MASTER RESEARCH PROPOSAL PRESENTATION

Name: Oluwaseun .O. OlogunNumber: S311080004WDepartment: Information and Communication Engineering

Supervisor: Liu Tong

Date: 25 – 12 -2012

Presentation Outline Proposed title of research Purposes and objectives Problem statement Background Scope Research Methodology Current state Proposed structure of final Thesis Time table Limitations References

Proposed title of research

Study of the key technology of Polar codes

Purposes and objectives ‘Every channel is described by a number called its Capacity, at any rate of

Transmission less than the capacity, you are bound to achieve an overly small error probability or BER as long as you code properly’

Arikan recently introduced the method of Channel polarization on which one can construct an efficient capacity achieving codes known as polar codes for a BDMC channel.

This research hopes to conduct an extensive work on the technologies involved in channel polarization or polar coding and also to show that polar codes are capacity-achieving using different channel models.

Two major aspects will be considered;

improving the decoding algorithm (SC) constructing an effective and efficient encoding

algorithm in an AWGN channel

Problem StatementOne major aspect or drawbacks of information theory is realizing an efficient

and reliable communication on an unreliable channel.

The answer or solution to this is Channel coding.

In a wireless communication link, coding is basically done to reduce the BER.

SNR^ⁿ = 1/BERWhere n is the parameter of the coding method used. n = 1 for an uncoded case, but n > 1 for a coded case.

‘n’ in this case means the type of coding scheme employed.

The above description simply explains how coding totally improves the reliability of any communication system.

Problem Statement cont…In every standard Channel Model let X denote Input Alphabets let Y denote Output Alphabets let W denote a Discrete Memory less channel let I (W) be the capacity of the DMC

W is defined as conditional or transitional probability distribution W(y|x)

W(y|x) represents probability that a channel output is y when x is transmitted.

Where x ∈ X and y ∈ Y

If we assume a Binary DMC, then X = {0 , 1} Assume that we have two independent copies of a B-DMC.

Now the question is that “Is there any way to combine these two channels to obtain at least one channel with more reliability?”

Problem Statement cont…

Fortunately, the answer to the question is YES.Channel polarization

This research hopes to study and concretize the issues relating to Channel polarization by;

Improving the decoding algorithm in a BDMC and

Create an encoding and decoding algorithm for polar codes if a channel is corrupted by AWGN

BackgroundWhat is Coding?

Coding is widely used in digital communication systems for a variety of purpose.

covers a wide range of techniques is one of the main advantages of digital over analog systems.

Definition: A systematic scheme for the replacement of the original information symbol sequence by a sequence of code symbols, in such a way as to permit its reconstruction.

Background cont…

Categories of codingCoding

Cryptography to preserve secrecySource coding

to compress data

Line coding to improve spectral

characteristicsError-control coding

to permit robust transmission of

dataError-detection coding

allows re-transmission of erroneous data

Forward Error Correction (FEC) coding

to correct errors even without a feedback

channel

‘channelcoding’

Error control coding

Bit energy to noise density ratio, dB

Capacity per unit bandwidth,

bits/s/Hz

Unattainable region

Practical systems

x convolutionalconcatenatedxx turbo

xuncoded

It was not approached until the invention of turbo-codes in 1993

Background cont…

Background cont…

Error correcting codes are basically used in different applications such as wireless communication, deep-space and satellite communications.

Polar code which is a family of error correcting code was discovered in 2009 by Erdal Arikan

It is a major breakthrough in coding history because of its explicit construction and efficient encoding and decoding algorithm

It is capacity achieving over binary input symmetric memory less channels

The major issue about polar codes is that it doesn't perform well at short or moderate block lengths ; reason for this might be because of the SC decoder used to decode the algorithm degrades with respect to maximum likelihood decoding performance

Background cont… Transmission over noisy channels

Digital Source

Digital Sink

Encoder

Decoder

Coding ChannelNoise

Background cont…Error control coding deals with reliable transmission of information over noisy channels while using power and bandwidth resource efficiently.

Assume message sequence is split into block of lengths K

The encoder adds redundancy to each message block and obtains a “codeword" of length N > K

The decoder makes use of this redundancy to estimate the original message block

Background cont… Channel polarization;

Arikan exploited loopholes in Shannon’s 1948 A Mathematical Theory of Communication and developed a coding method using Polar codes.

Channel polarization is a recursive method that is used to define polar codes, a class of codes that can provably achieve the capacity of several classes of channels.

The channel polarization phenomenon suggests to use the noiseless channels for transmitting information while fixing the symbols transmitted through the noisy ones to a value known both to the sender and receiver

Polar codes polarize the channel in such a way that;N channels with capacity I (W) polarize into N I(W) to transmit channels with capacity of 1s or N (1 – I(W)) to transmit channels with capacity of 0s.

Recall Shannon’s Capacity of a Channel equation; C = log N(T) / T

This equation was modified to arrive at the binary-input discrete memory less channels W, whose symmetric capacity is defined as;

Background cont…

Where X denote Input Alphabets Y denote Output Alphabets W denote a Discrete Memory less channel

W (y | x) is Transition probability of channel

ScopeChannel polarization or polar codes is relatively a new area in Coding theory and there is still a lot of work that needs to be done.

However this research will not cover all that there is in Channel polarization.

Areas interested in;

Channel polarization and how polar codes work,

Decoding algorithm using successive cancellation method (improvement of the method),

Different channel models and their capacities; Binary memory-less channels (BEC and BSC)

Binary input AWGN channel (emphasis will be on this)

Scope cont… ISI channel Z- channel

Simulations will be done using IT++IT++ is a C++  library of classes and functions for linear algebra,  numerical optimization,  signal processing, communications, and statistics.

It is currently gaining usage amongst users and researchers in the area of information theory.

The main reason for the use of IT++ is simply because of the speed of compiling and generating output for your program

Research Methodology Notations to denote sub vector

to denote sub vector with odd indices

to denote sub vector with even indices

denotes modulo – 2 addition

denotes Kronecker product of two matrices

Research Methodology cont…

represents Kronecker power

Where for all

Initial value of

We denote a DMC with input alphabet X and output alphabet Y and transition probability of W(y|x) ,Where x ∈ X and y

N independently use of DMC W as

Research Methodology cont… The Transition probability is given by;

The Symmetric capacity can therefore be calculated using

It measures the rate

Bhattacharyya parameter as

It measures the reliability

Research Methodology cont…

which is an upper bound on the probability of MAP decision error when we transmit a single bit with equal probability to be 0 or 1. It can be shown that this parameter is a convex function of the channel transition probabilities.

The channel polarization consists of two basic steps that are dependent on each other;

STEP 1;Channel combining phase

STEP 2;Channel splitting phase

Research Methodology cont… Channel combiningWe Synthesize or

combine the channels s.t N copies of DMC W in a recursive manner to produce a vector channel

The next recursive combination of channels will be s.t

Transition probabilities are given as

Research Methodology cont…

Recursive structures

Step 1 : Duplicate configurations

Step 2 : Construct another level of connectors

Step 3 : Connect levels s.t layer 2i – 1 is connected to the first copy and layer 2i is connected to the second copy

Research Methodology cont…

Channel splittingAt this stage we split the channel to construct

Channels

The channels are constructed using the transition probability ;

Research Methodology cont… Polar codes and how

they work.Let Channel inputs be

denoted asChannel inputs

satisfy that

Where is the generator matrix and its given as

Research Methodology cont… Example;

Let

Research Methodology cont… Polar code encoding

We have N channels

Transition probability is defined as

Research Methodology cont… Example, given the following values;

Assuming after some calculations, the following results are obtained;

and we want to transmit K= 4 bits of information, it will be most appropriate to transmit them on channels 4, 6, 7, and 8

Research Methodology cont… Successive

cancellation decoding

For eachIf is frozen,

set Or else generate a

decision

Research Methodology cont… BI-AWGN Channel Consider the discrete time channel model

The capacity can be shown to be

Current state Researchers are working tirelessly to improve the performance as it relates to short or moderate block lengths

Researchers in University of California, San Diego (USA) have developed new methods (including software implementations) to improve the error-correcting capability of polar codes

The method employs a new decoding method and a modification on the codes themselves.

The software is called ‘PolarList’ and the error correcting capability/performance is better than the current state of the art error correcting code.

Patent for this is however still pending.

Proposed structure of final Thesis

Chapter 1: Introduction

Chapter 2: Literature review

Chapter 3: Methodology/Theoretical Framework

Chapter 4: Results and Simulations

Chapter 5: Discussion of results

Chapter 6: Conclusion and recommendation

Chapter 7: References

Time table Present research proposal on December 31, 2012

Complete Literature review by March 15, 2013

Complete Methodology by April 28, 2013

Present results and simulations on June 21, 2013

Make corrections on report by July 5, 2013

Complete final report by July 31, 2013

Present and submit final report by August 9, 2013

However, all the dates listed up, except for the first one are all tentative dates. I might be able to handover at an earlier date or a later date.

The amount of work by the researcher and nature ,are important factors that should be considered.

Limitations Relatively a new area; few references can be accessed

The researcher’s usage of the programming language for simulations

Time insufficiency/constraints; Learning and mastering a new programming skill for a specific task

Formulating mathematical models

References [1] W. Ryan and S. Lin, Channel Codes; Classical and Modern.

Cambridge University press, 2009  [2] E. Arikan, “Channel polarization: A method for constructing

capacity achieving codes for symmetric binary-input memory less channels,” IEEE

Transactions o Information Theory, vol. 55, pp. 3051–3073, July 2009.  [3] E. Arikan, “A performance comparison of polar codes and reed-

Muller codes,” IEEE Communications Letters, vol. 12, no. 6, pp. 447–449,2008.

[4] N. Hussami, S. Korada, and R. Urbanke, “Performance of polar codes for channel and source coding,” in IEEE International Symposium on Information Theory (ISIT), 2009.

  [5] S. Korada, E. Sasoglu, and R. Urbanke, “Polar codes:

Characterization of exponent, bounds, and constructions,” in IEEE International Symposium on Information Theory (ISIT), pp. 1483 – 1487, 2009.

 

 

[6] S. Korada and R. Urbane, “Polar codes are optimal for lossy source coding,” IEEE Transactions on Information Theory, vol. 56, no. 4,pp. 1751 – 1768, 2010.

[7] H. Mahdavifar and A. Vardy, “Achieving the secrecy capacity of wiretap channels using polar codes,” in IEEE International Symposium on Information Theory (ISIT), June 2010. [8] M. Bakshi, S. Jaggi, and M. Effros, “Concatenated polar codes,” in IEEE International Symposium on Information Theory (ISIT), June 2010.

[9] E. Hof, I. Sason, and S. Shamai, “Polar coding for reliable communications over parallel channels,” in IEEE Information Theory Workshop, August 2010.

[10] A. Eslami and H. Pishro-Nik, “On bit error rate performance of polar codes in finite regime,” in 48th Annual Allerton Conference on Communication, Control, and Computing, August 2010.

[11] S. Lin and D. J. Costello, Error Control Coding: Fundamentals and Applications. Prentice-Hall, 1983.

 

 

References cont…

  [12] S. Korada, Polar Codes for Channel and Source Coding. PhD thesis, Ecole Polytechnique Fdrale de Lausanne (EPFL), 2009.

References cont…

Questions