Trigonometric Approximations for Bessel Functions

Bessel functions, used extensively in mathematical physics, electromagnetics, and communication-system theory, must often be approximated by appropriate formulas. The functions J?(x) and I?(x) for integer values of p are well approximated by the sum of a small number of circular and hyperbolic sines or cosines, respectively, when x is not too large, e.g., less than 10, and convenient expressions are obtained for their errors. These approximations are especially convenient for use on personal computers (PCs) and pocket calculators and in analytically evaluating integrals.     

Spectral Analysis with Sinusoids and Walsh Functions

The Walsh spectrum of a sinusoid of frequency f = p/2q with p odd consists entirely of lines at orders s that are odd multiples of 2/2q, and the Fourier spectrum of a Walsh function of order 2p/2q consists entirely of lines at frequencies that are odd multiples of 1/2q. For all other frequencies or orders, the spectrum contains no lines, but the power spectral density takes all values in the range [0,oo] in every interval, however short, while being almost everywhere zero. For detecting the presence of a sinusoid by means of Walsh analysis, the time scale should therefore be chosen so that the order of the Walsh function is a power of 2 and the Walsh function is a Rademacher function, i.e., a hard-limited sinusoid of the frequency being sought.     

On combining target-location ellipses

On the assumption of Gaussian statistics, expressions are obtained for the center, semiaxes, and the inclination of the major axis of a single location ellipse that combines the information in several location ellipses and/or bearings obtained from independent fixes on the same target. When all of the given ellipses have finite major axes and they are all shifted so as to have the same center, the resultant ellipse will not touch any of them     

Reception of pulsed signals by detuned receiver: addenda andcorrections

The parameters of a pulse perturbed by passage through a filter were examined in a variety of cases in a previous paper by the author (ibid., vol.AES-4, p.486-9, May 1968). One case not covered is that of a rectangular pulse passed through a detuned Gaussian filter; that case is presented, as is the case of a swept Gaussian pulse passed through such a filter, which may be dispersive. Corrections are included for two tabular entries in the original paper concerning a Gaussian pulse fed to an ideal bandpass filter when the time-bandwidth product and detuning are large     

Position Determination from Radio Bearings

A new, simpler derivation is presented for Stansfield's formulas for the most likely position ion and the distribution of its error when a number of bearings are measured from known locations. The simplification results from the use of complex numbers. Simple results are obtained for the case in which bearings are measured at regular, short intervals along a straight reconnaissance flight path, providing asymptotes for Butterly's curves.     

Extensions and critique of minimum-relative-entropy spectralestimation

Insight is provided into minimum-relative-entropy spectral estimation (MRESE), which has proved very successful in some applications. There are, however, shortcoming inherent in MRESE as it has so far developed; e.g. it would be desirable to incorporated objective, absolute (rather than subjective, relative) weighting for prior estimates. Such problems are pointed out, and solutions are offered for some of them. A few of these problems are common to MRESE and minimum-entropy spectral estimation. The unsolved problems are offered as topics for further research