An integral representation for the minimum mean-square error (MMSE) estimator for a random variable in an observation model consisting of a linear combination of two random variables is derived. The derivation is based on the moment-generating functions for the random variables in the observation model. The method generalizes so that integral representations for higher-order moments of the posterior of interest can be easily obtained. Two examples are presented that demonstrate how saddle-point approximation can be used to obtain accurate approximations for a MMSE estimator using the derived integral representation. However, the examples also demonstrate that when two saddle points are close or coalesce, then saddle-point approximation based on isolated saddle points is not valid. A saddle-point approximation based on two close or coalesced saddle points is derived and in the examples, the validity and accuracy of the derivation is demonstrated.

The effects of random shading, phasing errors, and element failures on the beampatterns of line and planar arrays are investigated. The results indicate that the effects of element failures and random phasing errors on the beam-pattern are more pronounced than the effect of shading errors. The deep sidelobe level, in both line and planar arrays, is critically affected by random errors which include shading, phasing and element failures. The combined effects of random shading, phase, and element failures on the beampattern of a line array is more dramatic than on the beampattern of a planar array.

Some integrals are presented that can be expressed in terms of theQ_Mfunction, which is defined as \begin{equation} Q_M(a,b) = \int_b^{\infty} dx x(x/a)^{M-1} \exp (- \frac{x^2 + a^2}{2}) I_{M-1}(ax), \end{equation} whereI_{M-1}is the modified Bessel function of orderM-1. Some integrals of theQ_Mfunction are also evaluated.

The minimum bandwidth ofMreal equi-energy signals is derived for the cases of either a specified code or a specified correlation matrix. The definition of bandwidth is in terms of the ratio of the energy passed by a linear time-invariant filter to the input energy, and incorporates a great variety of definitions. In one instance, the signals are time limited; in a second, a weighted integral over the infinite time interval is restrained. The extremum of the energy ratio depends only on the nonzero characteristic numbers of the correlation matrix of rankrand on the firstrcharacteristic numbers of a kernel related to the filter used in the bandwidth definition. The optimum signal waveforms also depend on the characteristic vectors of the correlation matrix and are given by linear sums of the firstrcharacteristic functions of the kernel. Bounds on the extrema are given, approximations in narrow-band situations are developed, and several useful examples are presented and solved.

The minimum rms (root-mean-square) bandwidth ofMreal equi-energy time-limited signals is derived for the case of either a specified code or a specified correlation matrix. The minimum bandwidth depends only on the nonzero characteristic roots of the correlation matrix, or equivalently, of a matrix derived from column properties of the code. The optimum waveforms depend additionally on the characteristic vectors of the respective matrix and are given by linear combinations of a time-limited fundamental sine wave and its harmonics. An upper bound on the minimum rms bandwidth is approximately proportional to the rank of either matrix. Applications to several codes and correlation matrices are given, including some best binary group codes, which indicate a worthwhile saving of bandwidth for the optimum waveforms as compared with conventional waveforms. The energy concentration of these optimum rms waveforms compares very favorably with that of the time-truncated prolate spheroidal wave functions, which concentrate most energy in an assigned frequency band. Furthermore, the optimum rms waveforms are easier to generate and process.

The minimum Gabor bandwidth ofMorthogonal real signals which together occupy a given time interval is derived. For special values ofM, one form of the optimum set ofMsignals is specified. Interrelations of the Gabor definition of bandwidth to other definitions are also discussed.

Formulas for the error probabilities of equicorrelatedM-ary signals under optimum phase-coherent and phase-incoherent reception are derived in the form of previously untabulated single and double integrals. These integrals are amenable to computer evaluation for arbitraryM. Two modes of reception are considered. In the first, one ofMequal energy equiprobable signals is known to be transmitted during a symbol interval ofTseconds through a nonfading channel with additive white Gaussian noise. The receiver is assumed to be synchronized in time and frequency with the incoming signal, and reception is on a per-symbol basis. Furthermore, the cross-correlation coefficients between all the signals are equal. The probability of correct decision in both phase-coherent and phase-incoherent reception is derived exactly, as a function of the signal-to-noise ratio, the common cross-correlation coefficient, and the size of the signal setM. In the second mode of reception, the only difference is that a threshold is incorporated in the receiver. The probability of false detection and the probability of detection and correct decision are derived exactly for both phase-coherent and phase-incoherent reception as a function of the threshold level, the signal-to-noise ratio, the common cross-correlation coefficient, and the size of the signal setM. The method of reduction of multiple integrals presented here can be generalized, and may find application in other statistical studies in which the Gaussian density form is encountered under an integral.