ECED 4504 Digital Transmission Theory

ECED 4504 Digital Transmission Theory Cyclic Codes and Convolutional Codes

Topics today • Cyclic codes • presenting codes: code polynomials • systematic and non-systematic codes • generating codes: generator polynomials • decoding: syndrome decoding • encoding/decoding circuits realized by shift registers • Convolutional codes • presenting codes • convolutional encoder • code trees and state diagram • generator sequences • Some topics in mod-2 arithmetic`s

Linear block codes: Cyclic codes • For practical applications rather large n and k must be used. This is because in order to correct up to t errors lhs and rhs of the following should be as close as possible: • Hence for large n and k must be used • Advantages of cyclic codes: • Encoding and syndrome computation easy by shift registers • Handy decoding schemes exist due to inherent algebraic structure • For most of the linear codes there exist cyclic codes, as for instance for Hamming, BCH and Golay codes number of error patters that can be corrected in an encoded word number of syndromes (or check bit error patterns)

Example • 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 Rc=0.8 and consider e with some values of larger n and k : • This demonstrates that long codes are more advantages when a high code rate and high error correction capability is required

Defining cyclic codes: code polynomial • 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. • Each cyclic code has the associated code vector with the polynomial • 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 the same code uniquely • 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 • For each cyclic code, there exist only one generator polynomial whose degree equals the number of check bits in the encoded word

An example of a (7,4) cyclic code, generator polynomial G(p)=1+p+p3

The common factor of cyclic codes • GF(2) operations (XOR and AND): • Cyclic codes have a common factor pn+1. In order to see this we consider summing two (unity shifted) cyclic code vectors: • Question is how to make the cyclic code from the multiplied code? Adding the last two equations together reveals the common factor: Unshifted Right rotated Right shifted by multiplication

Factoring cyclic code generator polynomial • Any factor of pn+1 with the degree q=n-k generates an (n,k) cyclic code • Example: Consider the polynomial p7+1. This can be factored as • For instance the factors 1+p+p3 or 1+p2+p3, can be used to generate an unique cycliccode. For a message polynomial 1+p2 the following encoded word is generated:and the respective code vector (of degree n-1, in this case) is

Obtaining a cyclic code from another cyclic code • Therefore unity cyclic shift is obtained by multiplication by p where after division by the common factor yields a cyclic codeand by induction, any cyclic shift is obtained by • Example:right shift 101 • Important point is that division by mod pn+1 and multiplication by the generator polynomial is enabled by tapped shift register. not a three-bit code, divide by the common factor

Using shift registers for multiplication • Figure shows a shift register to realize multiplication by 1+p2+p3 • In practice, multiplication can be realized by two equivalent topologies:

Example: multiplication by using a shift register adding dashed line would enable division by 1+pn determined by the tapped connections word to be encoded Encoded word

Examples of cyclic code generator polynomials • The generator polynomial for a (n,k) cyclic code is defined byand G(p) is a factor of pn+1. Any factor of pn+1 that has the degree q may serve as the generator polynomial. We noticed that a code is generated by the multiplication where M(p) is a block of k message bits. Hence this gives a criterion to select the generating polynomial, e.g. it must be a factor of pn+1. • Only few of the possible generating polynomials yield high quality codes (in terms of their minimum Hamming distance) Some cyclic codes:

Systematic cyclic codes • Define the length q=n-k check vector C and the length-k message vector M by • Thus the systematic n:th degree codeword polynomial is message bits check bits Check bits determined by: Question: Why these denote the message bits still the message bits are M(p) ???

Determining check-bits • Prove that the check-bits can be calculated from the message bits M(p) by must be a systematic code based on its definition (previous slide) message check Example: (7,4) Cyclic code:

Example: Encoding of systematic cyclic codes

Circuit for encoding systematic cyclic codes 0 1 • We noticed earlier that cyclic codes can be generated by using shift registers whose feedback coefficients are determined directly by the generating polynomial • For cyclic codes the generator polynomial is of the form • 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

Decoding cyclic codes • 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.) • Therefore dividing the R(p)/G(p) and considering the remainder as a syndrome can reveal if the error has happed and sometimes also to reveal in which bit (depending on code strength) • Division is accomplished by a shift registers with reversed tap order (why?) • The error syndrome of n-k-1 degree is therefore • This can be expressed also in terms of error E(p) and the code wordX(p) hence

Decoding cyclic codes: example Using denotation of this example:

Decoding cyclic codes (cont.) Table 16.6

Decoding circuit for (7,4) code syndrome computation 0 received code syndrome 1 • While first receiving the code, switch is set to “0” • The shift register is stepped until all the received code bits have entered the register • This results 3-bit syndrome (n - k = 3 ) that is then left to the register • Then the switch is turned to the direction “1” that drives the syndrome out of the register

Convolutional coding • Block codes are memoryless • Convolution codes have memory that utilizes previous bits to encode or decode following bits • Convolutional codes are specified by n, k and constraint length that is the maximum number of information symbols upon which the symbol may depend • Thus they are denoted by (n,k,L), where L is the code memory depth • Convolutional codes are commonly used in applications that require relatively good performance with low implementation cost • Convolutional codes are encoded by circuits based on shift registers and decoded by several methods as • Viterbi decoding that is a maximum likelihood method • Sequential decoding (performance depends on decoder complexity) • Feedback decoding (simplified hardware, lower performance)

Example: convolutional encoder (n,k,L) = (2,1,2) encoder • Convolutional encoder is a finite state machine processing information bits in a serial manner • Thus the generated code word is a function of input and the state of the machine at that time instant • In this (n,k,L)=(2,1,2) encoder each message bit influences a span of n(L+1)=6 successive output bits that is the code constraint length • Thus (n,k,L) convolutional code is produced that is a 2n(L-1) state finite-state machine

(3,2,1) Convolutional encoder Here each message bit influences a span of n(L+1)=3(1+1)=6successive output bits

Generator sequences • (n,k,L) Convolutional code can be described by the generator sequences that are the impulse responses for each coder output branch: • Generator sequences specify convolutional code completely by the generator matrix • Encoded convolution code is produced by matrix multiplication of the input and the generator matrix Note that the generator sequence length exceeds register depth by 1

Encoding equations • Encoder outputs are formed by modulo-2 discrete convolutions: • Therefore the l:th bit of the j:th output branch is • Input for extraction of generator sequences is • Hence for this circuit the following equations result: Encoder output:

Example of using generator matrix Verify that you can obtain the result shown!

Representing convolutional code: code tree Tells how one input bit is transformed into two output bits (initially register is all zero)

Shift register states Representing convolutional codes compactly: code trellis and state diagram Input state ‘1’ indicated by dashed line

ECED 4504 Digital Transmission Theory