A construction of periodically time-varying convolutional codes

Transcription

1 A construction of periodically time-varying convolutional codes Joan-Josep Climent, Victoria Herranz, Carmen Perea and Virtudes Tomás 1 Introduction Convolutional codes [3, 8, 12] are an specific class of error correcting codes that can be represented as time-invariant discrete linear systems over a finite field [17] In contrast to bloc codes, which have been deeply studied, very little is nown about the algebraic structure of convolutional codes They are used in phone data transmission, radio or mobile communication systems and in image transmissions from satellites (see [9, 11]) Convolutional coding is the main error correcting technique in data transmission applications due to its easy implementation and nice throughput in random error channels (see [10, 20]) While it is usual for bloc codes to have the minimum distance guaranteed, for convolutional codes it is common to find out the minimum free distance by a code search Since the required computation in a code search grows exponentially with the number of delay elements of a code, this code search of the minimum free distance can become difficult As well, the computational requirements of the decoding increase with the amount of delay elements δ In the construction proposed in this paper we will see that δ does not increase, what can reduce decoding computation (see [15]) The rest of the paper is organized as follows In Section 2 we introduce some This wor was partially supported by Spanish grant MTM Departament de Ciència de la Computació i Intel ligència Artificial, Universitat d Alacant, Apartat de correus 99, E Alacant, Spain Centro de Investigación Operativa, Departamento de Estadística, Matemáticas e Informática, Universidad Miguel Hernández, Avenida de la Universidad s/n, E Elche, Spain The wor of this author was supported by a grant of the Vicerectorat d Investigació, Desenvolupament i Innovació of the Universitat d Alacant for PhD students

2 basic concepts about convolutional codes and the notation we will use In Section 3 we give initial nowledge about periodically time variant convolutional codes and we construct the time-invariant equivalent one Then we develop the minimality conditions and we compute some bounds on the free distance Finally, we provide some conclusions and future research lines in Section 4 2 Preliminaries From now on we consider that F is a finite field A convolutional code C with rate /n is a submodule of F n [z] that can be described (see [18, 21]) as C = { v(z) F n [z] v(z) = G(z)u(z) with u(z) F [z] where u(z) is the information vector or information word, v(z) is the code vector or code word and G(z) is an n polynomial matrix with ran called generator or encoding matrix of C The complexity of C is the integer δ = i=1 ν i, with ν i being the maximum degree of the i-th column of G(z) We can describe a /n rate convolutional code C of complexity δ by means of the system x t+1 = Ax t + Bu t y t = Cx t + Du t v t = [ yt u t, t = 0, 1, 2, (1) ], x 0 = 0, where A F δ δ, B F δ, C F (n ) δ and D F (n ) We say then that the four matrices (A, B, C, D) are the input-state-output representation of the code C This representation was introduced in [18] and has been used in the last years to analyze and construct convolutional codes [1, 7, 17, 18, 16, 21] For each positive integer j let us consider the matrices Φ j (A, B) = [ B AB A j 2 B A j 1 B ] and Ω j (A, B) = C CA CA j 2 CA j 1 The pair (A, B) is controllable if ran(φ δ (A, B)) = δ The smallest integer κ such that ran(φ κ (A, B)) = δ is the controllability index of (A, B) The pair (A, C) is observable if ran(ω δ (A, C)) = δ The smallest integer ν such that ran(ω ν (A, C)) = δ is the observability index of (A, C) If the matrices A, B, C y D have the smallest possible size we say that (1) is a minimal representation of the code C Moreover, it is enough that the pair (A, B) is controllable to assure that the representation in (1) is minimal If

3 (A, B) is a controllable pair, then the code defined by the matrices (A, B, C, D) is an observable code if and only if (A, C) is an observable pair (see [16]) A representation of a convolutional code is said to be catastrophic if there exists some input sequence, u(z), with infinite Hamming weight, wt(u(z)) =, which generates a code sequence, v(z) = G(z)u(z), with finite Hamming weight, wt(v(z)) < This situation is not desirable since a finite number of errors can lead us to an infinite number of errors If we wor with observable codes we can assure the noncatastrophicity Important distance measures of a code are the column distance and the free distance By means of the input-state-output representation, we define the jth column distance of a code C as { j j d c j(c) = min wt(u t ) + wt(y t ) j = 0, 1, 2, u 0 0 t=0 where wt( ) represents the Hamming weight The free distance of a code C is { d free (C) = min wt(u t ) + wt(y t ) u 0 0 t=0 Moreover, it holds that d free (C) = lim j d c j Rosenthal and Smarandache [19] generalized the Singleton bound for convolutional codes If C is a (n,, δ)-code over any field F, then d free (C) (n ) t=0 ( δ + 1 t=0 ) + δ + 1 (2) This bound is called the generalized Singleton bound Similarly to bloc codes, we say that a (n,, δ)-code is maximum distance separable (MDS) if the free distance reaches the generalized Singleton bound Gluesing-Luerssen, Rosenthal and Smarandache [4] proved that the earliest column distance of a code C that can achieve the generalized Singleton bound is d c M, with M = δ + δ n Hutchinson, Rosenthal and Smarandache [7] introduced the following definitions Let C be an (n,, δ)-code with column distances d c j and free distance d free(c) 1 C has maximum distance profile (MDP) if d c j = (n )(j + 1) + 1 for j = 0, 1,, L with L = δ + δ n 2 C is a strongly MDS code if δ d c M = (n )( + 1) + δ + 1 If (n ) δ then the (n,, δ)-code has MDP if and only if is strongly MDS, since in this case M = L

4 3 Periodically time-variant convolutional codes We consider now a convolutional code with a code vector length n, an information vector length, and δ as the number of delay elements of the decoder We assume that the matrices A t, B t, C t and D t at time t are of sizes δ δ, δ, n δ and n, respectively We define the time-variant convolutional code by means of the system: x t+1 = A t x t + B t u t y t = C t x t + D t u t v t = [ yt u t ], x 0 = 0,, t = 0, 1, 2, (3) If this matrices change periodically with periods τ A, τ B, τ C and τ D respectively, (that is, A τa +t = A t, B τb +t = B t, C τc +t = C t and D τd +t = D t for all t) then we have a periodically time-varying convolutional code and the period is τ = lcm (τ A, τ B, τ C, τ D ) It is well nown, [2, 13, 14], that any periodic time-varying convolutional code is equivalent to an invariant one As we can always relate every state or every output with previous states, we can always rewrite any bloc of τ iterations at a given time j as x τ(j+1) = A τ 1,0 x τj + [ ] A τ 1,1 B 0 A τ 1,2 B 1 A τ 1,τ 1 B τ 2 B τ 1 y τj y τj+1 y τj+τ 1 + = C 0 C 1 A 0,0 C 2 A 1,0 C τ 1 A τ 2,0 x τj D 0 O O O C 1 B 0 D 1 O O C 2 A 1,1 B 0 C 2 B 1 O O C τ 2 A τ 3,1 B 0 C τ 2 A τ 3,2 B 1 D τ 2 O C τ 1 A τ 2,1 B 0 C τ 1 A τ 2,2 B 1 C τ 1 B τ 2 D τ 1 u τj u τj+1 u τj+τ 1, u τj u τj+1 u τj+τ 1 where A i,j = { Ai A i 1 A j+1 A j, if i j A i, if i = j

5 Observe that this system can be written as X j+1 = AX j + BU j Y j = CX j + DU j (4) where A = A τ 1,0, B = [ A τ 1,1 B 0 C 0 A τ 1,2 B 1 A τ 1,τ 1 B τ 2 B τ 1 ], C 1 A 0,0 C = C 2 A 1,0, D = C τ 1 A τ 2,0 D 0 O O O C 1 B 0 D 1 O O C 2 A 1,1 B 0 C 2 B 1 O O C τ 2 A τ 3,1 B 0 C τ 2 A τ 3,2 B 1 D τ 2 O C τ 1 A τ 2,1 B 0 C τ 1 A τ 2,2 B 1 C τ 1 B τ 2 D τ 1 y τj y τj+1 X j = x τj, Y j = and U j = y τj+τ 1, u τj u τj+1 u τj+τ 1 So the system in (4) will be the time-invariant convolutional code equivalent to the periodic time-variant For our particular construction we will tae A t = A an invertible matrix and D t = D for all t Then equation (3) is now x t+1 = Ax t + B t u t y t = C t x t + Du t v t = [ yt u t, t = 0, 1, 2, (5) ], x 0 = 0 and for the system in equation (4) we have A = A τ, B = [ ] A τ 1 B 0 A τ 2 B 1 AB τ 2 B τ 1, D O O O C 0 C 1 B 0 D O O C 1 A C =, D = C 2 AB 0 C 2 B 1 O O C τ 1 A τ 1 C τ 2 A τ 3 B 0 C τ 2 A τ 4 B 1 D O C τ 1 A τ 2 B 0 C τ 1 A τ 3 B 1 C τ 1 B τ 2 D In this concrete case, the controllability and observability matrices will be respectively Φ δ (A, B) and Ω δ (A, C)

6 31 Minimality conditions In this section we study the necessary conditions for the minimality of the new equivalent time-invariant convolutional code Theorem 1 If τ δ and the matrices B j, for j = 0, 1,, τ 1, are such that B j = A (τ j 1) E j, being E j F δ with E = [ E 0 E 1 E τ 1 ] a full ran matrix, then the system defined by equation (4) is controllable Proof According to the form of the controllable matrix in (5) we have that Φ δ (A, B) = [ B AB A δ 1 B ] = [ E A τ E A (δ 1)τ ] which is clearly a full ran matrix So, the system (4) is controllable In a similar way we have the following result Theorem 2 If τ(n ) δ and the matrices C j, for j = 0, 1,, τ 1, are such F 0 that C j = F j A j, being F j F (n ) δ F 1 with F = a full ran matrix, then F τ 1 the system defined by equation (4) is observable We conclude that choosing B j and C j as in Theorems 1 and 2 we can ensure the minimality of our new time-invariant system On the other hand, if we tae A = I, E = [ E 0 E 1 E τ 1 ] as the parity chec matrix of a bloc code and F = [ F 0 F 1 F τ 1 ] such that F j = I(j(n ) + 1 : (j + 1)(n ), :) for j = 0, 1,, τ 1, where I = [ I I I ] with I the identity matrix of size δ δ, then we obtain the periodically time-variant convolutional code proposed in [15] So our construction is a generalization of this case Let us now return to the general case given in expression (3) It can also be expressed as the following time invariant state-space form (see [5]): where u t (h) = u t+ht u t+ht +1 u t+ht +T 1 R(λ) x t (h) = A x t (h) + Bu t (h) y t (h) = C x t (h) + Du t (h), y t(h) = y t+ht y t+ht +1 y t+ht +T 1, x t(h) = x t+ht x t+ht +1 x t+ht +T 1

7 [ ] O I(T and R(λ) = 1)δ with λ denoting the one-step-forward time operator λi δ O in the variable h Further the matrices A, B, C and D are bloc-diagonal matrices defined as A = diag (A 0, A 1,, A T 1 ), B = diag (B 0, B 1,, B T 1 ), C = diag (C 0, C 1,, C T 1 ), D = diag (D 0, D 1,, D T 1 ) If we denote P i (λ) = [ A R(λ) B ] [ ] A R(λ) and P 0 (λ) =, then the C following theorem that we quote for further references holds also for finite fields Theorem 3 ([5]) The periodic system (3) is controllable (respectively, observable) if and only if the matrix P i (λ) (respectively, P o (λ)) has full row (respectively, column) ran for all λ C This result will be useful for the proof of the next theorem For the case where the subsystems forming the periodically time-varying convolutional code are (1, n, 1)-codes or (n 1, n, 1)-codes we state the following result Theorem 4 If the (1, n, 1)-codes (or (n 1, n, 1)-codes) forming the periodically time-varying convolutional code are controllable and observable, then the periodically time-varying convolutional code obtained is as well controllable and observable Proof We will develop the proof for the case (1, n, 1) In this case c t,0 d t,0 c t,1 A t = [a t ], B t = [b t ], C t =, D d t,1 t = c t,n 2 d t,n 2 Following the previous theorem, our periodic system is controllable if and only if the matrix 2 3 a b P i (λ) = ˆ A R(λ) B 0 a b 1 0 = λ a T b T 1 To ensure that this matrix has full ran we will chec that b i 0 for i = 0, 1,, T 1 For every subsystem we have that controllability holds Then by Popov- Belevitch-Hautus test [6] it holds that ran( [ ] zi a i b i ) = 1 for every eigenvalue z of [a i ] and i = 0, 1,, T 1 Since the only eigenvalue of [a i ] is z = a i then it must hold that b i 0 Thus the matrix P i (λ) has full ran and the periodic time-varying convolutional code is controllable

8 By the previous result, our periodic system is observable if and only if the matrix a a λ a T 1 c 0, c 0, [ ] P o A R(λ) (λ) = = C c 0,n c 1, c 1, c 1,n c T 1, c T 1, c T 1,n 2 To ensure that this matrix has full ran we will chec that for each i = 0, 1,, T 1 there exist at least one j such that c i,j 0 For every subsystem we assumed that observability holds Again by PopovzI a i c i,0 Belevitch-Hautus test it holds that ran c i,1 = 1 for every eigenvalue c 1,n 2 z of [a i ] and i = 0, 1,, T 1 Then it must exist at least one j such that c i,j 0 Thus the matrix P o (λ) has full ran and the periodic time-varying convolutional code is observable The proof for the case (n 1, n, 1) is analogous 32 Free Distance In this section we show some upper and lower bounds on the free distance of the code C For the construction shown in Section 31 and following the expression of the generalized Singleton bound in (2) we have that ( ) δ d free (C) τ(n ) δ + 1 τ To get an improvement of this bound respect to the bounds of the subsystems

9 forming the new equivalent time-invariant convolutional code, in order that the free distance of the new code can achieve higher values, it should hold that ( ) ( ) δ δ τ(n ) δ + 1 > (n ) δ + 1, τ that is and since n > 0, so we have ( ) ( ) δ δ τ(n ) + 1 > (n ) + 1 τ τ ( ) δ + 1 > τ τ δ + τ > τ ( ) δ + 1, δ + 1 Then, assuming that τ > 1, it is enough that the parameters of the code hold δ δ τ τ In the particular case that τ δ, the bound increases the amount of (n )(τ 1) Let ν be the observability index of the pair (A, C) As shown by Rosenthal [16], if there exist d Z + such that Φ dν (A, B) forms the parity-chec matrix of a bloc code of distance d, then the free distance of the code is greater or equal than d Theorem 5 If the matrix Φ dν (A, B) = [ B AB A dν 2 B A dν 1 B ] represents the parity-chec matrix of a MDS bloc code of distance d, then d free δ+1 Proof Let P = [ A τ 1 B 0 A τ 2 B 1 B τ 1 ] The matrix [ B AB A (dν 1) B ] = [ P A τ P A (dν 1)τ P ] is a matrix of size δ τdν Since the parity-chec matrix of a bloc code is a matrix of size (N K) N, we have that the parameters of the bloc code are (τdν, τdν δ, d) Now, as the bloc code is MDS and the Singleton bound of the bloc code is N K + 1, then we have d free (C) d = τdν τdν + δ + 1 = δ + 1 We should notice that if d free (C) = δ + 1, then C is not an MDS convolutional code To be MDS the code C should achieve the generalized Singleton bound In this case ( ) δ δ + 1 = τ(n ) δ + 1, τ

10 ( ) δ 0 = τ(n ) + 1 τ The only possibility in order that this equation holds is n =, ie, matrices C and D disappear 4 Conclusion In this paper we propose a new method for constructing periodically time-variant convolutional codes We study the minimality conditions of the new system and state some results about the dependence between the controllability and observability of the subsystems and the time-invariant equivalent one Some bounds related with the generalized Singleton bound and the free distance are as well given As it is shown in this construction, the number of delay elements remains as δ This fact maes that the decoding complexity of the new time-varying code does not increase respect to the complexity of the subsystems This give us a lower complexity of the arithmetic circuits when implementing the model As future research lines we will study the properties of the time-invariant convolutional code depending on the properties of the subsystems forming the periodically time-variant convolutional one; we will attempt to construct convolutional codes with optimal parameters as well as MDS and MDP convolutional codes

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