Performance evaluation of FSK and CPFSK optical communicationsystems: a stable and accurate method

The modified moments method for evaluating the performance of coherent optical FSK and CPFSK systems is presented. Since the classical procedure becomes ill-conditioned as the order of the moments increases, we consider the construction of Gaussian quadrature rules (GQR) from the modified moments. The analysis accounts for the influences of IF bandwidth, transmitter and local oscillator laser phase noise, postdetection filters, and additive Gaussian noise. It is found that the proposed approach is a highly reliable and efficient method for calculating the error probability. A comparison with results obtained from the Gaussian quadrature rule, Gaussian approximation method, and analytical approximation formulas shows that this technique is very accurate. Analytical expressions are derived for FSK and CPFSK receivers which include polarization and phase diversity techniques. The use of numerical programming to avoid many unnecessary computations is discussed. This evaluation method can be used to account for the effects of crosstalk in multichannel systems and the influence of error-control codes     

Performance evaluation of ASK phase-diversity lightwave systems

Phase diversity coherent systems are gaining increasing interest with the increase of the bit rate in optical transmissions. In this work, an accurate approach for the evaluation of the error probability in ASK phase-diversity lightwave systems is presented. The approach is based on a Gaussian quadrature rule integration, which takes into account the exact characterization of the filtered phase noise through its moments. The effects of the laser linewidth on the error probability for hybrid circuits with different number of ports are derived and discussed. Also, the effects of the frequency offset are considered     

Calculating the mean and variance of power sums with two log-normalcomponents

An efficient method for accurately calculating the mean and variance of power sums with two log-normal components is presented. It involves numerical quadrature with integrands consisting of Gaussian probability density functions. Only the technique of transformation and the trapezoidal rule are used in the evaluation. The whole computational work and roundoff error are much lower than the one by Schwartz and Yeh (1982). Because of its high efficiency, no least-squares fit is necessary     

Combined Effects of Intersymbol, Interchannel, and Co-Channel Interferences in M-ary CPSK Systems

An expression for the error probability of multilevel coherent phase-shift-keyed (CPSK) systems in the presence of intersymbol, interchannel, and co-channel interferences and additive Gaussian noise is derived. An exact expression is given for the binary CPSK system, whereas upper and lower bounds are presented for the multilevel systems. The approach proposed in this paper overcomes the difficulties of the exhaustive methods and allows accurate and fast evaluation of the character error probability. Only the impulse responses of the overall channels are needed, and the computational labor is reduced to the numerical evaluation of one particular type of integral, whose computation is based upon nonclassical Gaussian quadrature rules. The model of the system is general enough to allow the choice of a rectangular or otherwise-shaped modulating pulse and a constant or shaped envelope of the modulating carrier. Phase incoherence between adjacent carriers is also assumed, and a random misalignment among the modulating bit streams can be taken into account, if necessary. Extensive numerical results are presented for 2- and 4-level systems. The presentation of the results stresses a possible utilization of them in system design problems.     

Jensen-cotes upper and lower bounds on the gaussian Q-function and related functions

We present new families of lower and upper bounds on Q-functions. First, we consider the Craig form of the Gaussian Q-function Q(ξ) and shown that its integrand ϕ(ϕ; ξ) can be partitioned into a pair of complementary convex and concave segments. This property is then exploited in conjunction with the Jensen inequality and the Newton-Cotes' quadrature rule to produce a complete family of upper and lower bounds on Q(ξ), which can be made arbitrarily tight by finer segmentation. The basic idea is then utilized to derive families of upper and lower bounds also for the squared Gaussian Q-function Q/²(ξ), the 2D joint Gaussian Q-function Q(x, y, p), and the generalized Marcum Q-function Q_M(x, y). The bounds are shown to be tighter than alternatives found in the literature, and in some cases the lower bounds provided find no equivalent in current literature. The generality of the principle is the elegant point of the method and the resulting Jensen-Cotes bounds are easy to implement and evaluate since only elementary transcendental functions are involved. As an example of application to the analysis of communication systems, we consider the bit error rates (BER's) of decode-and-forward (DF) cooperative relaying schemes with coherent and differential phase-shift keying (PSK) modulations, which have been shown to have an intricate dependence on the Gaussian Q-function, complicated by crossproducts, irrational functional arguments and multiple numerical integrations. In that example the bounds substantially reduce the complexity required to evaluate the expressions, retaining tightness despite multiple numerical integrations with infinite limits.i     

Exponential error bounds for random codes on Gaussian arbitrarily varying channels

The main objective is to develop exponential bounds to the best error probability achievable with random coding on the Gaussian arbitrarily varying channel (GAVC) in the one case where a (strong) capacity exists (i.e., with peak time-averaged power constraints on both the transmitter and interference). The GAVC models a channel corrupted by thermal noise and by an unknown interfering signal of bounded power. The upper and lower bounds to the best error probability achievable on this channel with random coding are presented. The asymptotic exponents of these bounds agree in a range of rates near capacity. The exponents are universally larger than the corresponding exponents for the discrete-time Gaussian channel with the same capacity. It is further shown that the decoder can be taken to be the minimum Euclidean distance rule at all rates less than capacity.<>     

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     

数值积分的校正公式研究

讨论了积分校正公式的一般形式,利用代数精度的概念给出了确定了校正公式中的参数方法。给出了左矩形公式、梯形公式、辛甫生公式、一点高斯公式、二点高斯公式的校正公式,其它公式可类似处理。通过数值实例进行了比较。     

Performance analysis of asymptotically optimal noncoherentdetection of trellis-coded multi-amplitude/-phase modulation signals inGaussian noise and ISI channels

Performance upper bounds for noncoherent receivers employed in conjunction with single and multi-amplitude/-phase signals, transmitted over time dispersive and Gaussian noise channels are derived. Based upon a metric which has been previously derived by the authors, we present analytical expressions and computer generated results for the performance of asymptotically optimal noncoherent detection over such channels. As a typical application of the developed theoretical analysis, we consider wideband telecommunication systems. Where time dispersion resulting in intersymbol interference (ISI) is one of the significant sources of system performance degradation. Numerical evaluation of the optimal noncoherent decoding algorithms, shows the proposed bounds to be an effective and efficient means of evaluating the performance of the noncoherent receivers under investigation. Using the derived bounds, performance evaluation results for modulation schemes such as π/4-shift DQPSK (differential quadrature phase shift keying), 8- and 16-DQAM (differential quadrature amplitude modulation), at very low bit-error rates (BER), which would otherwise pose impractically high computational loads when using Monte-Carlo error counting techniques, are readily obtained. At BER>10^-4 evaluation results generated via computer simulation have verified the tightness of the bounds     

Modified Chernoff bounds for PAM systems with noise and interference (Corresp.)

By using principles of analytic continuation, upper and lower bounds on the error probability of a canonical binary system corrupted by additive interference and independent zero-mean Gaussian noise are derived. The bounds, which are simple functions, require only the evaluation or bounding of the moment generating function of the interference. For large signal-to-noise ratios (SNR) and even for moderately large interference, the bounds are shown to be tight and useful.     

Accurate performance evaluation of weakly coherent optical systems

An accurate performance evaluation approach which uses a closed-form exact analytical expression of the phase noise moments is presented. This enables one to derive a high-order Gaussian quadrature rule for the integrations needed to take into account the phase noise in the computation of error probability. A systematic comparison with results obtained through a Monte Carlo simulation shows that the approach is more accurate than previous methods. The analysis is performed on ASK and FSK heterodyne receivers with integrate-and-dump filtering, envelope detection, and optimized postdetection low-pass filtering. The feasibility of ASK and FSK heterodyne systems at bit rates comparable to the spectral line bandwidth of the laser sources is confirmed. The theory applied seems to be adequate to attack other problems, such as the evaluation of the effects of crosstalk between the FSK filters or among frequency division multiplexed channels     

Bit Error Probability of Trellis-Coded Quadrature Amplitude Modulation Over Cross-Coupled Multidimensional Channels

Convolutionally encodedM-ary quadrature amplitude modulation (M-QAM) systems operated over multidimensional channels, for example dual-polarized radio systems, are considered in this paper. We have derived upper bounds on the average bit-error probability for 4QAM (QPSK) with conventional convolutional coding by means of a truncated union bound technique and averaging over the cross-polarization interference by means of the method of moments. By modifying this technique, we have found approximate upper bounds on the average biterror probability for bandwidth efficient trellis-coded QAM systems. This is an extension of our previous work [1] that was based on one dominating error event probability as a performance measure. Our evaluations seem to indicate that bandwidth efficient trellis-codedMQAM schemes offer much larger coding gains in an interference environment, e.g., a cross-coupled interference channel, than in a Gaussian noise channel. In general, our findings point out that optimum codes for a Gaussian channel are not optimum when applied in an interference environment. We note that a rate 1/2 convolutional code for example, with a code memory greater than two, if applied to two of the bits in each signal point representation, can be utilized with a simple decoder to greatly improve the performance of a QAM signal in interference. Also, we have introduced a new concept referred to as dualchannel polarization hopping in this paper which can improve the system performance significantly for systems with nonsymmetrical interference.     

Computing and bounding the first-order Marcum Q-function: a geometric approach

A geometric interpretation of the first-order Marcum Q-function, Q(a,b), is introduced as the probability that a complex, Gaussian random variable with real mean a, takes on values outside of a disk C_O,b of radius b centered at the origin O. This interpretation engenders a fruitful approach for deriving new representations and tight, upper and lower bounds on Q(a,b). The new representations obtained involve finite-range integrals with pure exponential integrands. They are shown to be simpler and more robust than their counterparts in the literature. The new bounds obtained include the generic exponential bounds which involve an arbitrarily large number of exponential functions, and the simple erfc bounds which involve just a few erfc functions, together with exponential functions in some cases. The new generic exponential bounds approach the exact value of Q(a,b) as the number of exponential terms involved increases. These generic exponential bounds evaluated with only two terms and the new simple erfc bounds are much tighter than the existing exponential bounds in most cases, especially when the arguments a and b are large. Thus, in many applications requiring further analytical manipulations of Q(a,b), these new bounds can lead to some closed-form results which are better than the results available so far.     

Asymptotically tight bounds on the capacity and outage probabilityfor QAM transmissions over Rayleigh-faded data channels with CSI

In this article, novel quickly computable analytical upper and lower bounds are presented on the symmetric capacity for flat-faded Rayleigh channels with finite-size quadrature amplitude modulation constellations when perfect channel-state information at the receiving site is available; the proposed bounds are asymptotically tight both for high and low signal-to-noise ratios. Furthermore, an easily computable expression is also provided for a reasonably tight evaluation of the resulting outage probability     

Probability of an arbitrary wedge-shaped region of the MPSK system in the presence of quadrature error

The paper presents a new and simple expression for computing the probability on an arbitrary wedge-shaped region between two different phase angles when the signal vector is perturbed by two-dimensional Gaussian noise in the presence of a quadrature error. The two-dimensional Gaussian Q-function is used to characterize the expression. The expression can be very useful for computing the analytical error probability of the M-ary phase shift keying system operating in various channel environments.     

Evaluation of the exact union bound for trellis-coded modulationsover fading channels

An analytical technique is presented for computing the exact union bound on the average bit error probability of trellis coded modulation schemes over Rayleigh, Rician, or shadowed Rician-fading channels. To this end, an integral expression is derived for the pairwise error event probability (PEP). Existing bounds can be obtained as special cases of this expression. It turns out that a Gauss-Chebyshev quadrature rule offers excellent accuracy for this integral. By extension, the exact union bound (i.e., the weighted sum of an exact PEPs of a code) can readily be evaluated. This method has the same complexity as the union-Chernoff bound, and a few examples are given to show its application     

Moment space upper and lower error bounds for digital systems with intersymbol interference

A method for the evaluation of upper and lower bounds to the error probability of a linear pulse-amplitude modulation (PAM) system with bounded intersymbol interference and additive Gaussian noise is obtained via an isomorphism theorem from the theory of moment spaces. These upper and lower bounds are seen to be equivalent to upper and lower envelopes of some compact convex body generated from a set of kernel functions. Depending on the selection of these kernels and their corresponding moments, different classes of bounds are obtained. In this paper, upper and lower bounds that depend on the absolute moment of the intersymbol interference random variable, the second moment, the fourth moment, and an "exponential moment" are found by analytical, graphical, or iterative approaches. We study in detail the exponential moment case and obtain a family of new upper and a family of new lower bounds. Within each family, expressions for these bounds are given explicitly as a function of an arbitrary real-valued parameter. For two channels of interest, upper and lower bounds are evaluated and compared. Results indicate these bounds to be tight and useful.     

Discrete-Time Nonlinear Filtering Algorithms Using Gauss–Hermite Quadrature

In this paper, a new version of the quadrature Kalman filter (QKF) is developed theoretically and tested experimentally. We first derive the new QKF for nonlinear systems with additive Gaussian noise by linearizing the process and measurement functions using statistical linear regression (SLR) through a set of Gauss-Hermite quadrature points that parameterize the Gaussian density. Moreover, we discuss how the new QKF can be extended and modified to take into account specific details of a given application. We then go on to extend the use of the new QKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. A bank of parallel QKFs, called the Gaussian sum-quadrature Kalman filter (GS-QKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. The weights are obtained from the residuals of the QKFs. Three different Gaussian mixture reduction techniques are presented to alleviate the growing number of the Gaussian sum terms inherent to the GS-QKFs. Simulation results exhibit a significant improvement of the GS-QKFs over other nonlinear filtering approaches, namely, the basic bootstrap (particle) filters and Gaussian-sum extended Kalman filters, to solve nonlinear non- Gaussian filtering problems.     

New exponential bounds and approximations for the computation of error probability in fading channels

We present new exponential bounds for the Gaussian Q function (one- and two-dimensional) and its inverse, and for M-ary phase-shift-keying (MPSK), M-ary differential phase-shift-keying (MDPSK) error probabilities over additive white Gaussian noise channels. More precisely, the new bounds are in the form of the sum of exponential functions that, in the limit, approach the exact value. Then, a quite accurate and simple approximate expression given by the sum of two exponential functions is reported. The results are applied to the general problem of evaluating the average error probability in fading channels. Some examples of applications are also presented for the computation of the pairwise error probability of space-time codes and the average error probability of MPSK and MDPSK in fading channels.     

New Simple and Efficient Bounds on the Probability of Error in the Presence of Interference and Gaussian Noise

The problem of determining or of bounding the probability of error in the presence of intersymbol interference and Gaussian noise is an important one in data communications. In this concise paper we describe new upper and lower bounds for the probability of error when the peak distortion is finite (in practice, when the eye is open) and the data are independent. Over a range ofS/Nof interest in data communications; these bounds compare favorably with previously published bounds of similar complexity, especially when the eye opening is small. They can be applied to multilevel AM and CPSK transmission.