LP-decodable multipermutation codes

In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that may consist of duplicate entries. We first introduce a new class of matrices called multipermutation matrices. We characterize the convex hull of multipermutation matrices. Based on this characterization, we propose a new class of codes that we term LP-decodable multipermutation codes. Then, we derive two LP decoding algorithms. We first formulate an LP decoding problem for memoryless channels. We then derive an LP algorithm that minimizes the Chebyshev distance. Finally, we show a numerical example of our algorithm.Comment: This work was supported by NSF and NSERC. To appear at the 2014 Allerton Conferenc

Decomposition Methods for Large Scale LP Decoding

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory, specifically the Alternating Directions Method of Multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a "two-slice" characterization of the geometry of the parity polytope, which is the convex hull of all codewords of a single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by ADMM and allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher signal-to-noise ratio than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the 49th Annual Allerton Conference, September 2011. This version to appear in IEEE Transactions on Information Theor

Enhancing the Anti-Dispersion Capability of the AO-OFDM System via a Well-Designed Optical Filter at the Transmitter

This paper proposes a novel method to improve the anti-dispersion ability of the all-optical orthogonal frequency division multiplexing (AO-OFDM) system. By replacing the Sinc-shaped filter with a Gauss-shaped filter for sub-carrier generation and inserting a cyclic prefix (CP), the impact of dispersion on the system can be significantly mitigated. Formula derivation and numerical analysis of the pulse-shaping function of the AO-OFDM system in the time domain for each cycle indicated that the pulse-shaping function generated by the Gauss-shaped filter was less affected by the dispersion effect than that of the Sinc-shaped filter. Meanwhile, less inter-carrier crosstalk between carriers was also observed. After carrying out system transmission simulations employing these two different filters, we found that the AO-OFDM system based on the Gauss-shaped filter could greatly improve the anti-dispersion ability compared with the system based on a Sinc-shaped filter. When the parameter settings in both schemes were identical, that is, the number of subcarriers was 32 and the power of a single subcarrier was −13 dBm, the bit error rate (BER) of the system based on the proposed Gauss-shaped filter after 60 km SMF transmission was only 1.596 × 10−3, while the BER of the traditional Sinc-shaped filter based system scheme was as high as 8.545 × 10−2

Blind Modulation Format Identification Based on Principal Component Analysis and Singular Value Decomposition

As optical networks evolve towards flexibility and heterogeneity, various modulation formats are used to match different bandwidth requirements and channel conditions. For correct reception and efficient compensation, modulation format identification (MFI) becomes a critical issue. Thus, a novel blind MFI method based on principal component analysis (PCA) and singular value decomposition (SVD) is proposed. Based on square operation and PCA, the influence of phase rotation is removed, which avoids phase rotation-related discussions and training. By performing SVD on the density matrix about constellation, a denoise method is implemented and the quality of the constellation is improved. In the subsequent processing, the denoised density matrix is used as the feature of the support vector machine (SVM), and the identification of seven modulation formats such as BPSK, QPSK, 8PSK, 8QAM, 16QAM, 32QAM and 64QAM is realized. The results show that lower OSNR values are required for the 100% accurate identification of all modulation formats to be achieved, which are 5 dB, 7 dB, 8 dB, 11 dB, 14 dB, 14 dB and 15 dB. Moreover, the proposed method still retains the advantage, even when the number of samples decrease, which is beneficial for low-complexity implementation

Blind Modulation Format Identification Based on Principal Component Analysis and Singular Value Decomposition

As optical networks evolve towards flexibility and heterogeneity, various modulation formats are used to match different bandwidth requirements and channel conditions. For correct reception and efficient compensation, modulation format identification (MFI) becomes a critical issue. Thus, a novel blind MFI method based on principal component analysis (PCA) and singular value decomposition (SVD) is proposed. Based on square operation and PCA, the influence of phase rotation is removed, which avoids phase rotation-related discussions and training. By performing SVD on the density matrix about constellation, a denoise method is implemented and the quality of the constellation is improved. In the subsequent processing, the denoised density matrix is used as the feature of the support vector machine (SVM), and the identification of seven modulation formats such as BPSK, QPSK, 8PSK, 8QAM, 16QAM, 32QAM and 64QAM is realized. The results show that lower OSNR values are required for the 100% accurate identification of all modulation formats to be achieved, which are 5 dB, 7 dB, 8 dB, 11 dB, 14 dB, 14 dB and 15 dB. Moreover, the proposed method still retains the advantage, even when the number of samples decrease, which is beneficial for low-complexity implementation