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1 S Digital Communication Systems Cyclic Codes

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Timo O. Korhonen, HUT Communication Laboratory 2 S Digital Communication Systems u Lectures: Timo O. Korhonen, tel , Michael Hall, tel u Course assistants: Seppo Saastamoinen tel , Naser Tarhuni ), tel u Study modules: Examination /Tutorials /Project work u NOTE: Half of exam questions directly from tutorials u Project work guidelines published later in course homepage

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Timo O. Korhonen, HUT Communication Laboratory 3 Practicalities u References: –A. B. Carlson: Communication Systems (4th ed.) –J. G. Proakis, Digital Communications (4th ed.) –L. Ahlin, J. Zander: Principles of Wireless Communications u Prerequisites: S Transmission Methods in Telecommunication Systems u Homepage: u Timetables: –Lectures: Tuesdays S3, Wednesdays I346 –Tutorials: Mondays S2, starting

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Timo O. Korhonen, HUT Communication Laboratory 4 Targets today u Taxonomy of coding u How cyclic codes are defined? u Systematic and nonsystematic codes u Why cyclic codes are used? u How their performance is defined? u How practical encoding and decoding circuits are realized? u How to construct cyclic codes?

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Timo O. Korhonen, HUT Communication Laboratory 5

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6 Background u Coding is used for –error detection and/or error correction (channel coding) –ciphering (security) and compression (source coding) u In coding extra bits are added or removed in data transmission u Channel coding can be realized by two approaches –FEC (forward error coding) t block coding, often realized by cyclic coding t convolutional coding –ARQ (automatic repeat request) t stop-and-wait t go-back-N t selective repeat … etc. u Note: ARQ applies FEC for error detection

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Timo O. Korhonen, HUT Communication Laboratory 7 Block and convolutional coding u Block coding: mapping of source bits of length k into (binary) channel input sequences n (>k) - realized by cyclic codes! u Binary coding produces 2 k code words of length n. Extra bits in the code words are used for error detection/correction u (1) block, and (2) convolutional codes: –(n,k) block codes: Encoder output of n bits depends only on the k input bits –(n,k,L) convolutional codes: t each source bit influences n(L+1) encoder output bits –n(L+1) is the constraint length –L is the memory depth u Essential difference of block and conv. coding is in simplicity of design of encoding and decoding circuits (n,k) encoder k bitsn bits k input bits n output bits n(L+1) output bits input bit

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Timo O. Korhonen, HUT Communication Laboratory 8 Why cyclic codes? u For practical applications rather large n and k must be used. This is because in order to correct up to t errors it must be that u Hence for, large n and k must be used u Cyclic codes are –linear: sum of any two code words is a code word –cyclic: any cyclic shift of a code word produces another code word u Advantages: Encoding, decoding and syndrome computation easy by shift registers number of syndromes (or check-bit error patterns) number of error patters in encoded word (n,k) block coder k-bitsn-bits

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Timo O. Korhonen, HUT Communication Laboratory 9 Example u Consider a relatively high SNR channel such that only 1 or 2 bit errors are likely to happen. Consider the ration Take a constant code rate of R c =k/n=0.8 and consider with some values of larger n and k : u This demonstrates that long codes are more advantages when a high code rate and high error correction capability is required (n,k) block coder k-bitsn-bits Number of error patterns Number of check-bits

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Timo O. Korhonen, HUT Communication Laboratory 10 Some block codes that can be realized by cyclic codes u (n,1) Repetition codes. High coding gain, but low rate u (n,k) Hamming codes. Minimum distance always 3. Thus can detect 2 errors and correct one error. n=2 m -1, k = n - m, u Maximum-length codes. For every integer there exists a maximum length code (n,k) with n = 2 k - 1,d min = 2 k-1. Hamming codes are dual 1 of of maximal codes. u BCH-codes. For every integer there exist a code with n = 2 m -1, and where t is the error correction capability u (n,k) Reed-Solomon (RS) codes. Works with k symbols that consists of m bits that are encoded to yield code words of n symbols. For these codes and u Nowadays BCH and RS are very popular due to large d min, large number of codes, and easy generation u For further code references have a look on self-study material! 1: Task: find out from literature what is meant by dual codes!

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Timo O. Korhonen, HUT Communication Laboratory 11 Defining cyclic codes: code polynomial and generator polynomial u An (n,k) linear code X is called a cyclic code when every cyclic shift of a code X, as for instance X’, is also a code, e.g. u Each (n,k) cyclic code has the associated code vector with the n-bit code polynomial u Note that the (n,k) code vector has the polynomial of degree of n-1 or less. Mapping between code vector and code polynomial is one-to-one, e.g. they specify each other uniquely u Manipulation of the associated polynomial is done in a Galois field (for instance GF(2)) having elements {0,1}, where operations are performed mod-2. Thus results are always {0,1} -> binary logic circuits applicable u For each cyclic code, there exists only one generator polynomial whose degree equals the number of check bits q=n-k in the encoded word

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Timo O. Korhonen, HUT Communication Laboratory 12 Example: Generating of (7,4) cyclic code, by generator polynomial G(p)=p 3 +p+1 <- message <- encoded word <- generator

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Timo O. Korhonen, HUT Communication Laboratory 13 The common factor of cyclic codes u GF(2) operations (XOR and AND): u Cyclic codes have a common factor p n +1. In order to see this we consider summing two cyclic code vectors: u Question is how to make the n-1 degree cyclic code from the multiplied code? Adding them should yield a cyclic code: u Result can be forced to be a cyclic code of at most of n-1 degree by dividing by the underlined common factor! Left rotated Left shifted by multiplication Unshifted

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Timo O. Korhonen, HUT Communication Laboratory 14 Obtaining a cyclic code from another cyclic code u Unity cyclic shift is obtained by multiplication by p where after division by the common factor yields a cyclic code at the remainder and by induction, any cyclic shift is obtained by u Example: shift 101 u Important point of implementation is is that the division by p n +1 can be realized by a tapped shift register not a three-bit code (1010), divide by the common factor

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Timo O. Korhonen, HUT Communication Laboratory 15 Factoring cyclic code generator polynomial u Any factor of p n +1 with the degree of q=n-k generates an (n,k) cyclic code* u Example: Consider the polynomial p This can be factored as u For instance the factors p 3 +p+1 or p 3, +p 2 +1 can be used to generate an unique cyclic code. For a message polynomial p 2 +1 the following encoded word is generated: and the respective code vector (of degree n-1 or smaller) is u Hence, in this example (n,k) block coder k-bitsn-bits

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Timo O. Korhonen, HUT Communication Laboratory 16 Using shift registers for multiplication u Figure shows a shift register to realize multiplication by p 3 +p+1 u In practice, multiplication can be realized by two equivalent topologies: unit delay element alternate notation of XOR-circuit

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Timo O. Korhonen, HUT Communication Laboratory 17 Multiplication by a shift register generator polynomial determines connection of taps word to be encoded Encoded word

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Timo O. Korhonen, HUT Communication Laboratory 18 Determines tap connections Word to be rotated (divided by the common factor) Adding the dashed-line (feedback) enables division by p n +1 Remainder Calculating the remainder Remainder is left to the shift register

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Timo O. Korhonen, HUT Communication Laboratory 19 Examples of cyclic code generator polynomials u The generator polynomial for an (n,k) cyclic code is defined by and G(p) is a factor of p n +1. Any factor of p n +1 that has the degree q (the number of check bits) may serve as the generator polynomial. We noticed earlier that a cyclic code is generated by the multiplication where M(p) is the k-bit message to be encoded u Only few of the possible generating polynomials yield high quality codes (in terms of their minimum Hamming distance) Some cyclic codes:

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Timo O. Korhonen, HUT Communication Laboratory 20 Systematic cyclic codes u Define the length q=n-k check vector C and the length-k message vector M by u Thus the systematic n:th degree codeword polynomial is How to determine the check-bits?? Question: Why these denote the message bits still the message bits are M(p) ??? check bits message bits

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Timo O. Korhonen, HUT Communication Laboratory 21 Determining check-bits u Note that the check-vector polynomial is the remainder left over after dividing Example: (7,4) Cyclic code: > Definition of systematic cyclic code

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Timo O. Korhonen, HUT Communication Laboratory 22 Division of the generated code by the generator polynomial leaves no reminder This can be used for error detection/correction as we inspect later

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Timo O. Korhonen, HUT Communication Laboratory 23 Example: Encoding of systematic cyclic codes

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Timo O. Korhonen, HUT Communication Laboratory 24 Circuit for encoding systematic cyclic codes u We noticed earlier that cyclic codes can be generated by using shift registers whose feedback coefficients are determined directly by the generating polynomial u For cyclic codes the generator polynomial is of the form u In the circuit, first the message flows to the transmitter, and feedback switch is set to ‘1’, where after check-bit-switch is turned on, and the feedback switch to ‘0’, enabling the check bits to be outputted 1 0

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Timo O. Korhonen, HUT Communication Laboratory 25 Decoding cyclic codes u Every valid, received code word R(p) must be a multiple of G(p), otherwise an error has occurred. (Assume that the probability for noise to convert code words to other code words is very small.) u Therefore dividing the R(p)/G(p) and considering the remainder as a syndrome can reveal if an error has happed and sometimes also to reveal in which bit (depending on code strength) u Division is accomplished by a shift registers whose tap order is reversed u The error syndrome of q=n-k bits is therefore u This can be expressed also in terms of the error E(p) and the code word X(p) while noting that the received word is in terms of error hence

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Timo O. Korhonen, HUT Communication Laboratory 26 Decoding cyclic codes: example Using denotation of this example:

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Timo O. Korhonen, HUT Communication Laboratory 27 Table 16.6 Decoding cyclic codes (cont.)

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Timo O. Korhonen, HUT Communication Laboratory 28 Decoding circuit for (7,4) code syndrome computation u To start with, the switch is at “0” position u Then shift register is stepped until all the received code bits have entered the register u This results is a 3-bit syndrome (n - k = 3 ) that is then left to the register u Then the switch is turned to the direction “1” that drives the syndrome out of the register u Note that the tap order is reversed, why? 1 0 received codesyndrome

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Timo O. Korhonen, HUT Communication Laboratory 29 Lessons learned u You can construct cyclic codes starting from a given factored p n +1 polynomial by doing simple calculations in GF(2) u You can estimate strength of designed codes u You understand how to apply shift registers with cyclic codes u You understand how syndrome decoding works with cyclic codes u You can design encoder and decoder circuits for your cyclic codes

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