General radiation-pattern synthesis technique for array antennas of arbitrary configuration and element type, страница 3

To accommodate this type of fitting a search is made over all m to find the number of points p that infringe the conditions of eqns. 6 and 7. The set of equations solved is restricted to these p points, as will be seen. : Another contraint is often that of obtaining a certain degree of fit over certain parts of the pattern, so that

1^(^,'AJ--F'U,^)I^B.               (8)

Again a search is made for all points q that lie outside these tolerance limits. We then solve the set of equations:

depending on what type of fit is required. Experience has shown that for most practical cases, solving eqn 9 for d, = 0 allows sufficient tolerance in f{9^, <f>^ to allow convergence to an optimum solution.

The form of eqn. 9 is the same as eqn. 2, provided s = m; hence to save reinvertion of the matrix we can solve eqn. 9 for s = m, but for m -^ P, or m -^ q, or m + (peq) we let F^(Q, 4>) = 0.*

Solving eqn. 9 yields a far-field pattern that, when added to the initial solution, improves the fit to F^{9^, ^) under the conditions imposed by eqns. 6, 7, and 9. Successive iterations give convergence to a solution which fits F,, under these constraints. After K iterations we have:

where

£x(^,^)exp(^(0,,,<M is the solution of eqn 9 for m = s^ and

A^ exp Q'a^) = No /4,,o exp (ja^o)

K

+ E^t ^expo/a,*) »=i

By adjustment of &„ and or Fn various fits and compromises between conflicting parameters can be made.

42 Amplitude distribution fixed, only phase variable Consider eqn- 1 now that the amplitudes Ay are fixed. Let Now assume we increase the phases a, by a certain amount a, then we have

Now provided the a, are within a limit «r it can be shown that

where R and / are the real and imaginary parts of the zeroth iteration. Also R² + I² = Po the power of the zeroth iteration.

Suitable choices of 8^ 4>m ^wm mean that eqn. 11 constitutes a set of N — 1 linearly independent equations and hence can be solved, in the least squares sense, fsr ^v — 1 elements. If we can find a,, that satisfy the set of eqn. 11 then a better fit to Pn would be obtained. Eqn. 11, however, only holds for | a, | s$ a^. To overcome this problem we have to use an iterative solution such that for the kth iteration

condition of eqns 6, 7 and 8.

Again the solution may only be possible for W elements. & is a fractional difference between the required and obtained far-field such that the approximation holds. To find Q,, we solve eqn. 12 for Q^ = 1 and then normalize a^ so that

that convergence is complete. Experience has shown, however, that this is unnecessary for many practical

CSS65.

6.2 Extension to near-field and simultaneous pattern synthesis

The above solutions were for a far-field power pattern, but the method can easily be extended to the near-field by adding the distance p to those terms in eqn. 1 which are distance-dependent, i.e. h, C, c, F and/ The nsc.thod can also be used for synthesising patterns at many frequencies simultaneously, and for patterns F(y) which have a fixed phase shift A(y) between elements (such as certain sum and difference patterns). Incorporating all these factors into eqn. 1 we have n

£ a, exp 0'a.) exp (jb(n, 0^,<j)^, p, <r, y)) "=i

x C(n, 0^, <^,, p, a, y) exp (jc(n, Q^,(j>», p, a, y))

ix exp QA(y))

= \F{6», 4>m, P, o, y) exp {jf(n, 0^, <f>^, p, a, y))|²

for m = l-M (far or near far-field points) a =s 1-S (frequencies) y = 1-r (fixed phase shift patterns)

Obviously computer storage and run time constrain the magnitudes of M, N, S and r that can be processed.