Jason Bellorado

A Low-Complexity Method for Chase-Type Decoding of Reed-Solomon Codes

In this work we present a low-complexity implementation of Chase-type decoding of Reed-Solomon codes. In such, we first use the soft-information available at the channel output to construct a test-set of 2eta vectors, equivalent in all except the eta Lt eta least reliable coordinate positions. We then give an interpolation procedure to construct a set of 2eta bivariate polynomials, with the roots of each specified by its corresponding test-vector. Here, test-vector similarity is exploited to share much of the required computation. Finally, we obtain the candidate message from the single z-linear factor of each bivariate polynomial. Although we provide an expression for the direct computation of each candidate message, the complexity of repeating this computation for each interpolation polynomial is prohibitive. We, thus, also present a reduced-complexity factorization (RCF) method to select a single polynomial that, with high probability, contains the correctly decoded message in its z-linear factor. Although suboptimal, the loss in performance of RCF decreases rapidly with increasing code length. We provide extensive simulation results showing that a significant performance increase over traditional hard-decision decoding is achievable with a comparable computational complexity (as implemented with the Berlekamp-Massey algorithm)

Low-Complexity Soft-Decoding Algorithms for Reed–Solomon Codes—Part I: An Algebraic Soft-In Hard-Out Chase Decoder

In this paper, we present an algebraic methodology for implementing low-complexity, Chase-type, decoding of Reed-Solomon (RS) codes of length n . In such, a set of 2 ¿ test-vectors that are equivalent on all except ¿ ¿ n coordinate positions is first produced. The similarity of the test-vectors is utilized to reduce the complexity of interpolation, the process of constructing a set of polynomials that obey constraints imposed by each test-vector. By first considering the equivalent indices, a polynomial common to all test-vectors is constructed. The required set of polynomials is then produced by interpolating the final ¿ dissimilar indices utilizing a binary-tree structure. In the second decoding step (factorization) a candidate message is extracted from each interpolation polynomial such that one may be chosen as the decoded message. Although an expression for the direct evaluation of each candidate message is provided, carrying out this computation for each polynomial is extremely complex. Thus, a novel, reduced-complexity, methodology is also given. Although suboptimal, simulation results affirm that the loss in performance incurred by this procedure is decreasing with increasing code length n, and negligible for long (n > 100) codes. Significant coding gains are shown to be achievable over traditional hard-in hard-out decoding procedures (e.g., Berlekamp-Massey) at an equivalent (and, in some cases, lower) computational complexity. Furthermore, these gains are shown to be similar to the recently proposed soft-in-hard-out algebraic techniques (e.g., Sudan, Ko¿tter-Vardy) that bear significantly more complex implementations than the proposed algorithm.

Approaching the capacity of the MIMO Rayleigh flat-fading channel with QAM constellations, independent across antennas and dimensions

In this study we consider the challenge of reliable communication over a wireless Rayleigh flat-fading channel using multiple transmit and receive antennas. Since modern digital communication systems employ signal sets of finite cardinality, we examine the use of the quadrature amplitude modulation (QAM) constellation to approach the capacity of this channel. By restricting the channel input to the M-QAM subset of the complex-plane, the maximum achievable information rate (CM-QAM ) is strictly bounded away from the channel capacity (C). We utilize a modified version of the Arimoto-Blahut algorithm to determine CM-QAM and the probability distribution over the channel input symbols that achieves it. The results of this optimization procedure numerically indicate that the optimal input symbol distribution factors into the product of identical distributions over each real dimension of the transmitted signal. This is shown to vastly reduce the computational complexity of the optimization algorithm. Furthermore, we utilize the computed optimal channel input probability mass function (pmf) to construct capacity approaching trellis codes. These codes are implemented independent across all antennas and symbol dimensions and, if used as inner codes to outer low-density parity check (LDPC) codes, can achieve arbitrarily small error rates at signal-to-noise ratios very close to the channel capacity CM-QAM . Examples are given for a 2-transmit/2-receive antenna (2 times 2) system

Low-Complexity Soft-Decoding Algorithms for Reed–Solomon Codes—Part II: Soft-Input Soft-Output Iterative Decoding

In this paper, we present a practical approach to the iterative decoding of Reed-Solomon (RS) codes. The presented methodology utilizes an architecture in which the output produced by steps of belief-propagation (BP) is successively applied to a legacy decoding algorithm. Due to the suboptimal performance of BP conducted on the inherently dense RS parity-check matrix, a method is first provided for the construction of reduced-density, binary, parity-check equations. Iterative decoding is then conducted utilizing a subset of a redundant set of parity-check equations to minimize the number of connections into the least-reliable bits. Simulation results show that performance comparable to (and exceeding) the best known practical RS decoding techniques is achievable with the presented methodology. The complexity of the proposed algorithm is significantly lower than these existing procedures and permits a practical implementation in hardware.

Time-hopping sequence design for narrowband interference suppression

In this paper, we present a simple interference mitigation solution for the coexistence of ultra-wideband (UWB) systems with other wireless systems. Specifically, we consider the design of time-hopping (TH) codes for UWB systems that deploy an impulse radio architecture. We give a methodology for the construction of TH sequences that minimize the imposed UWB interference on a given narrowband system. To illustrate the effectiveness of our designed codes, we conduct physical layer simulations of the interference induced by UWB signals on close proximity wireless local area networks (WLANs) that deploy the IEEE 802.11a standard. Our performance measure is the IEEE 802.11a client rate degradation vs. distance from its access point. Our results show that, when using optimized TH codes, UWB systems can have no impact on the achievable data-rates and range of IEEE 802.11a WLAN devices; regardless of the position of the UWB system with respect to these devices.

Approaching the capacity of the mimo rayleigh flat-fading channel with qam constellations, independent across antennas and dimensions

In this study we consider the challenge of reliable communication over a wireless Rayleigh flat-fading channel using multiple transmit and receive antennas. Since modern digital communication systems employ signal sets of finite cardinality, we examine the use of the Quadrature Amplitude Modulation (QAM) constellation to approach the capacity of this channel. By restricting the channel input to the M-QAM subset of the complex-plane, the maximum achievable information rate (CM-QAM) is strictly bounded away from the channel capacity (C). We utilize a modified version of the Arimoto-Blahut al- gorithm to determine CM-QAM and the probability distribution over the channel input symbols that achieves it. The results of this optimization procedure numerically indicate that the optimal input symbol distribution factors into the product of identical distributions over each real dimension of the transmitted signal. This is shown to vastly reduce the computational complexity of the optimization algorithm. Furthermore, we utilize the computed optimal channel input probability mass function (pmf) to construct capacity approaching trellis codes. These codes are implemented independent across all antennas and symbol dimensions and, if used as inner codes to outer low-density parity check (LDPC) codes, can achieve arbitrarily small error rates at signal-to-noise ratios very close to the channel capacity CM-QAM. Examples are given for a 2-transmit/2-receive antenna (2 × 2) system.