Synthesis of an Array for Any Specified Azimuthal Pattern

Author: Edmund A. Laport

It is of interest to discuss a direct method for deriving the array specifications that will provide an arbitrarily prescribed azimuthal radiation pattern. The method will always provide an exact solution, though this solution is not always economically practical. It is frequently of value to apply this general method at first because a more practical array may be suggested from the final results. We shall indicate only the broad outlines of the mathematical procedure and refer the reader to more detailed sources for some of the elements of the method.

Wolf described a method of synthesizing an array for any arbitrary symmetrical pattern using the principles of complex Fourier analysis. In his method, the pattern from a pair of radiators supplies a term in an infinite series derived from the Fourier analysis of the desired pattern in terms of spherical harmonics, each of the form

The present method utilizes the method of Wolf in the course of its development.

Any radiation pattern F(β) in the equatorial plane of a multiplicity of parallel identical linear radiators is periodic in 2 π and can therefore be treated with the general methods of Fourier analysis. Any arbitrary pattern may be classed as even and symmetric if F(β) = F(-β), as odd and symmetric if F(β) = -F(-β), or as uneven (asymmetric) if in nonconformance with both of the foregoing. An even function can be obtained from a linear broadside array with symmetrical Fourier current distributions. An odd function can be obtained from a linear end-fire array with a symmetric but inverted Fourier current distribution. An uneven function can be obtained by the combination of the even and the odd functions.

Let F(β) in Fig. 2.50 represent a prescribed azimuthal radiation pattern for a particular application in the domain from -π to +π. This is shown to be an uneven function with respect to the reference azimuth β = 0. If we designate the function between 0 and +π as X and that between 0 and -π as Y, it can be demonstrated that the even and odd components of this uneven function are

and

and also that F_e(β) + F₀(β) = F(β). The even and the odd components of F(β) are shown dotted in Fig. 2.50. It is therefore obvious that since we have means for generating any desired even radiation function and also any desired odd radiation function, we can apply this method and obtain means for exactly generating any arbitrary asymmetric radiation pattern.

FIG. 2.50. Resolution of an asymmetric function F(β) into its component even Fe(β) and odd F0(β) functions.

obtain means for exactly generating any arbitrary asymmetric radiation pattern.

The application of Wolf's method is straightforward mathematically, though it remains for the engineer using it to decide on the minimum number of terms in the Fourier series that will give a satisfactory approximation to the desired result. By adding terms, which means adding pairs of radiators, one can approach the desired result as closely as economic considerations will permit.

The Fourier array, while it always provides a theoretical solution, frequently does not provide a practical or economical solution for an asymmetric pattern problem. One must then resort to the use of random, or nonsystematic, arrays. There are no known systematic methods for synthesizing such arrays except trial and error, reference charts of patterns for nonsystematic arrays of three or more radiators, or using an analog computer such as the RCA Antennalyzer.