Branch Metrics Research Articles

Equalization and detection for nonlinear recording channels with correlated noise

At high recording densities, signal nonlinearity due to both the medium and the magnetoresistive head and correlated noise due to the medium and the equalization process, substantially degrade the error-rate performance of the Viterbi algorithm (VA). This paper describes a method for dealing with both problems by simultaneously adapting the equalizer, its target and the VA branch metrics. A monic constraint is employed to locate a target that limits noise correlation at the input to the VA. Pattern-dependent offsets, rather than actual VA ideal values, are adaptively determined to account for channel nonlinearity in the VA. The effectiveness of these techniques is demonstrated on very high density data collected from a spin stand.

Adaptive state allocation algorithm in MLSD receiver for multipath fading channels: structure and strategy

This paper presents the adaptive state allocation (ASA) algorithm, a new scheme based on maximum likelihood sequence detection (MLSD) of signals transmitted over Rayleigh fading channels. Although MLSD is an optimal scheme, its computational complexity limits many applications. The ASA algorithm is a detection method whose performance is close to that of MLSD, but with greatly reduced computational complexity. Adaptive state partitioning in the trellis diagram is used in this algorithm by measuring the short-term received signal power. Also, an adaptive threshold for selecting only a few states of the trellis is employed in this algorithm based on the Chernoff distance between the probability density functions (PDFs) of correct and incorrect branch metrics. The ASA-DF, a special case of ASA using decision feedback, shows a very good tradeoff between the performance and computational complexity for selective fading channels. Using ASA with diversity reception not only improves the performance, but also decreases the computational complexity in comparison with the computational complexity of using MLSD with diversity reception.

A lower bound on the bit‐error‐rate resulting from mismatched viterbi decoding

AbstractA lower bound is derived on the bit error rate that results when a rate l/n convolutionally encoded binary data stream is transmitted over a noisy symmetric channel, and is then decoded using a mismatched Viterbi decoder, i.e., a Viterbi decoder that performs maximum‐likelihood sequence estimation using possibly incorrect branch metrics. The branch metrics are assumed to be symmetric, but are generally differrent from the log‐likelihood function. The proposed bound is expressed in terms of the pairwise error probability for a given error event normalized by the sum of the events length and the code's memory. While the bound can be somewhat loose for medium signal‐to‐noise ratios, it is usually asymptotically tight at high signal‐to‐noise ratios. For the special case where the branch metrics employed are equal to the log‐likelihood function our lower bound does not coincide with the standard bound that was derived by Forney and Mazo. It falls short by a multiplicative constant that depends on the code's memory and the shortest error event of minimum Hamming weight. We apply our lower bound to the study of convolutionally encoded direct‐sequence spread‐spectrum communication with Laplacian noise and show that nearest‐neighbor decoding, which is optimal for Gaussian noise, is sub‐optimal and asymptotically (for high processing gain) results in a 3 dB loss in performance when compared with the optimal maximum‐likelihood decoder. Extensions to rate c/n convolutional encoders, general trellis encoders and to symmetric channels with memory are also presented.

Performance analysis of Viterbi decoder using channel state information in COFDM system

This paper relates to the application method of channel state information (CSI) to the Viterbi (maximum likelihood) decoder in the digital terrestrial television broadcasting system. The proposed Viterbi decoder uses the CSI derived from the pilots inserted in the transmitter of the COFDM (coded orthogonal frequency division multiplexing) system. The CSI is calculated by interpolation using the pilots in the receiver. The active real (I) and imaginary (Q) data after equalization are transferred to the branch metric calculation block that decides the euclidean distance for soft decision decoding and also the estimated CSI values are transferred to the same block. After calculating the euclidean distance for the soft decision decoding, the euclidean distance of the branch metric is multiplied by CSI. To do so, new branch metric values that consider each carrier state information are obtained. We simulated this method in Rayleigh fading defined in the ETSI standard. From the simulation, this method has better performance of about 0.15 dB to 0.7 dB and 2.2 dB to 2.9 dB in the Rayleigh channel than that of conventional soft decision Viterbi decoding with or without a bit interleaver where the constellation is QPSK, l6-QAM and 64-QAM.

Reduced state Viterbi differential phase detectionof MDPSK signals

An improved differential phase detection scheme that uses a reduced state Viterbi algorithm (RSVDPD) is proposed for the reception of M-ary differential phase shift keying (MDPSK) signals. The transmitted sequence is estimated from the received signal phases only. The number of trellis states equals that of the signal constellation points M. The branch metric is computed by tracing back the surviving path connected to each state. The bit error rate (BER) performance with RSVDPD is evaluated using computer simulation for 4DPSK transmission in additive white Gaussian noise (AWGN) channels.

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 C/sub 1/ of smaller dimension and shorter length. Utilizing this fact, a two-stage optimum trellis decoding method is devised. The first stage decodes C/sub 1/ 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.

Decoding mechanism of a scarce state transition‐type viterbi decoder for general nonsystematic convolutional codes and its asymptotic behavior on a very noisy channel

AbstractWhen a scarce state transition (SST) Viterbi decoder is used to decode general nonsystematic convolutional codes (not included among quick look‐in (QLI) codes), metrics based on the maximum a posteriori probability (MAP) estimation method must be computed in order for the main decoder to operate under the criterion of minimizing the decoding error probability. In this case, however, it becomes necessary to extend the original code trellis structure by introducing augmented trellis states in order to incorporate the newly induced correlation among the apparent information sequence.This paper shows that the extended trellis degenerates into the original one as the channel error probability q approaches one‐half. By performing an equivalent transformation of metrics, it is shown also that the branch metrics derived based on the MAP estimation method converge to those computed using the conventional one as q approaches one‐half.

The pairwise error probability for decoding a convolutionally encoded sequence using integer metrics

AbstractThe performance of a communication system that operates over a binary‐input, output‐symmetric memoryless channel and employs a convolutional error correcting code is usually evaluated by applying the union bound to the node error and the bit error probabilities. These bounds depend on the probability that an incorrect path on the code trellis accumulates a larger metric than the correct path. The paper presents an easy method to compute this pairwise error probability that is applicable to the significant case of a Viterbi decoder that uses a finite set of integer branch metrics.

A blind sequence detection and its application to digital mobile communication

A method of sequence detection without the knowledge of the channel response is proposed. The Viterbi algorithm is extended to a blind form by introducing a new branch-metric, defined as the minimum of the short time-average of the squared error (y/spl circ/k-yk)/sup 2/, where yk and y/spl circ/k are the received signal and its replica, respectively. All possible candidate sequences contained in the short time squared error are defined as trellis states, for which the short time squared error is minimized in respect to a variable of the unknown channel response. The proposed implicit blind sequence detection need not keep each variable after the branch metric is calculated. The consistency of the algorithm is justified by proving that a unique sequence is detected in noise free case. The proof is accomplished under condition that the period of the time-average is longer than the channel response. If the additive white Gaussian noise is assumed, the short time squared errors are minimized beyond the desired minimum by the standard Viterbi algorithm using an apriori known channel response. We call this phenomenon over-minimization. The over-minimization is a major reason of the unavoidable error rate degradation in the blind receiver. An objective of this paper is to establish blind sequence detection for digital mobile communication. Provided the channel response can be regarded as time-invariant during the period of the short time-average of the squared error, the blind sequence detection keeps the same performance as in time-invariant case. In order to improve the error rate performance, an algorithm based on the fractional sampling scheme is introduced. Several error rate performances and behaviors of error events are investigated by computer simulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Approximate ML decoding of coded PSK with no explicit carrier phase reference

An approximate method is discussed for maximum likelihood estimation of coded PSK sequences in the presence of an unknown carrier phase. This method makes use of the Viterbi algorithm to find an optimum path in the encoder trellis diagram. Phase estimation is embedded into the structure of the detection algorithm and the branch metrics in the trellis are modified with respect to those employed with known carrier phase. Comparisons with traditional decoding schemes show that the proposed solution has better performance in terms of acquisition time and cycle slips.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Detection of TCM signals on severely distorted channelsusing survival path feedback equaliser

A new technique for detecting trellis code modulation signals in severely distorted channels is proposed. This technique is based on feedback of the survival paths from the Viterbi decoder and the equaliser uses these survival paths to update the branch metrics of the decoder. Simulation results clearly show that the system performance is significantly improved.

Design of a viterbi equalizer with field programmable gate arrays

The design of a Viterbi equalizer, suitable for the detection of binary data bursts received in the presence of intersymbol interferences, is presented in detail. The main advantages of this Viterbi equalizer are real-valued signal processing at a conveniently chosen intermediate frequency and its flexible signal processing due to the application of interchangeable EPROM look-up tables for the calculation of the branch metrics. The implemented Viterbi equalizer is versatile and therefore well suited for use in undergraduate practice on communications technology. The design is realized by using XILINX series XC3000 logic cell arrays, allowing rapid prototyping at a low price.

Quantization loss in convolutional decoding

The loss in quantizing coded symbols in the additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) or quadrature phase-shift keying (QPSK) modulation is discussed. A quantization scheme and branch metric calculation method are presented. For the uniformly quantized AWGN channel, cutoff rate is used to determine the step size and the smallest number of quantization bits needed for a given bit-signal-to-noise ratio (E/sub b//N/sub 0/) loss. A nine-level quantizer is presented, along with 3-b branch metrics for a rate-1/2 code, which causes an E/sub b//N/sub 0/ loss of only 0.14 dB. These results also apply to soft-decision decoding of block codes. A tight upper bound is derived for the range of path metrics in a Viterbi decoder. The calculations are verified by simulations of several convolutional codes, including the memory-14, rate-1/4 or -1/6 codes used by the big Viterbi decoders at JPL.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A Bayesian maximum-likelihood sequence estimation algorithm for a priori unknown channels and symbol timing

It is shown that the optimum demodulator for the case of an a priori unknown channel and symbol timing can be approximated using a modified Viterbi algorithm (VA), in which the branch metrics are obtained from the conditional innovations of a bank of extended Kalman filters (EKFs). Each EKF computes channel and timing estimates conditioned on one of the survivor sequences in the trellis. It is also shown that the minimum-variance channel and timing estimates can be approximated by a sum of conditional EKF estimates, weighted by the VA metrics. Simulated bit error rate (BER) results and averaged-squared channel/timing error trajectories are presented, with estimation errors compared to the Cramer-Rao lower bound. The BER performance of the modified VA is also shown to be superior to that obtained using a decision-directed channel/timing estimation algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Decoding of trellis-encoded signals in the presence of intersymbol interference and noise

A novel receiver for data-transmission systems using trellis-coded modulation is investigated. It comprises a whitened-matched filter and a trellis decoder which combines the previously separated functions of equalization and trellis-coded modulation (TCM) decoding. TCM encoder, transmission channel, and whitened-matched filter are modeled by a single finite-state machine with combined intersymbol interference and code states. Using ISI-state truncation techniques and the set-partitioning principles inherent in TCM, a systematic method is then developed for reducing the state complexity of the corresponding ISI and code trellis. A modified branch metric is used for canceling those ISI terms which are not represented by the trellis states. The approach leads to a family of Viterbi decoders which offer a tradeoff between decoding complexity and performance. An adaptive version of the proposed receiver is discussed, and an efficient structure for reduced-state decoding is given. Simulation results are presented for channels with severe amplitude and phase distortion. It is shown that the proposed receiver achieves a significant gain in noise margin over a conventional receiver which uses separate linear equalization and TCM decoding.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

High rate concatenated coding systems using bandwidth efficient trellis inner codes

High-rate concatenated coding systems with bandwidth-efficient trellis inner codes and Reed-Solomon (RS) outer codes are investigated for application in high-speed satellite communication systems. Two concatenated coding schemes are proposed. In one the inner code is decoded with soft-decision Viterbi decoding, and the outer RS code performs error-correction-only decoding (decoding without side information). In the other the inner code is decoded with a modified Viterbi algorithm, which produces reliability information along with the decoded output. In this algorithm, path metrics are used to estimate the entire information sequence, whereas branch metrics are used to provide reliability information on the decoded sequence. This information is used to erase unreliable bits in the decoded output. An errors-and-erasures RS decoder is then used for the outer code. The two schemes have been proposed for high-speed data communication on NASA satellite channels. The rates considered are at least double those used in current NASA systems, and the results indicate that high system reliability can still be achieved.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Performance of a viterbi decoder using simplified digital metric calculation

AbstractRecently, convolutional coding with Viterbi decoding has been used widely in digital satellite communication systems. This paper addresses the problem of simplifying the process of digital metric calculation in a Viterbi decoder to realize reduced hardware complexity and increased operating speed.An analytical expression is introduced for the bit error rate (BER) performance of such a Viterbi decoder, and effects of the simplifications are evaluated for a typical convolutional code. It is shown that measured data agree well with theoretical curves, and that the simplifications can be introduced with no significant performance degradation. Finally, a method is proposed to design Viterbi decoders so that the degree of soft decision is increased and the number of bits for branch metrics is reduced to a specified value using a nonlinear transformation. For a constant number of bits for branch metrics, the BER performance realized by the proposed design method can be superior to that realized by the conventional design method. Results in this paper would be useful in designing simple Viterbi decoders for high‐speed applications.

A branching process analysis of the average number of computations of the stack algorithm

A new analysis for bounding the average computational effort of sequential decoding is presented. It is based on a branching process model of the subset of incorrect paths explored by the decoder. Closed-form expressions formulated in terms of the set of branch metrics and their probability assignments as actually used by the decoder are derived. Code characteristics in the form of Hamming weights of the 'incorrect paths or column distance function are incorporated, making the analysis directly applicable to specific cases. Evaluation of the bounds is not difficult. Results are in good agreement with those obtained by lengthy computer simulations.

The contour extraction problem with biomedical applications

The problem of extracting a contour of some sort from noisy digitized images often arises within the context of the general pattern recognition problem. The approach presented here formulates the contour extraction problem as one of minimum cost tree searching. Branch costs or metrics are defined which are indicative of the likelihood that a particular branch lies on the true contour. The branch metrics incorporate both local and global or contextual information. The most likely path or contour is then extracted by application of a heuristic tree searching algorithm. In particular, use is made of the Zigangirov-Jelinek (ZJ) stack algorithm. This algorithm avoids the computational burden associated with structured or exhaustive search techniques and results in more manageable although random computational requirements. The approach is described first on an artificially contrived problem and then applied to two real biomedical image processing problems. The biomedical problems considered are concerned with the extraction of the left ventricular outline from serial angiocardiograms and the extraction of lung outlines from digitized chest X rays. It is demonstrated that the approach is fast, reliable, and versatile in the sense that it can be applied to a wide variety of contour extraction problems by appropriate definition of branch metrics.