IBP properties

This section discusses the properties of an inter-block permutation Πinter while an intra-block permutation Πintra is unknown, where Πintra,i = Πblock ∀i. Denote by C = {c⁰, c¹, c²} a turbo code codeword associated with an input information sequence u, where c^j is the output parity-bit sequence of the jth component code while c⁰ = u represents both the input sequence and the systematic (uncoded) output sequences. A sequence c^j also can be partitioned into N blocks corresponding to an input sequence u and the corresponding sequence is c^j = {c^j0, c^j₁, · · · , c^j_{N −1}}. These sequences are not necessary with the same length and various lengths do not influence our results.

Define two equivalent relations “∼” and “∼=” on the set of integers Z by

|i − j|^Tc = 0 ⇐⇒ i ∼ j

||i||^L = ||j||^L ⇐⇒ i ∼= j

where i, j ∈ Z, T^c is the period (to be defined later) of an RSC code. Clearly, i ≁ j or i ≇ j means i is not equivalent to j in either sense. The first relation i ∼ j indicates (i, j) pair causing a finite weight codeword corresponding to the RSC code with the period Tc, and distance property discussion mainly focuses on this kind of finite weight codeword. The second relation i ∼= j indicates that (i, j) pair is in the same block and simplifies our discussion.

3.3.1 First property: invariant permutation

The following theorem specifies the conditions under which the free distance of a C-IBPTC will be greater than or at least equal to that of its corresponding TC applying Πblock as its interleaver.

Theorem 3.1 For a classic TC CT C applying a Πblock as its interleaver, the corre-sponding C-IBPTC C_ibp has a free distance greater than or equal to that of C_{T C} if the Type I permutation is used and all sequences {¯c^ji, j = 0, 1, 2, i = 0, 1, · · · , N − 1} of a minimum weight codeword, Cmin = {¯c⁰, ¯c¹, ¯c²}, of C^ibp are also valid codewords of the corresponding component codes, where ¯c^j_i = {¯c^ji,k, 0 ≤ k < L}.

Proof: For a C-IBPTC Cibp, there exists at least a finite-weight data sequence umin = c⁰whose corresponding codeword has the minimum weight. Suppose the nonzero elements of umin are at positions {(i¹, k1), (i2, k2), . . . , (in, kn)} and the corresponding codeword is ¯Cmin. Denote by ˜c⁰ the IBP-interleaved version of ¯c⁰, where {˜c^ji, j = 0, 1, 2, i = 0, 1, · · · , N − 1} and ˜c^ji = {˜c^ji,k, 0 ≤ k < L}. The ¯c⁰j and ˜c⁰_j generate, for the two component codes, the encoded parity-bit sequences, ¯c¹_j and ¯c²_j with Hamming weights wt(¯c¹_j) and wt(¯c²_j), respectively. The systematic parts of both component codes are the same and the corresponding block are denoted by ¯c⁰_j. Let Ml be the permutation defined on the space of all blocks that moves a block to the lth block, i.e., Ml : cⁱ_k → cⁱl, ∀ k. If fb(j) = j and Πintra,i = Πblock, shifting each position (ni, ki) to the lth block by Ml gives Ml(˜cπibp(ni,ki)) = Ml(˜cfn(ni,ki),fb(πblock(ki))) = ˜cl,πblock(ki). (3.7) As the 2-tuple (¯c⁰_j, ¯c¹_j) is a valid codeword of the first component code of C_ibpaccording to our assumption,³L

jMl(¯c⁰_j),L

jMl(¯c¹_j)´ ,L

denoting addition of binary vectors, is also a codeword of the same component code. (3.7) implies that ³L

jMl(˜c⁰_j),L

jMl(¯x²_j)´ are valid codewords for the second component code of Cibp and CT C since the additional IBP does not change the relative positions of input bits within a block. The inequality

wt(cⁱ_j) + wt(cⁱ_l) = wt(Ml(cⁱ_j)) + wt(Ml(cⁱ_l)) ≥ w^t¡

Ml(cⁱ_j) ⊕ M^l(cⁱ_l)¢

, ∀j 6= l (3.8)

then implies that the free distance of Cibp, df ree(Cibp) satisfies df ree(Cibp) =X

i

X

j

wt(¯cⁱ_j) ≥X

i

wt

ÃM

j

Ml(¯cⁱ_j)

!

≥ d^{f ree}(Cblock) (3.9)

For a D-IBPTC, the sub-codewords ¯c^j_i associated with each input block automat-ically satisfy the requirement on all blocks. Since both tail-padding and tail-biting convolutional codes are linear codes, we have

Corollary 3.1 For a classic TC CT C, the corresponding D-IBPTC Cibp has a free dis-tance greater than or equal to that of Cblock if Πinter is a Type I permutation.

3.3.2 Second property: periodic permutation

The encoder of an RSC code acts like a scrambler and can be realized by using a shift register with both feedback and feedforward branches shown in Fig. 2.2. It is obvious that such an encoder would have a periodic impulse response. The rate 1/2 RSC code is specified by the transfer matrix [1, Q⁻¹(D)F (D)], where Q(D), F (D) ∈ F²(D) and Q(D) is usually a primitive binary polynomial of degree m. The period of the impulse response of the non-systematic part, Q⁻¹(D)F (D), is given by Tc whose maximum value is 2^m− 1. We denote by u^ij a weight-2 input sequence whose only nonzero elements are at coordinates i and j; the corresponding codeword is denoted by C^ij. Therefore, T_c is also the smallest integer such that u^ij, i ∼ j, will generate a finite-weight output parity sequence. It is thus easy to show [24]

Lemma 3.1 Let u^ij be an input sequence to a scrambler with period T_c and scrb(u^ij) be the corresponding output parity sequence. If i ∼ j, then there exist α ∈ N and β ∈ Z such that

w_t¡

scrb(u^ij)¢

= α|i − j|/Tc+ β, (3.10)

where N is the set of positive integers and α, β depend on the encoder (scrambler) struc-ture.

The puncturing may result in that the sequence generation of a scrambler becomes time-variant. In this case we can search α, β_U, β_Land the weight of sequence is bounded as

α|i − j|/Tc + β_L ≤ wt

¡scrb(u^ij)¢

≤ α|i − j|/Tc+ β_U, (3.11) where i ∼ j. In order to simply our discussion the puncturing case will not be considered in the following. We only remind the effect of the puncturing.

Obviously, if i ≁ j, u^ij will generate an infinite weight parity sequence if there is no termination at the end of a block. Lemma 3.1 implies that the codeword weight, wt(C^ij), of a turbo code satisfies

wt¡ C^ij¢

≥ 2 + α

µ|i − j| + |π(i) − π(j)|

Tc

¶

+ 2β, (3.12)

with equality holds iff

i ∼ j and π(i) ∼ π(j). (3.13)

Define ˜w2,min

def= min_(i,j)∈smwt(C^ij), where sm

def= {(i, j)|i ∼ j, π^ibp(i) ∼ π^ibp(j)} and let

δmin = min

(i,j)∈s^m[|i − j| + |π^ibp(i) − π^ibp(j)|] . (3.14) For the class of C-IBPTCs, ˜w2,min = w2,min

def= min_(i,j)wt(C^ij), therefore, maximizing the minimum weight of the codewords associated with the weight-2 input sequences is equivalent to maximizing δmin. The next theorem provides an upper bound of w2,min

any IBPTC can achieve, if choosing the intra-block permutation is not an option.

Theorem 3.2 For an IBPTC using an inter-block permutation Π_inter, there exists a Πintra such that w2,min ≤ 2 + α(T^s+ 1) + 2β, if L > Tc · T^s, where Ts is the number of inter-block permuted blocks.

Proof: Consider the partition {0, 1, · · · , L − 1} = STc−1 j=0

STs−1

l=0 Sjl, where Sjl = {g|g ∈ S^j, f_n^d(p, g) = l} for some p, S^j = {h|h ∼ j, 0 6 h < L}, 0 ≤ j < T^c and

0 ≤ l < T^s. Note that the decomposition Sj = STs−1

l=0 Sjl is induced by the function f_n^d(p, g) or equivalently, by f_n(r, g). Obviously, the codeword weight of the weight-2 information sequence u^{π−1ibp(g)π−1ibp(h)} is large, if u^′gh with g ≁ h. As we are concerned with w2,min, only those weight-2 sequences with nonzero coordinate pairs in the set, {(g, h)|g ∼ h ∼ j, for some j and g, h ∈ S^jl for some l} have to be considered.

Assume that ∀ j, l all pairs {(g, h) ∈ S^jl} satisfy the inequality |g − h| > T^c· T^s. For any pair (g, h) ∈ S^jl, g < h and the associated interior set V = {g+1, g+2, · · · , h−1}, we have |V | > T^c·T^s. If Sjl∩V 6= φ, there exists a pair (g^′, h^′) ∈ S^jl, where |g^′−h^′| < |g−h|.

Otherwise, if S_jl∩V = φ, by the pigeonhole principle [29], there exists a set Suvsuch that

|S^uvT

V | > 2, which implies that there is a pair (g^′, h^′) ∈ S^uv, where |g^′− h^′| < |g − h|.

As both cases lead to contradictions, we conclude that there exists a pair (g, h) ∈ S^jl for some j, l, such that |g − h| ≤ T^c · T^s. Since it is always possible to find Πintra

such that |πibp⁻¹(g) − πibp⁻¹(h)| = T^c, eqns. (3.12) and (3.13) then imply that w2,min ≤ 2 + α((Tc+ Tc· T^s)/Tc) + 2β = 2 + α(Ts+ 1) + 2β.

Theorem 3.2 indicates that lack of control on the intra-block permutation imposes an upperbound for w_2,min which an IBPTC can achieve. The coordinates of nonzero elements of the interleaved sequence u^′ij with i ∼= j will either remain in the same block or be in the different blocks with probabilities close to 1/Ts and (Ts − 1)/T^s when all coordinates in one block are evenly permuted to these blocks. The resulting codewords for the latter case are very likely to have large weights while those for the former case have smaller weights with the worst-case weight of 2 + 2α + 2β only.

To avoid generating low weight codewords for u^ij, we first notice that eqn. (3.12) implies ˜w2,min ≥ 2 + α(δ^min/Tc) + 2β. The IBP along with the intra-block permutation determines the relation between |i − j| and |πibp(i) − πibp(j)|, and their structures can be optimized to maximize δmin. For a pair of coordinates (i, j) ∈ s^m, if the integer-valued function f_n^d(p, k) is injective and satisfies the locally-periodic property for some p, f_n^d(p, k) = f_n^d(p, k + nTs) for 0 ≤ k + nT^s < L, then the requirements, i ∼= j and πibp(i) ∼=

πibp(j) imply |π^ibp(i) − π^ibp(j)|^Ts = 0 and therefore |π^ibp(i) − π^ibp(j)| ≥ lcm(T^c, Ts), where lcm(a, b) represents the least common multiple of a and b. In other words,

Lemma 3.2 An IBPTC that uses a Type III permutation satisfies

i∼=j,πibp(i)∼=πminibp(j),(i,j)∈s^m wt(C^ij) ≥ 2 + α · {[T^c+ lcm(Tc, Ts)] /Tc} + 2β. (3.15)

If Tc and Ts are relative prime, then

i∼=j,πibp(i)∼=πminibp(j),(i,j)∈sm

wt(C^ij) ≥ 2 + α · (T^s+ 1) + 2β. (3.16)