林茂昭 教授 台大電機系 96.2.15 個人專長 錯誤更正碼 數位通訊 大學部專題介紹 林茂昭 教授 台大電機系 96.2.15 個人專長 錯誤更正碼 數位通訊

Basic Coded Communication System Source Source Encoder Channel Encoder Channel User Source Decoder Channel Decoder Source encoder/decoder: For reducing redundancy without distortion or with distortion Channel encoder/decoder: for increasing transmission reliability by adding redundancy

Importance of Error-Control Coding (ECC) ECC can be used to increase the transmission rate and reliability in the digital communication (either wireless or wired communication) and digital storage system. Most powerful system can be obtained by integrating, ECC, modulation, channel information and synchronization.

Repetition Codes (I) Let C={(000),(111)} Received vector 0 0 0 0 0 1 0 1 0 1 0 0 Coding rate R=1/3 Minimum distance d=3 Error correcting capability decoding 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 1 1 1

Repetition Codes (II) Let c={(00000),(11111)}. Then d=5,R=1/5 and Let c={(00…0),(11…1)} and n=2t+1. Then d=2t+1 R=1/(2t+1) Apparently, as .Is it necessary that to achieve zero error probability, we can only have zero coding rate!

(7,4) Hamming code n=7 , k=4 , d=3, R=4/7. 0000 1000 0100 1100 0010 1010 0110 1110 0000000 1101000 0110100 1011100 1110010 0011010 1000110 0101110 0001 1001 0101 1101 0011 1011 0111 1111 1010001 0111001 1100101 0001101 0100011 1001011 0010111 1111111

Nearest-Neighbor Decoding: Let be the transmitted codeword, be the received vector and be the error vector. The codeword can be correctly recovered if . Equivalently

Error Control Coding (I) For any (n,k,d) code C, up to errors can always be corrected, or up to d-1 errors can always be detected. An (n,k,d) code C is a subset of with size , for which the minimum distance between any two codewords is d.

Error Control Coding (II) For a given d, it is desired to find a subset C of such that is as large as possible. For a given k, it is desired to find a subset C of such that and d is as large as possible Code construction Low decoding complexity

Binary Symmetric Channel (BSC) transition probabilities:

Additive White Gaussian Noise Channel (AWGN Channel) : one-side power spectral density of the AWGN Channel : Bandwidth

Hard Decision and Soft Decision Consider an AWGN channel with binary input A and –A. if hard decision in the output is made, then the AWGN channel is reduced to a BSC channel Example let C={(-1,-1),(1,1)}. Suppose that If hard decoding is used, no decision on the decoded codeword can be made. If soft decoding is used, (-1,-1) will be decoded.

Integrated Coding and Modulation Each signal point in the 8PSK signal set (or constellation) is labeled by 3 bits (a,b,c) (2) 010 (3) 110 100 (1) (4) 001 000 (0) 1 (5) 101 111 (7) 011 (6)

Trellis Coded MPSK over Rayleigh Fading Channel Let be the transmitted MPSK sequence. Let be the received sequence, where : zero mean complex Gaussian noise with variance : normalized random variable with Rayleigh distribution ,ie.

Current Topics Convolutional coding Coded OFDM Coded Equalization Noncoherent Coded Modulation Space-time coding Coding for Memory systems

Topics for Beginners Viterbi Decoder Turbo Decoder

Recent Publications (I) “Minimal Trellis Modules and Equivalent Convolutional Codes ,” IEEE Transactions on Information Theory, Vol. 52, No. 8, pp. 3738-3746, August 2006. “Recursive Clipping and Filtering with Bounded Distortion for PAPR Reduction,” IEEE Transactions on Communications, Vol. 55, No. 1, pp. 227-230, January 2007

Recent Publications (II) “Coded MIMO Using Interblock Memory,” IEEE Vehicular Technology Conference, Spring 2006, Melbourne, Australia, May 7-10, 2006. “Improved Decoding for Trellis Coded Modulation with A Delay Processor,” International Symposium on Information Theory and Its Applications, Seoul Korea, October 29-November 1, 2006 “Binary Turbo Coding with Interblock Memory,” IEEE Wireless Communications and Network Conference (WCNC 2007), March 11-15, Hong Kong.