-bit codeword.

is a bit-by-bit mod-2 addition without carry

Linear code: The sum of any two codewords is a codeword.

Observation: The all-zero sequence is a codeword in every Linear Block Code.

then

and If

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BASIC DEFINITIONS (CONT’D)

Def: The weight of a codeword ci , denoted by w(ci), is the

number of of nonzero elements in the codeword.

Def: The minimum weight of a code, wmin, is the smallest

weight of the nonzero codewords in the code.

Theorem: In any linear code, dmin = wmin

Systematic codes

Any linear block code can be put in systematic form

n-kcheck bits

kinformation bits

THE ERROR-CONTROL PARADIGMNoisy channels give rise to data errors: transmission or storage systems

Need powerful error-control coding (ECC) schemes: linear or non-linear

Linear EC Codes: Generated through simple generator or parity-check matrix

Binary information vector (length k)

Code vector (word): (length n)

Key property: “Minimum distance of the code”, , smallest separation between two codewords.

Rate of the code R= k/n

1 0 0 0 1 0 11 1 1 0 1 0 0

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

0 0 1 0 1 1 11 0 1 1 0 0 1

0 0 0 1 0 1 1

G H

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1 2 3 4( , , , )u u u u u

, 0Tx uG Hx

Binary linear codes:

LDPC CODESMore than 40 years of research (1948-1994) centered around

Weights of errors that a code is guaranteed to correct

“Bounded distance decoding” cannot achieve Shannon limit

Trade-off minimum distance for efficient decoding

Low-Density Parity-Check (LDPC) Codes

Gallager 1963, Tanner 1984, MacKay 1996

1. Linear block codes with sparse (small fraction of ones) parity-check matrix

2. Have natural representation in terms of bipartite graphs

3. Simple and efficient iterative decoding in the form of belief propagation (Pearl, 1980-1990)

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min( 1)/ 2d

THE CODE GRAPH AND ITERATIVE DECODING

Variable nodes Check nodes

(Irregular degrees/codes)1 1 1 0 1 0 0

0 1 1 1 0 1 0

1 0 1 1 0 0 1

H

Most important consequence of graphical description: efficient iterative decoding

Message passing:

Variable nodes: communicate to check nodes their reliability (log-likelihoods)

Check nodes: decide which variables are not reliable and “suppress” their inputs

Small number of edges in graph = low complexity

Nodes on left/right with constant degree: regular code

Otherwise, codes termed irregular

Can adjust “degree distribution” of variables/checks

Best performance over standard channels: long, irregular, random-like LDPC codes

Have proportional to length of code, but correct many more errorsmind

CONSTRUCTION LDPC codes are defined by a sparse 

parity-check matrix. This sparse matrix is often randomly generated, subject to the sparsity constraints. These codes were first designed by Gallager in 1962.

 In this graph, n variable nodes in the top of the graph are connected to (n−k) constraint nodes in the bottom of the graph. This is a popular way of graphically representing an (n, k) LDPC code.

Specifically, all lines connecting to a variable node (box with an '=' sign) have the same value, and all values connecting to a factor node (box with a '+' sign) must sum, modulo two, to zero (in other words, they must sum to an even number).

CONTD., This LDPC code fragment

represents a 3-bit message encoded as six bits. Redundancy is used, here, to increase the chance of recovering from channel errors. This is a (6, 3) linear code, with n = 6 and k = 3.

In this matrix, each row represents one of the three parity-check constraints, while each column represents one of the six bits in the received codeword.

CONT.,

• In this example, the eight codewords can be obtained by putting the parity-check matrix H into this form  through basic row operations.

• From this, the generator matrix G can be obtained as    (noting that in the special case of this being a binary code ), or specifically:

• Finally, by multiplying all eight possible 3-bit strings by G, all eight valid codewords are obtained. For example, the code word for the bit-string '101' is obtained by: