sign(z) =
0 ifz >0
0 with prob. 1/2 ifz = 0 1 with prob. 1/2 ifz = 0 1 ifz <0
Since the code Tanner graph has cycles, different scheduling methods yield in general non-equivalent BP algorithms. In this work we shall consider the following “classical” scheduling strategies:
• LDPC-like scheduling [15]. In this case, all bitnodes and all checkn-odes are activated alternately and in parallel. Every time a node is activated, it sends outgoing messages to all its neighbors. A decoding iteration (or “round” [39]) consists of the activation of all bitnodes and all checknodes.
• Turbo-like scheduling. Following [40], a good decoding scheduling con-sists of isolating large trellis-like subgraphs (or, more generally, normal realizations in Forney’s terminology) and applying locally the forward-backward BCJR algorithm [4] (that implements efficiently the BP al-gorithm on normal cycle-free graphs), as done for Turbo codes [3]. A decoding iteration consists of activating all the information bitnodes in parallel (according to (2.6)) and of running the BCJR algorithm over the entire accumulator trellis. In particular, the checknodes do not send messages to the information bitnodes until the BCJR iteration is completed.
These two scheduling methods arise from the fact that IRA codes are sub-classes of both LDPC codes and irregular turbo codes. Notice that for both of the above scheduling methods one decoder iteration corresponds to the activation of all information bitnodes in the graph exactly once.
2.4 Density Evolution and Stability
The bit error rate (BER) performance of BP decoding averaged over the IRA code ensemble can be analyzed, for any finite number ` of iterations and in the limit of k → ∞, by the DE technique [19]. For a given bitnode α and iteration `, the message sent over an outgoing edge (say edge e) is
a random variable that depends on the transmitted codeword, the channel noise and the interleaver (uniformly distributed over the set of permutations of N elements). The DE method finds the distribution of this random vari-able averaged over the channel noise and the interleaver, assuming that the block length goes to infinity. Under such an assumption, the probability that an oriented neighborhood of depth 2` of the edge econtains cycles vanishes.
Therefore, DE can be computed under the cycle-free condition, implying that the input messages at any node in the BP algorithm are statistically inde-pendent. For binary-input symmetric-output channels, the average message distributions do not depend on the transmitted codeword [39], so the trans-mission of the all-zero codeword can be assumed. The usefulness of the DE method stems from the Concentration Theorem [39, 18] which guarantees that, with high probability, the BER after ` iterations of the BP decoder applied to a randomly selected code in the ensemble and to a randomly gen-erated channel noise sequence is close to the BER computed by DE with high probability, for sufficiently large block length.
Next, we formulate the DE for IRA codes and we study the stability condition of the fixed-point corresponding to zero BER. As in [19, section III-B], we introduce the space of distributions whose elements are non-negative non-decreasing right-continuous functions with range in [0,1] and domain the extended real line.
It can be shown that, for a binary-input symmetric-output channel, the distributions of messages at any iteration of the DE satisfy the symmetry condition [19] Z
h(x)dF(x) = Z
e−xh(−x)dF(x) (2.10) for any function h for which the integral exists. IfF has densityf, (2.10) is equivalent to
f(x) = exf(−x) (2.11)
With some abuse of terminology, distributions satisfying (2.10) are said to be symmetric. The space of symmetric distributions will be denoted by Fsym.
The bit error probability operator Pe :Fsym →[0,1/2] is defined by Pe(F) = 1
2(F−(0) +F(0))
whereF−(z) is the left-continuous version of F(z). We introduce the “delta at zero” distribution, denoted by ∆0, for which Pe(∆0) = 1/2, and the “delta at infinity” distribution, denoted by ∆∞, for which Pe(∆∞) = 0.
2.4 Density Evolution and Stability 19 The symmetry property (2.10) implies that a sequence of symmetric dis-tributions {F(`)}∞`=0 converges to ∆∞ if and only if lim`→∞Pe(F(`)) = 0, where convergence of distributions is in the sense given in [19, Sect. III-F].
uα∼Fu
a 2
i
mo,β ∼Ql mo,β∼Q˜l
uα∼Fu
mo,α∼Pl mo,α∼P˜l
Figure 2.3: Message flow on the graph of a systematic IRA code The DE for IRA code ensembles is given by the following proposition.
Proposition 2.4 Let P` [resp., Pe`] denote the average distribution of mes-sages passed from an information bitnode [resp., parity bitnode] to a chec-knode, at iteration ` . Let Q` [resp., Qe`] denote the average distribution of messages passed from a checknode to an information bitnode [resp., parity bitnode], at iteration ` (see Fig. 2.3).
Under the cycle-free condition, P`,Pe`, Q`,Qe` satisfy the following recur-sion:
P` = Fu⊗λ(Q`) (2.12)
Pe` = Fu⊗Qe` (2.13)
Q` = Γ−1
Γ(Pe`−1)⊗2⊗Γ(P`−1)⊗(a−1)
(2.14) Qe` = Γ−1
Γ(Pe`−1)⊗Γ(P`−1)⊗a
(2.15) for ` = 1,2, . . ., with initial condition P0 = Pe0 = ∆0, where Fu denotes the distribution of the channel observation messages (2.5), ⊗denotes convolution of distributions, defined by
(F ⊗G)(z) = Z
F(z−t)dG(t) (2.16)
⊗m denotes m-fold convolution, λ(F) =∆ Pd
i=2λiF⊗(i−1), Γ(Fx) is the distri-bution of y=γ(x) (defined on F2×R+), when x∼Fx, and Γ−1 denotes the
inverse mapping of Γ, i.e., Γ−1(Gy) is the distribution of x = γ−1(y) when y∼Gy.
Proof: See Appendix 2.A.
The DE recursion (2.12 – 2.15) is a two-dimensional non-linear dynamical system with state-space Fsym2 (i.e., the state trajectories of (2.12 – 2.15) are sequences of pairs of symmetric distributions (P`,Pe`)). For this system, the BER at iteration`is given by Pe(P`). A property of DE, given in Proposition 2.4, is that Pe(P`) is a non-negative and non-increasing function of`[19, 39].
Hence, lim`→∞Pe(P`) exists.
It is easy to see that (∆∞,∆∞) is a fixed-point of (2.12 – 2.15). The local stability of this fixed-point is given by the following result:
Theorem 2.5 The fixed-point (∆∞,∆∞) for the DE is locally stable if and only if
λ2 < er(er−1)
a+ 1 +er(a−1) (2.17)
where r=−log(R
e−z/2dFu(z)).
Proof: See Appendix 2.B.
Here necessity and sufficiency are used in the sense of [19]. By following steps analogous to [19], it can be shown that if (2.17) holds, then there exists ξ > 0 such that if for some ` ∈ N, Pe(RP`(P0,Pe0) + (1−R)Pe`(P0,Pe0))< ξ then Pe(RP`+ (1−R)Pe`) converges to zero as ` tends to infinity. On the contrary, if λ2 is strictly larger than the RHS of (2.17), then there exists ξ >0 such that for all`∈N Pe(RP`(P0,Pe0) + (1−R)Pe`(P0,Pe0))> ξ.
Consider a family of channels C(ν) ={pνY|X :ν ∈R+}, where the channel parameter ν is, for example, an indicator of the noise level in the channel.
Following [39], we say that C(ν) is monotone with respect to the IRA code ensemble ({λi}, a) under BP-decoding if, for any finite `
ν ≤ν0 ⇔Pe(P`)≤Pe(P`0)
where P` and P`0 are the message distributions at iteration ` of DE applied to channels pνY|X and pνY0|X, respectively.
Let BER(ν) = lim`→∞Pe(P`), where{P`}is the trajectory of DE applied to the channel pνY|X. The threshold ν? of the ensemble ({λi}, a) over the
2.5 Conclusion 21