The method of saddle-point integration is applied to calculating the cumulative distribution of sums of independent random variables that either have a Rayleigh distribution or are sine waves with independently random phases in the presence of Gaussian random noise.

Minimum probability of bit error is difficult to achieve in a DS-CDMA receiver. Since multiple-access noise is the sum of many independent random processes, it is reasonable to approximate it by a Gaussian process of the same power spectral density. This leads to the criterion of maximizing signal-to-noise ratio (SNR). In this paper, receivers that maximize SNR in a particular DS-CDMA system model under various constraints are proposed and analyzed. The method proposed here does not require locking and despreading multiple arriving CDMA signals. The maximization of SNR is compared with the minimization of probability of error, when the receiver is constrained to operate bit-by-bit, in the absence of knowledge of the other users' spreading codes, timing, and phase.>

The error probability in a binary fiber-optical communication system whose receiver utilizes an avalanche-diode detector is calculated by applying the method of numerical saddlepoint integration to Laplace inversion integrals for the probabilities of errors of the first and second kinds. It is shown how to choose the contours of integration for efficient numerical quadrature. The effects of shot noise, intersymbol interference, and additive Gaussian noise are taken into account, and the decision level on the receiver output is set to minimize the average probability of error. Examples show how the method can be used in receiver design to select system parameters such as the output pulsewidth and the sampling time in order to minimize the probability of error.>

The cumulative distribution of the envelope of the sum of two sine waves and narrowband Gaussian noise, the difference of the phases of the sine waves being uniformly distributed over (0, 2 pi ), is expressed as an integral that is evaluated by numerical quadrature. Approximation for the distribution valid in the limit of large signal-to-noise ratio are derived. The probability density function of the envelope is similarly treated.>

A modified version of the method of saddlepoint integration is shown to compute the generalized Marcum Q-function accurately over a broad range of parameters. Bounds on the truncation error incurred by stopping the numerical quadrature at a certain point are presented.>

The cutoff rate R/sub 0/ of an optical channel transmitting M-ary phase-shift-keyed coherent states is calculated for a receiver using the optimum quantum detector. This is shown to be nearly equivalent, when M>>1, to a system that measures the phase of the received signal. A closed-form expression yielding a good approximation to R/sub 0/ is derived.>

A method is described for exactly computing the probability distribution of the sum of the output of an avalanche diode and Gaussian noise using the Personick-McIntyre model of the random electron multiplication in the diode. A saddlepoint approximation is also presented. Both are incorporated in an optimization procedure for efficiently calculating the minimum input signal strength and the decision level that are needed to attain a preassigned error probability when the diode is embodied in a binary optical communication receiver.>

The probability of error in a binary symmetric channel with intersymbol interference and additive noise is efficiently calculated by numerical quadrature of a Laplace inversion integral along a contour in the complex plane passing through a saddlepoint of the integrand. For Gaussian noise, a bound is set on the truncation error incurred by necessarily restricting the integration to a finite interval. The probability of error resulting from cochannel interference is calculated by a similar technique.

The cumulative distribution of the filtered output of a quadratic rectifier whose input is either narrow-band Gaussian noise or Gaussian noise with a low-pass spectral density is to be computed by numerical quadrature of a Laplace inversion integral along a contour in the complex plane chosen to economize the number of steps. The integrand contains the moment-generating function (mgf) of the output. It is expressed in terms of the Fredholm determinant and the resolvent kernel associated with an integral equation involving the autocovariance function of the input and the impulse response of the output filter. A special case is the power of a mean-zero Gaussian process averaged over a finite interval, and when this process has a rational spectral density, the mgf can be expressed as the ratio of certain finite determinants. By this method distributions are calculated for low-pass noise withRLCand second- and fourth-order Chebyshev spectral densities. For rational input spectral densities but arbitrary positive output filtering and an arbitrary additive input signal, the mgf can be calculated by integrating differential equations of the Kalman-Bucy type.

The performance of two-photon coherent-state (TCS) signals and integral-quantum signals in unimodal and bimodal free-space optical communications is compared with that of ordinary ,coherent signals under the constraint of fixed average error probability. For the unimodal channel of known phase the minimum attainable error probability for both on-off and antipodal TCS signals received in thermal noise is calculated by applying perturbation theory to the detection operator equation. For unimodal antipodal TCS signaling the threshold receiver is also considered. For channels of random phase the optimum photon-counting receiver is analyzed. In all cases the advantage of the nonclassical signal over an ordinary coherent signal vanishes as the transmittance of the channel goes to zero.

An iterative procedure is described for reducing the Bayes cost in decisions amongM>2quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses: the resulting decision strategy is a projection-valued measure and yields an upper bound to the minimum attainable Bayes cost. From it is derived an algorithm for finding the optimum measurement states for choosing amongMlinearly independent pure states with minimum probability of error. The method is also applied to decisions amongMunimodal coherent quantum signals in thermal noise.

The capacity of the photon counting channel with an average-power constraint equalsP_{s}/(kT \ln2)bit/s at all noise temperaturesT, whereP_{s}is the average signal power andkis Boltzmann's constant. The channel can carry arbitrary orthogonal signals and is not restricted to the poise-position modulation (PPM) signals adduced in the above paper.^{1}

The relation between the density operators of the aperture fields of a single-mode optical transmitter and a single-mode receiver is simply expressed in terms of their quasiprobability distributions. This formulation indicates that when the receiver is so far away that it picks up only a small fraction of the transmitted energy, the state induced in its aperture field by a nonclassical transmitted state differs only slightly from what it would be were a certain equivalent classical state transmitted instead. The implications for remote communications are discussed.

When the phase of the transmittance between a unimodal transmitter and a unimodal receiver is uniformly distributed over(0,2\pi), the density operator of the received signal field commutes with the photon number operator, and the optimum detector of the signal counts photons. A simple formula is given for the probability generating function of the number of photons in terms of the quasi-probability distribution of the transmitted signal. It is used to determine the average error probability of an on-off binary communication system transmitting two-photon coherent-state signals, received in the presence of thermal noise.

The nature of the conversion of a pure nonclassical transmitted quantum state into a mixed classical state at a remote receiver is discussed under assmnptions appropriate to a free-space optical communication system, with particular reference to a correspondence^{1}containing misconceptions about a paper of the author's [1].

Determining the minimum error probability of a binary communication receiver embodying a photoelectric detector requires calculating both the probability pkthatkoutput electrons are counted and the cumulative probability that the numberkexceeds a certain decision level. A saddlepoint method is described for both setting the decision level and approximating the error probability on the basis of the probability generating function of the pk's. The method is applied to the avalanche photodiode. The cumulative probabilities so calculated are compared with the Chernoff bound, which in typical cases is found to be too high by an order of magnitude or more. The error probabilities of a class of avalanche diodes having the same mean gain are compared with that of a receiver with no amplification at all.

A method is given for calculating the minimum average error probability attainable in deciding between a mixed state and a pure state of a quantum system. It is applied, first, to the detection of a noisy coherent pulse represented by a state with a Gaussian P-representation and, second, to the detection of a signal represented at the transmitter by a two-photon coherent state (TCS). In both problems the field of the receiver is free of background noise and, when no signal was transmitted, is in the ground state.

Methods are presented for calculating the minimum attainable probability of error in detecting a coherent quantum signal received amid thermal noise in the limits of Iow and high signal-to-noise ratio. In the latter limit, quantum-mechanical perturbation theory is applied to solving the detection operator equation approximately. Graphical results are furnished for binary signals transmitted with equal prior probabilities. Previous work on this problem is thus extended to a broader range of signal and noise strengths. The results are applied to the detection of antipodal two-photon-coherent-state signals.

The complex amplitude of a coherent quantum signal in the presence of thermal noise is to be estimated when its real and imaginary parts have a Gaussian prior distribution. Cost functions of Gaussian, power-law, and delta-function forms are shown all to lead to the same optimum estimator.

Basing decisions and estimates on simultaneous approximate measurements of noncommuting observables in a quantum receiver is shown to be equivalent to measuring commuting projection operators on a larger Hilbert space than that of the receiver itself. The quantum-mechanical Cramér-Rao inequalities derived from right logarithmic derivatives and symmetrized logarithmic derivatives of the density operator are compared, and it is shown that the latter give superior lower bounds on the error variances of individual unbiased estimates of arrival time and carrier frequency of a coherent signal. For a suitably weighted sum of the error variances of simultaneous estimates of these, the former yield the superior lower bound under some conditions.

The resolvability of point sources of incoherent thermal light is analyzed by quantum detection theory in terms of two hypothesis-testing problems. In the first, the observer must decide whether there are two sources of equal radiant power at given locations, or whether there is only one source of twice the power located midway between them. In the second problem, either one, but not both, of two point sources is radiating, and the observer must decide which it is. The decisions are based on optimum processing of the electromagnetic field at the aperture of au optical instrument. In both problems the density operators of the field under the two hypotheses do not commute. The error probabilities, determined as functions of the separation of the points and the mean number of received photons, characterize the ultimate resolvability of the sources.

An upper bound is calculated for the probability that a mean-zero Gaussian random process remains above a specified signal throughout a prescribed interval of time.

A quantum-mechanical form of the Cramér-Rao inequality is derived, setting a lower bound to the variance of an unbiased estimate of a parameter of a density operator. It is applied to the estimates of parameters such as amplitude, arrival time, and carrier frequency of a coherent signal as picked up by an ideal receiver in the presence of thermal noise. The estimation of parameters of a noise-like signal is also treated.

The detection integral equation is a Fredholm equation of the first kind whose kernel is the autocovariance function of a stationary random process. A simple technique is presented for solving this equation when the spectral density of the process is a rational function of frequency. The method uses the solutions of the corresponding Wiener-Hopf equations over the left- and right-hand semi-infinite intervals.

In order to discover what effect the quantum nature of radiation has on the detection of signals at optical frequencies, the detectability of coherent and, in the threshold limit, incoherent signals is calculated on the basis of an idealized quantum-mechanical model of a receiver. The detectabilities reduce at low frequencies to those found by ordinary signal-detection theory; at high frequencies they depend on Planck's constant. For incoherent signals of high frequency the threshold detector is equivalent to a system that focuses the radiation on a photoelectric surface of unit quantum efficiency and optimally processes the emitted electrons.

The principles of statistical signal-detection theory are applied to the detection of optical signals by postulating the photoelectric effect as the mechanism for the interaction of the electromagnetic field of the light with matter. The data with which the detection system works are the numbers of photoelectric events in various parts of a receptor on which the light falls after passing through an aperture. The theory prescribes how the data should be processed and measures the detectability of the signals in terms of an SNR. The theory is applied to the detection of a point source and of an array of incoherent point sources, all against a uniform background. The resolution of two weak close point sources of radiation observed against a uniform background is also treated as a problem in the testing of hypotheses, and the reliability of the system derived on this basis is measured by an SNR that is a function of the angular separation of the sources. The estimation of the parameters of an optical signal is discussed from the standpoint of statistical estimation theory. Estimation of the angular position of a point source against a uniform background is treated as an example.

Sequential analysis has been applied to detecting a radar target of unknown range by sampling the receiver output at instants separated by a Nyquist interval and conducting a sequential test at each sampling time. The antenna beam continues to point in a given direction until all tests have terminated, after which it is moved to a new position. It is shown that this system can search the sky in about half the time needed by an ordinary system of the same sensitivity.

Close-packed, single error-correcting codes are studied, the letters of which areN-tuples ofM-ary digits, whereMis the power of a prime. The lengthNof the letters must be given byN = (M^k - 1)/(M - 1), wherekis an integer. A balanced communication channel, for which all errors in a transmitted digit are equally likely, is defined and a physical model given. The probability of correct reception of the code letters and the rate with which they transmit information in a balanced channel are calculated. This involves deriving formulas for the numbers of code letters having various numbers of 0's. Numerical results are given for a quaternary code and are extended to the case where the quaternary channel has a null zone, so that erasures as well as errors may occur. For the type of signals and noise assumed, the balanced channel without a null zone is found to yield the better performance.

The construction of(n, k)group alphabets is discussed for the balancedM-ary channel, whereMis the power of a prime. In this channel allMdigits are equally likely to be in error, and an incorrect digit is equally likely to be any digit besides the one sent. The alphabets are formed by takingncolumns of the modular representation table of the Abelian group ofk-tuples of elements from the Galois fieldGF(M)under digitwise addition. The formation and properties of that table are described. Attention is focused on alphabets in which all letters except then-tuple of O's have the maximum number of non-null elements. Tables of such alphabets are given forM = 2, k = 2, 3, 4; M = 3, k = 2, 3; andM = 4, k = 2, 3.

It is shown how filtered Gaussian noise having a power spectrum which is a rational function of the square of the frequency can be represented as one component of a multidimensional Markov process. Methods are studied for obtaining the distribution of the number of times such a noise process crosses a given amplitude level in a fixed time interval. The generating function of this distribution is the solution of a Fokker-Planck type differential equation with appropriate boundary conditions. Integral equations are found for the generating function from which all the moments of the distribution can be calculated by iteration.