The singular value decomposition (SVD), a standard tool for theoretical and computational matrix analysis, represents a matrix as an ordered product of a unitary matrix, a nonnegative, real diagonal matrix, and a second unitary matrix. Our contribution is to extend the SVD representation to a matrix-valued impulse response. The representation involves a countably infinite number of vector-valued eigenfunctions and scalar singular values, and it provides the most concise and elegant description of the action of a matrix-valued filter (for example, a space-time communication channel) when driven by a finite-duration input signal.

This paper presents upper bounds on the bit-error rate (BER) of optimum combining in wireless systems with multiple cochannel interferers in a Rayleigh fading environment. We present closed-form expressions for the upper bound on the bit-error rate with optimum combining, for any number of antennas and interferers, with coherent detection of BPSK and QAM signals, and differential detection of DPSK. We also present bounds on the performance gain of optimum combining over maximal ratio combining. These bounds are asymptotically tight with decreasing BER, and results show that the asymptotic gain is within 2 dB of the gain as determined by computer simulation for a variety of cases at a 10/sup -3/ BER. The closed-form expressions for the bound permit rapid calculation of the improvement with optimum combining for any number of interferers and antennas, as compared with the CPU hours previously required by Monte Carlo simulation. Thus these bounds allow calculation of the performance of optimum combining under a variety of conditions where it was not possible previously, including analysis of the outage probability with shadow fading and the combined effect of adaptive arrays and dynamic channel assignment in mobile radio systems.

In many digital communications systems, crosstalk, rather than additive noise, is the primary channel impairment. For such systems, it is known that the spectral support of the optimum transmitter is not, in general, restricted to a Nyquist set, in contrast to the case for the additive-noise channel. Nevertheless, the problem of determining the optimum transmitter shaping function for the crosstalk channel without the Nyquist restriction is a difficult one, and has so far remained unsolved. Motivated by current interest in the high-speed digital subscriber line (HDSL) and related crosstalk-dominated applications, we explore a subcase of this problem in which only a single interferer is present. When applied to HDSL-like systems with a single (or dominant) interferer, our analysis and numerical results confirm that wider-than-Nyquist transmitters provide a large performance advantage over Nyquist-limited transmitters. Several interesting and counter-intuitive results also arise. For example, PAM and QAM systems operating at the same spectral efficiency do not, in general, perform identically over the crosstalk channel, despite their essential equivalence in additive noise. We explain why this is so, and show that for channels qualitatively similar to the HDSL wire-pair, QAM has a significant advantage over PAM at high data rates. Finally, we show how the characteristics of HDSL-like channels can be exploited by optimizing the symbol rate.>

We investigate the problem of decoding digital data when soft decisions are constrained to take on values from a finite set. We propose a physically reasonable objective function for selecting the desired assignment of metrics to the received analog signals. We develop a search algorithm for designing a table-look-up that is used by the decoder to select the appropriate intermediate metrics and show that an optimum solution exists. We provide a number of illuminating examples to elucidate our ideas and work out in detail some practical cases.>

The paper presents upper bounds on the bit error rate (BER) of optimum combining in wireless systems with multiple cochannel interferers in a Rayleigh fading environment. The authors present closed-form expressions for the upper bound on the bit error rate with optimum combining, for any number of antennas and interferers, with coherent detection of BPSK and QAM signals, and differential detection of DPSK. They also present bounds on the performance gain of optimum combining over maximal ratio combining. These bounds are asymptotically tight with decreasing BER, and results show that the asymptotic gain is within 2 dB of the gain as determined by computer simulation for a variety of cases at a 10/sup -3/ BER. The closed-form expressions for the bound permit rapid calculation of the improvement with optimum combining for any number of interferers and antennas, as compared with the cpu hours previously required by Monte Carlo simulation. Thus, these bounds allow calculation of the performance of optimum combining under a variety of conditions where it was not possible previously, including analysis of the outage probability with shadow fading and the combined effect of adaptive arrays and dynamic channel assignment in mobile radio systems.>

For a broad class of interference-dominated wireless systems including mobile, personal communications, and wireless PBX/LAN networks, the authors show that a significant increase in system capacity can be achieved by the use of spatial diversity (multiple antennas), and optimum combining. This is explained by the following observation: for independent flat-Rayleigh fading wireless systems with N mutually interfering users, they demonstrate that with K+N antennas, N-1 interferers can be nulled out and K+1 path diversity improvement can be achieved by each of the N users. Monte Carlo evaluations show that these results also hold with frequency-selective fading when optimum equalization is used at the receiver. Thus an N-fold increase in user capacity can be achieved, allowing for modular growth and improved performance by increasing the number of antennas. The interferers can also be users in other cells, users in other radio systems, or even other types of radiating devices, and thus interference cancellation also allows radio systems to operate in high interference environments. As an example of the potential system gain, the authors show that with 2 or 3 antennas the capacity of the mobile radio system IS-54 can be doubled, and with 5 antennas a 7-fold capacity increase (frequency reuse in every cell) can be achieved.>

A comprehensive theory for Nth-order space diversity reception combined with various equalization techniques in digital data transmission over frequency-selective fading channels is developed. The channels are characterized by N arbitrary impulse responses possessing random parameters as well as N additive Gaussian noise sources. Various combiner-equalizers that minimize the mean-squared error are determined. Formulas are presented for the attainable least-mean-squared errors and upper bounds on average probabilities of error. The theory is applied to optimize system parameters and to predict performance for QAM data transmission operating over a model for the mobile radio channel. For this model, estimates of average attainable error rates and outage probabilities are provided as functions of system parameters. In the channel models the uncoded data rates as well as Shannon capacity are regarded as random variables.>

For Pt.I, see ibid., vol.40, no.5, p.885-94 (1992). The probability distributions of the data rates that can be supported by optimum receiver structures as well as the distribution of the Shannon capacity are studied. The dependences among the important system parameters are exhibited graphically for several illustrative examples including QPSK. At outage probabilities>

Fast recursive algorithms for updating coefficients in digital echo cancellers can be derived from the well-known method of least squares. Unless zero initial conditions are assumed, the exact initialization of these algorithms is yet unknown. For random data of more than twice the order of the filter, the existence of a unique least squares solution is proven. A constructive recursive procedure in time and order for computing the pseudoinverse solution for the initial steps is derived. Since the data matrix is composed of integers, this technique facilitates the implementation of stable tap-update algorithms.>

The authors provide mathematical justification, using classical detection theory, for the observed superior performance of coherent dual optical detection over single-diode detection and establish a rationale for determining the intensity of the local oscillator. Viewing the dual detector as an optical channel with one input and two outputs, they show that in the limit of large local oscillator intensity, the optimal processor uses the difference of the two detector outputs, and consequently no bias subtraction needs to be carried out. Since phase is a major problem in applications of coherent optical communication, the authors propose modulation and detection formats that can possibly trade the effects of phase noise for the often less objectionable additive noise.>

The "computational cutoff rate," R0, represents a practical measure of the maximum reliable data rate that can be achieved by coding over a given communication channel using a given modulation format, in contrast with the "channel capacity,"C, which represents an idealized theoretical limit on the achievable data rate. Moreover, designing signal sets with good error probabilities using the R0criterion results in a mathematical problem that is much more tractable than that obtained by using the probability of error itself as a criterion. Both of the above reasons establish the importance of R0in communications theory. This paper starts with a brief tutorial background, which reveals the origin and the significance of R0. Next, the problem of achieving R0over the additive white Gaussian noise (AWGN) dispersive or nondispersive channel, using quadrature-amplitude modulation (QAM) with a peakpower constraint, is addressed. The major result is that, for both cases, the optimum transmission signal set is chosen from a discrete distribution. The solution is derived in detail for the peak-power-limited nondispersive channel, where it is shown that the optimum QAM symbols are selected independently from a probability distribution that is uniform in the phase and discrete in the radius. The solution for the corresponding peak-power-limited dispersive channel is obtained only asymptotically, for large signal-to-noise ratio (SNR), where it is shown that the QAM symbols are selected independently from a uniform distribution within a disk in the complex signal space.

Diffusion theory has sometimes been successful in providing excellent approximate solutions to difficult queueing problems. Here we explore whether such methods can be used to analyze a basic dynamic routing strategy associated with a single idealized node in a data network. We analyze a dynamic routing policy where messages, or packets, that arrive at a certain node are routed to leave the node on the link having the shorter queue. In the model, message or packet arrivals are Poisson and the service time is exponentially distributed. We explore a heavy traffic diffusion method and we also discuss the limitations of an ad hoc approach to applying diffusion. For a node withKoutgoing queues we find, under the assumption of heavy traffic, the optimum dynamic strategy, in the sense of minimizing the average delay. When this optimum dynamic strategy is compared to a static strategy where the outgoing traffic is split among theKqueues, we find that the average delay for the dynamic system is better by a factor ofK.

We analyze the effect of timing phase on the performance of digital data receivers which employ decision feedback equalization. It has long been conjectured, and verified by computer simulations, that decision feedback equalizers are considerably less sensitive to the choice of timing phase than are conventional transversal linear equalizers. We develop the theoretical machinery which provides a rationale for these observations. It follows from our results that for typical operating conditions a 1-2 dB penalty can be incurred by choosing a bad timing phase in decision feedback, while the penalty can be an order of magnitude greater than this (in dB) in conventional linear equalizers.

Asymptotically optimum (in the sense of minimum per-symbol error rate) receiver structures for data communication over the white Gaussian channel with unknown time delay and carrier phase jitter are developed. The receiver structures apply to the following suppressed-carrier modulation systems: double sideband (DSB), quadrature amplitude modulation (QAM) with an arbitrary constellation, vestigial sideband (VSB) and single sideband. The resulting minimum error probability receivers are asymptotically equivalent to maximum-likelihood digital {\em sequence}-estimating receivers. The optimum structures implicitly derive joint maximum-likelihood estimates of the unknown parameters and of the sequence of data symbols. It is shown that the parameter estimates can be obtained from two data-directed stochastic approximation algorithms. Unlike traditional theoretical treatments of this communication situation, which have separated the highly important carrier phase and timing recovery problem from the detection problem, a unified theory is presented from which the complete ideal receiver structure can be deduced.

The advantages with regard to noise enhancement of certain suboptimum nonlinear techniques for "equalization" of linear distortion on channels used for digital data transmission have recently received considerable attention. These nonlinear schemes fall into two classes: decision feedback and nonlinear precoding. Yet each of these methods has an associated peculiarity which has impeded attempts to give firm bounds on performance. Thus, for decision feedback error propagation has caused the analytical pains, while for nonlinear precoding of anL-level alphabet the unknown increase in transmitter power has been culpable. The former problem has recently received successful attention by one of the present authors and some colleagues. Here we address the concomitant problem for nonlinear precoding and its extension to quadrature amplitude modulation (QAM) and show that the transmitted normalized powerPsatisfies(L^{2} - 1)/3 \leq P \leq (L^{2} - 1)/3 + 1.

A model of nonlinear data wave regeneration without retiming is analyzed. Attention is focused on performance after regeneration and subsequent detection. The communication model consists of a pseudoternary signal that is regenerated at one data station, but not retimed prior to retransmission to another station. The signal regeneration is accomplished by a three-level slicer, and this nonlinear transformation degrades the overall system performance in the presence of noise. The extent of this degradation is our subject matter. The formulation and analysis are constructed so that one can put to effective use the fact that, for a certain Gaussian process, one can give definite expressions for the chance that its sample paths lie below a piecewise linear curve on an appropriately restricted time interval. We are able to use events of this type to give upper and lower bounds for the error rate. The bounds are compared with the error rate that would have been obtained had detection been accomplished before the nonlinear regeneration. We hope that the techniques employed here may be of use in other nonlinear communication problems.

We analyze the locking properties of a digital rate-locked loop that may be used to synchronize a slave oscillator with a master. Formulating the dynamics of this system by a nonlinear differentialintegral equation, we rigorously prove that in the absence of filtering in the loop the slave oscillator output waveform has periodic zero crossings occurring at exactly the same rate as those of the master oscillator. The main requirement for this is only that the quantization of the staircase comparator in the slave oscillator not be too coarse. A similar proof for a more general loop remains elusive.

A statistic often encountered in various estimation problems is the conditional ensemble average of the time derivative of a random signal given the signal. It turns out that for a very large class of random signals this statistic is equal to zero. This is a rather surprising result and as far as can be determined has not been precisely stated and rigorously proven. A precise statement and a rigorous proof of this theorem is the subject of this paper. Our result is the following. Lety(t)be a stationary random process possessing a mean-square derivative\dot{y}(t). Then the conditional ensemble averageE \{\dot{y}(t)\mid y(t) \}always vanishes.

This paper is concerned with some of the mathematical aspects of noise in solids. Attention is given to generation-recombination noise. This noise is due to current fluctuations arising from the generation, recombination, or trapping of carriers. We treat only homogeneous systems, and therefore, our descriptive random processes are independent of spatial coordinates. In our model, we associate a counting process\upsilon_{j}(t), j = 1, 2, \cdots , Nwith each energy levelE_{j}of the material, where\upsilon_{j}(t)represents the number of carriers inE_{j}at timet. Thus the statistics of the vector counting process\bar{V}(t) = (\upsilon_{1}(t), \upsilon_{2}(t), \cdots , \upsilon_{N}(t))completely determine the noise behavior of the material. Clearly the component processes of\bar{V}(t)are not statistically independent, and\Sigma \upsilon_{j}must be constrained since the total number of carriers is assumed to be fixed. Here we restrict ourselves to the study of the second-order statistics of\bar{V}(t). Making several physically reasonable assumptions and approximations, we derive the mean, covariance, and spectral density of\bar{V}(t), which agree with previously published results. While our basic results are not new, the underlying assumptions made in the derivations and the basic formulation of the problem are novel. Our approach is to expand time derivatives of the characteristic function for the vector random process\bar{V}(t)from which we extract systems of differential equations in the desired statistics. This method exhibits the results as solutions of linear state equations.

Compact formulas are derived for the spectral density function of an ensemble of continuous-phase constant-envelope multilevel FM waves. The results are general and are presented in terms of averages of elementary functions. Both continuous and discrete spectra are treated, and conditions in terms of the modulation parameters are given under which lines in the spectrum are present. The formulas are applicable in the study ofN-ary FM data transmission systems as well as in pulse frequency modulation.

A concise formula is derived for the probability distribution of the instantaneous frequency of an arbitrary bandpass signal plus Gaussian noise. The results are presented in terms of Marcum'sQ-functions, with well-known properties. The results are applicable in the performance analysis ofn-ary FM and PM data transmission systems. Illustrative examples of the application of the formula are worked out for binary FSK.