The constraints for intra-block permutation Πintraare also proposed under the derived rules, where Πintra,i = Πblock ∀i. The proposed bounds in eqn. (3.16) can not hold for all intra-block permutations Πintra due to the termination method for each block. The different methods provide exceptions on the boundary of each block. We derive some loose constraints for intra-block permutation Πintra to retain the bounds under these IBP rules and various termination methods.
Theorem 3.2 reminds us of the importance of a judicious choice of an intra-block permutation. For the question of how to choose an intra-block permutation whose associated w2,min is guaranteed to surpass the worst-case upperbound of Theorem 3.2, Lemma 3.2 gives only an unrefined answer. We need more elaborate constraints on the selection of the intra-block permutation to avoid producing a w2,min smaller than that bound. In general, any one of the four conditions, i ≇ j, πibp(i) ≇ πibp(j), i ≁ j, πibp(i) ≁ πibp(j), is very likely to result in large wt(Cij). However, there is still a small possibility that low weight codewords will be generated. Before presenting the requirements for eliminating these low weight codewords by using a proper intra-block permutation, we need to define a few new functions to facilitate our discussion.
We denote by uk the weight-1 sequence whose only nonzero element is at coordinate k and by gscrb(·) the RSC encoder that encodes a length-L sequence and terminates at the all-zero state using proper tail-bits. Based on the above definitions, we further define, for 0 ≤ i, j < L
f1(i, j) =
½ α|i − j| + β, if i ∼ j
wt( gscrb(uij)), otherwise (3.17) f2(i, j) = wt( gscrb(ui)) + wt( gscrb(uj)) (3.18) f3(i, j) =
½ |i − j|, if i ∼ j
∞, otherwise (3.19)
f4(i, j) = min(f3(i, j), f3(i, j + L), f3(i, j − L)). (3.20) As the way a low weight codeword is generated depends on how the encoder terminates its state at the end of a block, we begin with TP-IBPTC.
3.4.1 TP-IBPTC
For the class of TP-IBPTC, a weight-2 input sequence uij, (i, j) /∈ sm, can not generate an infinite-weight codeword because the encoder state is forced to be terminated at the all-zero state at the end of each block. On the other hand, low-weight codewords may be generated if
dL(i) + dL(j) + dL(πibp(i)) + dL(πibp(j)) < lcm(Ts, Tc) + Tc (3.21) where dL(n) = L − |n|Land in addition, (i) both i, j and πibp(i), πibp(j) are near the ends of different blocks, (ii) i ∼= j, πibp(i) ∼= πibp(j) and both pairs lie close to the end of a block, or (iii) i ≇ j or πibp(i) ≇ πibp(j) but both pairs lie close to the end of a block. To avoid generating low weight codewords out of case (i), we require that
f2(|i|L, |j|L) + f2(fb(πblock(|i|L)) , fb(πblock(|j|L))) ≥ B(Tc, Ts), (3.22) where B(Tc, Ts) = α [Tc+ lcm(Tc, Ts)/Tc] + 2β. Similarly, for cases (ii)-(iii), Πblock must satisfy
f1(|i|L, |j|L) + f1(fb(πblock(|i|L)) , fb(πblock(|j|L))) ≥ B(Tc, Ts), (3.23)
if i ∼= j and πibp(i) ∼= πibp(j), and
f1(|i|L, |j|L) + f2(fb(πblock(|i|L)) , fb(πblock(|j|L))) ≥ B(Tc, Ts) (3.24) if i ∼= j but πibp(i) ≇ πibp(j), and
f2(|i|L, |j|L) + f1(fb(πblock(|i|L)) , fb(πblock(|j|L))) ≥ B(Tc, Ts) (3.25) if i ≇ j but πibp(i) ∼= πibp(j).
Πintra may not meet the above conditions in eqns. (3.22)-(3.25). If we impose more constraints on Πinter and the conditions can be relaxed. It is straightforward to show Lemma 3.3 For a TP-IBPTC whose inter-block permutation Πinter is of Type IV
mini,j wt(Cij) ≥ B(Tc, Ts)
if each element in the set ΓTs = {(i, j) : 0 ≤ i, j ≤ L − 1, |πblock(i) − πblock(j)|Ts = 0}
satisfies
f2(i, j) + f2(πblock(i), πblock(j)) ≥ B(Tc, Ts)
f1(i, j) + f1(πblock(i), πblock(j)) ≥ B(Tc, Ts) (3.26) and ∀ (i, j) /∈ ΓTs the following two inequalities are satisfied
f1(i, j) + f2(πblock(i), πblock(j)) ≥ B(Tc, Ts)
f2(i, j) + f1(πblock(i), πblock(j)) ≥ B(Tc, Ts). (3.27)
Starting with an arbitrary intra-block permutation, say an s-random interleaver [44], we can apply the above criterion iteratively to find the smallest L for a given component code such that wt(Cij) ≥ 2 + B(Tc, Ts). When L is large enough, e.g., L > 2(Tc + lcm(Tc, Ts)), the constraints imposed by the above lemma are relatively easy to meet, i.e., an intra-block permutation that satisfies these constraints is easy to find. For example, it just has to permute the bits near both ends of a block to those coordinates far away from the ends.
3.4.2 TB-IBPTC
We discuss the constraints of intra-block permutation Πintra for TB-IBPTC. The tail-biting encoding results in the codeword weight of weight-1 and weight-2 input in-formation sequences determined by block length L. We give two definitions as follows.
Definition 14 scrbltb(u) is the weight of a length−L tail-biting convolutional code output for an input sequence u.
Definition 15
Sk = [∞ l=k
[S1(l) ∪ S2(l)] , (3.28) where
S1(l) = ©
M = α + 2scrbltb(ui) |0 ≤ i < lª S2(l) = ©
M = scrbltb(uij) + scrbltb(uπblock(i)πblock(j)) |i ≁ j, i ≁ j ± l, πblock(i) ≁ πblock(j),
πblock(i) ≁ πblock(j) ± l, 0 ≤ i, j < l} (3.29)
Let mk be the smallest integer of the set Sk. Obviously, {mk} is a nondecreasing series of k. Denote the least integer k such that mk≥ B(Tc, Ts) by kmin.
We observed that, for a TB-IBPTC whose block size L ≥ kmin, a weight-2 sequence uij generates a codeword whose weight is not larger than the bound B(Tc, Ts) only if (i, j) ∈ ΓTs and (i, j) satisfies the following conditions:
min {|(|i − j|)|Tc, |(L − |i − j|)|Tc} = 0 (3.30) min {|(|πibp(i) − πibp(j)|)|Tc, |(L − |πibp(i) − πibp(j)|)|Tc} = 0. (3.31) min {|i − j|, L − |i − j|}+min {|πibp(i) − πibp(j)|, L − |πibp(i) − πibp(j)|} < lcm(Tc, Ts)+Tc.
(3.32) Such (i, j) pairs will not exist if Πinter is of Type III and the corresponding Πblocksatisfies
f4(|i|L, |j|L) + f4(fb(πblock(|i|L)), fb(πblock(|j|L))) ≥ Tc+ lcm(Tc, Ts), 0 ≤ i, j < L,
∀ (i, j) ∈ ΓTs. In manner similar to the TP-IBPTC case, the above constraint on Πblock
can be further lessened when a Type IV permutation is used. In summary,
Lemma 3.4 For a TB-IBPTC that uses a Type IV permutation with a block length L ≥ kmin, w2,min ≥ B(Tc, Ts) if the corresponding Πblock satisfies
f4(|i|L, |j|L) + f4(πblock(|i|L), πblock(|j|L)) ≥ Tc + lcm(Tc, Ts). (3.33) for all (i, j) ∈ ΓTs.
Note that in designing the interleaver for the classic TCs that use the identical tail-biting convolutional code as the component codes, one must also consider the constraint similar to Lemma 3.4.
3.4.3 C-IBPTC
For the class of C-IBPTC, we only have to consider (i, j) ∈ sm. Low weight codewords are associated with those (i, j) pairs whose combined pre-interleaved and post-interleaved distance, |i − j| + |πibp(i) − πibp(j)|, is small. The upperbound promised by Theorem 3.2 can be achieved if
f4(|i|L, |j|L) + f4(fb(πblock(|i|L)), fb(πblock(|j|L))) ≥ Tc+ lcm(Tc, Ts), (3.34)
∀ (i, j) /∈ ΓTs, if Πinter is of Type III.
The constraint (3.34) is used to ensure that the pair (πibp(i), πibp(j)) though in dif-ferent blocks (since (i, j) /∈ ΓTs) are separated by a large distance.
In analogy to the case of TB-IBPTC, the constraint on Πblock can be relaxed if the corresponding Πinter is more restricted. It is easy to show
Lemma 3.5 For a C-IBPTC that uses a Type IV permutation, if the associated Πintra
is such that for all (i, j) /∈ ΓTs,
f4(|i|L, |j|L) + f4(πblock(|i|L), πblock(|j|L)) ≥ Tc + lcm(Tc, Ts), then w2,min ≥ B(Tc, Ts).
The above discussion shows that the Type IV permutation possesses some desired properties and should be used in conjunction with a proper intra-block permutation.
Hokfelt et al. [54] showed that, as the correlation function of the extrinsic output is exponentially decayed, the interleaver should separate neighboring bits as far as possible.
The local periodicity requirement of Type IV permutation is consistent with this intuition and let bits or samples within the neighborhood of Ts−1 blocks be moved to the different blocks.