General radiation-pattern synthesis technique for array antennas of arbitrary configuration and element type, страница 2

3   Relation to previous work

The methods given in this paper can be applied to arbitrary arrays of dissimilar elements, with various aperture constraints, and can cope with all the array and far-field types given in the preceding review.

The amplitude/phase-synthesis method is similar to the work of Mautz and Harrington [33] but extends it to arbitrary arrays with additional flexibility in denning the far-field power requirement.

The amplitude, or phase-only, synthesis methods linearise a set of nonlinear equations by using approximations, as do some of the methods in adaptive null steering. They can now however, be applied to arbitrary arrays with a given far-field defined in terms of power only, rather than maintaining a 'pencil beam' whilst cancelling the sidelobe level in a given direction.

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4   Method

Array radiation-pattern synthesis is a matter of solving the following set of equations:

r y

£ A» exp 0-a,) exp (jb{n, 0^, <^J) L«°i

x C(n, 0., <^J exp (jc(n, 9», <M

-[^.-M² (D

for m = \-M where n = element number

m = far-field point number

^m» ^m ^ angular coordinates of a far-field point m A, = excitation amplitude of the nth element a, = excitation phase of the nth element b{n, 0^, t^m) ^{ts tne} phase term for the nth element at {0^, <^J: this is a function of the element coordinates (x, y, z) C{n, ^, (j>^) = far-field amplitude of the nth element at

= far-field phase of the nth element at (0,,

is the amplitude of the far-field at (0^, (^j

It is useful to divide arrays into three different categories:

(a) Arrays where amplitudes A, and phases a, can have any practical values

(b) Arrays where amplitudes A, are fixed, and only phases a, are variable

(c) Arrays where phases a, are fixed, and only amplitudes A, are variable.

The following Sections summarise solutions to each of these cases in turn.

4.1 Amplitude and phase distribution variable Consider the equations:

for m = \-M where f(Q^, <^J if> the'tfhase of the far-field at (0^, 0J.

Now if M ^ N, and a suitable choice of (0, (p) is made, then eqns. 2 will comprise a set of N linearly independent equations, and can be solved in a least squares sense [51].

For power synthesis, the phase term /((?„, <f>^ is not constrained. In order to obtain an initial solution, however, we specify an arbitary phase term fn(9^, 0J and a required amplitude term Fg(0m, <i>m), where

the required far-field power distribution, normalised so that

Solving eqn. 2 for A, and a,, in the least squares sense yields an initial far-field solution Fo{6^, <^>J exp (jfo(8^,, ^J). Alternatively, we could specify /l,o and a,o to obtain the initial solution.

In order to obtain a valid comparison between F^(f)^, ^ and fo^ko <A«) ^we normalize Fo and A,o by factor No, such that

We then have the initial difference F^O.^, <^J between the required and the obtained far-field amplitude, given by

If we could synthesise F^Om, <Am) exp O'(/o(0», <^J +0, and add this to Fo(^ <^m)exp(Jo(^. <0 then the far-field approximation would be improved. The quantity </„ can be used to improve the convergence of the method. The angle must satisfy the limit \d»\ ^ 7,,, 'where 7,, is a function of the ratio

F^(0»,^) F^, <AJ The degree of fit, in the least squares sense, is given by

u

YJF^S^J^————————-——————'

An alternative synthesis requirement for main-beam and sidelobe contraints is as follows:

FX,,<^)^X,,<^)     "<£m^        (6) and

F\9», <^n) < Fi[e^, ^      m s£ u or m > v    (7)

where u and i; are angular ranges.