一种应用于LTE系统的Viterbi译码算法

LTE(long term evolution,长期演进)系统中采用了咬尾卷积码和Turbo码来实现前向纠错,Viterbi译码是卷积码的一种杰出的译码算法,它是一种最大似然译码方法。本文基于LTE系统中的咬尾卷积码,详细分析了几种较成熟的Viterbi译码算法,并综合现有算法,提出了一种改进算法,减小了译码计算的复杂度。仿真结果表明,改进算法在降低译码计算复杂度的同时还降低了译码误比特率,因此非常适合LTE系统的译码要求。     

Implementation of APP decoders on dual code trellis

For high rate k/n convolutional codes (k/n > 0.5), a trellis based implementation of a posteriori probability (APP) decoders is less complex on the dual code trellis owing to its branch complexity (2^n-k ) being lower than the code trellis (2^k). The log scheme used for APP decoders is not attractive for practical implementation owing to heavy quantisation requirements. As an alternative, presented is an arc hyperbolic tangent (AHT) scheme for implementing the dual- APP decoder. The trellis based implementation of this AHT dual APP decoder is discussed and some fundamental differences between primal APP and dual APP decoders that have an effect on a quantised implementation are reported.     

Reduced complexity decoding of convolutional codes based on the M-algorithm and the minimal trellis

In this paper, we propose three new sub-optimum, reduced complexity decoding algorithms for convolutional codes. The algorithms are based on the minimal trellis representation for the convolutional code and on the M-algorithm. Since the minimal trellis has a periodically time-varying state profile, each algorithm has a different strategy to select the number of surviving states in each trellis depth. We analyse both the computational complexity, in terms of arithmetic operations, and the bit error rate performance of the proposed algorithms over the additive white Gaussian noise channel. Results demonstrate that considerable complexity reductions can be obtained at the cost of a small loss in the performance, as compared to the Viterbi decoder.     

Two decoding algorithms for tailbiting codes

The paper presents two efficient Viterbi decoding-based suboptimal algorithms for tailbiting codes. The first algorithm, the wrap-around Viterbi algorithm (WAVA), falls into the circular decoding category. It processes the tailbiting trellis iteratively, explores the initial state of the transmitted sequence through continuous Viterbi decoding, and improves the decoding decision with iterations. A sufficient condition for the decision to be optimal is derived. For long tailbiting codes, the WAVA gives essentially optimal performance with about one round of Viterbi trial. For short- and medium-length tailbiting codes, simulations show that the WAVA achieves closer-to-optimum performance with fewer decoding stages compared with the other suboptimal circular decoding algorithms. The second algorithm, the bidirectional Viterbi algorithm (BVA), employs two wrap-around Viterbi decoders to process the tailbiting trellis from both ends in opposite directions. The surviving paths from the two decoders are combined to form composite paths once the decoders meet in the middle of the trellis. The composite paths at each stage thereafter serve as candidates for decision update. The bidirectional process improves the error performance and shortens the decoding latency of unidirectional decoding with additional storage and computation requirements. Simulation results show that both proposed algorithms effectively achieve practically optimum performance for tailbiting codes of any length.     

Design and decoding of optimal high-rate convolutional codes

This correspondence deals with the design and decoding of high-rate convolutional codes. After proving that every (n,n-1) convolutional code can be reduced to a structure that concatenates a block encoder associated to the parallel edges with a convolutional encoder defining the trellis section, the results of an exhaustive search for the optimal (n,n-1) convolutional codes is presented through various tables of best high-rate codes. The search is also extended to find the "best" recursive systematic convolutional encoders to be used as component encoders of parallel concatenated "turbo" codes. A decoding algorithm working on the dual code is introduced (in both multiplicative and additive form), by showing that changing in a proper way the representation of the soft information passed between constituent decoders in the iterative decoding process, the soft-input soft-output (SISO) modules of the decoder based on the dual code become equal to those used for the original code. A new technique to terminate the code trellis that significantly reduces the rate loss induced by the addition of terminating bits is described. Finally, an inverse puncturing technique applied to the highest rate "mother" code to yield a sequence of almost optimal codes with decreasing rates is proposed. Simulation results applied to the case of parallel concatenated codes show the significant advantages of the newly found codes in terms of performance and decoding complexity.     

Decoding of High Rate Convolutional Codes Using the Dual Trellis

This paper deals with a posteriori probability (APP) decoding of high-rate convolutional codes, using the dual code's trellis. After deriving the dual APP (DAPP) algorithm from the APP relation, its trellis-based implementation is addressed. The challenge involved in practical implementation of a DAPP decoder is then highlighted. Metric representation schemes similar to the log domain used for log-APP decoding are shown to be unattractive for DAPP decoding due to quantization requirements. After explaining the nature of the DAPP metrics, an arc hyperbolic tangent (AHT) scheme is proposed and its equivalent arithmetic operations derived. By using an efficient approximation, an addition is translated to an addition in the AHT domain. Efficient techniques for normalization and extrinsic log-likelihood ratio (LLR ) calculation are presented which reduce implementation complexity significantly. Simulation results with different high-rate codes are given to show that the AHT-DAPP decoder performs similarly to a log-APP decoder and at the same time performs better than a decoder for a punctured code. A fully fixed-point model of an AHT-DAPP decoder is shown to perform close to an optimum decoder. The decoding complexity of the log-APP and AHT-DAPP decoders are listed and compared for several rate-k/(k+1) codes. It is shown that an AHT-DAPP decoder starts to be less complex from a code rate of 7/8 . When compared against a max-log-APP decoder, the AHT-DAPP decoder is less complex at a code rate of 9/10 and above.     

Coset codes viewed as terminated convolutional codes

Coset codes are considered as terminated convolutional codes. Based on this approach, three new general results are presented. First, it is shown that the iterative squaring construction can equivalently be defined from a convolutional code whose trellis terminates. This convolutional code determines a simple encoder for the coset code considered, and the state and branch labelings of the associated trellis diagram become straightforward. Also, from the generator matrix of the code in its convolutional code form, much information about the trade-off between the state connectivity and complexity at each section, and the parallel structure of the trellis, is directly available. Based on this generator matrix, it is shown that the parallel branches in the trellis diagram of the convolutional code represent the same coset code C₁ of smaller dimension and shorter length. Utilizing this fact, a two-stage optimum trellis decoding method is devised. The first stage decodes C₁ while the second stage decodes the associated convolutional code, using the branch metrics delivered by stage 1. Finally, a bidirectional decoding of each received block starting at both ends is presented. If about the same number of computations is required, this approach remains very attractive from a practical point of view as it roughly doubles the decoding speed. This fact is particularly interesting whenever the second half of the trellis is the mirror image of the first half, since the same decoder can be implemented for both parts     

Low-complexity ML decoding for convolutional tail-biting codes

Recently, a maximum-likelihood (ML) decoding algorithm with two phases has been proposed for convolutional tailbiting codes [1]. The first phase applies the Viterbi algorithm to obtain the trellis information, and then the second phase employs the algorithm A* to find the ML solution. In this work, we improve the complexity of the algorithm A* by using a new evaluation function. Simulations showed that the improved A* algorithm has over 5 times less average decoding complexity in the second phase when E_b/N₀? 4 dB.     

Boosting the error performance of suboptimal tailbiting decoders

Tailbiting is an attractive method to terminate convolutional codes without reducing the code rate. Maximum-likelihood and exact a posteriori probability decoding of tailbiting codes implies, however, a large computational complexity. Therefore, suboptimal decoding methods are often used in practical coding schemes. It is shown that suboptimal decoding methods work better when the slope of the active distances of the generating convolutional encoder is large. Moreover, it is shown that considering quasi-cyclic shifts of the received channel output can increase the performance of suboptimal tailbiting decoders. The findings are most relevant to tailbiting codes where the number of states is not small relative to the block length.     

A new high rate code scheme with highly parallel and low complexity decoding algorithm

A new high rate code scheme is proposed in this paper. It consists of serial concatenated recursive systematic ordinary (nonpunctured) convolutional codes with only 8 states in the trellis of the corresponding reciprocal dual codes. With a low complexity and highly parallel decoding algorithm, over additive white Gaussian noise channels, the proposed codes can achieve good bit error rate (BER) performance comparable to that of turbo codes and low density parity check (LDPC) codes. At code rate R=16/17, the overall decoding complexity of the proposed code scheme is almost half that of the LDPC codes.     

An efficiently implementable maximum likelihood decoding algorithm for tailbiting codes

Convolutional tailbiting codes are widely used in mobile systems to perform error-correcting strategies of data and control information. Unlike zero tail codes, tailbiting codes do not reset the encoder memory at the end of each data block, improving the code efficiency for short block lengths. The objective of this work is to propose a low-complexity maximum likelihood decoding algorithm for convolutional tailbiting codes based on the Viterbi algorithm. The performance of the proposed solution is compared to that of another maximum likelihood decoding strategy which is based on the A* algorithm. The computational load and the memory requirements of both algorithms are also analysed in order to perform a fair comparison between them. Numerical results considering realistic transmission conditions show the lower memory requirements of the proposed solution, which makes its implementation more suitable for devices with limited resources.     

Symbol-by-symbol MAP decoding algorithm for high-rate convolutionalcodes that use reciprocal dual codes

A symbol-by-symbol maximum a posteriori (MAP) decoding algorithm for high-rate convolutional codes applying reciprocal dual convolutional codes is presented. The advantage of this approach is a reduction of the computational complexity since the number of codewords to consider is decreased. All requirements for iterative decoding schemes are fulfilled. Since tail-biting convolutional codes are equivalent to quasi-cyclic block codes, the decoding algorithm for truncated or terminated convolutional codes is modified to obtain a soft-in/soft-out decoder for high-rate quasi-cyclic block codes which also uses the dual code because of complexity reasons. Additionally, quasi-cyclic block codes are investigated as component codes for parallel concatenation. Simulation results obtained by iterative decoding are compared with union bounds for maximum likelihood decoding. The results of a search for high-rate quasi-cyclic block codes are given in the appendix     

On error exponents for woven convolutional codes with one tailbiting component code

An error exponent for woven convolutional codes (WCC) with one tailbiting component code is derived. This error exponent is compared with that of the original WCC. It is shown that for WCC with outer warp, a better error exponent is obtained if the inner code is terminated with the tailbiting method. Furthermore, it is shown that the decoding error probability decreases exponentially with the square of the memory of the constituent convolutional encoders, while the decoding complexity grows exponentially only with the memory.     

List Viterbi algorithms for continuous transmission

The conventional list Viterbi algorithm (LVA) produces a list of the L best output sequences over a certain block length in decoding a terminated convolutional code. We show in this paper that the LVA with a sufficiently long list is an optimum maximum-likelihood decoder for the concatenated pair of a convolutional code and a cyclic redundancy check (CRC) block code with error detection. The CRC is used to select the output. New LVAs for continuous transmission are proposed and evaluated, where no termination bits are required for the convolutional code for every CRC block. We also present optimum and suboptimum LVAs for tailbiting convolutional codes. Convolutional codes with Viterbi decoding were proposed for so-called hybrid in band on channel (hybrid IBOC) systems for digital audio broadcasting compatible with the frequency modulation band. For high-quality audio signals, it is beneficial to use error concealment/error mitigation techniques to avoid the worst type of channel errors. This requires a reliable error flag mechanism (error detection feature) in the channel decoder. A CRC on a block of audio information bits provides this mechanism. We demonstrate how the LVA can significantly reduce the flag rate compared to the regular Viterbi algorithm (VA) for the same transmission parameters. At the expense of complexity, a receiver optional LVA can reduce the flag rate by more than an order of magnitude. The difference in audio quality is dramatic. The LVA is backward compatible with a VA     

Convolutional Codes Under a Minimal Trellis Complexity Measure

We conduct a code search, restricted to the recently introduced class of generalized punctured convolutional codes, under the minimal trellis complexity measure defined by McEliece and Lin. For the same decoding complexity and the same code rate, new codes are compared to well-known existing classes of convolutional codes. Some of the best convolutional codes (in a distance spectrum sense) of existing and new trellis complexities are tabulated.     

Multilevel trellis MPSK modulation codes for the Rayleigh fadingchannel

The multilevel coding technique is used for constructing multilevel trellis M-ary phase-shift-keying (MPSK) modulation codes for the Rayleigh fading channel. In the construction of a code, all the factors which affect the code performance and its decoding complexity are considered. The error performance of some of these codes based on both one-stage optimum decoding and multistage suboptimum decoding has been simulated. The simulation results show that these codes achieve good error performance with small decoding complexity     

Two step SOVA-based decoding algorithm for tailbiting codes

In this work we propose a novel decoding algorithm for tailbiting convolutional codes and evaluate its performance over different channels. The proposed method consists on a fixed two-step Viterbi decoding of the received data. In the first step, an estimation of the most likely state is performed based on a SOVA decoding. The second step consists of a conventional Viterbi decoding that employs the state estimated in the previous step as the initial and final states of the trellis. Simulations results show a performance close to that of maximum-likelihood decoding.     

Graph-based iterative decoding algorithms for parity-concatenatedtrellis codes

We construct parity-concatenated trellis codes in which a trellis code is used as the inner code and a simple parity-check code is used as the outer code. From the Tanner-Wiberg-Loeliger (1981, 1996) graph representation, several iterative decoding algorithms can be derived. However, since the graph of the parity-concatenated code contains many short cycles, the conventional min-sum and sum-product algorithms cannot achieve near-optimal decoding. After some simple modifications, we obtain near-optimal iterative decoders. The modifications include either (a) introducing a normalization operation in the min-sum and sum-product algorithms or (b) cutting the short cycles which arise in the iterative Viterbi algorithm (IVA). After modification, all three algorithms can achieve near-optimal performance, but the IVA has the least average complexity. We also show that asymptotically maximum-likelihood (ML) decoding and a posteriori probability (APP) decoding can be achieved using iterative decoders with only two iterations. Unfortunately, this asymptotic behavior is only exhibited when the bit-energy-to-noise ratio is above the cutoff rate. Simulation results show that with trellis shaping, iterative decoding can perform within 1.2 dB of the Shannon limit at a bit error rate (BER) of 4×10^-5 for a block size of 20000 symbols. For a block size of 200 symbols, iterative decoding can perform within 2.1 dB of the Shannon limit     

Comparison of Convolutional and Turbo Coding for Broadband FWA Systems

It has been demonstrated that turbo codes substantially outperform other codes, e.g., convolutional codes, both in the non-fading additive white Gaussian noise (AWGN) channel as well as multiple-transmit and multiple-receive antenna fading channels. Moreover, it has also been reported that turbo codes perform very well in fast fading channels, but perform somewhat poorly on slow and block fading channels of which the broadband fixed wireless access (FWA) channel is an example. In this paper, we thoroughly compare the performance of turbo-coded and convolutional-coded broadband FWA systems both with and without antenna diversity under the condition of identical complexity for a variety of decoding algorithms. In particular, we derive mathematical expressions to characterize the complexity of turbo decoding based on state-of-the-art Log-MAP and Max-Log-MAP algorithms as well as convolutional decoding based on the Viterbi algorithm in terms of the number of equivalent addition operations. Simulation results show that turbo codes do not offer any performance advantage over convolutional codes in FWA systems without antenna diversity or FWA systems with limited antenna diversity. Indeed, turbo codes only outperform convolutional codes in FWA systems having significant antenna diversity.     

Parallel high-throughput limited search trellis decoder VLSI design

Limited search trellis decoding algorithms have great potentials of realizing low power due to their largely reduced computational complexity compared with the widely used Viterbi algorithm. However, because of the lack of operational parallelism and regularity in their original formulations, the limited search decoding algorithms have been traditionally ruled out for applications demanding very high throughput. We believe that, through appropriate algorithm and hardware architecture co-design, certain limited search trellis decoding algorithms can become serious competitors to the Viterbi algorithm for high-throughout applications. Focusing on the well-known T-algorithm, this paper presents techniques at the algorithm and VLSI architecture levels to design fully parallel T-algorithm limited search trellis decoders. We first develop a modified T-algorithm, called SPEC-T, to improve the algorithmic parallelism. Then, based on the conventional state-parallel register exchange Viterbi decoder, we develop a parallel SPEC-T decoder architecture that can effectively transform the reduced computational complexity at the algorithm level to the reduced switching activities in the hardware. We demonstrate the effectiveness of the SPEC-T design solution in the context of convolutional code decoding. Compared with state-parallel register exchange Viterbi decoders, the SPEC-T convolutional code decoders can achieve almost the same throughput and decoding performance, while realizing up to 56% power savings. For the first time, this work provides an approach to exploit the low power potential of the T-algorithm in very high throughput applications.