REPRESENTING THINNED AND NONUNIFORMLY SPACED
ARRAYS ON THE UNIT CIRCLE
Randy Haupt
Department of Electrical Engineering
USAF Academy, CO 80132
I. Introduction.
Low sidelobe amplitude tapers for arrays may be synthesized using a unit
circle representation. If the linear array has N equally spaced elements along the
x-axis, then the array factor may be written as
N- 1
f ($1
=Ea,ejnV
n=0
where
Y = 2xdcose
a,, = amplitude of element n
d = element spacing in h (wavelengths)
= plane wave incidence measured from x-axis
N = number of elements in the array
A simple substitution, z d Y , transforms this equation into a polynomial.
The roots of this equation can be represented on the unit circle.
11. Unit Circle Analysis of Thinned Arrays.
A thinned array has some of its uniformly spaced elements turned off to
simulate an amplitude taper across the aperture. A thinned array can still be
represented by a polynomial. An element that is off has its corresponding
polynomial coefficient equal to zero, while one that is on has its corresponding
polynomial coefficient equal to one.
The zeros or roots of the polynomial representation of the far field pattern
may be found using numerical methods developed specifically for polynomials.
For instance, a twenty element uniform array has equally spaced roots that lie on
the unit circle yields
If this same array is thinned to obtain a far field pattern with a minimum
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maximum sidelobe level, then elements 2 and 19 are turned off. The resulting far
field pattem has a maximum sidelobe level of -15.74dB and the zeros on the unit
circle are given by
Note that there are double roots at f60". All the roots have a magnitude of one,
so they lie on the unit circle. Most thinned arrays don't have roots on the unit
circle. Figure 1 shows a plot of the zeros on the unit circle, Figure 2 shows the
corresponding far field pattem. An example of a thinned array with roots off the
unit circle has elements X and 13 turned off (Figures 3 and 4).
111. Unit Circle Analysis of Nonuniformly Spaced Arrays.
Consider an N element linear array of equally weighted nonuniformly
spaced point sources with a far field pattem given by
where x, is the distance in wavelengths of element n from the first element. This
equation may be written in terms of 'I' as
jGv +
+e
+
...i
Making the substitution z=e" yields
This equation is not a polynomial, because the powers of z are not integers.
One way to solve for the roots of the far field pattem is to substitute
z=d2* where d is the largest common divisor for all the %. A ten element array
with x=[0, 0.9, 1.7, 2.4, 3.0, 3.5,4.1, 4.8, 5.6, 6.51 h. has d=0.1 h and
This polynomial has 64 roots. Only 14 have values of U between zkl and are kept
as the roots of the nonpolynomial equation. The far field pattem and unit circle
representation of the nonuniformly spaced array appear in Figures 5 and 6.
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Figure 1 Far field pattern of optimized array.
90
180
270
Figure 2 Unit circle for optimized array.
Figure 3 Far field patten of thinned array.
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Figure 4 Unit circle for thinned array.
Figure 5 Far field of nonuniformly spaced array.
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Figure 6 Unit circle for nonuniformly spaced array.
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