The Principle of Forward Error Correcting Codes and Its Application in the Internet and Wireless Ommunications

Li Mengli，Song Wei

（Department of telecommunication engineering with management，International school，Beijing University of Posts and Telecommunications，Beijing，102209，China）

Abstract： Error correction codes are important in protection data being transmitted.And the forward error correcting codes（FECs）has becomes a most important error-coding function.This paper introduces the principles of FEC and how they work and explains the FECs used in the Internet and Wireless communications.

Key words： FECs，Trellis，Internet communications，Wireless communications

1 Introduction

This paper concerns error control，or more generally with increasing the robustness of data transmission in the presence of channel perturbations such as noise.One approach is the use of error-detection coding，in which errors are detected and a re-transmission requestedVia some return，or feedback，channel.This is the technique widely used in computer networks，and corporate into the data link layer of the OSI stack.However，here we focus on forward error correction（FEC）coding，shown in Fig.1，which is able to correct transmission errors even without a feedback，channel.

2 Principle of FECs

Shannon showed that the method by which this capacity increaser can be achieved，paradoxically，is by the addition of redundant information to the transmitted data.This is done in such a way that the wanted information can be reconstructed from the received data，despite the corruption introduced by the channel.For a binary system this is done by inserting additional bits，called checked or parity bits，into the transmitted data.These check bits are obtained from the information bits by an appropriate algorithm.

Consider，for an example，a 2 bit message，to which three check bits are added.

There are four possible messages，giving rise to four possible 5 bit encoded blocks，called code words.As shown in Fig.2.

Suppose the second of these is transmitted，but an error occurs in the second bit，so that it is received as 00110.This error can be detected，because the resulting word is not one of the permitted code words.It can also be corrected，by comparing the received word with each of the code words in turn.It differs in one place from the second codeword，but in two or more form each of others，and thus the decoder can correctly select the codeword 01110 as the intended one.

The number of places in which two words differ is characterized as a ‘distance’ between them，called the Hamming distance.In these terms the operation of the decoder is to select the codeword closest in Hamming distance to the received word.As shown in the Fig.3

The distance between codewords is also a useful measure of the error correcting power of the code.Errors in a transmitted word will ‘move’ the received word a Hamming distance d equal to the number of errors，as shown in Fig.4.Provided this distance is less than halfway to the nearest alternative codeword，the decoder will still select the correct codeword.Thus the code will always correct a maximum number of errors：

where dmin is the minimum Hamming distance between any pair of code words.In the example 1，dmin is 3，so the code will indeed always correct single errors.Note that the inequality is strictly less than，since if the received word is exactly halfway between two codewords，the receiver will not be able to select the correct word reliably.

The cost of this error-correction capability，apart from the increased complexity of the receiver，is the need to transmit additional check bits in addition to the information bits.These additional bits constitute redundancy.Clearly，because of this redundancy a code can transmit information only at a rate somewhat lower than would be possible over the uncoded channel.

3 Different Types of FECs’ Applications in Wireless Communications

3.1 Example of Deep Space Communications（Cyclic Block Codes Application ）

One of the most extreme applications of radio communications is communication with deep space missions such as theVoyager mission to Saturn and Uranus.Free space path loss over the distance（several hundred million kilometers），and limitations on transmit power，are such that power efficiency of the communication system is of the utmost importance.（The inverse square law implies that a coding gain of 6dB due to the FEC coding scheme can double the range of the mission.）For the return link（spacecraft to Earth），very complex codes can be used，since powerful decoding hardware is available to Earth.

UntilVery recently these missions used concatenated codes in which once again Reed-Solomon codes formed the outer code.Here，however，the inner code was a half-rate convolutional code，with constraint length 7.Outer Reed-Solomon codes were used with block length 63-511（symbol size 6-9 bits）.Convolutional codes also produce error bursts at the inner decoder output，and so once again an interleaver is used between the inner and outer encoder/decoder.

3.2 Example ofVoiceband Modems（Trellis Coded Modulation Application ）

The application gives the initial impetus for the development of trellis coded modulation was theVoiceband modem：the device that allows digital services like FAX and Internet access over a conventional telephone line.This is designed to handle speech signals of trather restricted quality，and has a bandwidth extending from 200Hz TO 3.4KHz.As such it is clear that to achieve data rates greater than about 2.4kbits−1 requires multilevel modulation techniques.This was an ideal application for trellis coded modulation.

4 Conclusion

The basic function of error coding is reducing the number of reception errors in a digital communications system.Comparing the three types of FECs coding，analyzing the strengths and shortcomings of them and their applications in real life，we get a deep understanding of FECs.