The ternary alphabet is used by alternate mark inversion modulation; successive ones in data are represented by alternating ±1.

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1 Alphabets EE 387, Notes 2, Handout #3 Definition: An alphabet is a discrete (usually finite) set of symbols. Examples: B = {0,1} is the binary alphabet T = { 1,0,+1} is the ternary alphabet X = {00,01,...,FF} is the alphabet of 8-bit symbols (used in codes for compact discs, DVDs, and most hard disk drives) A channel alphabet symbol may be an indivisible transmission unit, e.g., one point from a signal constellation, or a sequence of modulation symbols encoded into a coding alphabet symbol The alphabets encountered in EE 387 usually have 2 m symbols. The ternary alphabet is used by alternate mark inversion modulation; successive ones in data are represented by alternating ±1. EE 387, September 23, 2015 Notes 2, Page 1

2 Block codes: definition The channel alphabet is the set of output symbols of the channel encoder same as set of input symbols to channel (modulator). The senseword alphabet is the set of output symbols from channel demodulator, i.e., input to channel decoder. The senseword alphabet may be larger than the channel alphabet, e.g., when the received symbols represent soft information. Examples: Binary erasure channel: input alphabet {0,1}, output alphabet {0,?,1}. Some disk drive read channel circuits quantize input signal to 6 bits; the channel alphabet has 64 symbols. Definition: A block code of blocklength n over an alphabet X is a nonempty set of n-tuples of symbols from X. C = {(c 11,...,c 1n ),...,(c M1,...,c Mn )} The n-tuples of the code are called codewords. We will think of codewords as vectors whose components are symbols in X. EE 387, September 23, 2015 Notes 2, Page 2

3 Block codes: rate Suppose that the channel alphabet has Q symbols. The rate of a block code of blocklengh n with M codewords is defined to be R = 1 n log QM. Codewords of length n are usually generated by encoding k information (data, message) symbols using an invertible encoding function. In this case, number of codewords is M = Q k. rate of code is R = 1 n log QQ k = 1 n k = k n. Such a code with blocklength n and rate k/n is called an (n,k) code. The rate is a dimensionless fraction (symbols per symbol). It is the fraction of transmitted symbols that carry information. EE 387, September 23, 2015 Notes 2, Page 3

4 Block codes: very simple examples C = { } = {SYN} Blocklength n = 8, M = 1, rate R = 1 8 log 21 = 0. Codes with rate 0 are called useless. This code could be used for error rate analysis or byte synchronization. C = {00,01,10,11} Blocklength n = 2, M = 4, rate R = 1 2 log 24 = 1. This code has no redundancy, so it can neither correct nor detect errors. C = {001,010,100} Blocklength n = 3, M = 3, rate R = 1 3 log 23 = This code might be used over a channel that drops bits (1 0 may occur but not 0 1), since any dropped 1 can be detected. C = {011,101,110} is a better code for this channel. Why? EE 387, September 23, 2015 Notes 2, Page 4

5 Block codes: more interesting examples Parity SIMMs have rate 8/9 and blocklength 9 or 36. They can detect one error per 8-bit byte. ECC DIMMs have blocklength 72 and rate 8/9. They can correct one error and detect two errors in 72 bits. Ethernet packet sizes range from 64 to 1518 bytes (12144 bits). Checksum is only 32 bits very high rate code for large packets. The number of binary 5-tuples of weight 2 or 3 (nearly DC balanced) is ( ) ( ) = = 20 > 16 = The 4B5B TAXI code for FDDI uses 16 of these 5-tuples to convey 4 bits of data information: {1E A 0B 0E 0F A 1B 1C 1D} (A few other 5-tuples are used for control purposes.) EE 387, September 23, 2015 Notes 2, Page 5

6 Hamming distance The Hamming distance d H between n-tuples is the number of components in which the n-tuples differ: { n 1 if xi y i d H (x,y) = d H (x i,y i ), where d H (x i,y i ) = 0 if x i = y i i=1 Hamming distance satisfies the axioms for a metric or distance measure: d(x,y) 0 with equality if and only if x = y (nonnegativity) d(x,y) = d(y,x) (symmetry) d(x,y) d(x,z)+d(z,y) (triangle inequality) Hamming distance is a coarse or pessimistic measure of difference. Other useful distances that occur in error control coding: Lee distance, distance on a circle, is applicable to phase shift coding. Euclidean distance is used with sensewords in R n. EE 387, September 23, 2015 Notes 2, Page 6

7 Minimum distance The minimum (Hamming) distance d of a block code is the distance between any two closest codewords: d = min{d H (c 1,c 2 ) : c 1,c 2 are codewords and c 1 c 2 } Obvious properties of minimum distance of a code of blocklength n: d 1 since Hamming distance between distinct codewords is a positive integer. d n if code has two or more codewords. d = n+1 or d = for the useless code with only one codeword. (This is a convention, not a theorem.) If C 1 C 2 then d (C 1 ) d (C 2 ) smaller codes have larger (or equal) minimum distance. The minimum distance of a code determines both its error-detecting ability and error-correcting ability. EE 387, September 23, 2015 Notes 2, Page 7

8 Error-detecting ability Suppose that a block code is used for error detection only. Let c be the transmitted codeword and let r the senseword the received n-tuple. If the r is not a codeword, a detectable error has occurred. If the r is a codeword but not the transmitted codeword, an error has occurred that cannot be detected. If d(c,r) < d, then the senseword cannot be an incorrect codeword. Otherwise c and r would be two codewords whose distance is less than the minimum distance. Conversely, let c 1,c 2 be two closest codewords. If c 1 is transmitted but c 2 is received, then an error of weight d has occurred that cannot be detected. Theorem: The guaranteed error-detecting ability is e = d 1. The error-detecting ability is a worst case measure of the code. Codes designed for error detection can detect the vast majority of errors when d or more symbols are incorrect. EE 387, September 23, 2015 Notes 2, Page 8

9 Error-correcting ability A block code is a set of M vectors in an n-dimensional space. The geometry of the space is defined by Hamming distance quite different from Euclidean geometry. Nonetheless, geometric intuition can be useful. d* decoding spheres decoding regiion The optimal decoding procedure is usually nearest-neighbor decoding: the senseword r is decoded to the nearest codeword ĉ: ĉ = argmin{d H (c,r) : c is a codeword} In R 2, decoding regions are Voronoi regions defined by perpendicular bisectors of lines connecting codewords. Hamming space is much more complicated combinatorial methods are needed. EE 387, September 23, 2015 Notes 2, Page 9

10 Error-correcting ability (cont.) Theorem: Using nearest-neighbor decoding, errors of weight t can be corrected if and only if 2t < d. Proof: The spheres of radius t surrounding the codewords do not overlap. Otherwise, there would be two codewords 2t distant. Therefore when t errors occur, the decoder can tell which codeword was sent. d* decoding spheres decoding regiion The maximal decoding region is usually larger than the decoding sphere. Most decoders correct only when senseword r belongs to a decoding sphere of radius t. They are called bounded-distance decoders. EE 387, September 23, 2015 Notes 2, Page 10

11 Decoding outcomes A block code can have more than one decoder, depending on the type of errors expected or observed the error rate the computational power available at the decoder Suppose that codeword c is transmitted, senseword r is received, and ĉ is the decoder s output. The table below classifies the outcomes. ĉ = c decoder success Successful correction (including no errors) ĉ =? decoder failure ĉ c decoder error Uncorrectable error detected, no decision (not too bad) Miscorrection (very bad) Important: the decoder cannot distinguish the outcome ĉ = c from ĉ c. However, it can assign probabilities to the possibilities; more bit errors corrected suggests a higher probability that the estimate is wrong. EE 387, September 23, 2015 Notes 2, Page 11

12 Decoding outcomes (cont.) The codeword transmitted and noise encountered are random variables, so the decoder outcomes are probabilistic events. r = c r c P e ĉ =? P ued ĉ c P mc ĉ c P ue No error Error occurred (error in codeword) Error detected but not corrected (decoder failure) Miscorrection (decoder error) Undetectable error (error detection only) Definition: A complete decoder is a decoder that decodes every senseword to some codeword; i.e., the decoder never fails (to make a hard decisions). For a complete decoder, Prued = 0. Definition: A bounded-distance decoder corrects all errors of weight t but no errors of weight > t. More than t errors results in failure or error. For a fixed code, if we reduce t then decoder failure becomes more common while decoder error becomes less likely. ued can be read as uncorrectable error detected, whereas ue is undetected error. EE 387, September 23, 2015 Notes 2, Page 12