CHAPTER ONE

INTRODUCTION

1.0 Background of Study.

Communication is the transference of information from one source

to another. It is said to be the process of sharing among two or more

sources i.e. knowledge or ideas in form of data. Communication is

carried out over a medium, however communication medium are in

different form, for this project the medium under study is the mobile

and wireless medium.

Wireless communication is the transmission of information i.e.

data using air as the medium of transfer (air is the transfer

channel). Wireless communications encompasses various types of fixed,

mobile, and portable two-way radios, cellular telephones, personal

digital assistants (PDAs), and wireless networking.

Transmission of data involves moving data from a source through

the transmission channel into the receiver which is the destination of

the information being transferred. Effective communication implies

that the receiver understands the message sent to it and takes the

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necessary action on the message. Other means of transmission channels

include:

i. Wireless channels

ii. Guided electromagnetic waves channels

iii. Optical channels

As communication revolution evolves, there is an increasing

number of communications thriving, although with a growing shortage of

bandwidth and the possibility of errors corrupting transmission

systems, there is a need to improve communication systems with regards

to protecting these systems from corruption of errors, while

simultaneously enhancing the efficiency of bandwidth use as well as

providing highly reliable and error free systems which should not be

taken for granted given the nature of wireless communication channels

It should be noted that communication system does not end in

transmission, i.e. it’s a thing to transmit data, its’ another to get

an uncorrupted message. Therefore, for communication to be effective,

the data received at the destination end must be uncorrupted.

Some of the information gets corrupted during transmission this

is because of the difficulties encountered which includes:

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Multipath propagation of signals

Interference from other communication devices.

Propagation path loss.

Noise.

For example when the data to be transmitted is of the form

01010101 and at the receiving end the data gotten is 11010101, this is

quite different from the data sent, this makes the system to be

incorrect.

A receiver must be able to detect and correct any irregularity in

the information it has received in other to give a perfect result to

the user of the information.

Therefore error detection in wireless communication is the

identification of unwanted or corrupted data in a transmitted message

or information. While error correction is the process of rectifying

the identified corrupted errors. Error correction coding makes

wireless communications more robust in the presence of noise. The

noise source is the main focus in implementing error correcting codes.

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Due to much errors found in the communication channel computer

scientist developed a number of techniques in identifying and

correcting these errors which include:

Parity check

Blocking code

Convolutional coding

Turbo coding

Reed Solomon

BCH code etc.

1.2 Statement of Problem

Errors usually happen at transmission or during storage of

information due to some interference such as noise. Detection and

correction of these errors at the receiving end is therefore essential

to having a quality and original information at the destination end.

1.3 Aim and Objectives

Due to the increase in the rate in which personal computers and

workstations are becoming portable, the desire to maintain

connectivity and high quality of data transmitted is important. The

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purpose of this project is therefore to have a data transmission over

a mobile and wireless medium that is free from errors and when there’s

an occurrence of error, the channel has the capability to detect and

do the appropriate correction required for the channel to an effective

one.

OBJECTIVES

The objectives of this project include:

To ensure that information or message sent and received

are of good quality..

To evaluate some error correction and detection

techniques.

To detect errors in the transmission line

Understand the concept of error correction and detection

in wireless communication systems.

1.4 Scope of Study

Since transmission can be done over several medium, this study is

limited to the detection and correction of errors in the mobile and

wireless communication.

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.

CHAPTER TWO.

LITERATURE REVIEW

2.1 Background of Study.

The error correction and detection started some centuries back

through the translation of one scroll in one language to another. The

Jewish scribe started the earliest famous error detection, by copying

the Jewish bible beginning before Christ. Emphasis were made on the

minute detail of the words and spelling which evolved into a perfect

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text in 135CE, that makes a deviation from the strictures render the

Torah scroll invalid.

The method of summing numbers of words per line and page and

checking the middle paragraph, words and letters against the original

text i.e. if a single mistake is found on a page, the page is rendered

invalid, while three mistake on a page renders the whole manuscript

invalid. It therefore has to be re-written i.e. retransmission. The

effectiveness of their method was verified by the accuracy of copying

through the centuries, demonstrated by the Discovery of the Dead Sea

Scrolls in 1947- 1956. (Wikipedia, 2013).

The idea of achieving error detection and correction is to add

some extra data to a message in other for the receiver to check the

consistency of the message delivered and to recover the data said to

be corrupted. (Wikipedia, 2013).

2.2 Error Detection and Correction.

The theory of error detection and correction is a branch of

engineering and mathematics that deals with reliable transmission and

storage of message. Information media are not 100% reliable in

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practice due to the interference such as noise which causes frequent

distortion of data. (Barbara Rosario, 2000). However, dealing with

this situation involves some form of redundancy being added to the

original data to do away with the undesired effect of interference.

Error detection refers to the class of techniques for detecting

errors i.e. garbled messages while error correction is the process of

correcting errors in data that have been corrupted during

transmission. ( Webopedia).

Error detection and correction scheme can either be systematic or

non-systematic. In systematic scheme, the transmitter sends the

original data and attaches a fixed check bits derived from data by the

deterministic algorithm. If only detection is required, then receiver

can simply apply the same algorithm to the received data bits and

compare its output with the received check bit; if the values do not

match then error as occurred at some point in transmission.

( Wikipedia 2013).

However, in non-systematic scheme, the original message is

transformed into an encoded message that has at least as many bits as

the original data.

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2.3 Error Correction

In electronic systems, information is represented in the binary

form. When binary information is passed from one point to another,

there are always the chances that error will occur, a “1” interpreted

as a “0” or otherwise. This can be caused difference type of

interference. When a bit is mistakenly misinterpreted, a bit error has

occurred. Therefore, there’s need for correction.

Error correction can either be done on the hardware or on the

software depending upon how fast it has to be done. In most cases, it

is usually done on the hardware such as magnetic disk drive or semi-

conductor memory. Errors are to be corrected at the same rate as data

is being read into the disk i.e. “on-the-fly”. The “on-the-fly”

performance requirement usually means the error correction

implementation must be done in digital logic. (Varnit, 2001).

2.4 Error Correcting Codes

Error correction codes are a means of including redundancy in a

stream of information bits to allow the detection and correction of

symbol errors during transmission. The approach to error correction

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coding taken by modern digital communication systems started in the

late 1940’s with the ground breaking work of Shannon, Hamming and

Golay (Valenti, 1999). The birth of error correction coding showed

that Shannon’s channel capacity could be achieved when transmitting

information through a noisy channel (Jemibewon, 2000).

Several researches have been carried out on error correction

codes for different types of systems. In Huang (1997), the evaluation

of soft output decoding for turbo codes was carried out in which the

performance of turbo codes were evaluated through computer simulation.

Coding theory related to this research was also studied, including

convolution encoding and Viterbi decoding. Simulation of error

correction codes on wireless communication systems investigates the

performance of error correcting codes for data signals used in

wireless systems; cellular phones, internet, geo location/GPS,

emergency services, radio and television broadcasting etc.

Error correcting codes add redundancy into bits to improve the

performance of a system. They are widely used in almost all digital

systems as they provide a method for dealing with the unknown, noise.

But this capability has several costs and trade-offs. Apart from

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needing an extra encoder and decoder, we have to transmit more

information in the same amount of time i.e. increased data rates which

translates into more bandwidth (Vats and Kacewicz, 2005).

Error Correction Codes (ECC) encodes data in such a way that a

decoder can identify and correct certain errors in the data. Channel

capacity is a theoretical measure of the fastest rate at which error

free transmission can be realized. Traditional modulation techniques

deliver performance significantly inferior to that predicted by

Shannon’s work, but when incorporated with error correction coding

most digital modulation schemes can achieve performance that

approaches channel capacity (Jemibewon, 2000). Shannon also

established the fundamental limits in transmission speeds in the

systems of digital communications and led to the search for coding

techniques for approaching the limit of this capacity (Morales, 2008).

When digital data is stored in a memory, it is crucial to have a

mechanism that can detect and correct a certain number of errors.

Usually data strings are encoded by adding a number of redundant bits

to them. When the original data is reconstructed, a decoder examines

the encoded message, to check for any errors (Valenti, 1999).

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Practically all systems that transmit or store digital data use

some form of error correction technique. In the presence of real-world

phenomenon, designers of these systems have to face the certain

outcome that some transmitted or stored bits of information will not

be received the way they were intended.

The choice of how to handle errors ranges from doing nothing to

using elaborate error detection and correction methods. Selecting

between the choices depends on the accuracy, speed, and latency

requirement of the information, and whether or not there is

simultaneous bidirectional communications between the sender and

receiver (Rossi, 1998).

2.5 Error Control Techniques.

There are two major error control techniques. They are the

Forward Error Correction and the Automatic Repeat Request. Also

there’s is the Hybrid Automatic Repeat Request.

2.5.1 Forward Error Correction.

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Forward Error Correction (FEC) which employs error correcting

codes to combat bit errors (due to channel imperfections) by adding

redundancy (henceforth parity bits) to information packets before they

are transmitted. This redundancy is used by the receiver to detect and

correct errors. They are indispensible due to the strict delay

requirement. The reconstructed data is what is said to be the original

data. FEC schemes maintain constant throughput and have bounded time

delay. A typical FEC schemes are stationary and must be implemented to

guarantee a certain Quality of Service (QOS requirement for the worst

case channel characteristics. As a consequence, FEC techniques are

associated with unnecessary overhead that reduces throughput when the

channel is relatively error free. (H. Liu et al.). There two main

types of correction methods under the FEC, which are the block codes

and the convolutional codes

2.5.2 Automatic Repeat Request

This can also be referred to as the backward error correction

wherein only error detection capability is provided and no attempt to

correct any packets received in error is made; instead it is requested

that the packets received in error be retransmitted. ARQ is simple

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and achieves reasonable throughput levels if the error rates are not

very large. However, in its simplest form, ARQ leads to variable

delays which are not acceptable for real-time services. (H. Liu et al.)

2.5.3 Hybrid Automatic Repeat Request

This is the combination of the forward error correction and the

automatic repeat request. It allows the minor errors to be corrected

without retransmission while the major ones are corrected through the

request for retransmission. This is in order to overcome their

individual drawbacks.

2.6 Block coding

Block coding schemes divide a bit stream into nonoverlapping

blocks and code each block independently. Block codes used in

practical applications today belong to the class of linear cyclic

codes, since these codes lend themselves to easier implementations. A

coding scheme is referred to as being linear if the sum of two code

vectors is also a code vector.

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Similarly, a coding scheme is referred to as being cyclic if all

cyclic shifts of a code vector results in a valid code vector. Binary

Bose–Chaudhuri–Hocquenghem (BCH) codes and non-binary Reed–Solomon

(RS) codes are two kinds of widely used linear cyclic block codes. (H.

Liu et al). Block codes are codes that require data being compiled

into blocks before coding or decoding can take place (Houghton, 1997).

They are based rigorously on finite field arithmetic and abstract

algebra. They can be used to either detect or correct errors. Each

codeword in the set is a linear combination of a set of generator code

words. They have very high code rates and are useful in situations

where Bit Error Rate (BER) of a channel is relatively low, bandwidth

availability is limited (Shelton, 1999). Block codes are also referred

to as (n, k) codes where a block of k data bits is encoded to become a

block of n bits called a code word (Valenti, 1999).

By predetermined rules, n-k redundant bits are added to the k

information bits to form the n coded bits. Block codes, also known as

redundancy codes, are thought of as giving external protection in the

encoding process. Some of the commonly used block codes are Hamming

codes, Golay codes, Bose–Chadhuri–Hocquenghem (BCH) codes, and Reed

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Solomon codes (which uses non binary symbols) (Vats and Kacewicz,

2005).

2.6.1 Bose–Chaudhuri–Hocquenghem (BCH) codes

For any positive integers, m > 3 and t < 2m−1, there is a binary

BCH code with the following parameters, (referred to as an (n, k, t)

BCH code): _ block length: n = 2m − 1, _ number of parity check bits:

n − k 6 mt, _ minimum distance: dmin > 2t + 1. Each binary BCH code (n,

k, t) can correct up to t-bit errors, and thus it is also referred to

as a t-error-correcting code. (H,Liu et al,1997).

2.6.2 Reed–Solomon (RS) codes

The binary BCH codes can be generalized to non-binary codes. If p

is a prime number and q is any power of p, there exist BCH codes with

q-ary symbols. For any choice of positive integer s and t, a q-ary BCH

code is of length n = qs−1, which is capable of correcting any

combination of t or fewer symbol errors and requires no more than 2s

parity-check symbols. RS codes are a subclass of non-binary BCH codes

with s = 1. A (n, k, t) RS code wit q-ary symbols has the following

parameters:

_ block length: n = q − 1,

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_ number of parity-check bits: n − k = 2t,

_ minimum distance: dmin = 2t + 1:

An (n, k, t) RS code is capable of correcting any combination of t

or fewer symbol errors. In practical applications, RS codes with code

symbols from q = 2m are chosen.

BCH and RS block coding schemes have a well-defined algebraic

structure, which has facilitated the development of efficient coding

and decoding schemes. In addition, RS codes have optimal “distance

properties”, i.e., provide optimal error correction capability given a

fixed number of parity bits, and excellent “burst error suppression”

capabilities. (H,Liu et al,1997).

2.6.3 Hamming Codes

A commonly known linear Block Code is the Hamming code. Hamming

codes can detect and correct a single bit-error in a block of data. In

these codes, every bit is included in a unique set of parity bits. The

presence and location of a single parity bit-error can be determined

by analyzing parities of combinations of received bits to produce a

table of parities each of which corresponds to a particular bit-error

combination. This table of errors is known as the error syndrome. If

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all parity is correct according to this pattern, it can be concluded

that there is not a single bit-error in the message (there may be

multiple bit-errors). If there are errors in the parities caused by a

single bit-error, the erroneous data bit can be found by adding up the

positions of the erroneous parities.

While Hamming codes are easy to implement, a problem arises if

more than one bit in the received message is erroneous. In some cases,

the error may be detected but cannot be corrected. In other cases, the

error may go undetected resulting in an incorrect interpretation of

transmitted information. Hence, there is a need for more robust error

detection and correction schemes that can detect and correct multiple

errors in a transmitted message. (vikas et al,2012)

2.6.4 Cyclic codes and Cyclic Redundancy Checks (CRC)

Cyclic Codes are linear block codes that can be expressed by the

following mathematical property. If C = [c n-1 cn-2 … c1 c0] is a code

word of a cyclic code, then [c n-2 cn-3 … c0 cn-1], which is obtained

by cyclically shifting all the elements to the left, is also a code

word.(Wikipedia,2001). In other words, every cyclic shift of a

codeword results in another codeword. This cyclic structure is very

useful in encoding and decoding operations because it is very easy to

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implement in hardware. A cyclic redundancy check or CRC is a very

common form of cyclic code which is used for error detection purposes

in communication systems. At the transmitter, a function is used to

calculate a value for the CRC check bits based on the data to be

transmitted. These check bits are transmitted along with the data to

the receiver. The receiver performs the same calculation on the

received data and compares it with the CRC check bits that it has

received. If they match, it is considered that no bit-errors have

occurred during transmission. While it is possible for certain

patterns of error to go undetected, a careful selection of the

generator function will minimize this possibility. Using different

kinds of generator polynomials, it is possible to use CRC‘s to detect

different kinds of errors such as all single bit-errors, all double

bit errors, any odd number of errors, or any burst error of length

less than a particular value. Due to these properties, the CRC check

is a very useful form of error detection. The IEEE 802.11 standard for

CRC check polynomial is the CRC-32. (Vikas,et al, 2012)

2.7 Convolution Codes

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Block coding schemes are frequently referred to as memoryless

since successive information blocks are coded independently.

Convolutional codes are a popular class of coders with memory, i.e.,

the coding of an information block is a function of the previous

blocks.

These are alternatives to block coding in which coding and

decoding can take place on a continuous data bit stream (Houghton,

1997). They are used in applications that require good performance

with low implementation cost. They operate on data stream not static

block. They are usually denoted by (n, k, l) (Er. Liu, 2004). The

code words produced depend on both the data message and a given number

of previously encoded messages. The encoder changes state with every

message processed. The length of the code word is usually constant

(Valenti, 1999).

In almost all applications of digital communications, convolution

codes are used as the main error correction code because of its ease

of implementation compared to an equivalent block code. Convolution

codes convert the entire data stream into one single codeword. The

encoded bits depend not only on the current input bits but also on

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past input bits. The main decoding strategy for convolution codes is

based on the widely used Viterbi algorithm (Huang, 1997).

2.8 PROPERTIES OF CONVOLUTION CODES

Passing the information sequence to be transmitted through a

linear finite state shift register generates convolution codes.

Convolution codes can be summed up as codes without a block structure

(Vats and Kacewicz, 2005). Convolution codes are one of the most

widely used channel codes in practical communication systems. These

codes are developed with a separate strong mathematical structure and

are primarily used for real time error correction. They are usually

denoted by (n, k, l) Where,

n- Number of output bits

k- Number of input bits

l- Code memory depth

To encode a block code, an infinite data stream is divided into

blocks and then the code is applied on each individual block. The

final output depends on the corresponding block it came from in the

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input stream. For convolution codes, the stream is again divided into

blocks, but the blocks are much smaller.

The final output for convolution codes depends on multiple blocks

from the input stream (Vats and Kacewicz, 2005). There are three

parameters which define the convolution code:

(a) Rate: This is the ratio of the number of input bits to the number

of output bits. That is to say that code rate R is determined by input

rate and output rate:

R=kn<1

Where,

K is the number of input bits and n is the number of

output bits

(b) Constraint length: Constraint length is defined as the number of

symbols in the data stream that have an effect on the output of the

encoder.

The constraint length K for a convolution code is defined as:

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k=m+1

(2.3)

Where m, is the maximum number of stages (memory size) in any shift

register. The shift registers store the state information of the

convolution encoder and the constraint length relates the number of

bits upon which the output depends.

(c) Generator polynomial: This is a polynomial involved in wiring of

the input sequence with the delay elements to form the output (Sankar,

2009).

According to Shannon, the band limited Additive White Gaussian

Noise channel with bandwidth B has the capacity C:

C=Blog2(¿1+s /N)bits /sec ¿ (2.4)

Where s and N are the average signal power and noise power

respectively.

The data transmission can be protected from errors induced by noise in

the channel provided that the information rate R is less than the

capacity C,

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R<C(2.5)

Since

N=NoB(2.6)

If the two-sided power spectrum density of the noise is:

No/2Watts/Hz

Equation (2.4) becomes:

C=Blog2(¿1+s /N)bits /sec ¿

¿Blog2(¿1+s /NoB)¿ (2.7)

Not considering its bandwidth efficiency, binary phase shift keying

(BPSK) is known to be the optimum binary modulation scheme, because it

is one kind of antipodal sampling. Consequently, it is often used in

communications theory as a benchmark for comparison. For the uncoded

BPSK. And BER denotes the bit error rates of the received sequence

(Cheng, 2005)

2.9 Iwadare Code

Iwadare code can be ranked among the systematic convolutional

codes. It does not therefore need to be divided into blocks for

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security reasons. The resulting transmitted code combination can be

divided into information elements and security elements. If the

threshold security capability is exceeded, the infinite error

intrusion into the information itself does not occur. Iwadare code is

capable of correcting error bursts b if there is security interval A

among these bursts where the respective information is transmitted

error-free.( Ing. Vítězslav Křivánek,2006).

There are basic version and extended version which has higher

security capability, but lower information rate. The extension level

is marked with letter i. If i = 1, then it is a basic code. Iwadare

code is described with the generator lock matrix for which the

following relationships are given (derivation in [1]):

B0=⟦m.n0;m.k0⟧

m and k0 are defined as follows:

m=n0.(n0−1 )

2+(2n0−1).i

k0=¿n0−1¿

where k0 and n0 means the number of partial input or output parallel

flows. Consequently, correction capability b and security interval A

are defined:

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b≤n0⋅i

A≥n0⋅m−1

2.9.1 Iwadare encoder

The basic code determines the general scheme of the block

matrix). The code matrix can be expanded so that its security

capabilities can be increased or security interval can be reduced. In

the case of i-multiple expansion, the number of (i - 1) zero lines are

substituted for all non-zero lines. The modelling of encoder and

decoder shall be also demonstrated in the basic Iwadare code which is

capable of the correction of the burst of 4 errors.

It is based on the generator block matrix defined in the formula

(1). To generate a connection design, you shall need the basic

syndrome equation of the code matrix from which individual elements of

encoder can be derived:

s13 = a1 + a4 + b5 + b7 + c8 + c9 + d13

The designed encoder connection is based on the syndrome equation

(6). On condition no errors occur it the encoder system itself, the

following relationship is valid: s13 = 0 , (7) consequently, the

relationship shall be simplified as follows to derive the security

element d13:d13 = a1 + a4 + b5 + b7 + c8 + c9 . (8) The connection

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diagram of the basic encoder (see Fig. 7) can be uniquely determined

from the equation (8). In the figure of the encoder and decoder, only

the first and the last memory cell of the respective section are

marked for the reason of clarity.

2.9.2 Iwadare decoder

The basic principle of the decoder consists in the use of the

threshold decoding principle. It is necessary to find the

orthogonality relation towards the element whose correctness after the

transmission is just being checked. To derive the decoder connection,

you shall need the syndrome equation system. The number of necessary

equations is determined by the requirement of the existence of at

least two orthogonality sums for one corrected bit. The basic

substance and condition are as follows: The corrected bit occurs in

both of these equations while the other bits do not and shall not

occur in both the equations.

The above-mentioned example of Iwadare code corrects the

erroneous bursts b ≤ 4 . Yet, it does not deal with the correction of

the security bit di because its correction is of no significance for

the transmitted information. The following three information bits ai,

bi, ci are only corrected. To design the Iwadare decoder, one can use

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the fact that it is a systematic code which is organized as follows:

security bits follow only after information bits of the respective

transmission. One more syndrome equation in relation to the basic

syndrome equation (6) shall be therefore found. If you define the

syndrome equation for specific time, you shall find bits participating

in security in previous time in relation to the bits contained in this

equation. The second syndrome equation

:s12 = a0 + a3 + b4 + b6 + c7 + c8 + d12 . (9)

Using these two syndrome equations (6) and (9), the correction of bit

c8 can be made:

s13 ⋅ s12 = 0 ∧ s13 ⋅ s12 = 1 , (10)

where the result equals zero if bit c8 is correct, and the syndrome

product equals one if there is an error. If you want to correct other

bits, you have to go “deeper in the history” and determine other

necessary syndrome equations for the correction of bits b5, a1. On the

basis of knowledge of other syndrome equations, it’s possible to model

a decoder.

2.10Code shortening

Often, a block code of desirable natural length or suitable

number of information digits may not exist. In this case code shortening

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is performed, which involves choosing a code with block length greater

than the required length and subsequently shortening it to meet the

requirement. Code shortening is most easily done by setting a selected

number of the information symbols to zero in the encoding operation.

For example, given an (n, k) code C, consider the se of code vectors

for which the b leading high-order information symbols are equal to

zero. Such code vectors form a subset of code C. If the b zero

information symbols are deleted from each of these code vectors, we

obtain a set of vectors of length n − b. These shortened vectors form

an (n−b, k−b) code. The error detection and correction capability of

the shortened code is at least as great as the code from which it was

derived. For RS codes, the minimum distance is unchanged after

shortening. (H lui, et al, 2007)

2.11Interleaving

An alternative to choosing long codes to combat the effect of

burst errors is interleaving. Interleaving simply involves

interleaving symbols from two or more codewords before transmission on

the channel. The number of codewords that are interleaved is referred

to as the depth of the interleaver, m. The reverse process is

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performed at the deinterleaver. Therefore, between successive symbols

of any given codeword there are m − 1 symbols that belong to

the m − 1 other codewords being interleaved. If the interleaver has

sufficient depth the fading processes that affect successive symbols

belonging to the same codeword will be uncorrelated.

Therefore, from the perspective of any single codeword,

interleaving makes a burst error channel appear as one which has only

random errors. Interleaving does not decrease the long-term bit error

rate but it is successful in decreasing the number of errors in each

codeword, therefore the codeword should have enough capability to

correct the erroneous symbols in it after deinterleaving. Result in

[34] show that the FEC and interleaving strategy is effective when tm

exceeds 1=r where t is the code error correction capability and 1=r is

the average burst length. Note that interleaving results in extra

delay because deinterleaving can be started only after all the

interleaved data is received. For the above example, a delay on the

order f 1=rt codewords is introduced. (H lui, et al, 2007).

2.12 Code puncturing

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The characteristics of a wireless channel typically vary with

time, and therefore to obtain optimal performance it is necessary to

adapt the error coding scheme to the changing channel characteristics.

Code puncturing allows an encoder/ decoder pair to change code rates,

i.e., code error correction capabilities, without changing their basic

structure. Code puncturing involves not transmitting (i.e., deleting)

certain code bits. It is important to note that both convolutional

codes and block codes can be punctured. Punctured convolutional codes

were first introduced by Clark et al. [9]. Hagenauer modified the

concept of punctured convolutional codes for the generation of a

family of rate compatible punctured convolutional (RCPC) codes by

adding a rate-compatibility restriction to the puncturing rule [12].

The rate-compatibility restriction implies that all he code bits of a

high rate code of the family are used by the lower rate codes. These

codes are attracting more and more attention because of their

flexibility. We now discuss in some detail the process of puncturing

codes. A low rate 1=n convolutional code (called mother code) is

periodically punctured with period p to obtain a family of codes with

rate p=v, where v can be varied between p + 1 and np. As an example,

we consider punctured convolutional codes obtained from a rate 1=3

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mother code. To generate a rate p=v punctured convolutional code (p=v

> 1=3), we delete (3p − v) bits from every 3p code bits corresponding

to the encoded output of p information bits by the original rate 1=3

code. The resulting rate is then equal to the desired rate r = p=v.

The punctured codes have the same number of states as the mother

code, i.e. the same memory length m. The deleted bit pattern must be

carefully chosen to obtain desirable performance. The elements of

puncturing matrices are only zeros and ones. A zero in a puncturing

matrix means that the corresponding code bit will not be transmitted,

a one means that it is inserted in the channel bit stream. For

example, to generate a 8=22 code, puncturing matrix p(8=22) is used.

Encoding of 8 information bits with the three generator polynomials

results in 38 intermediate bits at the three output branches. (H lui,

et al, 2007).

Every fourth and eighth output bit of the third branch is

deleted. Instead of transmitting 3,8 bits, only 22 bits are

transmitted per 8 information bits. Therefore, a rate 8/22 code is

generated. In general, the puncturing matrix p(r = p=v) = [pij ] for a

mother code of rate 1=n and a puncturing period of p has n rows and p

columns. The number of zeros in the puncturing table is equal to np−v.

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Two punctured convolutional codes, obtained from the same mother code,

are said to be rate-compatible if all the code bits in the higher rate

code are used in the lower rate codes.

2.13ERROR-CORRECTION CODE SELECTION

Through research, both theoretical and numerical, must precede the

design of error correction codes. There are many error-correcting

codes and their combinations, each fitting a particular task. The

basic issues to be taken into account in order to choose an error-

correcting code (ECC) are:

Physical channel characteristics (Signal to noise ratio, presence

of memory, e.g. fading)

Modulation type

Desirable channel characteristics (information rate, BER - bit

error probability).

A simple measure for the quality of a code is the information rate

which is the measure of how much of the code is the message. The rate

is given as:

log2C /n

33

Where,C represents the number of code words in the code.

Having discussed various coding schemes, we now consider criteria

that must be taken into account when selectin a FEC scheme for any

given application.

1. Probability of uncorrected errors: Since it is impossible for

any coding scheme to detect all errors and correct them, it is

important to choose coding schemes for which the probability of both

undetectable and uncorrectable (but detectable) errors is minimized

(or satisfies the application under consideration).

2. Overhead: The FEC codes should add as little as possible overhead

and maximize the code rate. However increased code capability

generally leads to lower code rate.

3. Complexity: The implementation complexity of the coding/decoding

scheme which typically increases with increase in code length and its

capability to detect and correct errors.

2.14Automatic Repeat Request (ARQ)

ARQ is an error control mechanism that relies on retransmitting

data that is received with errors [20]. In such schemes, messages are

divided into blocks of suitable size that are transmitted after a

34

small number of parity bits have been added. At the receiver these

parity bits are used to detect the presence of errors in the received

packet. In case errors are detected in a received packet, the receiver

requests a retransmission of the packet.

Automatic Repeat Request (ARQ) protocols roughly operate as

follows: The transmitter numbers the packets to be transmitted

sequentially (using numbers from a finite set) and maintains a timer

for each packet it transmits. The receiver acknowledges, at the very

least, the receipt of each successful packet (a packet that is

received with no errors) by transmitting a packet, referred to as an

ACK bearing the sequence number of the packet being acknowledged.

Packets that have not been successfully acknowledged, i.e., an ACK has

not been received, in a predetermined time interval, henceforth

referred to as timeout, are assumed to be lost (or corrupted) and are

retransmitted. In some cases, negative acknowledgements (NAKs) are

transmitted by the receiver for every packet received error. If NAKs

are employed, a packet is retransmitted following the receipt of a

negative acknowledgement. Since some of the transmitted packets can be

lost or misrouted NAKs cannot be transmitted for these lost packets.

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There three most popular ARQ protocols – Stop and Wait, Selective

Repeat, and Go-Back-N. (H, Liu et,al).

2.14.1 Stop and Wait

When using the Stop and Wait (SW) ARQ protocol, the DLC protocol

transmits a packet only when all previously transmitted packets have

been successfully acknowledged. Hence, when using SW, the transmitter

after transmitting a packet waits for its acknowledgement. Once its

acknowledgement has been received the next packet is transmitted.

However, if an acknowledgement does not arrive until a timeout

timer expires, the packet is retransmitted. Therefore, in SW there is

never more than a single packet that s unacknowledged at any given

instant of time. Since the transmitter does not use the available

channel during time intervals it waits for an ACK, the maximum data

transfer rate that can be supported is limited. This limits cases

where the SW ARQ protocol can be employed.

2.14.2 Selective Repeat

Unlike SW, when using Selective Repeat (SR), packets, if available,

are transmitted continuously by the DLC layer. As before, the receiver

36

acknowledges each successfully received packet by transmitting an ACK

bearing the sequence number of the packet being acknowledged. If an

acknowledgement is not received for a packet before the expiration of

the timeout, the packet is retransmitted.

Once a packet has been retransmitted the transmitter resumes

transmission of packets from where it left off, i.e., if a is the

packet with the largest sequence number that has been transmitted,

packet with sequence number a+1 is transmitted next (assuming that no

other timers have expired in the meantime). Since when the SR ARQ

protocol is employed, packets are continuously being transmitted the

inefficiency associated with SW is eliminated. Observe that when SR is

employe packets can be accepted out of sequence. Hence, packets

received out of sequence have to be buffered and sequenced before they

can be delivered. (Vikas et al, 2012)

2.14.3 Go-Back-N

When Go-Back-N (GBN) is employed, packets are transmitted

continuously as in SR. However, at the receiver, the DLC layer accepts

packets only in the order in which they were transmitted. Packets

received out of sequence are discarded and not acknowledged. Since the

receiver accepts packets only in-sequence, after a timeout, the

37

transmitter retransmits the packet that timed out and all packets with

sequence numbers that follow the one that was retransmitted. Hence,

each time a timeout occurs all packets that are yet to be acknowledged

are retransmitted. It is important to observe that GBN attempts to

combine the desirable features of SR and SW, i.e., packets are

transmitted continuously, as in SR, without the need to buffer out of

sequence packets and there is no re sequencing overhead.(H,liu et al)

2.15ERROR DETECTION

Error detection is much simpler than error correction and one or

more "check" digits are commonly embedded in credit card numbers in

order to detect mistakes. Early space probes like Mariner used a type

of error-correcting code called a block code, and more recent space

probes use convolution codes. Error-correcting codes are also used in

CD players, high speed modems, and cellular phones. Modems use error

detection when they compute, checksum which are sums of the digits in

a given transmission modulo some number. Error detection codes also

includes: repetition scheme, parity, ( Wolfram mathworld ).

38

2.15.1 Repetition Schemes

Repetition code is another mechanism that relates to error

detection. It is a coding schema that repeats bits across channels to

achieve error-free communication. Data bits in a stream of data are

divided into blocks of bits. Every block is transmitted a

predetermined number of times. They are not as effective as parity,

because the occurrence of errors in the same place leads to more

problems. However, they are simple and used in the transmission of

number stations.

Variations on this theme exist. Given a stream of data that is to

be sent, the data is broken up into blocks of bits, and in sending,

each block is sent some predetermined number of times. For example, if

we want to send "1011", we may repeat this block three times each.

Suppose we send "1011 1011 1011", and this is received as "1010 1011

1011".

As one group is not the same as the other two, we can determine

that an error has occurred. This scheme is not very efficient, and can

be susceptible to problems if the error occurs in exactly the same

place for each group e.g. "1010 1010 1010" in the example above will

39

be detected as correct in this scheme. The scheme however is extremely

simple, and is in fact used in some transmissions of numbers stations.

(Dinesh Thakur)

2.15.2 Checksum

Checksum is an error detection method that is a modular

arithmetic sum of message code words of fixed word length. Checksum

schemes involve longitudinal redundancy checks, parity bits and check

digits. A checksum of a message is an arithmetic sum of message code

words of a certain word length, for example byte values, and their

carry value. The sum is negated by means of ones-complement, and

stored or transferred as an extra code word extending the message. On

the receiver side, a new checksum may be calculated, from the extended

message. If the new checksum is not 0, error is detected. Checksum

schemes include parity bits, check digits, and longitudinal redundancy

check. Suppose we have a fairly long message, which can reasonably be

divided into shorter words (a 128 byte message, for instance). We can

introduce an accumulator with the same width as a word (one byte, for

instance), and as each word comes in, add it to the accumulator.

40

When the last word has been added, the contents of the

accumulator are appended to the message (as a 129th byte, in this

case). The added word is called a checksum. Now, the receiver performs

the same operation, and checks the checksum. If the checksums agree,

we assume the message was sent without error. (Dinesh Thakur)

2.15.3 Parity Method for Error Detection

The oldest method of error correction involves using parity. It

works by adding an additional bit to each character word transmitted.

The state of the bit is determined by a number of factors such as the

type of parity and the number of logic-one bits in the data character.

The movement of digital data from one location to another can

result in transmission errors, the receiver not receiving the same

signal as transmitted by the transmitter as a result of electrical

noise in the transmission process. Sometimes a noise pulse may be

large enough to alter the logic level of the signal. For example, the

transmitted sequence 1001 may be incorrectly received as 1101. In

41

order to detect such errors a parity bit is often used. A parity bit is

an extra 0 or 1 bit attached to a code group at transmission. In

the even parity method the value of the bit is chosen so that the total

number of 1s in the code group, including the parity bit, is an even

number. For example, in transmitting 1001 the parity bit used would be

0 to give 01001, and thus an even number of 1s. In transmitting 1101

the parity bit used would be 1 to give 11101, and thus an even number

of 1s. With odd parity the parity bit is chosen so that the total number

of 1s, including the parity bit, is odd. Thus if at the receiver the

number of 1s in a code group does not give the required parity, the

receiver will know that there is an error and can request that the

code group be retransmitted.

An extension of the parity check is the checksum in which a block

of code may be checked by sending a series of bits representing their

binary sum. Parity and checksums can only detect single errors in

blocks of code, double errors go undetected. Also, the error is not

located so that correction by the receiver can be made. Multiple-error

detection techniques and methods to pinpoint errors have been devised.

42

CHAPTER THREE.

METHODOLOGY.

3.1 INTRODUCTION.

The purpose of this work is to evaluate the performance of some

error detection and correction codes used in detecting and correcting

errors in mobile and wireless communication. Particularly, this work

will be focused on block codes (Reed-Solomon code) and the

convolutional codes. This will help in the designing and understanding

of digital communication systems that are reliable.

The methodologies applied throughout in this project would be

purely analytical and computer simulation. This chapter includes the

description of the algorithm for generating random binary data, using

Galois array, encoding the data, passing the data through a noise

channel, detecting the errors and correcting the errors.

The encoders and decoders would be modeled using MATLAB; these

are then used in simulating a communication system with view of

evaluation to demonstrate the benefits and limitations.

3.2 ALGORITHMS.

3.2.1 Description of the algorithm.

43

The steps involved in simulating a communication channel using

Galois array block and convolutional decoding from the transition of

the original message to the final message are below the Galois field

array is a function in MATLAB that has 2^m element where m is an

integer between 1 and 16 the other variable must also be an integer

between 0 and 2^m-1.

3.2.1.1 Generating the data.

Generating the data to be transmitted through the channel can

simply be done using a random number generator which would produce a

uniform distribution of numbers on the internal 0 to a maximum value

using RANDINT provided in MATLAB. Using the Galois array, it is

specified that the non-zero element should be used.

Data= randint (nw,k).

3.2.1.2 Encoding the data.

The encoder used at this stage is the RSENC. The Reed- Solomon

encoder encodes the message in MSG using an (N, K) Reed-Solomon

encoder with the narrow sense generator polynomial. MSG is a Galois

array of symbols over GF (2m). Each k-element row of the MSG

represents a message word, where the leftmost is the most significant

44

symbol. If N is smaller than 2m-1, and then RSENC uses a shortened RS

code. Parity symbols are at the end of each word in the output Galois

array code

Simplex syntax for encoding

Coded = rsenc (msg_galois, n, k)

3.2.1.3 Adding noise to the data.

Coding with this code is to be treated to a channel in which the

transmitted signal is corrupted mainly by Additive White Gaussian

Noise (AWGN). White noise is a random signal (or process) with a flat

power spectral density. Gaussian noise is statistical

noise that has a probability density function of the

normal distribution (also known as Gaussian distribution).

In other words, the values that the noise can take on are

Gaussian-distributed. It is most commonly used as additive white

noises to yield additive white Gaussian noise (AWGN).

Noise = randerr (nw, n.t);

Cnoisy=coded + noise;

3.2.1.4 Decoding the data

45

A RSDEC decoder is used at this stage. The Reed-Solomon decoder

attempt to decode the received signal in the code using an (N, K)

Reed-Solomon decoder with the narrow sense generator polynomial. The

code is a Galois array of symbols over GF (2m) where m is the number

of bits per symbol. Each N-element row of code represents a corrupted

systematic codeword, where the parity symbols are at the end and the

leftmost is the most significant symbol. If N is smaller than 2^m-1,

then RSDEC assumes that the code is a corrupted version of a shortened

code.

[corrcode, nerrs, decoded] = rsdec (cnoisy, n, k).

3.3 Algorithm for convolutional code.

3.3.1 Convolutional code encoder.

Convolution codes protect information by adding redundant bits.

This accepts a sequence of binary input vectors to produce a sequence

of binary output. The encoder takes k input bit streams i.e 2k

possible input symbol, the block input vector length will be L*K for

some positive integer. And will produce n output bit streams. A

trellis structure is used in describing convolution codes. A trellis

description of a convolution encoder shows how each possible input to

the encoder will influence both the output and the state transitions

46

of the encoder. To specify the encoder using its constraint length,

generator polynomial, the poly2trellis command is used. It parameters

are the constraint length and code generator polynomial. A convolution

encoder is called so because it performs a convolution of the input

stream with encoder's impulse responses: Where x is an input sequence,

y is a sequence from output and h is an impulse response for output.

3.3.2 Convolutional decoder.

Convolution code decoding algorithms infer the values of the

input information sequence from the stream of received distorted

output symbols. The major decoding algorithm for convolutional code is

the Viterbi algorithm. The goal of the Viterbi algorithm is to find

the transmitted sequence (or codeword) that is closest to the received

sequence. It finds the maximum likelihood path to the trellis. As long

as the distortion is not too severe, this will be the correct

sequence.

To evaluate the performance of the convolution codes in the noisy

channel, an Additive White Gaussian Noise (AWGN) channel is modeled.

Adding Gaussian noise to the encoded data is done by generating

Gaussian random numbers with desired energy per symbol to noise ratio.

47

48

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