Linear Block Code ppsx

Linear Block Codes 2006-02-16 Lecture 9 2 Last time we talked about:  Evaluating the average probability of symbol error for different bandpass modulation schemes  Comparing different modulation schemes based on their error performances. 2006-02-16 Lecture 9 3 Today, we are going to talk about:  Channel coding  Linear block codes  The error detection and correction capability  Encoding and decoding  Hamming codes  Cyclic codes 2006-02-16 Lecture 9 4 Block diagram of a DCS Format Source encode Format Source decode Channel encode Pulse modulate Bandpass modulate Channel decode Demod. Sample Detect C h a n n e l Digital modulation Digital demodulation 2006-02-16 Lecture 9 5  Channel coding:  Transforming signals to improve communications performance by increasing the robustness against channel impairments (noise, interference, fading, )  Waveform coding: Transforming waveforms to better waveforms  Structured sequences: Transforming data sequences into better sequences, having structured redundancy.  “Better” in the sense of making the decision process less subject to errors. What is channel coding? 2006-02-16 Lecture 9 6 Error control techniques  Automatic Repeat reQuest (ARQ)  Full-duplex connection, error detection codes  The receiver sends a feedback to the transmitter, saying that if any error is detected in the received packet or not (Not-Acknowledgement (NACK) and Acknowledgement (ACK), respectively).  The transmitter retransmits the previously sent packet if it receives NACK.  Forward Error Correction (FEC)  Simplex connection, error correction codes  The receiver tries to correct some errors  Hybrid ARQ (ARQ+FEC)  Full-duplex, error detection and correction codes 2006-02-16 Lecture 9 7 Why using error correction coding?  Error performance vs. bandwidth  Power vs. bandwidth  Data rate vs. bandwidth  Capacity vs. bandwidth (dB) / 0 NE b B P A F B D C E Uncoded Coded Coding gain: For a given bit-error probability, the reduction in the Eb/N0 that can be realized through the use of code: [dB][dB] [dB] c 0 u 0         −         = N E N E G bb 2006-02-16 Lecture 9 8 Channel models  Discrete memory-less channels  Discrete input, discrete output  Binary Symmetric channels  Binary input, binary output  Gaussian channels  Discrete input, continuous output 2006-02-16 Lecture 9 9 Linear block codes  Let us review some basic definitions first which are useful in understanding Linear block codes. 2006-02-16 Lecture 9 10 Some definitions  Binary field :  The set {0,1}, under modulo 2 binary addition and multiplication forms a field.  Binary field is also called Galois field, GF(2). 011 101 110 000 =⊕ =⊕ =⊕ =⊕ 111 001 010 000 =⋅ =⋅ =⋅ =⋅ Addition Multiplication [...]... Lecture 9 is a basis for V4 14 Linear block codes  Linear block code (n,k)  A set with cardinality C ⊂ Vn is called a linear 2k block code if, and only if, it is a subspace of the vector space Vn    2006-02-16 Vk → C ⊂ Vn Members of C are called code- words The all-zero codeword is a codeword Any linear combination of code- words is a codeword Lecture 9 15 Linear block codes – cont’d Vk Vn mapping... 2006-02-16 Lecture 9 16 Linear block codes – cont’d     The information bit stream is chopped into blocks of k bits Each block is encoded to a larger block of n bits The coded bits are modulated and sent over channel The reverse procedure is done at the receiver Data block Channel encoder k bits n-k Codeword n bits Redundant bits k Rc = Code rate n 2006-02-16 Lecture 9 17 Linear block codes – cont’d ... parity bits 2006-02-16 Lecture 9 message bits 25 Linear block codes – cont’d  For any linear code we can find an matrix , which H rows are orthogonal to rows of its ( n − k )×n   : G GH = 0 T H is called the parity check matrix and its rows are linearly independent For systematic linear block codes: H = [I n − k 2006-02-16 Lecture 9 PT ] 26 Linear block codes – cont’d Data source m Format Channel encoding... vk 2  Lecture 9 v1n  v2 n     vkn  22 Linear block codes – cont’d  Encoding in (n,k) block code U = mG   V1  V  (u1 , u 2 , , u n ) = (m1 , m2 , , mk ) ⋅  2      Vk  (u1 , u 2 , , u n ) = m1 ⋅ V1 + m2 ⋅ V2 +  + m2 ⋅ Vk The rows of G, are linearly independent 2006-02-16 Lecture 9 23 Linear block codes – cont’d  Example: Block code (6,3) Message vector  V1  1 1 0 1 0 0... calculation are easily performed using feedback shift-registers   Hence, relatively long block codes can be implemented with a reasonable complexity BCH and Reed-Solomon codes are cyclic codes 2006-02-16 Lecture 9 34 Cyclic block codes  A linear (n,k) code is called a Cyclic code if all cyclic shifts of a codeword are also a codeword  U = (u0 , u1 , u2 , , un −1 ) “i” cyclic shifts of U Example: (i ) U =... minimum distance of a block code is d min = min d (U i , U j ) = min w(U i ) i≠ j 2006-02-16 i Lecture 9 18 Linear block codes – cont’d  Error detection capability is given by e = d min − 1  Error correcting-capability t of a code, which is defined as the maximum number of guaranteed correctable errors per codeword, is  d min − 1 t=  2   2006-02-16 Lecture 9 19 Linear block codes – cont’d  For... The corrected vector is estimated ˆ ˆ U = r + e = (001110) + (100000) = (101110) Lecture 9 31 Hamming codes  Hamming codes   Hamming codes are a subclass of linear block codes and belong to the category of perfect codes Hamming codes are expressed as a function of a single integer m≥2 n = 2m − 1 Code length :  Number of information bits : k = 2 m − m − 1 Number of parity bits : n-k = m Error correction... non-zero binary m-tuples 2006-02-16 Lecture 9 32 Hamming codes  Example: Systematic Hamming code (7,4) 1 0 0 0 1 1 1 H = 0 1 0 1 0 1 1 = [I 3×3   0 0 1 1 1 0 1   0 1 1 1 0 0 0  1 0 1 0 1 0 0  = [P G= 1 1 0 0 0 1 0   1 1 1 0 0 0 1  2006-02-16 Lecture 9 PT ] I 4×4 ] 33 Cyclic block codes   Cyclic codes are a subclass of linear block codes Encoding and syndrome calculation are easily...  V2  = 0 1 1 0 1 0      V3  1 0 1 0 0 1     2006-02-16 Lecture 9 Codeword 000 100 010 000000 110100 011010 110 001 101 011 1 11 1 011 1 0 1 01 0 0 1 0 111 0 1 1 1 0 011 0 0 0 1 11 24 Linear block codes – cont’d  Systematic block code (n,k)  For a systematic code, the first (or last) k elements in the codeword are information bits G = [P I k ] I k = k × k identity matrix Pk = k × (n...  1-p 0 Note that for coded systems, the coded bits are modulated and transmitted over channel For example, for M-PSK modulation on AWGN channels (M>2):  2( log 2 M ) Ec  2( log 2 M ) Eb Rc 2 2  π   π   =  p≈ Q sin    Q sin    log 2 M  N0 N0  M   log 2 M   M   where Ec is energy per coded bit, given by Ec = Rc Eb 2006-02-16 Lecture 9 21 Linear block codes –cont’d Vk  Vn . 9 15 Linear block codes  Linear block code (n,k)  A set with cardinality is called a linear block code if, and only if, it is a subspace of the vector space .  Members of C are called code- words.  The. code- words.  The all-zero codeword is a codeword.  Any linear combination of code- words is a codeword. n V n VC ⊂ k 2 nk VCV ⊂→ 2006-02-16 Lecture 9 16 Linear block codes – cont’d n V k V C Bases. C mapping 2006-02-16 Lecture 9 17 Linear block codes – cont’d  The information bit stream is chopped into blocks of k bits.  Each block is encoded to a larger block of n bits.  The coded bits are modulated