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.     

A new twist on the Marcum Q-function and its application

A new form of the Marcum (1950) Q-function is presented that has both computational and analytical advantages. The new form is particularly useful in simplifying and rendering more accurate the analysis of the error probability performance of uncoded and coded partially coherent, differentially coherent, and noncoherent communication systems in the presence of fading. It also enables simple upper and lower bounds to be found analogous to the Chernoff bound on the Gaussian Q-function     

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     

An efficient technique for evaluating direct-sequencespread-spectrum multiple-access communications

A technique is presented for obtaining bounds on the average probability of error for direct-sequence spread-spectrum multiple-access (DS/SSMA) communications. The technique is of interest because it yields arbitrarily right bounds, involves a small amount of computation, avoids numerical integrations, and applies to many types of detection. As an illustration, the technique is applied to binary DS/SSMA communications, an additive white Gaussian noise channel, and a coherent correlation receiver. It is assumed that all the signature sequences are deterministic. Each transmitter is assumed to have the same power, although the approach can accommodate the case of transmitters with unequal powers. Expressions are given for the density functions of the random variables that model the multiple-access interference. These expressions are used to obtain arbitrarily tight upper and lower bounds on the average probability of error without making a Gaussian approximation or performing numerical integrations to incorporate the effects of multiple-access interference     

General Capacity Bounds for Spatially Correlated Rician MIMO Channels

This paper considers the capacity of spatially correlated Rician multiple-input multiple-output (MIMO) channels. We consider the general case with double-sided correlation and arbitrary rank channel means. We derive tight upper and lower bounds on the ergodic capacity. In the particular cases when the numbers of transmit and receive antennas are equal, or when the correlation is single sided, we derive more specific bounds which are computationally efficient. The bounds are shown to reduce to known results in cases of independent and identically distributed (i.i.d.) and correlated Rayleigh MIMO channels. We also analyze the outage characteristics of the correlated Rician MIMO channels at high signal-to-noise ratio (SNR). We derive the mean and variance of the mutual information and show that it is well approximated by a Gaussian distribution. Finally, we present numerical results which show the effect of the antenna configuration, correlation level (angle spreads), Rician$K$-factor, and the geometry of the dominant Rician paths.     

Performance Analysis of SEIGA in WCDMA Systems Over Weibull Fading Channels

In this letter, the performance of a Wideband CDMA (W-CDMA) system modeled by a Simplified Expression for Improved Gaussian Approximation (SEIGA), is studied in the presence of a Weibull fading channel. The simulated Bit Error Rate (BER) for W-CDMA systems over Weibull fading channels is computed for the transmission of 10⁶ input bits, and is compared with the corresponding theoretical upper and lower bounds provided by the Q-function. The performance measures are compared for different channel memories.     

Data fusion with minimal communication

Two sensors obtain data vectors x and y, respectively, and transmit real vectors m&oarr;₁(x) and m&oarr;₂(y), respectively, to a fusion center. The authors obtain tight lower bounds on the number of messages (the sum of the dimensions of m&oarr;₁ and m&oarr;₂) that have to be transmitted for the fusion center to be able to evaluate a given function f&oarr;(x,y). When the function f&oarr; is linear, they show that these bounds are effectively computable. Certain decentralized estimation problems can be cast in the framework and are discussed in some detail. In particular, the authors consider the case where x and y are random variables representing noisy measurements and f&oarr;(x,y)=E[z|x,y], where z is a random variable to be estimated. Furthermore, it is established that a standard method for combining decentralized estimates of Gaussian random variables has nearly optimal communication requirements     

An alternative expression for the symbol-error probability of MPSK in the presence of I/Q unbalance

This letter provides a new expression for the probability of the phase angle between two vectors perturbed by Gaussian noise with in-phase/quadrature-phase (I/Q) unbalance, in order to simplify the analysis of the error probabilities of M-ary phase-shift keying (MPSK). The error probabilities, symbol-error rates, or bit-error rates for MPSK with I/Q balance or unbalance, can be presented straightforwardly from the derived expression. Because the newly derived result is provided in terms of the conventional first-order Gaussian Q-function and the joint Gaussian Q-function with a correlation coefficient dependent on M, it readily allows rapid evaluation for various cases of practical interest.     

Integral representation and bounds for Marcum Q-function

A new expression for the Marcum Q-function involving an integral over a fixed interval is given. Tight upper and lower bounds are then derived and applied to the performance evaluation of noncoherent and differentially coherent detection of digital modulation over Nakagami fading channels     

New bounds for the Marcum Q-function

New bounds are proposed for the Marcum Q-function, which is defined by an integral expression where the 0th-order modified Bessel function appears. The proposed bounds are derived by suitable approximations of the 0th-order modified Bessel function in the integration region of the Marcum Q-function. They prove to be very tight and outperform bounds previously proposed in the literature. In particular, the proposed bounds are noticeably good for large values of the parameters of the Marcum Q-function, where previously introduced bounds fail and where exact computation of the function becomes critical due to numerical problems     

Hard Bounds on the Probability of Performance With Application to Circuit Analysis

In this paper, we address the problem of analyzing the performance of an electrical circuit in the presence of uncertainty in the network components. In particular, we consider the case when the uncertainties are known to be bounded and have probabilistic nature, and aim at evaluating the probability that a given system property holds. In contrast with the standard Monte Carlo approach, which utilizes random samples of the uncertainty to estimate “soft” bounds on this probability, we present a methodology that provides “hard” (deterministic) upper and lower bounds. To this aim, we develop an iterative algorithm, based on a property oracle, which is shown to converge asymptotically to the true probability of property satisfaction. Construction of the property oracles for specific applications in circuit analysis is explicitly presented. In particular, we study in full detail the problems of assessing the probability that the gain of a purely resistive network does not exceed a prescribed value, and of evaluating the probability of stability of an uncertain network under parameter variations. The paper is accompanied by illustrating examples and extensive numerical simulations.      

Bounds on the pairwise error probability of coded DS/SSMAcommunication systems in Rayleigh fading channels

Arbitrarily tight upper and lower bounds on the pairwise error probability (PEP) of a trellis-coded or convolutional-coded direct-sequence spread-spectrum multiple-access (DS/SSMA) communication system over a Rayleigh fading channel are derived. A new set of probability density functions (PDFs) and cumulative distribution functions (CDFs) of the multiple-access interference (MAI) statistic is derived, and a modified bounding technique is proposed to obtain the bounds. The upper bounds and lower bounds together specify the accuracy of the resulting estimation of the PEP, and give an indication of the system error performance. Several suboptimum decoding schemes are proposed and their performances are compared to that of the optimum decoding scheme by the average pairwise error probability (APEP) values. The approach can be used to accurately study the multiple-access capability of the coded DS/SSMA system without numerical integrations     

Binary Signaling over Channels Containing Quadratic Nonlinearities

This paper examines the transmission of binary data signals over channels which contain quadratic nonlinearities and additive Gaussian noise. We consider the case where the channel is nonlinear with memory and where the signal is passed through an input receiver filter and sampled once every signaling interval. The samples are represented by a discrete Volterra series and a special case where the received sample contains a single quadratic distortion term is examined. The optimum (maximum-likelihood) receiver (processor) is derived and upper and lower performance bounds obtained. The performance of a practical, suboptimum receiver is examined by means of computer simulation and is shown to be very close to the lower bound of the optimum receiver. Next we examine the case where the received sample contains two quadratic distortion terms. Again, upper and lower performance bounds are obtained. The performance of a suboptimum receiver which uses nonlinear decision feedback is evaluated by computer simulation. Its performance is shown to be superior to an optimum linear receiver.     

Characterizing the statistical properties of mutual information in MIMO channels

We consider Gaussian multiple-input multiple-output (MIMO) frequency-selective spatially correlated fading channels, assuming that the channel is unknown at the transmitter and perfectly known at the receiver. For Gaussian codebooks, using results from multivariate statistics, we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal-to-noise ratio (SNR). We show that our bound is tighter than previously reported analytical lower bounds, and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity. Based on a closed-form approximation of the variance of mutual information in correlated flat-fading MIMO channels, we provide insights into the multiplexing-diversity tradeoff for Gaussian code books. Furthermore, for a given total number of antennas, we consider the problem of finding the optimal (ergodic capacity maximizing) number of transmit and receive antennas, and we reveal the SNR-dependent nature of the maximization strategy. Finally, we present numerical results and comparisons between our capacity bounds and previously reported bounds.     

Exponential-type bounds on the generalized Marcum Q-function withapplication to error probability analysis over fading channels

Strict upper and lower bounds of exponential-type are derived for the generalized (mth order) Marcum Q-function which enable simple evaluation of a tight upper bound on the average bit-error probability performance of a wide class of noncoherent and differentially coherent communication systems operating over generalized fading channels. For the case of frequency selective fading with arbitrary statistics per independent fading path, the resulting upper hound on performance is expressed in the form of a product of moment generating functions of the instantaneous power random variables that characterize these paths     

On the capacity of MIMO relay channels

We study the capacity of multiple-input multiple- output (MIMO) relay channels. We first consider the Gaussian MIMO relay channel with fixed channel conditions, and derive upper bounds and lower bounds that can be obtained numerically by convex programming. We present algorithms to compute the bounds. Next, we generalize the study to the Rayleigh fading case. We find an upper bound and a lower bound on the ergodic capacity. It is somewhat surprising that the upper bound can meet the lower bound under certain regularity conditions (not necessarily degradedness), and therefore the capacity can be characterized exactly; previously this has been proven only for the degraded Gaussian relay channel. We investigate sufficient conditions for achieving the ergodic capacity; and in particular, for the case where all nodes have the same number of antennas, the capacity can be achieved under certain signal-to-noise ratio (SNR) conditions. Numerical results are also provided to illustrate the bounds on the ergodic capacity of the MIMO relay channel over Rayleigh fading. Finally, we present a potential application of the MIMO relay channel for cooperative communications in ad hoc networks.     

On the true Cramer-Rao lower bound for data-aided carrier-phase-independent frequency offset and symbol timing estimation

In this letter we present new and simple closed form expressions for the true Cramer-Rao lower bound (CRB) for data-aided (DA) joint and individual carrier frequency offset and symbol timing estimation from a linearly modulated waveform transmitted over an AWGN channel. The bounds are derived under a carrier-phase-independent (CPI) estimation strategy wherein the carrier phase is viewed as a nuisance parameter and assumed to have a worst-case noninformative uniform distribution over [-ππ]. The computation of these CRBs requires only a single numerical integration. In addition, computationally simpler yet highly accurate asymptotic lower bounds are presented. As particularizations, new bounds for individual CPI frequency estimation with known symbol timing from M-PSK and continuous wave (CW) signals are also reported.     

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.     

Error bounds for the nonlinear filtering of signals with smalldiffusion coefficients

New upper and lower bounds for the nonlinear filtering problem are presented. The lower bounds are especially useful in the region of small diffusion coefficients where previously known bounds are inefficient. The upper and lower bounds are shown to be tight. An example demonstrating the tightness of the bounds is presented     

Error bounds for the white Gaussian and other very noisy memoryless channels with generalized decision regions

For a class of generalized decision strategies, which afford the possibility of erasure or variable-size list decoding, asymptotically tight upper and lower error bounds are obtained for orthogonal signals in additive white Gaussian noise channels. Under the hypothesis that a unique signal set is asymptotically optimal for the entire class of strategies, these bounds are shown to hold for the optimal set in both the white Gaussian channel and the class of input-discrete very noisy memoryless channels.