Spatially coupled codes

Spatial coupling is a technique introduced by Kudekar et. al. to explain the performance of convolutional LDPC codes. In this article we only consider transmission over a binary erasure channel using $(l,r)$ regular LDPC codes as the base ensemble. The MAP threshold of the ensemble can be computed from the BP GEXIT curve using the area theorem. It is known that the MAP threshold of the $(l,r)$ ensemble approaches the channel capacity with increasing $l$. Spatial coupling allows us to construct ensembles from the base ensemble, whose BP and MAP thresholds are close to the MAP threshold of the underlying ensemble. This phenomenon is called threshold saturation. So spatially coupled codes can be used to transmit information at rates arbitrarly close to capacity for the BEC. Threshold saturation has also been observed for more general channels.

Introduction
Spatially coupled codes provide an entirely new way to achieve channel capacity. Unlike other capacity achieving LDPC ensembles, e.g. optimized irregular ensembles, irregular-repeat accumulate (IRA) or accumulate-repeat-accumulate (ARA) LDPC codes, spatially coupled codes can be easily designed to avoid error floors. In other words, they achieve capacity with large minimum distance.

The $(l,r,L)$ ensemble




Rate Loss
The design rate of the ensemble $(l,r,L,\gamma)$ with $\gamma\leq 2L$ is given by

\begin{equation} R(l,r,L,\gamma)=1-\frac{l}{r}-\frac{l}{r}\frac{\gamma+1-2\sum_{i=0}^{\gamma}(\frac{i}{\gamma})^{r}}{2L+1}. \end{equation}

Hence, threshold saturation comes at the cost of a reduction in the rate with respect to the underlying regular $(l,r)$ LDPC ensemble.

Threshold Saturation over the BEC with BP decoding
\begin{equation} \lim_{\gamma\rightarrow\infty}\lim_{L\rightarrow\infty}\epsilon^{\text{BP}}(l,r,L,\gamma)=\epsilon^{\text{MAP}}(l,r) \end{equation}

Finite-length Scaling Behavior
Density evolution analysis of spatially coupled codes typically assumes that the chain-length $L$ is kept fixed while the lifting factor $M$ tends to infinity. In practice, we are interested in optimizing the performance for finite $L$ and $M$. To first order, one might wonder for which scaling $L=f(M)$ we optimize the asymptotic performance. Empirical observations indicate that the threshold saturation phenomenon happens even when the number of sections grows considerably faster than $M$, which suggests that the threshold saturation phenomenon is very robust.

Windowed Decoding
The good performance of spatially coupled codes is apparent when both the lifting factor $M$ and the coupling length $L$ become large. However, as either of these parameters becomes large, belief propagation (BP) decoding becomes complex. The windowed decoder (WD) exploits the structure of coupled codes to reduce the decoding complexity while maintaining the advantages of the BP decoder in terms of performance. An additional advantage of the windowed decoder is the reduced latency of decoding. In a nutshell, the WD performs BP over a subgraph consisting of $W$ sections of the spatially coupled code.

Consider windowed decoding of the $(l, r, L, \gamma)$ spatially coupled ensemble over the binary erasure channel. Then for a target fraction of unrecovered variables $\delta < \delta_{*}$, there exists a positive integer $W_{\min}$ such that when the window size $W>W_{\min}$ the WD threshold satisﬁes:

\begin{equation} \epsilon^{\mathrm{WD}}(l, r, \gamma, W, \delta) \geq \Big(1 - \frac{lr}{2}\delta^{\frac{l-2}{l-1}}\Big)\Big(\epsilon^{\mathrm{BP}}(l, r, \gamma) - e^{-\frac{1}{b}(\frac{W}{\gamma - 1} - a\ln\ln\frac{d}{\delta} - c)}\Big) \end{equation}

where $a$, $b$, $c$, $d$ and $\delta_{*}$ are strictly positive constants that depend only on code parameters $l,r$ and $\gamma$. This implies that the threshold obtained with windowed decoding approaches the BP threshold at least exponentially in the window size.