II. Linear Block Codes

© Tallal Elshabrawy 2 Digital Communication Systems Source of Information User of Information Source Encoder Channel Encoder Modulator Source Decoder Channel Decoder De-Modulator Channel

© Tallal Elshabrawy 3 Motivation for Channel Coding B B* Pr{B*≠B}=p For a relatively noisy channel, p (i.e., probability of error) may have a value of For many applications, this is not acceptable Examples: Speech Requirement: Pr{B*≠B}<10 -3 Data Requirement: Pr{B*≠B}<10 -6 Channel coding can help to achieve such a high level of performance

© Tallal Elshabrawy 4 Channel Coding B 1 B 2.. B k Channel Decoder Channel Encoder Channel Encoder: Mapping of k information bits in to an n- bit code word Channel Decoder: Inverse mapping of n received code bits back to k information bits Code Rate r=k/n r<1 W 1 W 2.. W n W* 1 W* 2.. W* n B* 1 B* 2.. B* k Physical Channel

© Tallal Elshabrawy 5 What are Linear Block Codes? Information sequence is segmented into message blocks of fixed length. Each k-bit information message is encoded into an n-bit codeword (n>k) Linear Block Codes Binary Block Encoder 2 k k-bit Messages 2 k n-bit DISTINCT codewords

© Tallal Elshabrawy 6 What are Linear Block Codes? Modulo-2 sum of any two codewords is ……… also a codeword Each codeword v that belongs to a block code C is a linear combination of k linearly independent codewords in C, i.e., Linear Block Codes

© Tallal Elshabrawy 7 Linear Independence A set of vectors g 0, g 1,…, g k-1 are linearly independent if there exists no scalars u 0, u 1,…, u k-1 that satisfy Unless u 0 =u 1 =…= u k-1 =0 Examples [0 1 0 ], [1 0 1], [1 1 1] are ……… Linearly Dependent [0 1 0 ], [1 0 1], [0 0 1] are ……… Linearly Independent

© Tallal Elshabrawy 8 Why Linear? Encoding Process Store and Index 2 k codewords of length n Complexity Huge storage requirements for large k Extensive search processing for large k Linear Block Codes Stores k linearly independent codewords Encoding process through linear combination of codewords g 0, g 1,…, g k-1 based on input message u=[u 0, u 1,…, u k-1 ] Generator Matrix

© Tallal Elshabrawy 9 Example MessageCodeword g0g0 g1g1 g2g2 g3g3 u= [ ] Linear Block Encoder (v=u.G) v= g 1 +g 2 v= [ ]

© Tallal Elshabrawy 10 Example u= [ ] Block Encoder (v=u.G) v= g 1 +g 2 +g 3 v= [ ] Linearly Dependent u= [ ] Block Encoder (v=u.G) v= g 0 +g 3 v= [ ] NOT DISTINCT

© Tallal Elshabrawy 11 Linear Systematic Block Codes Redundant Checking Part Message Part n-k bits k bits p-matrix kxk- identity matrix

© Tallal Elshabrawy 12 The Parity Check Matrix For any k x n matrix G with k linearly independent rows, there exists an (n-k) x n matrix H (Parity Check Matrix), such that G.H T =0

© Tallal Elshabrawy 13 Example

© Tallal Elshabrawy 14 Encoding Circuit u0u0 u1u1 u2u2 u3u3 Input u To channel + ++ v0v0 v1v1 v2v2 Parity Register Message Register [u 0 u 1 u 2 u 3 ] [v 0 v 1 v 2 u 0 u 1 u 2 u 3 ] Output v Encoder Circuit

© Tallal Elshabrawy 15 Syndrome Characteristic of parity check matrix (H) Channel v r + v r=v+e e Error Pattern Syndrome

© Tallal Elshabrawy 16 Error Detection r is NOT a codeword An Error is Detected: What Options do we have? Ask for Retransmission of Block Automatic Repeat Request (ARQ) Attempt the Correction of Block Forward Error Correction

© Tallal Elshabrawy 17 Undetectable Error Patterns Can we be sure that r=v ?? NO! WHY? How many undetectable error patterns exist? 2 k -1 Nonzero codeword means 2 k -1 undetectable error patterns

© Tallal Elshabrawy 18 Syndrome Circuit r1r1 r2r2 r3r3 r4r4 r5r5 r6r6 r0r s0s0 s1s1 s2s2