A note on the computation of the bit error rate for binary block codes

Space time block coding is a modulation scheme recently discovered for the transmit antenna diversity to combat the effects of wireless fading channels. Using the equivalent Single-Input Single-Output (SISO) model, this paper presents closed-form expressions for the exact Symbol Error Rate (SER) and Bit Error Rate (BER) of Orthogonal Space-Time Block Codes (OSTBCs) with M-ary Phase-Shift Keying (MPSK) and M-ary Quadrature Amplitude Modulation (MQAM) over flat uncorrelated Nakagami-m and Ricean fading channels.

This paper considers the optimal generator matrices of a given binary cyclic code over a binary symmetric channel with crossover probability p → 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. Our goal here is to seek the optimal realization of encoding functions by taking advantage of the structure of the codes, and to derive the probability of information bit error when possible. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many cyclic codes with such properties. We also present some binary cyclic codes whose optimal generator matrices are non-systematic under complete decoding.

Parallel Concatenated Gallager Code (PCGC) is a class of concatenated error correcting codes designed to reduce the encoding and decoding complexity of long block length Low Density Parity Check (LDPC) codes. In this paper, we present a novel class of PCGC in which one constituent code satisfies zero syndrome criteria and the other one satisfies non-zero syndrome criteria. Simulation results show that proposed non-zero syndrome PCGC outperforms existing non-zero syndrome PCGC in terms of its bit error rate (BER) performance in Binary Symmetric Channel (BSC). We also show that proposed concatenation method gives better performance than dedicated LDPC of same frame length and code rate over BSC.

Mobile satellite broadcast systems have the potential of delivering entertainment, information and other data services with outstanding coverage and reliability. However, the long signal interruptions due to obstacles in the terminal environment that characterizes the land mobile satellite channel make the design of efficient air interfaces challenging. In this paper, a file based mobile satellite broadcast system is studied where, to overcome the signal blockages, packet time diversity is applied. Based on some simple assumptions, an analytical expression for the file error rate of the system is derived. Using this result, a design scheme is proposed that, for a given channel and file, minimizes the total number of transmitted symbols required to satisfy quality of service (QoS) constraints in terms of acceptable delay and file receive probability. The results can also be used to estimate the reduction in bandwidth requirements that can be achieved if users in demanding environments can be served using terrestrial repeaters or other means.

ARQ methods of error control can considerably improve the reliablity of data transmission in such areas as satellite communications, computer networks, etc. A number of ARQ schemes using both block and convolutional codes have appeared in the literature. In this paper, the following problem is addressed. Given two different implementations of an ARQ scheme, one using a block code and the other using a convolutional code, such that the bit error probability of both implementations does not exceed some specific value, which implementation has the higher throughput and under what conditions will it be attained? The comparison is made for three basic retransmission schemes using both hybrid and pure ARQ: stop-and-wait, go-back- N , and selective repeat. Numerical estimates of the throughput were obtained using approximate theoretical expressions for BCH codes and simulation results for sequential decoding of rate 1/2 convolutional codes. Parameters optimizing the performance of both block and convolutional codes for different channel conditions and round trip delays were found and were used to obtain these numerical estimates. Comparison of the quantitative results indicates a trend toward preferring convolutional codes as delay and/or block length increases. A binary symmetric channel with noiseless feedback was assumed. Possible implications for the Gaussian channel are also discussed.

Analytical expressions for symbol error rate (SER) of M-ary dual ring star shaped quadrature amplitude modulation (QAM), when corrupted by additive white Gaussian noise (AWGN), have been derived. Utilizing the result for AWGN channel, SER expressions for various frequency non-selective slow fading channels (e.g. Rayleigh, Rician, and Nakagami-m) are also found. The results obtained are in the form of summation of single integrals which can be easily calculated through numerical methods. More importantly, compared to the erroneous results published earlier, the derived expressions are accurate. Extensive Monte Carlo simulations were performed to validate the analytical framework. The inadequacy of Gray's approximation for calculation of bit error rate (BER) has also been pointed out.

Tighter upper bounds on the error event and the bit error probabilities, respectively, for maximum-likelihood decoding of binary convolutional codes on the binary symmetric channel are derived from upper bounds previously published by Viterbi [1]. The measured bit error rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_b</tex> for a constraint length 3 decoder has been plotted versus the channel transition probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> and shows close agreement with the improved bound on the bit error probability.

In this paper, we investigate the effects of RF impairments and imperfect channel estimation (ICE) on the performance of Orthogonal Frequency Division Multiplexing (OFDM) wireless systems with transmit diversity for Space-Time Block Codes (STBC) employing Nt : Nt ∈ (2, 3, 4) transmit antennas and one receive antenna in a Rayleigh fading environ- ment. The bit error rate (BER) expressions of the OSTBC-OFDM system is derived for each transmit-receive antenna configuration in the presence of phase noise and channel estimation errors. With emphasis on complex signal constellations, we further investigate the effect of doppler on the performance of OSTBC- OFDM for different code rates R :1 /2 ≤ R ≤ 1, including maximal-rate OSTBC codes. To illustrate the extent of system degradation due to these impairments on a proposed practical system, the presented results are verified through simulations by making use of the system parameters from the IEEE802.16e standard.

The estimation of the error rates is of vital importance in classification problems, as this is used as a basis to choose the best discriminant function, i.e. the one with a minimum misclassification error. The quadratic discriminant function (QDF) has been in use for a long time for the purpose of classification. However, there is no exact analytical expression for the misclassification error rate based on QDF. Consequently, researchers depend on use of simulation trials to gauge the misclassification error. This paper provides exact formulae for the misclassification error rates associated with QDF involving two multivariate normal populations. The results are obtained for the general case where the means and the covariance matrices for the two populations are unequal.

In this work, coherent modulation Terahertz communication link with pointing errors are analyzed for the Gamma-Gamma turbulence channels. The high-precision bit error rate and outage probability expressions for the Gamma-Gamma turbulent channel using coherent quadrature phase-shift keying signal are derived, and the bit error rate and outage probability are analyzed for its asymptotical property. The results show that our series error rate expression can accurately evaluate the system performance. It has high precision in estimating and predicting actual error rates.

A model-based framework of classification error rate estimation is proposed for speech and speaker recognition. It aims at predicting the run-time performance of a hidden Markov model (HMM) based recognition system for a given task vocabulary and grammar without the need of running recognition experiments using a separate set of testing samples. This is highly desirable both in theory and in practice. However, the error rate expression in HMM-based speech recognition systems has no closed form solution due to the complexity of the multi-class comparison process and the need for dynamic time warping to handle various speech patterns. To alleviate the difficulty, we propose a one-dimensional model-based misclassification measure to evaluate the distance between a particular model of interest and a combination of many of its competing models. The error rate for a class characterized by the HMM is then the value of a smoothed zero-one error function given the misclassification measure. The overall error rate of the task vocabulary could then be computed as a function of all the available class error rates. The key here is to evaluate the misclassification measure in terms of the parameters of environmental-matched models without running recognition experiments, where the models are adapted by very limited data that could be just the testing utterance itself. In this paper, we show how the misclassification measure could be approximated by first computing the distance between two mixture Gaussian densities, then between two HMMs with mixture Gaussian state observation densities and finally between two sequences of HMMs. The misclassification measure is then converted into classification error rate. When comparing the error rate obtained in running actual experiments and that of the new framework, the proposed algorithm accurately estimates the classification error rate for many types of speech and speaker recognition problems. Based on the same framework, it is also demonstrated that the error rate of a recognition system in a noisy environment could also be predicted.

Infinite integrals involving the products of two and three Gaussian Q-functions of different arguments are solved in closed-form. These solutions provide novel expressions in the error rate analyses of signals such as general rectangular quadrature amplitude modulation, differentially encoded quaternay phase shift keying and new solutions for the error rate of the repetition code. As an example, closed-form expressions for the average symbol error rates of general rectangular quadrature amplitude modulation and the repetition code are obtained for a Nakagami-m fading channel.

New closed form expressions for the average error rate of MC-CDMA systems in Nakagami-m fading channels are derived based on the exponential approximation of the Q-function. Numerical results by substituting in these expressions are presented and compared to results provided by other researchers using computer simulations or numerical integrations. From the comparison, it is concluded that the derived expressions can be used with good accuracy.

This paper considers the optimal generator matrices of a binary cyclic code over a binary symmetric channel with crossover probability p rarr 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many binary cyclic codes satisfying these properties.

A complete decoder is one that guarantees to decode a received vector to its nearest codeword. For the design of an efficient complete decoding algorithm, storage demand and decoding complexity are two essential factors. Along this consideration, the Zero-Guards algorithm provides a significant improvement as the number of vectors required to be stored has been made much less than what is required by the conventional standard array decoding. When being applied to certain block codes, however, it may be inefficient when code length is short. At this background, a new step-by-step complete decoding algorithm for cyclic codes is proposed in this work. Based on a newly devised transfer function, our algorithm can achieve a better decoding efficiency than the Zero-Guards algorithm, while keeping a smaller number of vectors for most of the cyclic codes examined.