It is known that for terminated (TE) spatially coupled low-density parity-check (SC-LDPC) codes [1, 2, 3, 4] the belief propagation (BP) threshold of the underlying code ensemble approaches the maximum a posteriori (MAP) threshold in the limit of large coupling lengths. When the coupling length is moderate or small, the termination (which is necessary to trigger the decoding wave) entails a non negligible rate loss. The rate loss can be avoided by resorting to tailbiting (TB) SC-LDPC codes [5, 6, 7] at the cost of a (potentially large) threshold degradation. In fact, the BP threshold of a TB SC-LDPC code ensemble is the same as the one of the underlying uncoupled code ensemble. Approaches to improve the BP decoding threshold of TB SC-LDPC code ensembles were introduced in [5, 6, 7]. They rely on the possibility of either mapping a specific portion of the codeword to the most reliable bit levels in a bit interleaved coded modulation scheme, or on fixing some of the codeword bits to known values. The latter case, which can be adopted on any binary input channel, still entails a rate loss due to the code shortening. Both approaches aim at triggering the wave-like decoding phenomenon that is at the base of the threshold saturation effect. For the case of TE SC-LDPC code ensembles, this is enabled by the stronger protection provided by the termination. In , a way to mitigate the rate loss is introduced, which relies on appending variable nodes with suitable degree distributions to the SC-LDPC graph.
In this letter, we investigate an alternative approach that enables large improvements of the BP threshold of TB SC-LDPC code ensembles over the binary input additive white Gaussian noise (bi-AWGN) channel. The technique, which preserves the rate of the TB SC-LDPC code ensemble, relies on distributing different energies to the codeword bits. We refer to this approach as energy shaping.111The term “shaping” shall not be intended in the sense of signal shaping . As it will be described in Section II, here a time-sharing approach is considered, where the transmission energy is allowed to vary over time. By doing so, different bit reliabilities are achieved. The ensemble BP threshold is analyzed by means of protograph extrinsic information transfer (P-EXIT)  analysis. Thanks to its capability of dealing with VNs associated to channels with different signal-to-noise ratio (SNR) , we show how, under the restriction of admitting only two energy values, the boosting level and the length of the boosted portion can be optimized to attain the lowest possible threshold for a given TB SC-LDPC code ensemble.
The idea of improving the BP threshold of protograph-based low-density parity-check (LDPC) code ensembles  by means of different transmission energies was originally proposed in . The approach in  performs a VN-by-VN optimization over the protograph, where the optimization of the energy allocated to each protograph VN is performed through a downhill optimization approach. While very general, the approach may become costly for large protographs. In this letter, we focus on the simplified case where the transmission energy of a bit can be chosen among two values, and where the largest one is used for the transmission of a contiguous portion of the codeword. The intuition is that, by localizing the energy boost on a contiguous portion of the codeword bits, the wave-like decoding effect is triggered. With respect to , which performs the energy optimization under the assumption that the decoder adopts a parallel message passing schedule, our approach is specifically tailored to operate with the sliding window decoding schedule typically employed in SC-LDPC code decoders to reduce the decoding complexity .
We consider transmission over the bi-AWGN channel of a (modulated) codeword , where is the block length and . The channel output is denoted by , where
with. Here, is a parameter proportional to the energy used for the transmission of , such that
We denote the code rate by . The average SNR, defined as the ratio between the average energy per information bit and the single-sided noise power spectral density , is given by . We focus on the case where two values for are allowed. We denote them by and , with . The ratio is referred to as boosting factor. We assume next that for the first channel uses the parameter is set to , whereas for the remaining channel uses . We refer to the parameter as the boosting length, and by as the normalized boosting length. Note that for the first channel uses the SNR is while for the remaining channel uses the SNR is . Also, observe that (1) can now be restated as . Hence the transmission parameters are fully specified by the energy shaping parameters . Furthermore, we have that
with , which yields
where the dependency on and is implicit due to (3).
Ii-a Protograph-Based Spatially Coupled LDPC Codes
Here, we consider protograph-based SC-LDPC codes . In particular, in the following sections we will consider for simplicity a special class of rate- regular TB SC-LDPC ensembles, where is the VN degree and is the check node (CN) degree. A protograph  is a small bipartite graph comprising a set of VNs (also referred to as VN types) and a set of CNs (i.e., CN types) . A VN type is connected to a CN type by edges. A protograph can be equivalently represented in matrix form by an matrix . The -th column of is associated to VN type and the -th row of is associated to CN type . The element of , , indicates the number of edges connecting and . A larger graph (derived graph) can be obtained from a protograph by applying a copy-and-permute procedure. The protograph is copied times ( is commonly referred to as lifting factor), and the edges of the different copies are permuted preserving the original protograph connectivity: If a type- VN is connected to a type- CN with edges in the protograph, in the derived graph each type- VN is connected to distinct type- CNs (observe that multiple connections between a VN and a CN are not allowed in the derived graph). The derived graph is the Tanner graph of an LDPC code with length . A protograph can be used to define a code ensemble . For a given protograph , consider all its possible derived graphs with VNs. The ensemble is the collection of codes associated to the derived graphs in the set.
Iii Extrinsic Information Transfer Analysis
Iii-a Analysis over Parallel Bi-AWGN Channels
The performance of protograph-based LDPC codes over parallel channels can be analyzed by the P-EXIT analysis. Following , we consider next the case where the codeword bits corresponding to the protograph VNs are transmitted over parallel bi-AWGN channels. We denote by the mutual information (MI) between the message sent at iteration by the -th VN to the -th CN and the corresponding codeword bit. Similarly, denotes the MI between the message sent at iteration by the -th CN to the -th VN and the corresponding codeword bit. We further define the SNRvector with if and otherwise. The evolution of the MI can be tracked by applying the recursion
where by convention we set if . In (6) and are the variable and check extrinsic information transfer (EXIT) functions, whose expression can be found in [11, Sec. IV.A]. We finally introduce as the MI between the logarithmic a posteriori probability (APP) ratio at the -th VN in the -th iteration and the corresponding codeword bit.
Iii-B Achievable Regions for Decoding Convergence
For a regular TB SC-LDPC ensemble and for a pair we say that the SNR pair is achievable under BP decoding if, for a sufficiently large number of iterations and for large , a code picked at random from the ensemble exhibits (on average) a vanishing small bit error probability, i.e., if converges to for all as . The region of pairs for which converges to for all , as , is referred to as the achievable region . Formally,
For an arbitrary value of , that we denote by , one may define the minimum value for , that we denote by , such that is achievable under BP decoding. Note that each pair is unequivocally associated with a specific boosting factor as . The set of pairs , denoted by , determines the boundary of the achievable region .
Iii-C Decoding Thresholds
In Fig. 1, a graphical interpretation of (7) is provided. The figure displays the achievable region and its boundary for a ensemble with and . For a given , one has to find the straight line defined by (2), with minimal , intersecting the boundary . For the example in the figure, the minimum is found at dB, which corresponds to dB and dB, yielding . By optimizing over the normalized boosting length , we finally find
Fig. 2 depicts as a function of the normalized boosting length for the ensemble with . On the same chart, the average SNR required to achieve a rate equal to according to (4) is depicted. The minimum decoding threshold is attained for (with ).
Iv Performance Analysis
In the following, results for regular TB SC-LDPC code (ensembles) with energy shaping are presented through EXIT analysis and Monte Carlo simulations. In both cases, windowed decoding  has been considered.
We analyze the performance of the proposed scheme by comparing its iterative decoding thresholds with
the MAP threshold222
The estimate of theMAP threshold is obtained by exploiting the threshold saturation effect of SC-LDPC code ensembles . In particular, we estimate the MAP threshold of the uncoupled ensemble by performing an EXIT analysis of the TE SC-LDPC code ensemble for very large and by setting . of the corresponding uncoupled ensemble with uniform energy (UE);
The analysis is summarized in Table I for the case of . The table provides the thresholds for the different ensembles, for , , and . For large , the threshold achieved with the proposed approach matches the one of the terminated ensemble with UE, saturating to the MAP threshold of the corresponding uncoupled ensemble with UE. For moderate-to-small , the proposed approach achieves a larger threshold with respect to the terminated case. However, while for the terminated ensemble the actual coding rate is less than (it reduces to for ), the proposed scheme keeps the code rate to the nominal rate. The difference in coding gain between the terminated case and the TB SC-LDPC code ensemble with energy shaping is limited to about dB for , and reduces to dB for . The gain with respect to the uncoupled ensemble threshold exceeds dB for all cases summarized in Table I. Table II compares the thresholds achieved by various ensembles for the case of . Remarkably, for energy shaping achieves the smallest threshold for the ensemble, with a gain of almost dB over the corresponding block ensemble. This fact has to be attributed to the moderate number of spatial positions , whereas for growing large the decoding threshold shall improve with increasing VN and CN degrees.
We simulated the performance of the TB SC-LDPC code with , with and without energy boosting. For the case where energy boosting is employed, the boosting parameters have been set to the values minimizing the BP threshold, i.e., (with ). The protograph has been expanded with a lifting factor , yielding a block length . For decoding, we employed windowed decoding with iterations per window position (local iteration), and a window spanning over VNs. The window is shifted by positions at the end of the local iteration steps, circling twice around the tailbiting Tanner graph. The results, in terms of bit error rate (BER) and codeword error rate (CER), are provided in Fig. 3. A gain close to dB at over the UE case is achieved.333We may observe that the gain at finite block lengths is smaller than the one predicted by the EXIT analysis. In particular, the gain is expected to diminish as the lifting factor becomes small.
We analyzed the convergence behavior of TB SC-LDPC codes over the bi-AWGN channel with different reliabilities assigned to the codeword bits by shaping the transmission energies. By focusing on the case where only two energy levels are allowed, and a contiguous portion of the codeword is transmitted with the largest energy, we showed that large coding gains (e.g., up to for the
ensemble) can be attained with reference to the case where a uniform energy is employed. The results of the asymptotic analysis are confirmed by finite-length simulations. We conjecture that additional coding gains might be achieved by jointly optimizing the protograph ensemble and the shaping parameters.
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