Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes

The performance of maximum-likelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binary-input and output-symmetric communication channel, and their effectiveness is exemplified for various ensembles of turbo-like codes over the additive white Gaussian noise (AWGN) channel. An expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds     

Explicit evaluation of Viterbi's union bounds on convolutional code performance for the binary symmetric channel (Corresp.)

An explicit method is given to evaluate Viterbi's union bounds on both the first event error probability and the bit error probability of binary convolutional codes on a binary symmetric channel. These bounds are evaluated for the rate1/2code with generators1 + D + D^{2}and1 + D^{2}. Comparison is made with the bounds and experimental results of van de Meeberg.     

Bounds on performance for the noisy reference PSK channel

A lower bound is derived for the uncoded error probability of coherent digital phase-shift-keying without making any assumptions on phase-locked loop speed. An improved upper bound on bit error probability is also derived. The results are applicable to both lightwave homodyne detection, in which the laser's phase noise must be taken into account, and to general partially coherent detection of (binary phase-shift-keying) over the additive white Gaussian noise channel when a noisy phase reference is derived. Bit-error-rate and cut-off bounds are presented     

Tight bounds on Rician-type error probabilities and someapplications

Consider the classic problem of evaluating the probability that one Rician random variable exceeds another, possibly correlated, Rician random variable. This probability is given by Stein (1964) in terms of the Marcum's Q-function, which requires numerical integration on the computer for its evaluation. To facilitate application in many digital communication problems, we derive here tight upper and lower bounds on this probability. The bounds are motivated by a classic result in communication theory, namely, the error probability performance of binary orthogonal signaling over the Gaussian channel with unknown carrier phase. Various applications of the bounds are reported, including the evaluation of the bit error probabilities of MDPSK and MPSK with differential detection and generalized differential detection, respectively. The bounds prove to be tight in all cases. Further applications will be reported in the future     

Tight exponential upper bounds on the ML decoding error probability of block codes over fully interleaved fading channels

We derive tight exponential upper bounds on the decoding error probability of block codes which are operating over fully interleaved Rician fading channels, coherently detected and maximum-likelihood decoded. It is assumed that the fading samples are statistically independent and that perfect estimates of these samples are provided to the decoder. These upper bounds on the bit and block error probabilities are based on certain variations of the Gallager bounds. These bounds do not require integration in their final version and they are reasonably tight in a certain portion of the rate region exceeding the cutoff rate of the channel. By inserting interconnections between these bounds, we show that they are generalized versions of some reported bounds for the binary-input additive white Gaussian noise channel.     

Finite-Dimensional Bounds on Zm and Binary LDPC Codes With Belief Propagation Decoders

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zopf_m and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zopf _m LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zopf_m LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional noniterative bound, the latter of which is the best known bound that is tight for binary-symmetric channels (BSCs), and is a strict improvement over the existing bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels     

Error Rates for Fading NCFSK Signals in an Additive Mixture of Impulsive and Gaussian Noise

The effects of slow and nonselective signal fading on the performance of multilevel noucoherent FSK (NCFSK) systems in an additive mixture of Gaussian and highly impulsive noise are analyzed. For binary systems the bit error rate is derived; forM-ary NCFSK systems upper and lower bounds of the character error probability are obtained. The analysis is performed considering the maximum likelihood receiver for additive white Gaussian noise.     

On the capabilities of codes to correct synchronization errors

A synchronization error is said to occur when either a bit which does not belong appears, or is detected in a channel between bits which were transmitted; or a bit which was transmitted is lost or not detected. A model for such a channel will be proposed, and a lower and upper bound on the redundancy necessary to correct a given error rate will be derived. We will consider the case of single synchronization error correction in detail, and stronger bounds will be derived for that case. We will consider multiple adjacent synchronization errors as a special case, and show that the bounds can be tightened in this case as well.     

Upper Bound on Error Exponent of Regular LDPC Codes Transmitted Over the BEC

The error performance of the ensemble of typical low-definition parity-check (LDPC ) codes transmitted over the binary erasure channel (BEC) is analyzed. In the past, lower bounds on the error exponents were derived. In this paper, a probabilistic upper bound on this error exponent is derived. This bound holds with some confidence level.      

Performance comparison of different decoding strategies for abandwidth-efficient block-coded scheme on mobile radio channels

The performance of bandwidth-efficient Reed-Solomon (RS)-coded MPSK schemes is evaluated on a shadowed Rician fading channel using different decoding strategies, namely, errors-only, errors-and-erasures, and soft-decision decoding. The lower bounds of the bit error probability are found for errors-only and for errors-and-erasures decoding. For the soft-decision decoding the upper bound of the bit error rate is derived. The error bounds are calculated and examined by simulation for some RS-coded MPSK schemes on a shadowed Rician channel. It is shown that their performance is significantly improved compared to uncoded QPSK. The amount of improvement depends on the signal-to-noise ratio (SNR), the decoding strategy, and the degree of shadowing. A comparison between different decoding techniques, for one of the RS-coded schemes, for different degrees of shadowing shows that the use of channel measurement information in the decoding process is more effective for heavy shadowed channels     

On the relationship between mutual information and bit error probability for some linear dispersion codes

In this paper, we derive the relationship between the bit error probability (BEP) of maximum a posteriori (MAP) bit detection and the bit minimum mean square error (BMMSE). By using this result, the relationship between the mutual information and the BEP is derived for multiple-input multiple-output (MIMO) communication systems with the bit-linear linear-dispersion (BLLD) codes for the Gaussian channel. From the relationship, the lower and upper bounds on the mutual information can be derived.     

Bounds on the a priori index crossover probabilities fortrellis-based channel codes

This paper derives truncated union bounds on the a priori index crossover probabilities p(j|i) that result when an n-bit data index i is convolutionally encoded, transmitted over a noisy channel, and decoded with the Viterbi algorithm, giving received index j. The bounds are derived with a modified transfer function technique, using n-stage state transition matrices with symbolic labels. The technique is easily automated with commercial symbolic algebra packages. Bounds are obtained for convolutional and trellis-coded modulation (TCM) codes, over binary symmetric and additive white Gaussian noise (AWGN) channels. A joint source channel coding example demonstrates that the bounds on p(j|i) developed in this paper can give a 13-dB accuracy improvement in end-to-end signal-to-noise ratio (SNR) predictions, when compared to predictions based on bounds on the delivered bit error probability P_b     

A new upper bound on the first-event error probability for maximum-likelihood decoding of fixed binary convolutional codes (Corresp.)

An upper bound on the first-event error probability for maximum-likelihood decoding of fixed binary convolutional codes on the binary symmetric channel is derived. The bound is evaluated for rate1/2codes, and comparisons are made with simulations and with the bounds of Viterbi, Van de Meeberg, and Post. In particular, the new bound is significantly better than Van de Meeberg's bound for rates aboveR_{comp}.     

On general Gilbert bounds

We define a distance measure for block codes used over memoryless channels and show that it is related to upper and lower bounds on the low-rate error probability in the same way as Hamming distance is for binary block codes used over the binary symmetric channel. We then prove general Gilbert bounds for block codes using this distance measure. Some new relationships between coding theory and rate-distortion theory are presented.     

Bit Error Rate Evaluation for Spread-Spectrum Multiple-Access Systems

A unified method for approximating and bounding the average bit error probability for spread-spectrum multiple-access communication systems is presented. Various forms of direct-sequence spreadspectrum modulation are considered including binary phase-shift keying, quadriphase-shift keying, and minimum-shift keying. The analysis of the multiple-access interference makes use of a number of moments sufficent to evaluate the error probability with a high degree of accuracy. A computationally efficient algorithm for computing the moments is also given. The subsequent transformation from the moments to the average bit error probability is carried out by means of Gauss-type numerical integration formulas. It is shown that the same approach can be exploited for evaluating two classes of upper and lower bounds on the bit error rate. Finally, some results and comparisons are reported.     

Performance bounds of rate-? convolutional codes with QPSK on Rayleigh fading channel

An analytic expression for the bit error probability upper bounds of rate-? convolutional codes in conjunction with QPSK modulation and maximum-likelihood Viterbi decoding on the fully interleaved Rayleigh fading channel is presented. The given expression is evaluated numerically for selected rate-? optimum convolutional codes together with QPSK.     

Probability of bit error for MPSK modulation with diversityreception in Rayleigh fading and log-normal shadowing channel

In an additive white Gaussian noise channel subject to Rayleigh fading and log-normal shadowing, consideration is given to diversity reception with K-port macroscopic selection and L-branch microscopic maximal-ratio combining, and analytical expressions are derived for upper and lower bounds on the bit error probabilities for BPSK, QPSK, 8-PSK, and 16-PSK modulations. The derived expressions can be evaluated at any location in the serving cell. Error-probability can be calculated by averaging over all possible locations within the serving cell. By evaluating these bounds, numerical results for the following cases are presented: without diversity reception, microdiversity combining, macrodiversity selection, and simultaneous use of macrodiversity selection and microdiversity combining. For symmetrical arrangement of macrodiversity ports against the lognormal shadowing, error probability at the equidistant point gives upper bounds on the error probabilities for most of the area in the serving cell. Error probability at the equidistant point is a good estimate of the error probability obtained by averaging over all possible locations in the cell     

Soft decision decoding of block codes using received signalenvelope in digital mobile radio

The codeword error rate (WER) performance of noncoherent frequency-shift keying with soft decision decoding of block codes using Chase's second algorithm (1972) is investigated in a Rayleigh fading channel. The received signal envelope is sampled and used as channel measurement information. The theoretical upper and lower bounds of the WER are derived, assuming independent Rayleigh envelope samples in a received block. When the Golay (23, 12, 7) code is used, soft decision decoding with 6-bit error correction capability (3-bit error and 3-bit erasure) requires an average signal-to-noise power ratio about 5 dB lower than that for minimum distance decoding with 3-bit error correction to obtain a WER=10^-3. The effects of bit interleaving on the WER performance when fading envelope variation is slow compared to the bit rate are investigated through computer simulations     

Multiple-access capability of frequency-hopped spread-spectrumrevisited: an analysis of the effect of unequal power levels

A method for the evaluation of the probability of error of uncoded asynchronous frequency-hopped spread-spectrum multiple-access communications is presented. For systems with binary FSK modulation this method provides an accurate approximation and a tight upper bound to the bit error probability; for systems with M-ary FSK modulation, it provides tight upper bounds to the symbol error probability. The method enables the computationally efficient averaging of the error probability with respect to the delays, phase angles, and data streams of the different users. It relies on the integration of the product of the characteristic function of the envelope of the branch of the BFSK demodulator, which carries the desired signal, and of the derivative of the characteristic function of the envelope of the other branch. For sufficient frequency separation between the BFSK tones, the method can achieve any desirable accuracy. Moreover, the computational effort required for its evaluation grows linearly with the number of interfering users. In the M-ary case, tight upper bounds based on the union bound and the results of the binary case are derived. The method allows the effect of unequal power levels on other-user interference in FH/SSMA systems to be quantified accurately for the first time. The results indicate that the FH/SSMA systems suffer from the near-far problem, although less seriously than direct-sequence SSMA systems     

Bounds on the error performance of coding for nonindependentRician-fading channels

New upper bounds on the error performance of coded systems for Rician channels are presented. The fading channels need not be fully interleaved to obtain meaningful performance results. These bounds hold for coherent, differentially coherent and noncoherent demodulation of binary signals. They provide a useful analytical approach to the evaluation of the error performance of convolutional or block coding and they may be generalized to M-ary signals and trellis modulation. The approach allows for complex bounds using the fine structure of the code, for simpler bounds similar to those on memoryless channels and finally for a random coding bound using the cutoff rate of the channel. The analysis thus permits a step by step evaluation of coded error performances for Rician-fading channels