Branch Metrics Research Articles

Performance of BICM-SC and BICM-OFDM systems with diversity reception in non-gaussian noise and interference

In this paper, we present a general mathematical framework for performance analysis of single-carrier (SC) and orthogonal frequency division multiplexing (OFDM) systems employing popular bit-interleaved coded modulation (BICM) and multiple receive antennas. The proposed analysis is applicable to BICM systems impaired by general types of fading (including Rayleigh, Ricean, Nakagami-m, Nakagami-q, and Weibull fading) and general types of noise and interference with finite moments such as additive white Gaussian noise (AWGN), additive correlated Gaussian noise, Gaussian mixture noise, co-channel interference, narrowband interference, and ultra-wideband interference. We present an approximate upper bound for the bit error rate (BER) and an accurate closed-form approximation for the asymptotic BER at high signal-to-noise ratios for Viterbi decoding with the standard Euclidean distance branch metric. For the standard rate-1/2 convolutional code the proposed approximate upper bound and the asymptotic approximation become tight at BERs of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> and 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-12</sup> , respectively. However, if the code is punctured to higher rates (e.g. 2/3 or 3/4), the asymptotic approximation also becomes tight at a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> . Exploiting the asymptotic BER approximation we show that the diversity gain of BICM systems only depends on the free distance of the code, the type of fading, and the number of receive antennas but not on the type of noise. In contrast their coding gain strongly depends on the noise moments. Our asymptotic analysis shows that as long as the standard Euclidean distance branch metric is used for Viterbi decoding, BICM systems optimized for AWGN are also optimum for any other type of noise and interference with finite moments.

Application of Sequential Estimation Algorithm Based on Branch Metric to Frequency-Selective Channel Equalization

In this work, a sequential estimation algorithm based on branch metric is used as channel equalizer to combat intersymbol interference in frequency-selective wireless communication channels. The bit error rate (BER) and computational complexity of the algorithm are compared with those of the maximum likelihood sequence estimation (MLSE), the recursive least squares (RLS) algorithm, the Fano sequential algorithm, the stack sequential algorithm, list-type MAP equalizer, soft-output sequential algorithm (SOSA) and maximum-likelihood soft-decision sequential decoding algorithm (MLSDA). The BER results have shown that whilst the sequential estimation algorithm has a close performance to the MLSE using the Viterbi algorithm, its performance is better than the other algorithms. Beside, the sequential estimation algorithm is the best in terms of computational complexity among the algorithms mentioned above, so it performs the channel equalization faster. Especially in M-ary modulated systems, the equalization speed of the algorithm increases exponentially when compared to those of the other algorithms.

Effects of elevated atmospheric CO 2 and tropospheric O 3 on tree branch growth and implications for hydrologic budgeting

The forest hydrologic budget may be impacted by increasing CO 2 and tropospheric O 3. Efficient means to quantify such effects are beneficial. We hypothesized that changes in the balance of canopy interception, stem flow, and through-fall in the presence of elevated CO 2 and O 3 could be discerned using image analysis of leafless branches. We compared annual stem flow to the results of a computerized analysis of all branches from the 2002, 2004, and 2006 annual growth whorls of 97 ten-year-old trees from the Aspen Free-Air CO 2 and O 3 Enrichment (Aspen FACE) experiment in Rhinelander, WI. We found significant effects of elevated CO 2 and O 3 on some branch metrics, and that the branch metrics were useful for predicting stem flow from birch, but not aspen. The results of this study should contribute to development of techniques for efficient characterization of effects on the forest hydrologic budget of increasing CO 2 and tropospheric O 3.

A novel PAPR reduction with trellis shaping for NC-OFDM in cognitive radio

In this paper, trellis shaping is applied to reducing the Peak-to-Average Power Ratio (PAPR) of NC-OFDM signals due to the fact that the total number of active subcarriers and their locations might change continuously in a dynamic spectrum access network. A new branch metric of trellis shaping is proposed which reduces the PAPR of OFDM signals through minimizing the variance of the power by VA decoder. The overall performance considerably depends on the type of signal mapping, and therefore a new type of 64QAM mapping method is designed. Simulation results show that this scheme significantly reduces the PAPR. Finally, a significant phenomenon is demonstrated that the PAPR of NC-OFDM signals in which the deactivated carriers is distributed in the Bernoulli distribution has better performance than that of OFDM signals.

Near constant envelope trellis shaping for PSK signaling

In this paper, we propose a powerful trellis shaping for peak-to-average power ratio (PAR) reduction of pulse-shaped phase shift keying (PSK) systems. The proposed approach consists of 1) memory extension of trellis shaping encoder for capturing the full effect of pulse shaping filter impulse response, and 2) simple branch metrics that quantify envelope fluctuation of the filter output. Simulation results demonstrate that achieving near constant envelope is possible even if a pulse shaping filter with a roll-off factor as low as 0.1 is employed. Furthermore, due to the circular nature of the resulting signal trajectory, the proposed system also exhibits a robustness against sampling timing jitter at the receiver.

MAP Receiver with Spatial Filters for Suppressing Cochannel Interference in MIMO-OFDM Mobile Communications

This paper proposes joint maximum a posteriori (MAP) detection and spatial filtering for MIMO-OFDM mobile communications; it offers excellent receiver performance even over interference-limited channels. The proposed joint processor consists of a log likelihood generator and a MAP equalizer. The log likelihood generator suppresses cochannel interference by spatially filtering received signals and provides branch metrics of transmitted signal candidates. Using the branch metrics, the MAP equalizer generates log likelihood ratios of coded bits and performs channel decoding based on the MAP criterion. In the first stage, the log likelihood generator performs spatio-temporal filtering (STF) of the received signals prior to the fast Fourier transform (FFT) and is referred to as preFFT-type STF. Estimation of parameters including tap coefficients of the spatio-temporal filters and equivalent channel impulse responses of desired signals is based on the eigenvalue decomposition of an autocorrelation matrix of both the received and transmitted signals. For further improvement, in the second stage, the generator performs spatial filtering (SF) of the FFT output and is referred to as postFFT-type SF. Estimation of both tap coefficients of the spatial filters and channel impulse responses employs the recursive least squares (RLS) with smoothing. The reason for switching from preFFT-type STF into postFFT-type SF is that preFFT-type STF outperforms postFFT-type SF with a limited number of preamble symbols while postFFT-type SF outperforms preFFT-type STF when data symbols can be reliably detected and used for the parameter estimation. Note that there are two major differences between the proposed and conventional schemes: one is that the proposed scheme performs the two-stage processing of preFFT-type STF and postFFT-type SF, while the other is that the smoothing algorithm is applied to the parameter estimation of the proposed scheme. Computer simulations demonstrate that the proposed scheme can achieve excellent PER performance under interference-limited channel conditions and that it can outperform the conventional joint processing of preFFT-type STF and the MAP equalizer.

Tail Extrapolation in MLSE Receivers Using Nonparametric Channel Model Estimation

This paper presents a method for determining the probability of rare events, in particular for probability density function (pdf) and bit error rate (BER) estimation. The derivation of the method is based on the presumption that the pdf is a member of a family of distributions very often named as the generalized exponential (GE) class of distributions. Based on high reliability estimations obtained in short simulation/measurement times, the low probably events are estimated accurately by extrapolation. The suggested method can be applied to some distributions that are different from GE distributions, such as noncentral chi-square distributions, to extrapolate to low probability events, with some extrapolation error. It can also be applied to BER estimation. The method is in particular helpful for estimating channels suffering from both severe signal distortion causing undesired intersymbol interference (ISI) of several symbols, and from severe noise. Such conditions prevail, for example, in metro and long haul high-speed optical fiber communication systems. So the method may be implemented in particular in maximum-likelihood sequence estimation (MLSE) optical receivers using nonparametric channel model estimation. A special use of the extrapolation method is explained for practical systems using trellis branch metrics derived from the estimated pdf to decode the transmitted sequence of symbols.

Adaptive Selection of Surviving Symbol Replica Candidates for Quasi-Maximum Likelihood Detection Using M-Algorithm with QR-Decomposition for OFDM MIMO Multiplexing

This paper proposes an adaptive selection algorithm for the surviving symbol replica candidates (ASESS) based on the maximum reliability in maximum likelihood detection with QR decomposition and the M-algorithm (QRM-MLD) for Orthogonal Frequency Division Multiplexing (OFDM) multiple-input multiple-output (MIMO) multiplexing. In the proposed algorithm, symbol replica candidates newly-added at each stage are ranked for each surviving symbol replica from the previous stage using multiple quadrant detection. Then, branch metrics are calculated only for the minimum number of symbol replica candidates with a high level of reliability using an iterative loop based on symbol ranking results. Computer simulation results show that the computational complexity of the QRM-MLD employing the proposed ASESS algorithm is reduced to approximately 1/4 and 1/1200 compared to that of the original QRM-MLD and that of the conventional MLD with squared Euclidian distance calculations for all symbol replica candidates, respectively, assuming the identical achievable average packet error rate (PER) performance in 4-by-4 MIMO multiplexing with 16QAM data modulation. The results also show that 1-Gbps throughput is achieved at the average received signal energy per bit-to-noise power spectrum density ratio (Eb/N0) per receiver antenna of approximately 9dB using the ASESS algorithm in QRM-MLD associated with 16QAM modulation and Turbo coding with the coding rate of 8/9 assuming a 100-MHz bandwidth for a 12-path Rayleigh fading channel (root mean square (r.m.s.) delay spread of 0.26µs and maximum Doppler frequency of 20Hz).

Decoding the golden space-time trellis coded modulation

This letter presents new techniques enabling low complexity implementation of a decoder for 2 times 2 MIMO using an inner space-time block code (STBC) concatenated with an outer trellis coded modulation (TCM). The receiver includes a Viterbi decoder and sphere decoders to compute branch metrics. We present a technique to reduce the number of sphere decoding processes at each step of the trellis still retaining maximum- likelihood (ML) frame-error rate (FER) performance as well as a sub-optimal technique to speed up the decoding, with a negligible FER performance loss.

Performance Analysis of MLSE Receivers Based on the Square-Root Metric

In this paper, we analyze the performance of IMDD-NRZ ASE-noise-limited systems using MLSE receivers in the presence of CD and DGD. We study the performance of different approximated branch metrics and compare them to the optimum metric evaluated using the Karhunen-Loeve (KL) method. We specifically concentrate on the SQRT metric, because of its low complexity and potential ease of channel estimation. We test it at 10.7 Gb/s, up to about 12000 ps/nm of CD (700 km of SSMF) and about 150 ps of DGD, showing that the SQRT metric performance is virtually coincident with that of the optimum metric. This result holds true over a broad range of TX and RX configurations, using various different filtering optimization strategies. The effects of a finite resolution of the A/D converter on all considered metrics are also analyzed, showing the impact of quantization over the SQRT metric to be the same as for the optimum metric. Finally, the statistical properties of the SQRT metric are discussed. In particular, the probability density functions of signal and noise related to the SQRT metric are compared with the exact ones.

Parallel Detection Algorithm Using Multiple QR Decompositions With Permuted Channel Matrix for SDM/OFDM

Space division multiplexing (SDM)/orthogonal frequency division multiplexing (OFDM) systems transmit different data using the same frequency, so it is necessary to separate the simultaneously received signals in the receiver. Previous studies have shown that maximum likelihood detection (MLD) provides the best bit error rate (BER) performance. However, the complexity of MLD exponentially increases with the constellation size and the number of transmit antenna branches. Therefore, it is impractical to use a full MLD without reducing its computational complexity, because it would be prohibitively large for implementation. Recently, the use of QR decomposition with an M-algorithm (QRD-M) has been proposed to reduce the system complexity while maintaining the performance of the system. However, the QRD-M performance depends on the number of surviving symbol replica candidates. When QRD-M is used with a small number of surviving symbol replica candidates, the performance declines, but when there is a large number of surviving symbol replica candidates and the transmitter antenna branches, QRD-M requires a large memory to maintain their branch metrics, and a long latency time is also required. To reduce these problems, in this paper, we propose a parallel detection algorithm using multiple QR decompositions with permuted channel matrices for SDM/OFDM systems.

Quasi-Synchronous Sampling and Adaptive Partial-Response Maximum-Likelihood Read-Channel System

We propose a read-channel system with quasi-synchronous sampling and an adaptive partial-response maximum-likelihood (PRML) method to achieve accurate timing recovery and correct data extraction under high-density recording with a high transfer rate. The proposed system can extract data correctly in the state of either oversampling or undersampling using first-in-first-out flip-flops (FIFOs) and an asynchronously sampled data processable maximum-likelihood detector (ASML). In other words, it allows a sampling clock to be incompletely synchronized with a channel bit. Furthermore, the ASML adaptively learns the references required for calculating branch metrics even though the input data are not synchronized with the channel bit. This system achieves stable data extraction under quasi-synchronous sampling, improves the bit-error-rate performance, and can be applied to next-generation, high-density, and high-transfer-rate storage systems.

Read Channel for Multilevel Run-Length-Limited Read-Only Disc

A novel multilevel run-length-limited read channel which employs an interpolating timing recovery system and an adaptive partial-response maximum-likelihood (PRML) detector is developed. In proposed adaptive PRML detection system, we adopt a 112-levels adaptive Viterbi detector which uses every RF sample to adjust the reference levels for the calculation of branch metrics. The reference levels of adaptive Viterbi detector are also used to regenerate RF signals for phase error calculation. The usage of adaptive PRML detector makes the performance more robust, and the combination of Viterbi reference levels and RF generator makes the calculated phase errors more accurate.

New Adaptive Partial Response Maximum Likelihood Detection for High-Density Optical Recording

Adaptive partial response maximum likelihood (PRML) detection is proposed to improve the detection performance for nonlinearity of the readout signal, such as an asymmetry compared with normal PRML detection. However, the target levels of the equalizer are defined according to predetermined PR polynomial, the same as in the normal PRML. Such an equalization causes an undesirable noise enhancement when the readout signal characteristics is quite different from the selected PR. To resolve this problem, we have proposed a new adaptive PRML detection in which both of the target levels of the equalizer and the reference levels of branch metrics are defined by the equalizer input. We have confirmed that our proposed method can effectively improve bit error rate (bER) compared with the conventional adaptive PRML and the normal PRML detections. The bER of the new PRML detection is lower than one-half of those of previous methods for a high-density recording signal including a significant nonlinearity.

Combining Coded Signals with Arbitrary Modulations in Orthogonal Relay Channels

We consider a relay channel for which the following assumptions are made. (1) The source-destination and relay-destination channels are orthogonal (frequency division relay channel). (2) The relay implements the decode-and-forward protocol. (3) The source and relay implement the same channel encoder, namely, a convolutional encoder. (4) They can use arbitrary and possibly different modulations. In this framework, we derive the best combiner in the sense of the maximum likelihood (ML) at the destination and the branch metrics of the trellis associated with its channel decoder for the ML combiner and also for the maximum ratio combiner (MRC), cooperative-MRC (C-MRC), and the minimum mean-square error (MMSE) combiner.

Viterbi Decoding of Convolutional Codes in Symmetric α -Stable Noise

Algorithms developed with a Gaussian noise assumption perform poorly in impulsive noise, such as that described by the symmetric alpha-stable (SalphaS) distribution. We investigate the performance of antipodal signaling and Viterbi decoding of convolutional codes in SalphaS noise. We demonstrate that the p-norm branch metric is robust in SalphaS noise.

Refinements of Multi-Track Viterbi Bit-Detection

In optical storage, data can be arranged on the disc in a meta-spiral consisting of a large number of bit-rows with a small track-pitch. Successive revolutions of the meta-spiral are separated by a narrow guard band. For high storage densities, such a system results in severe 2-D inter-symbol interference. In the multi-track Viterbi algorithm (MVA) of Krishnamoorthi and Weeks, the complex problem of 2-D bit-detection is broken down into a number of smaller bit-detection problems on sets of adjacent bit-rows called stripes. We improve the bit error rate (bER) performance of such a 2-D bit-detector via a number of measures such as: 1) weighing of the separate contributions to the branch metrics from each bit-row in a stripe and 2) inclusion of an additional contribution to the branch metrics from a bit-row adjacent to a stripe. In addition, we reduce the computational complexity by varying the number of bit-rows per stripe during successive iterations of the MVA, and through the use of local sequence feedback.

Golden Space–Time Trellis Coded Modulation

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

A new sequential decoding algorithm based on branch metric

A new sequential decoding algorithm with an adjustable threshold and a new method of moving through the decoding tree is proposed. Instead of the path metric of the conventional sequential decoding algorithms, the proposed algorithm uses a branch metric based on maximum-likelihood criterion. Two new parameters, the jumping-back distance and going-back distance, are also introduced. The performance of the algorithm for long constraint length convolutional codes is compared to those of the other sequential decoding algorithms and the Viterbi algorithm. The results show that the proposed algorithm is a good candidate for decoding of convolutional codes due to its fast decoding capability and good bit error rate (BER) performance.

Super-orthogonal space-time trellis codes with eight dimensional phase-shift keying signal constellations

We propose a new method of designing super-orthogonal space-time trellis codes (SOSTTCS) by labeling each state transition of the trellis diagram with an ordered pair of 2×2 orthogonal matrices instead of only a single 2×2 orthogonal matrix. An 8D instead of a 4D signal constellation is thus used for modulation and the delay elements of the trellis encoder are 4T instead of 2T, where T is the 2D symbol duration. Therefore, at the receiver, the sequence decoding using Viterbi algorithm requires two times less frequent updating of the branch metrics. We investigate by computer simulation the performance of the new sosttcs compared to sosttcs designed by the known method. The frame error rate (fer) and the bit error rate (ber) simulation results show that the performance of the new sosttcs is practically as good as that of the sosttcs of similar complexity designed along classical lines, with the advantage that the number of arithmetic and logic operations performed by the decoder per time unit is smaller, which is crucial in high data-rate applications.