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Forward-backward algorithm, used by watermark decoder for correcting non-binary synchronization errors, requires to traverse a very large scale trellis in order to achieve the proper posterior probability, leading to high computational complexity. In order to reduce the number of the states involved in the computation, an adaptive pruning method for the trellis is proposed. In this scheme, we prune the states which have the low forward-backward quantities below a carefully-chosen threshold. Thus, a wandering trellis with much less states is achieved, which contains most of the states with quite high probability. Simulation results reveal that, with the proper scaling factor, significant complexity reduction in the forward-backward algorithm is achieved at the expense of slight performance degradation.

This letter presents a prediction algorithm that foresees the possibility of irreducible error floor for a particular design of a generator matrix G in space time trellis code. This is done without the need of complex and time-consuming simulation, which removes the demand of considerable computational resource. To note, the prediction achieves approximately 93% accuracy and reduces roughly 99% of the temporal cost of simulation. It capitalizes on error prone substructure, a well-known concept from low density parity check code to enable the predictive mechanism. This can be useful in code design where problematic generator matrices can be omitted from consideration.

Trellis codes, including Low-Density Parity-Check (LDPC), turbo, and convolutional code (CC), are widely adopted in advanced wireless standards to offer high-throughput forward error correction (FEC). Designing a multistandard FEC decoder is of great challenge. In this paper, a trellis application specified instruction-set processor (TASIP) is presented for multistandard trellis decoding. A unified forward–backward recursion kernel with an eight-state parallel trellis structure is proposed. Based on the kernel, a datapath for multialgorithm and a shared memory subsystem are introduced. The flexibility and the compatibility are guaranteed by a programmable decoding flow and the trellis decoding instruction set. Synthesis results show that the area consumption is $2.12~\text {mm}^{2}$ (65 nm). TASIP provides trimode FEC decoding ability with the throughput of 533, 186, and 225 Mb/s for LDPC, turbo, and 64 states CC under the clock frequency of 200 MHz, which outperforms other trimode proposals both in area efficiency and recursion efficiency. TASIP provides high-throughput decoding for current standards, including 3rd Generation Partnership Project-Long Term Evolution, 802.16e, and 802.11n, with unified architecture and high compatibility.

We present a novel class of space time trellis codes based on super quasi-orthogonal space time codes for a reconfigurable antenna system. We call them space time state trellis codes (STSTCs). We design for 4 trellis states and both 2 and 4 branch trellis structures for BPSK and QPSK constellations. The proposed codes take advantage of the additional degrees of diversity offered by reconfigurable antennas and changing propagation states. They achieve full rate, full diversity and high coding gain. Simulation results demonstrate their good performance. STSTCs achieve the diversity order expected of four transmit antennas while using only two reconfigurable transmit antennas. Our code out performs the existing super orthogonal space time trellis code, which fails to achieve full diversity in a reconfigurable antenna system. The proposed 4 trellis state STSTC is designed based on the following performance criteria: minimum coding gain distance of the minimum return path (CGD MRP), performance factor (PF), rank of the partial valid path (RPVP) on top of traditional minimum coding gain distance of parallel path (CGD PP). We show through analysis and simulations that minimum CGD PP is not a sufficient criteria and that additional performance gains are provided by considering the CGD MRP.

In this paper, we propose a new type of code called Trellis-based Quasi-Cyclic (TQC)-LDPC convolutional code, which is a special case of protograph-based LDPC convolutional codes. The proposed TQC-LDPC convolutional code can be derived from any QC-LDPC block code by introducing trellis-based convolutional dependency to the code. The main advantage of the proposed TQC-LDPC convolutional code is that it allows reduced decoder complexity and input granularity (which is defined as the minimum number of input information bits the code requires to generate a codeword) while maintaining the same bit error-rate as the underlying QC-LDPC block code ensemble. We also propose two related power-efficient encoding methods to increase the code rate of the derived TQC-LDPC convolutional code. The newly derived short constraint length TQC-LDPC convolutional codes enable low complexity trellis-based decoders and one such decoder is proposed and described in this paper (namely, QC-Viterbi). The TQC-LDPC convolutional codes and the QC-Viterbi decoder are compared to conventional LDPC codes and Belief Propagation (BP) iterative decoders with respect to bit-error-rate (BER), signal-to-noise ratio (SNR), and decoder complexity. We show both numerically and through hardware implementation results that the proposed QC-Viterbi decoder outperforms the BP iterative decoders by at least 1 dB for same complexity and BER. Alternatively, the proposed QC-Viterbi decoder has 3 times lower complexity than the BP iterative decoder for the same SNR and BER. This low decoding complexity, low BER, and fine granularity makes it feasible for the proposed TQC-LDPC convolutional codes and associated trellis-based decoders to be efficiently implemented in high data rate, next generation mobile systems.

The performance and complexity of turbo decoding using rate k/n constituent codes are investigated. The conventional, minimal and sectionalized trellis modules of the constituent convolutional codes are utilized. The performance metric is the bit error rate (BER), while complexity is analyzed based on the number of multiplications, summations and comparisons required by the max-log-MAP decoding algorithm. Our results show that the performance depends on how the systematic bits are grouped in a trellis module. The best performance is achieved when the k systematic bits are grouped together in the same section of the module, so that the log-likelihood ratio (LLR) of the k-bit vector is calculated at once. This is a characteristic of the conventional trellis module and of some of the sectionalizations of the minimal trellis module. Moreover, we show that it is possible to considerably reduce the decoding complexity with respect to the conventional trellis if a particular sectionalization of the minimal trellis module is utilized. In some cases, this sectionalization is found within the best performing group, while in some other cases a small performance loss can be traded off for a large complexity reduction.

In this paper, we present an improvement and a new implementation of a simplified decoding algorithm for non-binary low density parity-check codes (NB-LDPC) in Galois fields GF(q). The base algorithm that we use is the Extended Min-Sum (EMS) algorithm, which has been widely studied in the recent literature, and has been shown to approach the performance of the belief propagation (BP) algorithm, with limited complexity. In our work, we propose a new way to compute modified configuration sets, using a trellis representation of incoming messages to check nodes. We call our modification of the EMS algorithm trellis-EMS (T-EMS). In the T-EMS, the algorithm operates directly on the deviation space by considering a trellis built from differential messages, which serves as a new reliability measure to sort the configurations. We show that this new trellis representation reduces the computational complexity, without any performance degradation. In addition, we show that our modifications of the algorithm allows to greatly reduce the decoding latency, by using a larger degree of hardware parallelization.

In this paper, we propose a trellis-search based soft-input soft-output detection algorithm and its very large scale integration (VLSI) architecture for iterative multiple-input multiple-output (MIMO) receivers. We construct a trellis diagram to represent the search space of a transmitted MIMO signal. With the trellis model, we evenly distribute the workload of candidates searching among multiple trellis nodes for parallel processing. The search complexity is significantly reduced because the number of candidates is greatly limited at each trellis node. By leveraging the trellis structure, we develop an approximate Log-MAP algorithm by using a small list of largest exponential terms to compute the LLR (log-likelihood ratio) values. The trellis-search based detector has a fixed-complexity and is very suitable for parallel VLSI implementation. As a case study, we have designed and synthesized a trellis-search based soft-input soft-output MIMO detector for a 4 × 4 16-QAM system using a 1.08 V TSMC 65 nm technology. The detector can achieve a maximum throughput of 1.7 Gb/s with a core area of 1.58 mm2.

Spatial modulation (SM), in which multiple antennas are used to convey information besides the conventional M-ary signal constellations, is a new multiple-input multiple-output (MIMO) transmission technique, which has recently been proposed as an alternative to V-BLAST (vertical Bell Labs layered space-time). In this paper, a novel MIMO transmission scheme, called spatial modulation with trellis coding (SM-TC), is proposed. Similar to the conventional trellis coded modulation (TCM), in this scheme, a trellis encoder and an SM mapper are jointly designed to take advantage of the benefits of both. A soft decision Viterbi decoder, which is fed with the soft information supplied by the optimal SM decoder, is used at the receiver. A pairwise error probability (PEP) upper bound is derived for the SM-TC scheme in uncorrelated quasi-static Rayleigh fading channels. From the PEP upper bound, code design criteria are given and then used to obtain new 4-, 8- and 16-state SM-TC schemes using quadrature phase-shift keying (QPSK) and 8-ary phase-shift keying (8-PSK) modulations for 2,3 and 4 bits/s/Hz spectral efficiencies. It is shown via computer simulations and also supported by a theoretical error performance analysis that the proposed SM-TC schemes achieve significantly better error performance than the classical space-time trellis codes and coded V-BLAST systems at the same spectral efficiency, yet with reduced decoding complexity.

Two rate-one differential quasi-orthogonal space-frequency trellis codes (DQOSFTCs) for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) channels are proposed. The DQOSFTCs are systematically constructed within an OFDM symbol period, by combining unitary quasi-orthogonal trellis codes and differential modulation over the frequency domain (DF-QOSFTC) or time domain (DT-QOSFTCs). Besides multipath diversity, our DQOSFTCs achieve high-coding gain and simple decoding when the channel state information is not available at both the transmitter and the receiver. Simulation results show that our proposed codes significantly outperform the existing differential space-frequency trellis codes.

In this letter, we propose two full-rate space-time-frequency trellis code (STFTC) designs for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. The first proposal based on rotating constellation is called extended super-orthogonal STFTC (Ex-SOSTFTC). The second proposal, called Quasi-Orthogonal STFTCs (QOSTFTCs), combines set partitioning and the structure of quasi-orthogonal space-frequency designs in a systematic way. In addition to spatial diversity, the proposed codes provide multipath diversity and achieve high-coding gain over a frequency selective fading channel. Simulation results show the proposed codes significantly outperform the existing STFTC designs.

Space-time trellis code (STTC) has been widely applied to coded multiple-input multiple-output (MIMO) systems because of its gains in coding and diversity; however, its great decoding complexity makes it less promising in chip realization compared to the space-time block code (STBC). The complexity of STTC decoding lies in the branch metric calculation in the Viterbi algorithm and increases significantly along with the number of antennas and the modulation order. Consequently, a low-complexity algorithm to mitigate the computational burden is proposed. The results show that more than 70%, 78%, and 83% of the computational complexity is reduced for 2 × 2, 3 × 3, and 4 × 4 MIMO configurations, respectively. Based on the proposed algorithm, a reconfigurable MISO STTC Viterbi decoder is designed and implemented using 0.18 ?m 1P6M CMOS technology. The decoder achieves 11.14 Mbps, 8.36 Mbps, and 5.75 Mbps for 4-PSK, 8-PSK, and 16-QAM modulations, respectively.

This paper presents a technique to obtain an expurgated union bound on the frame error rate (FER) of space-time trellis codes (STTCs) on quasi-static Rician fading channels. We first present an algorithm to compute the distance spectrum of STTCs. The proposed algorithm iteratively manipulates the entries of the transition matrix of the product state transition diagram, which is convenient for symbolic computation. Then, a new method to identify the dominant quasi-static fading error events is proposed. Comparisons with simulated results reveal that the expurgated union bound on the FER (obtained using the limiting-before-average technique) using these dominant error events is tight.

In the wireless local area network (WLAN) standard IEEE 802.11b, complementary code keying (CCK) modulation has been adopted for the high data rate transmission mode. In this paper, complexity reduction for block reduced-state sequence estimation (bRSSE), tailored for CCK transmission over frequency-selective channels, is considered. A trellis diagram for the chip phases of the codewords fully describes the CCK code properties. Subset trellises are derived from the full CCK trellis diagram based on set partitioning of the multidimensional CCK code set. The CCK subset trellises connect consecutive bRSSE states forming a compound trellis. The Viterbi algorithm (VA) with per-survivor processing is applied to the compound trellis to take the inter-chip interference into account. Inter-codeword interference is also accounted for by state-dependent decision feedback. The resulting scheme is denoted as bRSSE-pS and has a significantly lower complexity than bRSSE with bruteforce search over the entire CCK code set. By introducing a sphere constraint on the overall decoding trellis (SC-bRSSEpS), the complexity of bRSSE-pS can be further reduced. Omitting states in the CCK subset trellises that violate the sphere constraint, edges that emanate from such states can be pruned, and the average number of metric calculations per CCK trellis segment can be reduced. All presented schemes are also specialized to the case of a single bRSSE trellis state, resulting in block decision-feedback equalization (bDFE) algorithms with per-survivor processing and sphere decoding (bDFE-pS and SC-bDFE-pS, respectively). Simulation results show that the performance of bRSSE-pS (bDFE-pS) and SC-bRSSE-pS (SCbDFE-pS), respectively, is essentially equivalent to that of bRSSE (bDFE) with brute-force search over the entire CCK code set, while complexity is significantly reduced.

A simple design method using trellis search is proposed for good low-density parity-check (LDPC) codes with relatively low code rates. By applying a trellis search technique to the design of a pre-assigned part of the parity-check matrix that allows a simple encoding, we improve the distribution of cycles formed by the entries contained in the parity-check part of the parity-check matrix. In addition, we extend the proposed algorithm to a class of structured LDPC codes, which have been recently preferred in many practical applications. Simulation results show that the codes designed by the proposed method outperform those constructed by conventionally used greedy design algorithms.

Space-time trellis coding is an established diversity technique that reduces the effects of multipath fading over wireless communication channels. Here, we consider high rate space-time trellis codes (STTC) with continuous phase frequency shift keying (CPFSK). We present optimized rate-2/3 STTC implemented with 3 transmit antennas. These codes provide system throughputs of 4 and 6 bits per channel use with 4-ary and 8-ary CPFSK respectively. Simulated error rate performance of the optimized codes with receive diversity is presented. We show that although the schemes do not achieve full transmit diversity, they provide excellent coding gains compared to full rank schemes that have equivalent throughput, but higher order modulations and greater complexity.

We present constructions of space-time (ST) codes based on lattice coset coding. First, we focus on ST code constructions for the short block-length case, i.e., when the block length is equal to or slightly larger than the number of transmit antennas. We present constructions based on dense lattice packings and nested lattice (Voronoi) shaping. Our codes achieve the optimal diversity-multiplexing tradeoff (DMT) of quasi-static multiple-input multiple-output (MIMO) fading channels for any fading statistics, and perform very well also at practical, moderate values of signal-to-noise ratios (SNR). Then, we extend the construction to the case of large block lengths, by using trellis coset coding. We provide constructions of trellis coded modulation (TCM) schemes that are endowed with good packing and shaping properties. Both short-block and trellis constructions allow for a reduced complexity decoding algorithm based on minimum mean-squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding and a combination of this with a Viterbi TCM decoder for the TCM case. Beyond the interesting algebraic structure, we exhibit codes whose performance is among the state-of-the art considering codes with similar encoding/decoding complexity.

In most cooperative communication works, perfect synchronization among relay nodes is assumed in order to achieve the cooperative diversity. However, this assumption is not realistic due to the distributed nature of each relay node. In this paper, we propose a family of distributed space-time trellis code (DSTTC) that does not require the synchronization assumption. It is shown that the proposed DSTTC has minimum memory order, and the construction of such DSTTCs is equivalent to using the generator matrices that have full row rank, regardless of sub-matrices shifting problem. Here, a sub-matrix corresponds to the generator matrix of a relay node under frequency-selective channels. We derive sufficient conditions on the code design such that the full cooperative and multipath diversities can be achieved. By further studying the diversity product, we design the DSTTCs which can achieve full diversity and the maximum coding gain through the exhaustive computer search. The newly proposed codes exhibit good properties, e.g., high energy efficiency and low synchronization cost, and can be applied to distributed wireless networks. Finally, various numerical examples are provided to corroborate the analytical studies.

We consider the maximum likelihood (ML) receiver design, performance analysis and code design for space-time trellis codes (STTC) over non-identical, rapid fading channels with imperfect channel state information (CSI). The exact pairwise error probability (PEP) and PEP bounds for the ML receiver are obtained. A new code design criterion exploiting the statistical information of the channel estimates is proposed, which can minimize the performance loss caused by channel estimation error. New codes are obtained via an iterative search algorithm with reduced complexity. Under actual channel estimation conditions, our codes perform better than the existing codes in the literature which are designed on the assumption of identical channels, and perfect CSI at the receiver. More performance gain can be achieved by our codes when the degree of imbalance among the links is higher.

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.

This paper proposes improved space-time multiple TCM (ST-MTCM) codes via a systematic expansion of space- time block code (STBC) for two transmit antennas. Starting from the orthogonal STBC, the STBC set is expanded into super- orthogonal STBC, and further expanded up to spatial multiplexing in the case of BPSK and QPSK constellations. Exploiting the expanded set of STBCs as the number of states increase, improved full diversity ST-MTCM codes can be designed by increasing the coding gain and Euclidean distance of the pairwise errors.

This paper presents a reduced-complexity soft-input soft-output trellis/tree multiuser equalizer for an iterative DS-CDMA system undergoing Rayleigh frequency selective fading. The algorithm first expands the equalizer-trellis to an equivalent trellis/tree structure. Then it applies the M-algorithm to the equivalent structure twice, once to reduce the number of states in the trellis and the other to reduce the number of branches emanating from each state. To compute soft-information, the algorithm utilizes not only those fully-extended paths reaching the end of the trellis but also paths that are traversed and discarded in the pruned trellis. Through a simple update-and-discard procedure, reliable soft-information is extracted from discarded paths which enables an extremely large trellis to be successfully decoded with modest complexity. BER performance is presented for a convolutional-coded DS-CDMA system employing random spreading sequences. Our results demonstrate that the proposed algorithm is capable of achieving single-user performance with a much reduced complexity. The proposed algorithm can also be applied to reduce the complexity of multiuser detection where the transmission channel is frequency flat. Single-user performance can also be achieved with the proposed technique.

New space-time trellis codes with four- and eight-level phase-shift keying (PSK) and 16-phase quadrature amplitude modulation (QAM) for two transmit antennas in slow-fading channels are presented in this paper. Unlike most of the codes that are reported in the literature, the proposed codes are specifically designed to minimize the frame error probability from a union-bound perspective. The performance of the proposed codes with various memory orders and receive antennas is evaluated by simulation. It is shown that the proposed codes outperform previously known codes in all studied cases.

Shift-full-rank (SFR) matrices are matrices that have full row rank no matter how their rows are shifted. SFR matrices have been used lately as generator matrices for a family of space-time trellis codes to achieve full diversity in asynchronous cooperative communications, where the numbers of columns of the SFR matrices correspond to the memory sizes of the trellis codes. A systematic construction of SFR matrices, including the shortest (square) SFR (SSFR) matrices, has been also previously proposed. In this paper, we study a variation of SFR matrices with a relaxed condition: limited-shift-full-rank (LT-SFR) matrices, i.e., the matrices that have full row rank no matter how their rows are shifted as long as the shifts are within some range called delay tolerance. As the generator matrices for the previously proposed space-time trellis codes, LT-SFR matrices can guarantee asynchronous full diversity of the corresponding codes when the timing errors are within the delay tolerance. Therefore, due to the relaxed condition imposed on LT-SFR matrices, more eligible generator matrices than SFR matrices become available.

Reduced-state trellis detection with decision feedback is widely used to reduce the energy consumption of trellis detectors, particularly for soft-output trellis detectors that are energy-hungry by nature. However, the decision feedback tends to increase the circuit critical path and, more important, makes it difficult to apply some well-proven high-speed trellis detector design techniques such as bit-level pipelining. This paper presents a method, referred to as quasi-reduced-state trellis detection, to tackle such speed bottlenecks. The basic idea is to simply obviate the use of decision feedback by mapping only the data storage block of the trellis detector onto a reduced-state trellis and keeping the trellis state metric computation on the original full-state trellis. This makes sense because the data storage block tends to dominate the overall energy consumption while the decision feedback is due to the reduced-state trellis metric computation. Therefore, it is intuitive that such quasi-reduced-state detectors may largely maintain the energy saving potentials of reduced-state trellis detection without being subject to decision-feedback-induced speed bottlenecks. We demonstrated the effectiveness of this proposed design method by using soft-output Viterbi algorithm (SOVA) detection for a magnetic recording read channel as a test vehicle.

Rate-distortion trellis source codes are developed for quantizing memoryless independent and identically distributed (i.i.d.) sources. The codes are generated by simple linear congruential recursions. The method generates codes at a variety of rates including fractional ones; reproducer sets can be large, a crucial advantage with certain sources. Axioms for good code construction are developed that are based on recursion properties and certain trellis symmetries. These axioms are justified by the outcome of random searches for good codes. It is found that the trellis code design breaks into two problems: the trellis labels should have certain properties regardless of the source distribution; and the reproducer values depend on the source. Encoders are simulated for a number of continuous amplitude sources. For the same computational complexity, the new codes, in most cases, perform better than the best codes in the literature, including trellis-coded quantization and fake process approaches.

In this letter, we propose a new family of space-time trellis codes, which are constructed by combining a super set of quasi-orthogonal space-time block codes with minimum decoding complexity with an outer multiple trellis coded modulation encoder. A systematic set-partitioning method for quadratic amplitude modulation constellations is given. The proposed scheme can be used for systems with four or more than four transmit antennas. Furthermore, its decoding complexity is low because its branch metric calculation can be implemented in a symbolwise way. Simulation results demonstrate that the proposed scheme has a comparable performance as super quasi-orthogonal space-time trellis codes proposed by Jafarkhani and Hassanpour while providing a lower decoding complexity.

In this paper, we present a multidimensional trellis coded modulation scheme for a high rate 2times2 multiple-input multiple-output (MIMO) system over slow fading channels. Set partitioning of the Golden code is designed specifically to increase the minimum determinant. The branches of the outer trellis code are labeled with these partitions and Viterbi algorithm is applied for trellis decoding. In order to compute the branch metrics, a sphere decoder is used. The general framework for code design and optimization is given. Performance of the proposed scheme is evaluated by simulation and it is shown that it achieves significant performance gains over the uncoded Golden code

In current cooperative communication schemes, to achieve cooperative diversity, synchronization between terminals is usually assumed, which may not be practical since each terminal has its own local oscillator. In this paper, based on the stack construction proposed by Hammons and El Gamal, we first construct a family of space-time trellis codes for BPSK modulation scheme that is characterized to possess the full cooperative diversity order without the synchronization assumption. We then generalize this family of the space-time trellis codes from BPSK to higher order QAM and PSK modulation schemes based on the unified construction proposed by Lu and Kumar. Some diversity product properties of space-time trellis codes are studied and simplified decoding methods are discussed. Simulation results are given to illustrate the performance of the newly proposed codes

Super-orthogonal space-time trellis codes (SOSTTCs) designed by hand can significantly improve the performance of space-time trellis codes. This paper introduces a new representation of SOSTTCs based on a generator matrix that allows a systematic and exhaustive search of all possible codes. This will verify that some of the known codes are optimal, and provides a means to easily implement encoders and decoders with a large number of states without relying on a graphical representation. New codes with up to 256 states that outperform previously known codes are presented

In this letter, super-orthogonal space-time trellis codes (SOSTTCs) are extended for quadrature amplitude modulation (QAM) constellations. A systematic set-partitioning method for QAM constellations is given. Furthermore, trellis shaping based on set partitioning is incorporated in SOSTTCs with QAM symbols to achieve extra shaping gain. Peak constraints can be used to limit the constellation expansion ratio and peak-to-average power ratio

Transmission efficiency in radio channels can be considerably improved by using multiple transmit and receive antennas, and employing a family of schemes called space–time coding. Both extended range and/or improved bandwidth efficiency can be achieved, compared with a radio link with a single transmit and receive antenna. Bit-interleaved coded modulation schemes give diversity gains on fading channels with higher order modulation constellations combined with conventional binary convolutional codes, also for the case of a single transmit and receive antenna radio link. In this paper, we study a family of flexible bandwidth-efficient space–time coding schemes which combine these two ideas in a narrowband flat-fading channel and single-carrier modems. We address receiver complexity for the case of a large number of transmit antennas and higher order modulation constellations. Especially, we focus on practical configurations where the number of transmit antennas is greater than that of receive antennas. Simplified receivers using tentative decisions are proposed and evaluated by means of simulations. Tradeoffs between complexity reduction and performance loss are presented. We emphasize systems that are of particular interest in applications where the number of transmit antennas exceeds the number of receive antennas. A system with four transmit antennas with an eight-fold complexity reduction and a performance loss of about 1 dB is demonstrated.

International audience; Variable length codes exhibit de-synchronization problems when transmitted over noisy channels. Trellis decoding techniques based on Maximum A Posteriori (MAP) estimators are often used to minimize the error rate on the estimated sequence. If the number of symbols and/or bits transmitted are known by the decoder, termination constraints can be incorporated in the decoding process. All the paths in the trellis which do not lead to a valid sequence length are suppressed. This paper presents an analytic method to assess the expected error resilience of a VLC when trellis decoding with a sequence length constraint is used. The approach is based on the computation, for a given code, of the amount of information brought by the constraint. It is then shown that this quantity as well as the probability that the VLC decoder does not re-synchronize in a strict sense, are not significantly altered by appropriate trellis states aggregation. This proves that the performance obtained by running a length-constrained Viterbi decoder on aggregated state models approaches the one obtained with the bit/symbol trellis, with a significantly reduced complexity. It is then shown that the complexity can be further decreased by projecting the state model on two state models of reduced size.

Space-time coding (STC) has been proposed recently for multiple-antenna wireless communication systems. Most of the proposed STC schemes use the assumption that either no channel-state information, or the channel mean/covariance information, is available at the transmitter. In this paper, we propose a new STC scheme for a closed-loop transmission system, where quantized channel-phase information is available at the transmitter. A new performance criterion is derived for the quasi-static fading channel. This design criterion is then used to construct a new class of space-time trellis codes (STTCs). The proposed code construction is based on the concatenation of a standard multiple trellis-coded modulation outer code with an inner code. The inner code is selected from a series of inner codes using the channel-phase feedback. The series of inner codes are constructed based on the systematic set partitioning of several classes of space-time signal designs. Simulation results show significant performance improvement over the other STTCs in the literature. In addition, the proposed coding scheme enjoys low peak-to-average-power ratio, simple decoding, and power-efficient low-cost implementation

In this paper, we present two differential trellis coded space-time modulation schemes when the channel state information is not available at the transmitter and at the receiver. First, we extend the design of super-orthogonal space-time trellis codes to improve their performance. Then, based on these extended super-orthogonal codes, we propose the first differential scheme. Compared with the trellis coded differential unitary space-time modulation scheme, not only does the proposed differential scheme achieve full diversity, but also it provides identical or higher coding gain and much lower decoding complexity. In order to achieve further improved coding gain in addition to full diversity for more than two transmit antennas, we propose a second differential scheme which is based on super-quasi-orthogonal codes. Such a differential scheme outperforms the differential modulation scheme based on super-orthogonal codes with simpler decoding. Simulation results demonstrate the good performance of our schemes

Multiple antennas can be used for increasing the amount of diversity (diversity gain) or increasing the data rate (the number of degrees of freedom or spatial multiplexing gain) in wireless communication. As quantified by Zheng and Tse, given a multiple-input-multiple-output (MIMO) channel, both gains can, in fact, be simultaneously obtained, but there is a fundamental tradeoff (called the Diversity-Multiplexing Gain (DM-G) tradeoff) between how much of each type of gain, any coding scheme can extract. Space-time codes (STCs) can be employed to make use of these advantages offered by multiple antennas. Space-Time Trellis Codes (STTCs) are known to have better bit error rate performance than Space-Time Block Codes (STBCs), but with a penalty in decoding complexity. Also, for STTCs, the frame length is assumed to be finite and hence zeros are forced towards the end of the frame (called the trailing zeros), inducing rate loss. In this correspondence, we derive an upper bound on the DM-G tradeoff of full-rate STTCs with nonvanishing determinant (NVD). Also, we show that the full-rate STTCs with NVD are optimal under the DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros. Next, we give an explicit generalized full-rate STTC construction for any number of states of the trellis, which achieves the optimal DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros

A new differential space-frequency trellis coding (DSFTC) scheme is proposed for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. The transmitter differentially encodes the message drawn from a set of unitary matrices following a trellis encoding rule. At the receiver, where no channel state information (CSI) is available, differential decoding and soft-input Viterbi algorithms are employed in the decoding process. The proposed codes can achieve full rate, 1 symbol/subcarrier, and full spatial diversity. Simulation results show that the proposed codes provide significant improvements over differential space-frequency block codes (DSFBC) due to increased coding gains

In this correspondence, it is shown that some convolutional codes with distinct memory sizes of minimal encoders are equivalent in the sense that the minimal trellises of these codes are the shifted versions of one another. For an (n,k) binary convolutional code, the weight spectrum obtained from the minimal trellis may be slightly different from that obtained from the conventional code trellis with n-bit branches. Code search is conducted to find some good (n,n-1) binary convolutional codes. Bounds on the trellis complexity, measured by the number of states and the number of branches in the minimal trellis module, of any convolutional code and its equivalent codes are also derived.

In broadcast scenarios or in the absence of accurate channel probability distribution information, code design for consistent channel-by-channel performance, rather than average performance over a channel distribution, may be desirable. Root and Varaiya's compound channel theorem for linear Gaussian channels promises the existence of universal codes that operate reliably whenever the channel mutual information (MI) is above the transmitted rate. This paper presents two-dimensional trellis codes that provide such universal performance over the compound linear vector Gaussian channel when demultiplexed over two, three, and four transmit antennas. The presented trellis codes, found by exhaustive search, guarantee consistent performance on every matrix channel that supports the information transmission rate with an MI gap that is similar to the capacity gap of a well-designed additive white Gaussian noise (AWGN)-specific code on the AWGN channel. As a result of their channel-by-channel consistency, the universal trellis codes presented here also deliver comparable, or, in some cases, superior, frame-error rate and bit-error rate performance under quasi-static Rayleigh fading, as compared with trellis codes of similar complexity that are designed specifically for the quasi-static Rayleigh-fading scenario

We demonstrate super-orthogonal space-time (ST) trellis codes with rectangular signal constellations for wireless communications with a spectral efficiency of 4 bits/s/Hz. Considering a mobile wireless communications system with two transmit antennas, we form a 4D signal constellation as Cartesian product of two 2D rectangular signal sets, a 4D point being transmitted by the first antenna as two concatenated 2D points in two consecutive channel uses. The 2D symbols transmitted by the second antenna are not independent, but so chosen as to form, together with the symbols transmitted by the first antenna, the entries of a 2 times 2 orthogonal matrix. This can be done in four different ways, resulting in four sets of 2 times 2 orthogonal matrices. To obtain a higher data rate, the union of two of them forms a super-orthogonal signal set. With 2 times 2 orthogonal matrices, we then label the state transitions of a trellis diagram describing the operation of the encoder. We perform simulations using a model that takes into account the Doppler effect as well as the effect of spatial antenna correlation at both the base station (BS) and at the mobile station (MS). The frame error rate (FER) and the bit error rate (BER) performance results show an excellent behaviour of our proposed super-orthogonal ST trellis code, which clearly outperforms all known simple ST trellis codes of same spectral efficiency using 16QAM or 16PSK

A novel full diversity space-time trellis code, referred to as an assembled space-time trellis code (ASTTC), is presented in this letter. For this scheme, space-time trellis coded signals are first linearly transformed, and then the transformed signals are coded by using the Alamouti space-time block code. A new design criterion is proposed. It is shown that the ASTTCs can achieve not only the full diversity order but also a significant coding gain determined by the minimum weighted code distance (MWCD). Based on this design criterion, a new 4-state code is derived by a systematic code search. Simulation results show that the new ASTTC is superior by about 1.7 dB to the TSC (TarokhSeshadri-Calderbank) code at the frame error rate (FER) of 1.0e-2 over quasi-static fading channels.

An inner space–time code, i.e., one that is complemented by an outer error-control code, calls for vastly different design strategies than a space–time code that stands alone. This letter investigates the design of a linear inner space–time code for a $t$ -input $r$ -output Rayleigh fading channel by examining its outage capacity, which assumes an idealized outer code. We show that a linear space–time code with rate $R ≪ min(t, r)$ can achieve at most a fraction $R/min(t, r)$ of the underlying channel's outage capacity at high signal-to-noise ratio (SNR). Conversely, we find that a space–time code with low raw diversity order (as calculated using the rank rule) does not necessarily suffer a capacity penalty. Under very general conditions, a rate of $R = min(t, r)$ is sufficient to ensure that the outage capacity of the space–time code approaches that of the underlying channel at high SNR. Simulation results are presented to support the claims.

This paper introduces a low-complexity full-rate full-diversity space-time (ST) code for wireless communication systems with four transmit antennas. This novel code is designed by combining delay-diversity transmission with Alamouti's orthogonal ST block code. Analytical and computer simulation results show that this code yields significant frame-error-rate (FER) performance gains over some previously reported full-rate full-diversity codes for four transmit antennas.

A novel full rate space-time turbo trellis code, referred to as an assembled space-time turbo trellis code (ASTTTC), is presented in this paper. For this scheme, input information binary sequences are first encoded using two parallel concatenated convolutional encoders. The encoder outputs are split into four parallel streams and each of them is modulated by a QPSK modulator. The modulated symbols are assembled by a predefined linear function rather than punctured as in the standard schemes. This results in a lower code rate and a higher coding gain over time-varying fading channels. An extended two-dimensional (2-D) log-MAP (maximum a posteriori probability) decoding algorithm, which simultaneously calculates two a posteriori probabilities (APP), is developed to decode the proposed scheme. Simulation results show that, under the same conditions, the proposed code considerably outperforms the conventional space-time turbo codes over time-varying fading channels.

Super-orthogonal space-time trellis codes recently proposed in the literature are space-time trellis codes of full diversity and high rates systematically constructed by concatenating orthogonal space-time codes and multiple trellis coded modulation (MTCM). However, the existing MTCM only has designs for phase-shift keying (PSK) signals. In this paper, MTCM is extended from PSK signals to non-PSK signals in some special cases. With obtained constellations of MTCM, several super-orthogonal space-time trellis codes for two transmit antennas are presented. The 2- and 4-state codes have a simple mathematical expression for the coding gain distance (CGD), or diversity products. At rates 2.5, 3, 3.5, 4 bits/s/Hz, the newly proposed codes outperform the existing ones.

In this paper, we design capacity-approaching codes for partial response channels. The codes are constructed as concatenations of inner trellis codes and outer low-density parity- check (LDPC) codes. Unlike previous constructions of trellis codes for partial response channels, we disregard any algebraic properties (e.g., the minimum distance or the run-length limit) in our design of the trellis code. Our design is purely probabilistic in that we construct the inner trellis code to mimic the transition probabilities of a Markov process that achieves a high (capacity-approaching) information rate. Hence, we name it a matched information rate (MIR) design. We provide a set of five design rules for constructions of capacity-approaching MIR inner trellis codes. We optimize the outer LDPC code using density evolution tools specially modified to fit the superchannel consisting of the inner MIR trellis code concatenated with the partial response channel. Using this strategy, we design degree sequences of irregular LDPC codes whose noise tolerance thresholds are only fractions of a decibel away from the capacity. Examples of code constructions are shown for channels both with and without spectral nulls.

We introduce a new family of space-time trellis codes that extends the powerful characteristics of super-orthogonal space-time trellis codes to four transmit antennas. We consider a family of quasi-orthogonal space-time block codes as building blocks in our new trellis codes. These codes combine set partitioning and a super set of quasi-orthogonal space-time block codes in a systematic way to provide full diversity and improved coding gain. The result is a powerful code that provides full rate, full diversity, and high coding gain. It is also possible to maintain a tradeoff between coding gain and rate. Simulation results demonstrate the good performance of our new super-quasi-orthogonal space-time trellis codes.

During the last few years a number of space-time block codes have been proposed for use in multiple transmit antennas systems. We propose a method to extend any space-time code constructed for m transmit antennas to m p transmit antennas through group-coherent codes (GCCs). GCCs make use of very limited feedback from the receiver (as low as 1 bit). In particular the scheme can be used to extend any orthogonal code (e.g., Alamouti code) to more than two antennas while preserving low decoding complexity, full diversity benefits, and full data rate.

A space-time trellis code with linear transformation (STTC-LT) is presented. The upper bound on the pairwise error probability (PEP) is derived and a new design criterion is proposed. It is shown that the performance of STTC-LT over fast Rayleigh fading channels depends on the weighted square product distance (WSPD). Simulation results are provided for QPSK signal sets. They show that the proposed scheme is superior by about 8 dB and 10 dB to the conventional beamforming and Tarokh-Seshadri-Calderbank codes (TSC) at the bit error rate (BER) of 1.0e-4 over fast fading channels.

The potential for capacity increase in multiple-antenna wireless communication systems has drawn considerable attention to space-time codes. However, most of the existing space-time code construction methods have assumed ideal channel models: either quasi-static fading or fast fading. In this work, we propose a systematic space-time trellis code construction method for correlated fading channels. We derive the performance criteria that take into account both spatial and temporal channel correlation, assuming that the space-time correlation matrix is of full rank. Moreover, we introduce a new design criterion and analyze the properties of the space-time trellis codes satisfying the proposed criterion. Using the design criterion, we develop a code construction procedure that jointly considers diversity advantage and coding advantage for an arbitrary number of transmit antennas and any memoryless modulation. The flexibility of the proposed code construction method is demonstrated by designing space-time trellis codes for two, three, and four transmit antennas with quaternary phase-shift keying (QPSK), 8PSK (phase-shift keying), and 4ASK (amplitude-shift keying) modulations. The simulation results demonstrate that not only can our method generate new codes for an arbitrary number of antennas, but in some special cases, where previously proposed codes exist, our codes also perform very well.