On the error performance analysis of space-time trellis codes

We present analytical performance results for space-time trellis codes over spatially correlated Rayleigh fading channels. Bit-error-probability estimates are obtained based on the derivation of an exact pairwise error probability expression using a residue technique combined with a characteristic function approach. We investigate both quasi-static and interleaved channels and demonstrate how the spatial fading correlation affects the performance of space-time codes over these two different channel models. Simulation results are also included to confirm the accuracy of analytical estimates.     

关于非线性码的格子复杂度

研究了非线性码的格子复杂度.给出了非线性码的维数/长度轮廓的定义,并利用这一定义,将Forney所给出的线性码格子复杂度的新下界推广到了非线性码上去.然后推出了非线性码的Berger-Be′ery上界.     

On the trellis representation of the Delsarte-Goethals codes

In this correspondence, the trellis representation of the Kerdock and Delsarte-Goethals codes is addressed. It is shown that the states of a trellis representation of DG(m,δ) under any bit-order are either strict-sense nonmerging or strict-sense nonexpanding, except, maybe, at indices within the code's distance set. For δ⩾3 and for m⩾6, the state complexity, s_max[DG(m,δ)], is found. For all values of m and δ, a formula for the number of states and branches of the biproper trellis diagram of DG(m, δ) is given for some of the indices, and upper and lower bounds are given for the remaining indices. The formula and the bounds refer to the Delsarte-Goethals codes when arranged in the standard bit-order     

On the performance of space time turbo trellis codes with adaptive power allocation

It has been shown that when partial or full channel state information (CSI) is available at the transmitter, the performance of a space time coded system can be dramatically improved by weighting the transmitted signals. In this letter, we evaluate the performance of space time turbo codes with adaptive power allocation when only partial CSI is available at the transmitter. An, optimum power allocation strategy for space time turbo trellis coded systems (STTTC) is derived by simulations. Simulation results show that the proposed power allocation can bring a gain of about 0.8 dB gain relative to the conventional STTTC with equal power allocation.     

On constructing embedded multilevel trellis codes

A design technique to reduce the search time for trellis codes with multilevel phase modulation is presented. Codes are constructed by connecting trellis diagrams for codes with fewer states in parallel. For example, an N-state code can be constructed by connecting two N/2-state codes. The way in which the embedded codes are connected increases the upper limit on minimum free distance otherwise imposed by parallel transitions between states. In some cases, this technique can reduce the number of codes in a code search by a factor of approximately 2^ν, the number of coder states. A computer search incorporating this technique for eight-level amplitude modulation (8-AM) codes having 2¹¹ and 2¹² states produced codes with greater minimum free distance than reported previously (i.e. greater than 6 dB coding gain). New eight-level phase-shift-keying (8-PSK) codes, which have a different structure from previously reported codes, are also presented     

On the BCJR trellis for linear block codes

In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algorithm is Θ(e). This result suggests that the “best” trellis representation for a given linear block code is the one with the fewest edges. We will then show that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf (1978), Massey (1978), and Forney (1988), uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine (1995), we will also discuss “trellis-oriented” or “minimal-span” generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it     

Better phase-modulation error performance using trellis phase codes

We construct phase codes for use in phase- and frequency-shift modulation. A trellis structure allows simple decoding using the Viterbi algorithm. Even short codes give a large improvement in error performance.     

On the design and maximum-likelihood decoding of space-time trellis codes

In this letter, we present a simple generalization of the maximum ratio combining principle for space-time coded systems. This result leads to a maximum-likelihood decoder implementation that does not depend on the number of receive antennas and avoids the loss in performance incurred in the decoders proposed by Tarokh and Lo (1998) and Biglieri et al. The insights offered by this decoding rule allow for a simple and elegant proof for the space-time code design criterion in systems with large number of receive antennas. We further present an upper bound on probability of error that captures the dependence of space-time code design on the number of receive antennas. Finally, we present a computationally efficient approach for constructing space-time trellis codes that exhibit satisfactory performance in systems with variable number of receive antennas.     

Algorithmic construction of trellis codes

An algorithmic approach is proposed whereby long convolutional codes of rate r_c=k/n can easily be constructed for any chosen signal constellation in signal space. These algorithms are iterative, and in each step a number of candidate codes are found which locally maximize the distance (Hamming or Euclidean) between the codewords. The result is not necessarily a free-distance-optimizing code. However, since the construction complexity can be chosen, optimal codes are quite frequently found. The codes ae constructed such that a rapid growth of the column distance is achieved. A method of combining two codes into a single code of twice the constraint length is also presented     

Selective trellis pruning for block codes: theory and application to iterative decoding

In this paper we present a new method to simplify trellis‐based decoding of linear block codes. Our method is based on removing (pruning) some edges and states from the trellis representation of the code; the trellis is pruned through some hard‐decisions that are made on received bits whose likelihood exceeds a predefined threshold. We show in this paper how this simplification can be accounted for by a new generator matrix (and sometimes a coset leader) that totally parametrize the new trellis, or, equivalently, the set of allowed codewords after simplification. Extensive simulations show that significant computational savings can be achieved, at a very small loss in coding performance, as long as the operating point and threshold are carefully chosen. Moreover, the application of this technique to iterative decoding of product codes is outlined, and our results show that the simplifications do not hinder the convergence, again, as long as proper parameters are chosen. Copyright © 2007 John Wiley & Sons, Ltd.     

On trellis codes with a delay processor and a signal mapper

It is already known that a trellis code T, which is constructed by using the encoder of a convolutional code C with short constraint length followed by a delay processor and a signal mapper, is equivalent to a trellis code with large constraint length. In this paper, we derive a new lower bound on the free distance of T, which, in some cases, is better than the previously derived bound. Moreover, instead of the decoding used in earlier publications, we apply iterative decoding on both tailbiting and zero-tail representations of T to take advantage of the new lower bound and, in the meantime, to decrease the associated error coefficient caused by the decoding used in earlier publications. Comparisons among various designs of such a trellis code and some well-known coding methods are also provided.     

Distance spectra and performance bounds of space-time trellis codes over quasistatic fading channels

This correspondence presents a general approach to upper bounding coded system performance over quasistatic fading channels (QSFC). This approach has the advantage of yielding a closed-form upper bound that converge for all signal-to-noise ratios (SNRs). The proposed approach is used to upper-bound the performance of space-time trellis codes (STTC) over QSFCs. The resulting upper bounds for STTCs are better adapted to the QSFC and present an improvement over worst case pairwise error probability (PEP) analysis used so far. In its second part, this correspondence investigates several ways to reduce the complexity of computing the distance spectrum of STTCs. The combined result obtained from using the new upper bounds and the computed distance spectra are shown to be close to simulated performance for all SNRs.     

On constructing trellis codes with large free distances and lowdecoding complexities

We propose a trellis-coding scheme for which large free distances and low decoding complexities can be easily achieved. The proposed trellis-coding scheme is a generalization of the trellis-coding scheme proposed by Hellstern (1993). With the proposed trellis-coding scheme, we have the flexibility of controlling the decoding delay     

On saving decoder states for some trellis codes and partialresponse channels

It is shown that under certain conditions, when convolutional codes with precoding are used in conjunction with a partial response channel, the number of decoder states in a maximum-likelihood decoder matched to both the code and channel is 1/2 of that predicted by J.K. Wolf and G. Ungerboeck (ibid., vol.COM-34, no.7, p.751-72, 1986)     

Decoding performance of linear block codes using a trellis indigital mobile radio

Trellis decoding of linear block codes in a Rayleigh fading channel is discussed. Two methods for calculating metric values for each bit in a received block are considered: the values are calculated from the received signal envelope sample and from the demodulator output. Bit error rate (BER) performances of hard decision and trellis decoding are compared using Hamming (7, 4) and Golay (24, 12) codes in computer simulations and laboratory experiments. A simplified trellis decoding algorithm, in which the hard decision output of a bit with an envelope sample greater than the threshold value is accepted as correct, is presented. Laboratory experimental results for trellis decoding in combination with Gaussian minimum-shift-keying (GMSK) modulation and frequency detection are shown. The effect of n-bit A/D-conversion in signal envelope sampling is investigated experimentally. The results show that the trellis decoding algorithm improves BER performance     

Joint synchronization and demodulation of trellis codes

This paper examines the joint synchronization and detection of Ungerboeck coded modulation. Estimation theory is used to derive a synchronization structure that is efficient in estimating carrier phase and symbol timing. The maximum likelihood receiver generates estimates of carrier phase and timing that are free of data-dependent jitter for any continuous pulse shape. Various feedback schemes to be used with Ungerboeck codes are presented and simulated. Simulations show that efficient estimates of carrier and clock can be found when joint data and parameter estimation of an Ungerboeck coded signal is performed     

Erasure-free sequential decoding of trellis codes

An erasure-free sequential decoding algorithm for trellis codes, called the buffer looking algorithm (BLA), is introduced. Several versions of the algorithm can be obtained by choosing certain parameters and selecting a resynchronization scheme. These can be categorized as block decoding or continuous decoding, depending on the resynchronization scheme. Block decoding is guaranteed to resynchronize at the beginning of each block, but suffers some rate loss when the block length is relatively short. The performance of a typical block decoding scheme is analyzed, and we show that significant coding gains over Viterbi decoding can be achieved with much less computational effort. A resynchronization scheme is proposed for continuous sequential decoding. It is shown by analysis and simulation that continuous sequential decoding using this scheme has a high probability of resynchronizing successfully. This new resynchronization scheme solves the rate loss problem resulting from block decoding. The channel cutoff rate, demodulator quantization, and the tail's influence on performance are also discussed. Although this paper considers only the decoding of trellis codes, the algorithm can also be applied to the decoding of convolutional codes     

Differential trellis decoding of convolutional codes

This paper investigates the principle of metric differences for trellis decoding of convolutional codes. Based on this differential method, a new algorithm, referred to as differential trellis decoding (DTD), is proposed. DTD offers an alternative to the conventional “add-compare-select” (ACS) method for implementing the Viterbi algorithm     

Bounds on the minimum distance of trellis codes

The performance of trellis codes is determined by their minimum Euclidean distance. Upper bounds on this minimum distance valid for phase-shift-keyed (PSK) signals that improve on previously derived bounds are derived. Although the bound is valid only for PSK signals, the bounding techniques developed here can be extended to other equal-energy configurations and hence could pave the way to obtaining more general results