In this paper, we present a sparse neural network decoder (SNND) of polar codes based on belief propagation (BP) and deep learning. At first, the conventional factor graph of polar BP decoding is converted to the bipartite Tanner graph similar to low-density parity-check (LDPC) codes. Then the Tanner graph is unfolded and translated into the graphical representation of deep neural network (DNN). The complex sum-product algorithm (SPA) is modified to min-sum (MS) approximation with low complexity. We dramatically reduce the number of weight by using single weight to parameterize the networks. Optimized by the training techniques of deep learning, proposed SNND achieves comparative decoding performance of SPA and obtains about 0.5 dB gain over MS decoding on (128,64) and (256,128) codes. Moreover, 60% complexity reduction is achieved and the decoding latency is significantly lower than the conventional polar BP.
In this paper, we present a sparse neural network decoder (SNND) of polar codes based on belief propagation (BP) and deep learning. At first, the conventional factor graph of polar BP decoding is converted to the bipartite Tanner graph similar to low-density parity-check (LDPC) codes. Then the Tanner graph is unfolded and translated into the graphical representation of deep neural network (DNN). The complex sum-product algorithm (SPA) is modified to min-sum (MS) approximation with low complexity. We dramatically reduce the number of weight by using single weight to parameterize the networks. Optimized by the training techniques of deep learning, proposed SNND achieves comparative decoding performance of SPA and obtains about 0.5 dB gain over MS decoding on (128,64) and (256,128) codes. Moreover, 60% complexity reduction is achieved and the decoding latency is significantly lower than the conventional polar BP.
Compared with binary Polar codes, non-binary Polar codes exhibit better performance, which is suitable for short packet services of Internet of Things. But its decoding complexity is high(O(2Nlog2(N)), whereNis the length of the symbol sequence). To make it practical, this paper proposes a sparse neural network (SNN) decoder. Firstly, the Polar factor graph is transformed into a bipartite graph which contains variable node and check node. Then, we propose a pruning algorithm to prune the bipartite graph so as to get a sparse graph. After that, the sparse graph is expanded to get the structure of the deep neural network. Optimized by the training of deep learning, the proposed SNN decoding scheme has competitive performance to the traditional Belief-Propagation (BP) decoding scheme. Moreover, the SNN decoder achieves a reduction of 51.7% in the number of connections compared with BP decoding scheme and its decoding latency is significantly lower than the traditional BP.
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional well-established LDPC decoding, e.g., using a fully parallel sum-product algorithm (SPA). This facilitates the systematic analysis and design of polar codes using the well-established tools known from analyzing LDPC codes. We show that the proposed iterative polar decoder has a negligible performance loss for short-to-intermediate codelengths compared to Arikan's original BP decoder. Finally, the proposed decoder is shown to benefit from both reduced complexity and reduced memory requirements and, thus, is more suitable for hardware implementations.
Deep learning recently shows strong competitiveness to improve polar code decoding. However, suffering from prohibitive training and computation complexity, the conventional deep neural network (DNN) is only possible for very short code length. In this paper, the main problems of deep learning in decoding are well solved. We first present the multiple scaled belief propagation (BP) algorithm, aiming at obtaining faster convergence and better performance. Based on this, deep neural network decoder (NND) with low complexity and latency, is proposed for any code length. The training only requires a small set of zero codewords. Besides, its computation complexity is close to the original BP. Experiment results show that the proposed (64,32) NND with 5 iterations achieves even lower bit error rate (BER) than the 30-iteration conventional BP and (512, 256) NND also outperforms conventional BP decoder with same iterations. The hardware architecture of basic computation block is given and folding technique is also considered, saving about 50% hardware cost.
For the scenarios with high throughput requirements, the belief propagation (BP) decoding is one of the most promising decoding strategies for polar codes. By pruning the redundant variable nodes (VNs) and check nodes (CNs) in the original factor graph, the graph is condensed to a sparse bipartite graph which is similar to the graph for low-density parity-check (LDPC) codes. In this paper, we introduce the bit-flipping scheme into the LDPC-like BP (L-BP) decoding and propose two methods to identify the error-prone VNs. By additional decoding attempts, the L-BP flip (L-BPF) decoding improves the error-rate performance with a similar average complexity for high Eb=N0values. The simulation results show that the L-BPF decoding achieves 0:25 dB gain compared with the L-BP decoding.
This paper presents deep learning (DL) methods to optimize polar belief propagation (BP) decoding and concatenated LDPC-polar codes. First, two-dimensional offset Min-Sum (2-D OMS) decoding is proposed to improve the error-correction performance of existing normalized Min-Sum (NMS) decoding. Two optimization methods used in DL, namely back-propagation and stochastic gradient descent, are exploited to derive the parameters of proposed algorithms. Numerical results demonstrate that there is no performance gap between 2-D OMS and exact BP on various code lengths. Then the concatenated OMS algorithms with low complexity are presented for concatenated LDPC-polar codes. As a result, the optimized concatenated OMS decoding yields error-correction performance with CRC-aided successive cancellation list (CA-SCL) decoder of list size 2 on length-1024 polar codes. In addition, the efficient hardware architectures of scalable polar OMS decoder are described and the proposed decoder is reconfigurable to support three code lengths (N = 256, 512, 1024) and two decoding algorithms (2-D OMS and concatenated OMS). The polar OMS decoder implemented on 65 nm CMOS technology achieves a maximum coded throughput of 5.4 Gb/s at Eb/N0= 4 dB for code length 1024 and 7.5 Gb/s at Eb/N0= 3.5 dB for code length 256, which are comparable to the state-of-the-art polar BP decoders. Moreover, a 5.1 Gb/s throughput at Eb/N0= 4 dB is achieved under concatenated OMS decoding mode for code length 1024 with a latency of 200 ns, which is superior to existing CA-SCL decoders that have similar error-correction performance.
The recent development of deep learning methods provides a new approach to optimize the belief propagation (BP) decoding of linear codes. However, the limitation of existing works is that the scale of neural networks increases rapidly with the codelength, thus they can only support short to moderate codelengths. From the point view of practicality, we propose a high-performance neural min-sum (MS) decoding method that makes full use of the lifting structure of protograph low-density parity-check (LDPC) codes. By this means, the size of the parameter array of each layer in the neural decoder only equals the number of edge-types for arbitrary codelengths. In particular, for protograph LDPC codes, the proposed neural MS decoder is constructed in a special way such that identical parameters are shared by a bundle of edges derived from the same edge-type. To reduce the complexity and overcome the vanishing gradient problem in training the proposed neural MS decoder, an iteration-by-iteration (i.e., layer-by-layer in neural networks) greedy training method is proposed. With this, the proposed neural MS decoder tends to be optimized with faster convergence, which is aligned with the early termination mechanism widely used in practice. To further enhance the generalization ability of the proposed neural MS decoder, a codelength/rate compatible training method is proposed, which randomly selects samples from a set of codes lifted from the same base code. As a theoretical performance evaluation tool, a trajectory-based extrinsic information transfer (T-EXIT) chart is developed for various decoders. Both T-EXIT and simulation results show that the optimized MS decoding can provide faster convergence and up to 1dB gain compared with the plain MS decoding and its variants with only slightly increased complexity. In addition, it can even outperform the sum-product algorithm for some short codes.