1. 1 Channel Coding (II) Cyclic Codes and Convolutional Codes

2. ECED4504 2 of 20 Topics today u Cyclic codes –presenting codes: code polynomials –systematic and non-systematic codes –generating codes: generator polynomials –encoding/decoding circuits realized by shift registers u Convolutional codes –presenting codes t convolutional encoder t code trees and state diagram t generator sequences

3. ECED4504 3 of 20 Defining cyclic codes: code 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 cyclic code has the associated code vector with the 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 the same code 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 u For each cyclic code, there exist only one generator polynomial whose degree equals the number of check bits in the encoded word

4. ECED4504 4 of 20 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 (unity shifted) cyclic code vectors: u Question is how to make the cyclic code from the multiplied code? Adding the last two equations together reveals the common factor: Right rotated Right shifted by multiplication Unshifted

5. ECED4504 5 of 20 Factoring cyclic code generator polynomial u Any factor of p n +1 (Note: decompose it into factors) with the degree q=n-k generates an (n,k) cyclic code u Example: Consider the polynomial p 7 +1. This can be factored as u For instance the factors 1+p+p 3 or 1+p 2 +p 3, can be used to generate an unique cyclic code. For a message polynomial 1+p 2 (I.e. 110), the following encoded word is generated: and the respective code vector (of degree n-1, n=7, in this case) is

6. ECED4504 6 of 20 Obtaining a cyclic code from another cyclic code u Therefore unity cyclic shift is obtained by (1) multiplication by p where after (2) division by the common factor yields a cyclic code and by induction, any cyclic shift is obtained by u Example: right shift 101 u (n=3) u Important point is that division by mod p n +1 and multiplication by the generator polynomial is enabled by tapped shift register. not a three-bit code, divide by the common factor

7. ECED4504 7 of 20 Using shift registers for multiplication u Figure shows a shift register to realize multiplication by 1+p 2 +p 3 u In practice, multiplication can be realized by two equivalent topologies:

8. ECED4504 8 of 20 Example: multiplication by using a shift register determined by the tapped connections word to be encoded adding dashed line would enable division by 1+p n Encoded word

9. ECED4504 9 of 20 Examples of cyclic code generator polynomials u The generator polynomial for a (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 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 generator polynomial, e.g. it must be a factor of p n +1. u Only few of the possible generating polynomials yield high quality codes (in terms of their minimum Hamming distance) Some cyclic codes:

10. ECED4504 10 of 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 Check bits determined by: check bits message bits

11. ECED4504 11 of 20 Determining check-bits u Prove that the check-bits can be calculated from the message bits M(p) by checkmessage Example: (7,4) Cyclic code: must be a systematic code based on its definition (previous slide)

12. ECED4504 12 of 20 Example: Encoding of systematic cyclic codes

13. ECED4504 13 of 20 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 the error has happened and sometimes also to reveal in which bit (depending on code strength) u The error syndrome of n-k-1 degree is therefore u This can be expressed also in terms of error E(p) and the code word X(p) error syndrome S(p) is:

14. ECED4504 14 of 20 Decoding cyclic codes: example Using denotation of this example:

15. ECED4504 15 of 20 Table 16.6 Decoding cyclic codes (cont.) msg code error error syndrome

16. ECED4504 16 of 20 Part II. Convolutional coding u Block codes are memoryless u Convolution codes have memory that utilizes previous bits to encode or decode following bits u Convolutional codes are specified by n, k and constraint length that is the maximum number of information symbols upon which the symbol may depend u Thus they are denoted by (n,k,L), where L is the code memory depth u Convolutional codes are commonly used in applications that require relatively good performance with low implementation cost u Convolutional codes are encoded by circuits based on shift registers and decoded by several methods as t Viterbi decoding that is a maximum likelihood method t Sequential decoding (performance depends on decoder complexity) t Feedback decoding (simplified hardware, lower performance)

17. ECED4504 17 of 20 Example: convolutional encoder u Convolutional encoder is a finite state machine processing information bits in a serial manner u Thus the generated code word is a function of input and the state of the machine at that time instant u 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 u Thus (n,k,L) convolutional code is produced that is a 2 n(L-1) state finite- state machine (n,k,L) = (2,1,2) encoder

18. ECED4504 18 of 20 (3,2,1) Convolutional encoder Here each message bit influences a span of n(L+1)=3(1+1)=6 successive output bits

19. ECED4504 19 of 20 Tells how one input bit is transformed into two output bits (initially register is all zero) Representing convolutional code: code tree

20. ECED4504 20 of 20 Representing convolutional codes compactly: code trellis and state diagram Shift register states Input state ‘1’ indicated by dashed line