Modulation and Coding. Laboratory. Channel Coding Hamming code

Transcription

1 Modulation and Coding Laboratory Channel Coding Hamming code Hamming code is a set of error-correction code s that can be used to detect and correct bit errors that can occur when computer data is moved or stored. Hamming code is named for R. W. Hamming of Bell Labs. Like other error-correction code, Hamming code makes use of the concept of parity and parity bits, which are bits that are added to data so that the validity of the data can be checked when it is read or after it has been received in a data transmission. In data transmission, the ability of a receiving station to correct errors in the received data is called forward error correction (FEC) see fig 1:. Figure 1 General scheme of FEC (forward Error correction) 1. Hamming code on an example Hamming (7,4) In the Hamming code, k parity bits are added to an n-bit data word, forming a new word of n+k bits. The bit positions are numbered in sequence from 1 to n+k. Those positions numbered with powers of two are reserved for the parity bits. The remaining bits are the data bits. The code can be used with words of any length.

2 Figure 2: Configuration of control bits in Hamming(7,4) code The 3 parity bits P 1 through P 4 are in positions 1, 2 and 4, respectively. Each parity bit is calculated as follows: where: b n is number of (or sum) of bits in given positions. Next step is calculate a parity bits: A parity bit, or check bit, is a bit added to a string of binary code to ensure that the total number of 1-bits in the string is even or odd. Parity bits are used as the simplest form of error detecting code. There are two variants of parity bits: even parity bit and odd parity bit. Odd parity version: The parity bit is called the control bit, which takes the value 1 when the number of ones ('1'), in the transmitted, message is odd Even parity version: The parity bit is called the control bit, which takes the value 1 when the number of ones ('1'), in the transmitted, message is even (we will use a odd parity version of parity bits) Example: Input message: 1010 b 4 = b 5 + b 6 + b 7 b 2 = b 3 + b 6 + b 7 b 1 = b 3 + b 5 + b 7 Bit position: p 1 p 2 p 3 p 4 p 5 p 6 p 7 Message: Controls Bits: Codec Message:

3 7 - total number of bits (n-bits) 4 - number of information bits (our message, k-bits ) n k - number of control bits (m-bits) Hamming codes can be computed in linear algebra terms through matrices because Hamming codes are linear codes. For the purposes of Hamming codes, two Hamming matrices can be defined: the code generator matrix G and the parity-check matrix H: rows 1, 2, and 4 of matrix G map the data bits to their parity bits. Decoding : where: T S = v H S Syndrom Matrix v- received message H T - Transposed parity check matrix if S <> 0 then the message contains an error 2. Exercise: Hamming coding and decoding

4 n = 7; k = 4; msg = [1, 0, 1, 0].' % Code length % Message Length % Message (in binary representation) % coding message in Hamming(7,4) code encdata = encode(msg,n,k,'hamming/binary') % decoding message (compare with example!) decdata = decode(encdata,n,k,'hamming/binary') % check number of error numerr = biterr(binmessage,decdata) Now we can add some errors... Let say, on 4th bit encdata (4) = ~ encdata (4); % decoding message (compare with example!) decdata = decode(encdata,n,k,'hamming/binary') % check number of error numerr = biterr(binmessage,decdata) Code with minimum distance Dmin can detect D min -1 errors and correct (Dmin-1) / 2 or Dmin / 2-1. Decode function automatically correct errors. 2.1 Simple data transmission over noisy-channel We try s a text over a noisy-channel without and with error correction. Source data preparation n = 15; k = 11; % Code length % Message Length data = dec2bin( Some testing message,k); (the message could be longer ) errors = 0; errorsnp = 0; % number of errors for protection channel % number of errors for non-protection channel

5 additional functions: % function for convert from string to number array: function [ numarray ] = str2numarray( data,dim ) datalength = size(data,dim); numarray = zeros(1,datalength); for i = 1 : datalength numarray(i) = str2num(data(i)); % simple channel simulator: % Pe Error Propability function [ message ] = simplychannelmodel( message, Pe ) for i = 1 : size(message,2) if (rand <= Pe) message(i) = ~message(i); Main function: for i = 1 : size(data,1) binmessage = str2numarray(data(i,:), 1); encdata = encode(binmessage,n,k,'hamming/binary'); Pe = 0.05; encdata = simplychannelmodel( encdata, Pe ); dataencoded(i,:) = encdata; nonprotectiondata = simplychannelmodel( binmessage, Pe ); decdata = decode(encdata,n,k,'hamming/binary'); numerr = biterr(binmessage,decdata); numerrnp = biterr(nonprotectiondata,decdata); if (numerr > 0) errors = errors + 1; if (numerrnp > 0) errorsnp = errorsnp + 1; BER = errorsnp / size(data,1); message = strcat('ber [No Error Correction] = [',num2str(ber),']'); disp(message);

6 BER = errors / size(data,1); message = strcat('ber [Error Coorection] = [',num2str(ber),']'); disp(message); Check BER level for different P e. Compare BER coefficient for both transmissions: with and without Error Correction. Try draw a chart with BER for different P e values (for both transmissions). 2.2 Calculate BER coefficient in more realistic case: Transmission with FSK modulation and noisy channel model (AWGN Model) FSK Modulation and demodulation: M = 2; % Modulation order k = log2(m); % Bits per symbol EbNo = 5; % Eb/No (db) Fs = 16; % Sample rate (Hz) nsamp = 8; % Number of samples per symbol freqsep = 10; % Frequency separation (Hz) % Modulation txsig = fskmod(datatx,m,freqsep,nsamp,fs); % Noisy-Channel-Model rxsig = awgn(txsig,ebno+10*log10(k)-10*log10(nsamp),... 'measured',[],'db'); % Demodulation dataout = fskdemod(rxsig,m,freqsep,nsamp,fs); % get BER [num,ber] = biterr(datatx,dataout); BER_theory = berawgn(ebno,'fsk',m,'noncoherent'); [BER BEREC BER_theory]

7 Message Data Preparation: data = dec2bin( Some testing message,k); for i = 1 : size(data,1) binmessage = str2numarray(data(i,:), 1); encdata = encode(binmessage,n,k,'hamming/binary'); dataencoded(i,:) = encdata; % Preparing of 1-dimensional data stream... [nrmessages, messagelength] = size(data); datatxec = reshape(dataencoded, [nrmessages * n, 1]); Check BER coefficient for different E b /N 0 Compare BER coefficient for transmission with and without Error correction Draw a chart. On OY axis put the BER coefficient. On Ox axis put The E b /N 0 ratio. (for both transmissions).