An Upper Bound on the Minimum Distance of Serially Concatenated Convolutional Codes

Transcription

1 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY An Upper Boun on the Minimum Distance o Serially Concatenate Convolutional Coes Alberto Perotti, Member, IEEE, an Sergio Beneetto, Fellow, IEEE Abstract This paper presents an upper boun on the minimum istance o serially concatenate convolutional coes with interleaver employing rate k 0/n 0 constituent encoers. The resulting expression shows that their minimum istance cannot grow more than approximately K 1 1/(O), where K is the inormation wor length, an (O) is the ree istance o the outer coe. This result can also be applie to serial concatenations where the outer coe is a linear block coe. The obtaine upper boun is shown to agree with an, in some cases, improve over previously known results. Inex Terms Serial, concatenate, convolutional, minimum istance, combinatorial, perormance bouns. I. INTRODUCTION Serially concatenate convolutional coes with interleaver [1] are known to perorm better than parallel turbo coes in the error loor region [2]. However, when heavy puncturing is applie to obtain higher coe rates, it is not uncommon to observe a signiicant error loor. In orer to estimate the error probability in such a region, the minimum istance is a crucial parameter. In this paper, a metho to obtain an upper boun on the minimum istance o serially concatenate convolutional coes is escribe. The present result also applies to serial concatenations where the outer coe is a linear block coe in general. Moreover, it can be applie to rate k 0 /n 0 general convolutional constituent coes. Results on the minimum istance o serially concatenate coes have been presente in [3] an [4]. Both papers show an exponential epenence o the minimum istance on the block length: in [3] the exponent epens on the minimum istance o the outer coe, while, in [4], it epens on the memory an rate o the outer encoer. Our result, although similar to the cite ones, improves over [4], where a broaer class o constituent encoers is consiere. Moreover, it is coherent with [3]. The metho use here to erive the upper boun has been inspire by that use in [5] an [6] to obtain an upper boun on the minimum istance o parallel turbo coes. Alberto Perotti an Sergio Beneetto are with the Center or Multimeia Raio Communications at Politecnico i Torino, Torino (Italy).

2 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY Sec. II summarizes the previously known results an Sec. III escribes the coing scheme use as a reerence in orer to erive the boun. In Sec. IV the upper boun is erive. Finally, in Sec. V the obtaine upper boun is applie to ierent coing schemes an compare with the previously known results. In the ollowing, we will interchangeably use the ollowing notations or symbol vectors: v = {v i, i = 0,..., N 1} v(d) = II. KNOWN UPPER BOUNDS Result on the minimum istance o serially concatenate coes have been presente in [3] an [4]. In [4], the general class o nonlinear constituent encoers is consiere. These encoers are characterize by a state-transition graph. Each state transition correspons to k 0 = 1 input bits an n 0 output bits. From [4], Theorem 4, the minimum istance obeys the ollowing boun: [ min (BMS) min = 3 n (O) 0 ] 2 ( ) (I) n 0 ν (O) + 2 K 1 1 n (O) N 1 i=0 v i D i. 0 (ν (O) +2) [ ν (I)] 1 n (O) 0 (ν (O) +2) (1) where quantities relate to the outer (inner) encoer are ientiie by a ( ) (O) (equivalently, ( ) (I) ) superscript, the ν parameter is the number o memory bits o the constituent encoers, an K is the inormation wor length. In [3], Theorem 2, an asymptotic result is presente as an average over the set o all serially concatenate coes: { P N } min (C) < N 1 2/(O) { P min (C) > N 1 2/(O) N } = 0 (2) = 0 (3) where N is the coewor length. We can eine the asymptotic minimum istance o serially concatenate convolutional coes as where (O) (KU) min 1 2/(O) N (4) is the ree istance o the outer encoer. This result, while giving the average behavior o the minimum istance, oes not provie inormation on the largest achievable minimum istance. III. REFERENCE CODING SCHEME We consier the serial concatenation o recursive, systematic convolutional coes o rate k 0 /n 0 (see Fig. 1 an Fig. 2). The two constituent encoers are connecte through an interleaver which reorers the outer coewor bits accoring to the permutation Π : i π i, i = 0,..., K (I) 1, where K (I) is the interleaver length. A sequence o K (O) inormation bits enters the outer encoer CC (O). The sequence generate by CC (O) is puncture an an outer coewor o length N (O) = K (I) bits is prouce. The outer coewor is permute by the interleaver an sent to the inner encoer CC (I), which generates, ater puncturing, an inner coewor o length

3 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY N (I) bits. Ater the input symbols have been processe, each constituent encoer is terminate by aing ν trellis steps an orcing its state to zero. ig/sccc.eps Fig. 1. Serially concatenate encoer. The outer coe is the set o outer coewors resulting rom the operations o outer encoing (CC (O) ) an outer puncturing (P UNCT (O) ): Moreover, we eine C (O) = {c k }, k = 0,..., 2 K(I) 1. C (O) C (O), C (O) = {c C (O) : w H (c) = } i.e., C (O) is the subset o outer coewors with Hamming weight. A. Constituent encoers We eine a recursive convolutional encoer o rate k 0 /n 0 as a evice which processes a sequence o K symbols o k 0 bits proucing a sequence o K +ν symbols o n 0 bits. The evice is characterize by the ollowing equations: c i = Fs i Gu i (5) s i+1 = Hu i Ls i where F, G, H an L are matrices corresponing to the encoer connections, an u i, s i an c i inicate the inormation symbol, encoer state an coe symbol at time instant i. Fig. 2 shows the scheme o a rate k 0 /n 0 recursive convolutional encoer. It is worth noting that this moel is a generalization o both systematic recursive encoers an rate 1/n 0 convolutional encoers. It coul also be consiere as a generalization o puncture recursive encoers, when the puncturing pattern is perioic, but here we will consier puncturing as a separate unction. The eeback connections, corresponing to vector L, are here eine accoring to primitive polynomials over GF (2 ν ) [7]. This results in the ollowing property: i the encoer state at time instant i is s i, an the input sequence u j = 0, j = i,..., i + p, then s i+p = s i, where p is the perio o the convolutional encoer. A primitive eeback polynomial results in a perio p = 2 ν 1. Inee, non-primitive polynomials can be consiere as well, but the corresponing encoers eature shorter perios. This property is exploite in the ollowing lemma. Lemma 1: An inormation sequence o type:

4 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY ig/rsc-kn.eps Fig. 2. Rate k 0 /n 0 recursive convolutional encoer. u(d) = u 0 (D)(D k0i + D k0(i+jp) ) i.e., the same inormation symbol is repeate twice at ierent time instants (u i = u i+jp = u 0 ), results in a single error event o length jp + 1 trellis steps. Proo: We assume that the encoer is initialize by orcing the initial state s 0 = 0 ν. Thereore: s 0 =... = s i = 0 ν. By applying (5) at time instant i: c i = Gu 0 (6) s i+1 = Hu 0 Now, moving to instant i + jp, an consiering that u l = 0, l = i + 1,..., i + jp 1: c i+jp = Fs i+jp (7) s i+jp+1 = Hu i+jp Ls i+jp = Hu 0 Ls i+jp Consiering that, or the mentione property o primitive-eeback encoers, Ls i+jp = s i+1, rom (6): s i+jp+1 = 0 ν (8) Thereore, the number o states 0 ν is i + jp i = jp, an the number o trellis steps o the error event is jp + 1. An upper boun to the minimum istance o serially concatenate convolutional coes is erive by assuming that the constituent encoers are ixe, an consiering the set o all permutations: where C C is a subset o the coewors o the concatenate coe. m max min w H(c) (9) Π c C

5 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY We begin our erivation by analyzing the characteristics o the outer encoer an puncturer. Although the main part o our erivation is perorme inepenently o the outer coe type (the only requirement is linearity), the assumption that the outer coe is convolutional (an, eventually, recursive an systematic) allows to better highlight the epenence o the minimum istance on the block length. The cascae o the recursive convolutional encoer an the puncturer can be consiere as a perioic evice, an the resulting perio is an integer multiple o the puncturing perio an n 0 (LCM(p q, n 0 )). This is mae clear in the ollowing Theorem 1. Theorem 1: Let C be the coe generate by a rate k 0 /n 0 recursive convolutional encoer whose output is puncture accoring to the puncturing sequence q. Let p q be the perio o the puncturing sequence, an p = LCM(p q, n 0 ). Let w be the Hamming weight o any subsequence 1 o q with length p. Then: c(d) C ( ) D jw c(d) C, i m w j < N i M w (10) where N is the coewor length an: i M (c) = sup{i : c i = 1} (11) i m (c) = in{i : c i = 1}. (12) In other wors, i a coewor c(d) = N 1 i=0 c id i belongs to the outer coe, all its shits o multiples o w positions belong to the outer coe as well, provie that the coewor bounaries are not crosse. Proo: Consier a rate k 0 /n 0 convolutional encoer CC. When the inormation wor u(d) is sent to the encoer, the output coewor is CC[u(D)] = c(d). It is easy to prove that: CC[D sk0 u(d)] = D sn0 c(d), s Z : 0 ĩ m + sn 0 ĩ M + sn 0 < Ñ where Ñ is the coewor length beore puncturing, an The puncturing operation is perorme in the ollowing way: ĩ M (c) = sup{i : c i = 1} (13) ĩ m (c) = in{i : c i = 1}. (14) c(d) q(d) i M c i q i D i an the symbols corresponing to q i = 0 are einate. Now, eine p = LCM(n 0, p q ): i=i m 1 We eine here a subsequence o a sequence p as a subset o contiguous elements o p.

6 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY since p is an integer multiple o p q. ( D p c(d) ) q(d) = = ĩ M +p i=ĩ m+p c i p q i D i ĩ M i=ĩ m c i q i+p D i+p = D p c(d) q(d) In orer to complete our proo, consier that the symbols corresponing to q i = 0 are iscare, thereore any subsequence o p symbols in c(d) q(d) is reuce to w symbols, then a shit o p positions beore puncturing correspons to a shit o w positions ater puncturing. This proves (10). A irect consequence o Theorem 1 is the ollowing corollary, where we etermine the number o coewors in C resulting rom shits o a given coewor c. Corollary 1: Shiting a coewor o multiples o w positions to obtain another coewor is subject to the ollowing constraints: i M + jw < N i m + jw 0 j Z : i m w j < N i M w. (15) Thereore, given a coewor c C, there are at least n c coewors in C that can be obtaine by shiting c by multiples o w positions, where: n c N i M (c) w + i m(c) w where γ c = 1/(R C w ), an R C = K/N is the coe rate. 1 = N w i M (c) i m (c) w 1 = γ c K γ 0,c (16) In the ollowing corollary, we count the number o coewors with given Hamming weight w. Corollary 2: Let n w be the number o coewors with Hamming weight w in C. The ollowing inequality hols: n w K γ c γ 0,c (17) c:w H (c)=w c:w H (c)=w = Kγ w γ 0,w (18) where K is the length o the inormation wor, an γ w an γ 0,w epen on the encoer connections an puncturing sequence. Corollary 2 is particularly useul in our erivation when use to count the number o nearest neighbors, i.e., when w =. Tables I to III show values o (γ w, γ 0,w ) or ierent recursive systematic convolutional encoers an various coe rates obtaine by puncturing. The rate o the consiere mother constituent encoers is 1 / 2. Moreover, each encoer is characterize by its eeback an eeorwar connections: commonly, the connections

7 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY are expresse as a ratio o octal numbers, where the numerator reers to the eeorwar branch an the enominator reers to the eeback branch. As an example, the encoer o Fig. 3 is eine by 15 8 /13 8. TABLE I VALUES OF (γ w, γ 0,w ) FOR THE 5 8 /7 8 (ν = 2) ENCODER. DIFFERENT CODE RATES ARE OBTAINED BY PUNCTURING. R\w ( ) 1/ , 3 (2, 7) 2 ( 2/3-1 2, 0) ( 2, 22 ) ( ) ( ) 3 7, , ( 3/4-4 3, 3) ( 32 3, 99) ( 100 3, 383) - ( 6/7 5 6, 41 ) ( , 449 ) ( , 2763 ) ( ) ( 7/8 1, , 89) ( 43, 4243 ) TABLE II VALUES OF (γ w, γ 0,w ) FOR THE 15 8 /13 8 (ν = 3) ENCODER OF FIG. 3. DIFFERENT CODE RATES ARE OBTAINED BY PUNCTURING. R\w / (2, 8) 2/3 - - ( 3 2, 22 ) 3 ( 9 2, 95 ) 3 ( 31 2, 162) 3/4 - ( 2 3, 9 ) ( , 101 ) 4 (17, 183) - 6/7 ( 1 2, 25 ) ( ) ( 7 3, , 2259 ) /8 ( 4 7, 5) ( 30 7, 163 ) ( , 1441 ) TABLE III VALUES OF (γ w, γ 0,w ) FOR THE 35 8 /23 8 (ν = 4) ENCODER. DIFFERENT CODE RATES ARE OBTAINED BY PUNCTURING. R\w / (2, 11) 2/3 - - ( 1 2, 1) (0, 0) ( 27 2, 377 ) 3-3/4 - ( 1 3, 3 ) ( 2, 69 ) 4 ( 34 3, 131) - 6/7 7/8 ( ) ( 2, , 913 ) ( , 656) - - ( 1 7, 9 ) ( , 26) ( 104 7, 369 ) The analysis perorme in Theorem 1 an ollowing corollaries will be use in our erivation to characterize the outer coe. The inner encoer is characterize ollowing the work carrie out in [8] an extene in [9].

8 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY ig/rsc8.eps Fig. 3. Recursive convolutional encoer (15 8 /13 8 ). Lemma 2: Let CC be a puncture rate k 0 /n 0 recursive convolutional encoer with memory ν an perio p = 2 ν 1. I the inormation wor u is o the ollowing type: u(d) = D k0i + D k0(i+jp) the ollowing inequality hols: w H [CC(u)] αj + β (19) where α an β epen on the encoer connections, memory an puncturing sequence. Proo: A proo o this lemma is a irect consequence o Lemma 1 when u 0 (D) = D m, 0 m < k 0, i.e., the consiere symbol u 0 has Hamming weight 1. Table IV reports values o α an β or various memory values an rates or a rate 1/2 puncture recursive systematic convolutional encoer with memory ranging rom 2 bits to 4 bits. The consiere puncturing sequence is perioic: the systematic bits are never puncture, while the coe bits are puncture accoring to the ollowing sequence: which results in a rate l/(l + 1) encoer. q(d) = K 1 i=0 K/l 1 D 2i + j=0 D 1+jl (20) Finally, in orer to complete the characterization o the inner encoer, we eine its bit perio as: k 0 (2 ν 1). (21) It ollows rom Lemma 1 that an input wor o type u(d) = D i + D i+jp(i) j/k 0 trellis steps. results in a single error event o length IV. THE UPPER BOUND The upper boun on the minimum istance ollows in Theorem 2.

9 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY ν Connections p R 1/2 2/3 3/4 6/7 7/ / ,2 1,2 1,2 1,2 1, / ,2 2,2 2,2 1,2 1, / ,2 5,2 4,2 3,2 2,2 TABLE IV PARAMETERS α, β OF THE CONSIDERED INNER ENCODERS. THE CONNECTIONS COLUMN SHOWS THE FEEDFORWARD AND FEEDBACK CONNECTIONS OF THE CONVOLUTIONAL ENCODERS. Theorem 2: The minimum istance o a serially concatenate convolutional coe satisies the ollowing inequality: m min (O) w K (I) [α M (w) + (2 + β)w] (22) where α an β erive rom the characterization o the inner encoer perorme in Lemma 2, an M (w), whose meaning will be clariie in the proo, is: { { K (I) } 1 M (w) sup 0,..., w : (γ wk (O) γ 0,w ) w δ w w! [ K (I) 1 ] w } + provie that δ w w! < γ w K (O) γ 0,w. In (23), K (O) is the inormation wor length, γ w an γ 0,w characterize the outer encoer (see Theorem 1 an relate corollaries, an Tables I to III), an (23) ( w + ) 1 δ w = w is the number o istinct unorere sets o w elements rawn rom a set o elements. Proo: For the proo o this theorem, see Appenix I. From the inequality in the RHS o (23) we erive the ollowing constraint on K (O) : (24) K (O) K (O) = δ ww! + γ 0,w γ w (25) Otherwise, M (w), an the upper boun becomes useless. Since w 2, this constraint also inclues the one eriving rom (39) in Appenix I, which results in Table V shows the values o K (O) K (O) > δ w + γ 0,w γ w. or ierent coing schemes. K(O) is generally low or high-rate coes, but it becomes unacceptably large or low-rate coes an large memory o the constituent encoers.

10 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY VALUES OF K (O) (THEOREM 2), K (O) TABLE V (THEOREM 3) AND K (O) (THEOREM 4) FOR DIFFERENT CODING SCHEMES. Outer coe rate Outer mother Inner encoer K (O) (ater puncturing) encoer connections perio: K (O) K (O) 1/2 5 8 / , /4 5 8 / /7 5 8 / / / , ,572 3/ / ,012 6/ / / / ,025,606 58,146 7,441,926 3/ / ,250 2,050 16,330 6/ / Another rawback o Theorem 2 is the lack o a close-orm solution or the minimum istance o the coe. The ollowing Theorem 3 provies in general a slightly looser upper boun in close-orm with a weaker itation on K (O). Theorem 3: The minimum istance o a serially concatenate convolutional coe satisies the ollowing inequality: m min (O) w K (I) [α M (w) + (2 + β)w] (26) where α an β erive rom the characterization o the inner encoer perorme in Lemma 2, an M (w) 2 where δ w has been eine in Theorem 2. [ K (I) ] w 1 w [ w!δ w R (O) C γ w γ 0,w /K (I) Proo: The proo o this theorem partially coincies with that o Theorem 2 (see Appenix I up to (55)). [ ] w: Consier now the intersection between a sphere U (c) an the region S K = 0, (I) 1 U (c) [ K (I) 1 0, ] w ] 1 w (27) 1 2 w V w() c C (O) w (28) This inequality correspons to the case where c coincies with a vertex o S. In this case, the intersection between U (c) an S ixes the signs o the w coorinates o U (c). Since there are w coorinates, an U eatures symmetry with respect to origin, we can compute the volume o such intersection in a way similar to Appenix II. The only

11 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY ierence is that we compute each o the w integrals on an interval which is hal the size, thereore we obtain a actor o 1/2 w. This result allows to rewrite (57) in Appenix I in a slightly ierent way: (γ w K (O) γ 0,w ) δ w which, ater ew rearrangements, results in (27). w M K (I) w [ ] w! 2 w 1 K (I) w (29) A constraint eriving rom (39) in Appenix I results in Table V shows the values o K (O) coes. K (O) > K (O) = δ w + γ 0,w γ w. or ierent coing schemes. K (O) is signiicantly lower than K(O) or low-rate By constraining w = (O), Theorem 3 allows to highlight the asymptotic behavior o the minimum istance with the block length: since m (C) N N = 0 where N = R (O) C K (I), we can state that serially concatenate convolutional coes are asymptotically ba, in the traitional meaning o this statement [10]. A. A thir version o the upper boun A urther observation on the position o coewors in the region S allows to erive the ollowing theorem. Theorem 4: The minimum istance o a serially concatenate convolutional coe satisies the ollowing inequality: m min [α (O) M (w) + (2 + β)w] (30) q K (I) where α an β erive rom the characterization o the inner encoer perorme in Lemma 2, an provie that M(w) [ K (I) ] w 1 w [ w!δ w 2 w 1 R (O) C γ w (γ 0,w + (2 w 1 )δ w )/K (I) ] 1 w (31) an δ w has been eine in Theorem 2. Proo: γ w K (O) γ 0,w δ w 2 w (32) This theorem is erive by consierations similar to those o Theorem 3. In aition, we note that the region S has 2 w vertices. Suppose there are 2 w spheres with centers in such vertices. Their intersection with

12 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY S is V w ()/2 w. Moreover, the remaining spheres intersect S eining a region whose volume is V w ()/2 w 1. Thereore: [ 2 w w M w! 2 w + γw K (O) ] γ 0,w 2 w w M K (I) w [ ] δ w w! 2 w 1 1 K (I) w (33) which, ater ew rearrangements, results in (31). Similarly to Theorem 2, the constraint (32) results in a range o inormation wor lengths where Theorem 4 hols: K (O) K (O) = 2w δ w + γ 0,w γ w. As in Theorem 2, this constraint inclues the one eriving rom (39) in Appenix I. Table V shows the values o K (O) or ierent coing schemes. Comparing the 3 rightmost columns, we see that Theorem 2 results in the narrower range o inormation wor lengths, while Theorem 3 provies the wier range an Theorem 4 results in an intermeiate range. Moreover, we note that the stronger itation o Theorem 2 correspons to a generally tighter boun, while the weaker itation o Theorem 3 correspons to a looser boun. Finally, Theorem 4 eatures intermeiate values. Since the computation o (23), (27) an (31) might be impractical ue to the large number o values to be consiere or w, we provie a simpliication in the ollowing corollary. Corollary 3: The minimum istance o a serially concatenate convolutional coe satisies the ollowing inequality: m α M ( (O) ) + (2 + β) (O). ( ) (34), rom (27) where M can be substitute with M, M or M accoring to the use boun. By computing M we obtain: [ K M ( (O) (I) ]1 1 (O) ) 2 R (O) C γ (O) (O)!δ (O) γ 0, (O) /K (I) where the exponential relationship between the coewor length an the ree istance o the outer coe is evient. Here, the set over which we perorm the minimization o (22) has been restricte to the lowest possible value w = (O). A reason or this choice is that, or low values o w, the computation o the γ w parameters relative to the outer encoer is easier ue to the low number o error events with Hamming weight (O). Another reason or restricting the range o values or w to (O) w = (O) results in the lowest value o K (O). 1 (O) (O) (35) is the constraint on K (O) given in (25). In act,

13 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY ig/chart_342_232.eps ig/chart_122_122.eps (a) Rate 1/4. Outer coe rate: 1/2; inner coe rate: 1/2; number o states: 4. (b) Rate 1/2. Outer coe rate: 3/4; inner coe rate: 2/3; number o states: 4. ig/chart_343_233.eps ig/chart_672_782.eps (c) Rate 3/4. Outer coe rate: 7/8; inner coe rate: 6/7; number o states: 4. () Rate 1/2. Outer coe rate: 3/4; inner coe rate: 2/3; number o states: 8. Fig. 4. in (). Results. The constituent encoers are both 4-state systematic recursive convolutional encoers in (a), (b) an (c), an 8-state encoers B. Extension to general linear outer coes The result o Theorem 2 an successive theorems can be extene to the case where a general linear block coe is employe as the outer coe. In act, the convolutional nature o the outer coe has been use only in Theorem 1 an ollowing corollaries to relate the number o coewors with given Hamming weight to the inormation wor length K. I a linear block coe is employe as the outer coe, in orer to apply the upper boun it is suicient to in the number o coewors with Hamming weight w or some w (O). V. RESULTS Some results have been obtaine or coing schemes consisting o the serial concatenation o two recursive systematic convolutional encoers. Both encoers are puncture to obtain variable coing rates. Results have been obtaine by using the simpliie result o Corollary 3 applie to Theorem 2, Theorem 3 an Theorem 4. Our results have been compare with the well known Singleton boun an Gilbert-Varshamov boun [11, ch. 10]. Moreover, the result o [3], Theorem 2 has been reporte in the charts. Finally, the result presente in [4], Theorem 4 has also been reporte. Since this result reers to rate 1/n 0 constituent encoers, here it has been extene to the case rate k 0 /n 0 constituent encoers by replacing n 0 with n 0 /k 0.

14 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY ig/chart_nu.eps Fig. 5. Results or increasing number o states o the constituent encoers. Overall coe rate: 1/2; outer coe rate: 3/4; inner coe rate: 2/3. ig/chart_w.eps Fig. 6. Results or w = (O) to w = (O) + 2. Overall coe rate: 1/2; outer coe rate: 3/4; inner coe rate: 2/3; number o states: 8. Fig. 4 shows the obtaine results compare to some previously known results on linear block coes an serially concatenate coes. The chart shows that our result improves over the Singleton boun an even over the Gilbert- Varshamov boun, which is an existence boun on the minimum istance o linear block coes. As expecte, the Kahale-Urbanke result [3], which is an average result over the set o all the serially concatenate coes with given coewor length an outer ree istance, is always lower than our results. In particular, when (O) we have (KU) min = 1. = 2 (as in 4(c)), Fig. 5 shows the upper boun or a rate 1/2 coe (rate 3/4 outer coe an rate 2/3 inner coe) with increasing number o states. As expecte, a larger number o states results in a larger upper boun an in a reuce range o inormation wor sizes. This is a irect consequence o the act that a larger number o states results in a larger ree istance o the constituent encoers. Fig. 6 shows the upper boun o Theorem 4 or a rate 1/2 coe (rate 3/4, 8-state outer coe an rate 2/3, 8-state inner coe). The boun has been compute or increasing values o w, starting rom w = (O). The chart shows that the best result is obtaine or w = (O). VI. CONCLUSION We have presente an upper boun on the minimum istance o serially concatenate convolutional coes with interleaver employing rate k 0 /n 0 constituent encoers. The result shows that the minimum istance cannot grow

15 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY more than a power o the inormation wor length, where the exponent is approximately 1 1 (O), where (O) the ree istance o the outer coe. This result can be easily extene to serial concatenations where the outer coe is a linear block coe. Improvements over the previously known results have been shown. is Given the set o inices at the interleaver output: APPENDIX I PROOF OF THEOREM 2 I = {ι i }, i = 0,..., K (I) 1 we eine a partition o I in parts I j, j = 0,..., 1 corresponing to the resiue classes mo : ι i I j i mo = j, j = 0,..., 1 (36) an we reer to these parts as the inner classes o I. Clearly: Consier the set C (O) w K (I) K (I) I j, j = 0,..., 1. o outer coewors whose Hamming weight is w. For each o the coewors c C (O) w, we eine a label L(c) consisting o an unorere set o integers eine as ollows: L(c) = {i mo, i : (Π(c)) i = 1} where (Π(c)) i inicates the i + 1-th element o the permute outer coewor Π(c). There are exactly ( w + ) 1 δ w = w ierent values or L(c), i.e., the number o unorere combinations o w elements rawn rom a set o elements with repetition. As a irect consequence o the pigeonhole principle [12, ch. 2], we can state that: L : C (O) w,l C w (O) where C (O) is the set o outer coewors with Hamming weight w an label L. w,l I δ w w! < C w (O) : L : C (O) 2 (39) w,l Now pick at ranom two coewors c 1 an c 2 C (O) w,l. Since we assume that the outer coe is linear, c 1 c 2 C (O) (where inicates the moulo-2 sum). Since c 1 an c 2 have equal labels, Π(c 1 c 2 ) eatures an even number δ w (37) (38)

16 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY ig/classes.eps Fig. 7. Classiication o inices at the input o the inner encoer ( = 3). Two outer coewors with Hamming weight w = 3 an equal label generate short inner error events. o 1 in each inner class (in act, the label values inicate to which inner class each 1 belongs ater interleaving). This situation is shown in Fig. 7. I we make the urther assumption that the values in labels L(c 1 ) an L(c 2 ) are all ierent: L j (c i ) L k (c i ), i = 1, 2, j, k {0,..., w 1}, j k (40) we can state that v = Π(c 1 c 2 ) results in exactly two 1 in each inner class. This assumption results in a simpler calculation o the boun; moreover, as we will see later, it oes not invaliate our result. We can then write the input wor to the inner encoer v as the moulo-2 sum o w wors : v = w 1 i=0 v i where the symbol inicates a moulo-2 summation, an v i contains ones only in positions belonging to the same inner class: Clearly: v i = D ji + D ji+kip(i), i = 0,..., w 1. (41) w H (v i ) = 2, i = 0,..., w 1 an CC (I) (v i ) consists o a single error event o length k i + 1 trellis steps. Since the inner coe is linear, the inner coewor (beore inner puncturing) is w = CC (I) (v) = w 1 i=0 CC(I) (v i ) (42)

17 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY an its weight is w H (w) This is a consequence o the triangle inequality: w 1 i=0 ) w H (CC (I) (v i ) (43) w H (a b) w H (a) w H (b). In orer to evaluate this upper boun, we must in an approximation or the RHS o (43). From (41), applying Lemma 2: Thereore, rom (43) we obtain: ) w H (CC (I) (v i ) αk i + β + w H (v i ) (44) which can be evaluate i the quantities k i are known. w 1 w H (w) w(β + 2) + α k i (45) A urther step in our proo is perorme by consiering that each outer coewor whose Hamming weight is w can be represente as a point in a w-imensional space S R w with each imension corresponing to one label value L i (c). To this purpose, we eine the ollowing set o inices: i=0 (Π[c]) J c = {j 0,..., j w 1 } : ji = 1, i = 0,..., w 1 j i mo j i+1 mo, i = 0,..., w 2 (46) where the sign can be substitute with < i c satisies (40). We associate to this outer coewor the ollowing set o coorinates: s(c) = { ji, j i J c } (47) which etermines the position o point s S. It is worth noting that the permutation etermines the position o points s(c) since it is involve in the einition o J c. We eine the istance between two points s 1 an s 2 in S using the l 1 -norm: s(c 1 ) s(c 2 ) 1 w 1 i=0 s i (c 1 ) s i (c 2 ) = Using einition (48), we can reormulate (45) in the ollowing way: w 1 i=0 k i (48) w H (w) w(β + 2) + α s(c 1 ) s(c 2 ) 1 (49)

18 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY Thereore, the problem o maximizing (9) over all the permutations, reuces to ining the maximum allowable minimum istance between any two points in S corresponing to ierent outer coewors. This task can be easily accomplishe ater some consierations: the coorinates o the points associate to the outer coewors are K (I) 1 0 s i hence these point lie in a rectangular region S S whose volume is: (50) K S (I) w 1 = (51) Using the mentione einition o istance, we eine the sphere o iameter centere in point s(c) as the region U (c) = { p S : s(c) p 1 } 2 (52) Fig. 8 shows a sphere in S. ig/sphere.eps Fig. 8. A sphere in l 1 -norm (w = 2). Given two outer coewors c 1 an c 2, each surroune by a sphere U (c i ), i = 1, 2, we can state that: U (c 1 ) U (c 2 ) p U (c 1 ) U (c 2 ) : p s(c i ) 1 /2, i = 1, 2 (53) an, rom the triangle inequality applie to l 1 norm: s(c 1 ) s(c 2 ) 1 s(c 1 ) p 1 + p s(c 2 ) 1 = (54)

19 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY Suppose now that the points s(c), c C (O) satisy the ollowing istance constraint or a given value o : w,l s(c 1 ) s(c 2 ) 1 > U (c 1 ) U (c 2 ) =, c 1, c 2 C (O) w,l. (55) Otherwise, rom (53) an (54), the istance between s(c 1 ) an s(c 2 ) woul be. Consiering that c C (O) w,l we can erive an upper boun on the value o : where V w () is the volume o U. Applying (17) an (38), we obtain: C (O) w,l C (O) w,l [ K (I) w 1 U (c) + ] (56) [ K (I) 1 V w () n(o) w δ w K(O) γ w (O) an, consiering that the spheres are isjoint, we can rewrite (57) as ollows: n (O) w V w () δ w [ K (I) 1 where V w () = w /w! is the volume o U (see Appenix). + ] w (57) γ (O) 0,w (58) δ w + ] w (59) In orer to complete our proo, we must show that our result is vali even when the consiere label value L has two or more equal elements, i.e., constraint (40) is not satisie. In this case, i two label elements are equal, the number o involve inner classes reuces rom w to w 1, thereore we shoul perorm our erivation on a (w 1)-imensional space. However, we can assign at ranom the coewor bits corresponing to equal label values to ierent coorinates an hence obtain the w inner inormation wors v i o (42) an (43). Then or the linearity o the inner coe, the remaining part o the proo remains unchange. This consieration implies the valiity o our result also when w >. APPENDIX II VOLUME OF THE SPHERE IN l 1 -NORM The region U (n) (p) R n eine by the ollowing inequality: x 1 + x x n /2 is the n-imensional sphere o iameter with center in p. The volume o U (n) is:

20 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY V (n) Proo: The problem o computing the volume V (n) = n n! (60) o U (n) can be solve recursively. Clearly, when n = 2: We will prove the ollowing recursive expression: V (2) = 2 2. We have: V (n) = V n 1 n. an this proves the result. V (n) = 2 = 2 /2 0 /2 0 V (n 1) ( 2z)z ( 2z) n 1 z (n 1)! = 1 [ 2z]/2 0 = n n! n! (61) REFERENCES [1] S. Beneetto, D. Divsalar, G. Montorsi, an F. Pollara, Serial concatenation o interleave coes: Perormance analysis, esign an iterative ecoing, IEEE Transactions on Inormation Theory, vol. 44, no. 3, pp , May [2] R. Garello, P. Pierleoni, an S. Beneetto, Computing the ree istance o turbo coes an serially concatenate coes with interleavers: Algorithms an applications, IEEE Journal on Selecte Areas in Communications, vol. 19, no. 5, pp , May [3] N. Kahale an R. Urbanke, On the minimum istance o parallel an serially concatenate coes, submitte to IEEE Transactions on Inormation Theory. [Online]. Available: rueiger/papers/weight.ps [4] L. Bazzi, M. Mahian, an D. Spielman, The minimum istance o turbo-like coes. [Online]. Available: mahian/tc3.ps [5] M. Breiling an J. B. Huber, Upper boun on the minimum istance o turbo coes, IEEE Transactions on Communications, vol. 49, no. 5, pp , May [6], Combinatorial analysis o the minimum istance o turbo coes, IEEE Transactions on Inormation Theory, vol. 47, no. 7, pp , Nov [7] S. Beneetto an G. Montorsi, Design o parallel concatenate convolutional coes, IEEE Transactions on Communications, vol. 44, no. 5, pp , May [8] M. Breiling an J. B. Huber, A logarithmic upper boun on the minimum istance o turbo coes. [Online]. Available: breiling/research/publications/bre01b.p [9] A. Perotti an S. Beneetto, A new upper boun on the minimum istance o turbo coes. [Online]. Available: [10] N. J. A. Sloane an F. J. MacWilliams, The Theory o Error-Correcting Coes. Amsteram: North-Hollan, [11] S. Beneetto an E. Biglieri, Principles o Digital Transmission with Wireless Applications. New York: Kluwer Acaemic/Plenum Publishers, [12] N. L. Biggs, Discrete Mathematics. Oxor: Oxor Science Publications, 1987.