ELG 5372 Error Control Coding. Claude D Amours Lecture 2: Introduction to Coding 2

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1 ELG 5372 Error Control Coding Claude D Amours Leture 2: Introdution to Coding 2

2 Deoding Tehniques Hard Deision Reeiver detets data before deoding Soft Deision Reeiver quantizes reeived data and deoder uses likelihood information to deode. Magnitude of deision variable usually indiates the likelihood that the detetion is orret. Maximum Likelihood Deoding MLD.

3 Hard Deision vs Soft Deision Hard Deision Soft Deision 2 bit quantization

4 Maximum Likelihood Deoding Transmit odeword Reeive word r. Deode. Deoding error ours if.

5 Maximum Likelihood Deoding 2 P E r = P ' r P E = P ' r r P r The optimum deoding rule is one that minimizes PE. Pr is independent of the deoding rule, therefore we must minimize P r, whih is equivalent to maximizing P = r.

6 Maximum Likelihood Deoding 3 / r r r P P P P = Assuming all odewords are equally likely, maximizing P r is the same as maximizing Pr. Assuming hard deision with disrete memoryless hannel DMC: log, log, log,, p d n p d P p p r P P d n d i i i + = = = r r r r r r Sine p < -p, Pr is maximized by the odeword for whih dr, is minimized. This is known as a minimum distane deoding rule. 2

7 Hamming Distane vs Eulidean Distane Hamming Distane = number of positions in whih two vetors differ. Hard deision deoding deodes using minimum Hamming distane rule previously shown Eulidean distane between r and is r- Soft deision deoding uses minimum Eulidean distane approximately

8 Deision Variables f ri r i - f ri r i r i P i =/P i =-=f ri r i i =/f ri r i i =-

9 Example: HD vs SD Consider Hamming 7,4 ode used with the following hannels 0 0 P0 0=P =0.9 P 0=P0 = P0 0=P =0.6 P0 0=P =0.3 P 0=P0 =0.099 P 0=P0 =0.00 a HD b SD

10 Example: HD vs SD Suppose we reeive r = For HD, there is no quantization, so r = and will be deoded as using minimum Hamming distane rule. In the SD ase, using with = , we get Pr = However, for =0000, we get Pr = This means that for the given r, it is almost 7 times more probable that 0000 was transmitted than

11 Errors and Channel Models Memoryless hannels: Noise affets eah transmitted symbol independently. Tx symbol has probability p of being reeived inorretly and probability -p of being reeived orretly. Transmission errors our randomly in the reeived sequene. Memoryless hannels are often referred to as random-error hannels.

12 Errors and Channel Models 2 Examples AWGN: r i = s i +n i, E[n i ]=0, E[n i2 ]=σ n2 and E[n i n j ]=0 for i j. DMC: 0 P[0 0] P[ 0] 0 P[0 ] P[ ]

13 Errors and Channel Models 3 Channels with memory Errors do not our randomly. Either noise is not independent from transmission to transmission oloured noise Or slow time varying signal to noise ratio auses time dependent error rates fading hannels.

14 Errors and Channel Models 4 Gilbert and Frithman model q -q Good state Bad State -q 2 q 2

15 Errors and Channel Models 5 Channels with memory lead to error bursts. Burst-error orreting odes Random error orreting odes with interleaving-deinterleaving to randomize errors.

16 Performane Measures Probability of deoding error PE. Probability that odeword at output of deoder is not the transmitted one. Also referred to as word error rate WER or Blok error rate BLER. Bit error rate BER P b Probability that message bit at output of deoder is inorret. Coding Gain measured in db Savings in transmitted power to ahieve a speifi BER using oding ompared to unoded ase

17 Performane Measures 2 Asymptoti oding gain Coding gain when E b /N o

18 Performane Measures 3.00E-02.00E-03.00E-04.00E-05 BER.00E-06.00E-07 unoded oded.00e-08.00e-09.00e Eb/No in db

19 Coded Modulation Use of ECC reates bandwidth expansion due to redundant symbols. Combining ECC and modulation allows the redundany to be ontained in the modulation 0 s t+s 0 t-t+s t-2t+s t-3t s t+s 2 t-t+s t-2t+s 3 t-3t Memory is reated without adding redundant bits by using a higher order modulation sheme and using a bit in two suessive symbols.

20 Trellis Coded Modulation State mahine adds redundant bits and reates memory State hange is enoded by seleting a symbol from a larger than needed onstellation, thus no bandwidth expansion ours and signifiant oding gains are ahieved.

On the Bit Error Probability of Noisy Channel Networks With Intermediate Node Encoding I. INTRODUCTION

On the Bit Error Probability of Noisy Channel Networks With Intermediate Node Encoding I. INTRODUCTION 5188 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 11, NOVEMBER 2008 [8] A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood estimation from inomplete data via the EM algorithm, J.