On the Convergence Speed of Spatially Coupled LDPC Ensembles
Spatially coupled low-density parity-check codes show an outstanding performance under the low-complexity belief propagation (BP) decoding algorithm. They exhibit a peculiar convergence phenomenon above the BP threshold of the underlying non-coupled ensemble, with a wave-like convergence propagating through the spatial dimension of the graph, allowing to approach the MAP threshold. We focus on this particularly interesting regime in between the BP and MAP thresholds.
On the binary erasure channel, it has been proved  that the information propagates with a constant speed toward the successful decoding solution. We derive an upper bound on the propagation speed, only depending on the basic parameters of the spatially coupled code ensemble such as degree distribution and the coupling factor . We illustrate the convergence speed of different code ensembles by simulation results, and show how optimizing degree profiles helps to speed up the convergence.
Low-density parity-check (LDPC) codes are widely used due to their outstanding performance under low-complexity belief propagation (BP) decoding. However, an error probability exceeding that of maximum-a-posteriori (MAP) decoding has to be tolerated with (sub-optimal) BP decoding. However, it has been empirically observed for spatially coupled LDPC (SCLDPC) codes – first introduced by Felström and Zigangirov as convolutional LDPC codes  – that the BP performance of these codes can improve dramatically towards the MAP performance of the underlying code under many different settings and conditions, e.g. [3, 4, 5]. This phenomenon – termed threshold saturation – has been proven by Kudekar, Richardson and Urbanke in [6, 7]. In particular, they proved that the BP threshold of a coupled LDPC ensemble tends to its MAP threshold on any binary symmetric memoryless channel (BMS). The principle behind threshold saturation seems to be very general and has been applied to a variety of more general scenarios in information theory and computer sciences.
More recently, a new proof technique of threshold saturation has been given by introducing a potential function [8, 9]. Independently, a similar technique has been used in  to study one-dimensional continuous coupled systems. It was proven that below the MAP threshold, the information propagates in a wave-like manner with a constant propagation speed by progressing density evolution. This result has also been extended to discrete spatially coupled one-dimensional systems.
We can distinguish between two convergence regions for SCLDPC codes. If the channel entropy is below the BP threshold of the underlying non-coupled ensemble, the convergence is governed by the ensemble degree distribution. We call this convergence intra-graph convergence. If the channel entropy is between the BP and MAP thresholds of the non-coupled ensemble, the wave-like solutions manifests itself after some iterations and the convergence – which we denote inter-graph convergence – is dominated by the coupling of the graphs. We particularly focus on this latter convergence phenomenon and are interested in the inter-graph convergence (propagation) speed of the wave-like solution. Knowing this speed has several important practical implications: The degree distribution can be optimized in order to maximize the speed for a given target channel and consequently, to minimize the number of required iterations to decode successfully. Additionally, if we employ windowed decoding , the convergence speed gives the correct timing to shift the decoding window.
In this paper, we derive upper bounds on the convergence speed of spatially coupled LDPC ensembles for the BEC. Moreover, we compare the convergence speeds of different ensembles and show that the speed is sensitive to the choice of the degree distribution. We also compare the speed of a spatially coupled ensemble on different types of BMS channels.
We briefly explain the graphical structure of LDPC ensembles and spatially coupled LDPC ensemble. Then, we describe the potential functions associated to these ensembles.
Ii-a LDPC Ensembles
LDPC codes are a subset of block codes with sparse parity check matrices. Let be the block length of the code and be the number of constraints to satisfy. The design rate of the code is . We usually represent LDPC codes by a bipartite graph called factor graph. To each of the bits, we assign a node, called variable node and we assign a node to each of the constraints, called check node. We connect variable node to check node by an edge if and only if the bit participates in the corresponding constraint. To construct a random LDPC code, we sample the degree of node according to a given degree distribution. We represent the degree distribution of the variable nodes by a polynomial and the degree distribution of check nodes by . For a node of degree , we consider sockets from which the edges emanate. Thus, There are sockets in both variable side and check side. After labeling the sockets, we randomly choose a socket from the variable side and connect to a randomly chosen socket in the check side. Finally, we have a random instance of a LDPC ensemble. For the detail of construction, we refer to .
To study the performance of belief propagation algorithm on sparse graph codes, it is common to use density evolution. Let denote the erasure probability of the channel and let denote the erasure probability flowing from variable side to the check side at iteration , then
where and . We set . The error probability of decoding vanishes if
Definition 1 (BP Threshold).
The BP threshold of the LDPC ensemble is defined as:
The MAP threshold is defined as the maximum in which the decoding error probability of MAP decoding is equal to zero. In general, .
Ii-B Spatially Coupled LDPC Ensemble
We first lay out a set of positions indexed by integers on a line. This line represents a spatial dimension. We fix a coupling factor which is an integer . Consider sets of variable nodes each having nodes, and locate the sets in positions to . Similarly, locate sets of check nodes each, in positions to . The degree of each variable (check) node is randomly sampled according to the degree distribution () leading to sockets at each position.
To construct a random instance of the SCLDPC ensemble, we connect the variable nodes to the check nodes in the following manner: Each of sockets of variable nodes at position is connected uniformly at random to a socket of check nodes within the range . At the end of this procedure, all sockets of check nodes in position are occupied except for the check nodes in , where only a fraction of them are connected. Symmetrically, for the check nodes in , a fraction of the sockets are connected. The rest of the sockets are free and we can assume that they are connected to virtual variable nodes with zero erasure probability. For the detail of construction, we refer to .
Let be the erasure probability incoming to check nodes in position at iteration . The density evolution (DE) equation is
where for , and zero otherwise. We initialize for . For the boundary values, , we set for all .
As density evolution progresses, the perfect boundary information from the left and right sides propagates inward. It was shown in [6, 7] that is non-decreasing sequence for and becomes non-increasing sequence afterward. The symmetric initialization and symmetric boundary conditions induce symmetry on all the erasure probabilities, i.e. . This system has been termed two-sided spatially coupled LDPC ensemble. One half of the spatially coupled ensemble is enough to describe the system.
Definition 2 ().
Let . The one-sided spatially coupled LDPC ensemble is a modification of (2) defined by fixing for .
Lemma 1 ().
For the one-sided spatially coupled LDPC ensemble, the densities resulting from density evolution over the BEC satisfy
for all and ,
Ii-C Potential Function
We define the potential function of the LDPC ensemble as in ,
It is indeed equal to the normalized (by ) replica-symmetric free energies in  and it has the following properties.
Depending on the degree distributions and , the potential function can have many stationary points. The potential function of regular LDPC ensemble for is depicted in Fig. 1. Each local minimum corresponds to a stable fixed-point of DE equation and each local maximum corresponds to an unstable fixed-point. For there is only one local minimum which is .
We define the area threshold as
It is shown in [8, Lemma 6] that the area threshold is equal to the Maxwell threshold. The Maxwell threshold is equal to MAP threshold on regular ensembles. It has been conjectured that they are equal in general and the equality was recently justified for a large class of LDPC codes in . Thus, for and becomes zero at .
Now consider the SCLDPC ensemble. We use the extension of potential function introduced in  as follows:
where . We retrieve the DE equation from the partial derivatives of the potential function, i.e.
Thus, the stationary points of are the fixed-points of DE equation.
Define the vector as for . By using the first order Taylor expansion,
where is the remainder term. Define
Numerical evaluation shows that the remainder term is negative and small in comparison with and then,
To upper-bound the speed, we must show that there is such that for all and .
We prove in the next section and in Appendix A for large . However, our simulation results suggest . The extension to is currently work in progress.
Iii Bounds on The Convergence Speed
Consider the DE equation of one-sided SCLDPC ensemble. Denote the BP threshold and the MAP threshold of the underlying LDPC ensemble by and , respectively. It has been proven in [6, 8] that there is such that for and ,
for all . The question is how fast converges to zero. Assume that . We distinguish two distinct phases during DE progress. In the first few iterations, all except the ones close to converges to the forward DE fixed-point of the underlying ensemble, (intra-graph convergence). Then, in the next iterations, the information propagates from the boundary and becomes zero successively (inter-graph convergence).
In this section, we bound the speed of information propagation. First we consider LDPC ensembles whose DE equation has three fixed-points (two stable and one unstable fixed-point). The potential function of one such ensemble is shown in Fig. 1. Many LDPC ensembles including regular LDPC codes have such property. Then we consider the ensembles with more than three DE fixed-points.
Iii-a DE Equation with Three Fixed-points
Fig. 2 shows the solution of DE equation of one-sided SCLDPC ensemble in different iterations. We observe that the sequence moves uniformly forward with a fixed speed by progressing density evolution. More precisely, there is such that
for all and . The existence of such a DE solution is proven in  for the one-sided SCLDPC ensemble on in which the DE equation of the underlying ensemble has three fixed-points: zero, an unstable fixed-point and a stable fixed-point , the forward DE fixed-point.
We define the propagation speed of such a DE solution:
For , define
Additionally, we define the propagation speed .
One can show that is an increasing sequence and for
We are mostly interested in knowing the speed for close to in which the DE solution travels very slowly, i.e. . In the following theorems, we upper bound (or equivalently, lower-bound ). We assume that at , the wave-like solution is already formed and for ,
for some iteration .
Consider (5). Assume that , then there exists such that for , .
The sketch of the proof is given in Appendix A.
Theorem 1 (Upper bound).
Assume that the DE solution moves forward uniformly and for . Then, for any ,
Proof: Appendix B.
As we see later, the above bound is a tight upper bound for large . For the calculation, we must run DE until the wave-like solution is formed, which in practice, appears after a few iterations. We can additionally state the following (looser) upper bound which eliminates the need to run density evolution:
Assume that . Let . Then,
which simplifies for to
Proof: Appendix C
Theorem 3 (Lower Bound [1, Theorem 2] ).
The speed is lower-bounded by
For , .
Note that is equal to the area defined in .
Iii-B DE Equation with Many DE Fixed-points
In general, the DE equation of irregular LDPC ensembles can have more than three fixed-points in some . In this case, the DE solution of coupled ensemble can become more complex. It can be a mixture of wave-like solutions with distinct speeds, in which the solution with larger speed overlaps the solutions with lower speed. For instance, consider one-sided SCLDPC ensemble with and . For , the underlying ensemble has three DE fixed-points and thus, the DE solution is as explained in the previous section. For , the underlying ensemble has five DE fixed-points, namely 0, and the stable fixed-points and and the unstable ones. In this case, two wave-like DE solutions appear. In Fig. 3, the DE solution for is depicted. We observe that for each , first reduces from to by the first wave-like solution and then, reduces from to by the second wave-like solution.
Let and denote the propagation speed of the upper and the lower solutions (). determines the total number of iterations required for decoding. We can still apply to upper bound with the following assumption. The waves are well-separated such that the right tail of the lower wave converges to from below and the left tail of upper wave converges to from above (see Fig. 3). If this assumption holds at some , we can separately study the waves by following the proof of Theorem 1. Shortly,
and by summing up both inequalities, and noting that , we have
One can also adapt for the general case by following the similar proof.
Iv Simulation Results
As we mentioned before, the numerical results support the conjectured claim . In fact, it is the speed given by the first order Taylor approximation. We therefore set in the sequel. Consider the SCLDPC ensemble. We give and for and different in Tab. I. We observe that the speed increases almost linearly in terms of . Note that is an increasing sequence in terms of and saturates to the real propagation speed. However, for , for any . In table I, We also compute the upper bounds and the lower bound for each . We observe that becomes tighter (to with large ) by increasing . seems to be larger than the real speed by an almost constant factor which does not vary significantly with .
Now consider the asymptotically regular SCLDPC ensembles. We plot and for and for and in Fig. 4. The curves of are shown by the dashed lines. We compute in two distinct ways: first by the DE equation which are shown by solid lines. We observe that it is a decreasing function and becomes zero at . Since is an integer value and hence, it is not so sensitive to small changes in , is a staircase function. Second, we compute the average by running BP over 100 code instances. These curves are shown by dotted lines. As we expect, the empirical is close to derived by DE. However, the curves deviate from each other for close to the MAP threshold. The reason is due to the error floor that appears for such as a result of finite length effects (finite ). The dashed-dotted tail of each curve shows where the error-floor appears.
Figure 4 illustrates that the order of ensembles in terms of speed changes by . It suggests that for a desired , we can change the degree distributions in order to obtain a faster convergence speed. To be more fair, we compare two LDPC ensembles with the same average degree. In Fig. 5, we depict for the SCLDPC ensemble and for the regular SCLDPC ensemble. We observe that the former has much larger speed than the latter over a wide range of . For instance, the speed is about times larger at . However, the regular ensemble has a larger speed for very close to its .
These examples show the need for optimizing the speed for a given . For this purpose, one can use since it is tight for close to the MAP threshold.
Iv-a Other Types of BMS Channels
Consider the SCLDPC ensemble. Similar to the BEC, simulations results suggest that belief propagation also leads to a wave-like solution on the BSC and on the AWGN channel. In Fig. 6, we compare the empirical speeds for the BEC, BSC and AWGN channel. We compute for the channel entropy between BP threshold and MAP threshold in each case. Each point is averaged over 100 code instances. We refer to [11, Chapter 4] for the definition of channel entropy. As discussed before, we observe an error floor close to the MAP threshold due to finite length effects. The dashed tail of each curve indicates where the error-floor appears.
The BP threshold of the underlying ensemble is larger for the BEC than for the AWGN channel and it is furthermore larger for the AWGN channel than for the BSC. The MAP thresholds and the speed preserve the same order. Thus, we conjecture that the upper bound of the speed on BEC is also an upper bound of the speed on the other channels with the same channel entropy.
In this paper we have analyzed the convergence speed of the wave-like solution that dominates the convergence of spatially coupled codes if the channel entropy lies between the BP and MAP thresholds. Using the Taylor expansion of the spatially coupled code’s potential function, we have derived a new upper bound on the convergence speed. This upper bound is based on the degree distribution of the code ensemble and requires the wave-like DE solution. We have additionally derived a looser upper bound that solely depends on the potential function, i.e., on the degree distribution of the ensemble and on the coupling factor. Finally, we have compared the bound with the convergence speeds obtained by density evolution and by simulation of code instances. We have observed that the upper bound seems to be an upper bound for any binary input memoryless symmetric channel.
Appendix A Sketch of Proof of Lemma 3
where the last inequality is obtained because for all . Now consider to be for . Thus,
We write the second order Taylor expansion of around :
where are defined in what follows. We have
is the remainder term. According to Taylor’s theorem, there exists for some that
where The partial derivatives , denoted by , are given in equations (11) to (15). By rearranging the terms, we can write in the form of (16) in which there are two positive terms and two negative terms. Now, we show for a large enough ,
We know that and then,
We have the result if we show that
For each , the second term and the third term of this expression are negative. Furthermore, first term and the last term of in (16) are also negative (note that ). Finally, we consider the positive terms of , which are, for each ,
Now compare (18) and
For a large the above term dominates (18) since