Letting ζ = exp (jψ), we can write the array factor of an equally spaced array as a polynomial, A(ψ), in ψ, and many characteristics of the array pattern can be estimated by examining the distribution of the zeros of the array polynomial on a unit circle. In general, an N-element linear array has N – 1 zeros, ψ0m (m = 1,2,..., N – 1), distributed around the unit circle. Find A(ψ) and locate all ψ0m on a unit circle for the following linear arrays:
a) a two-element array,
b) a three-element binomial array,
c) a five-element uniform array,
d) a five-element array having amplitude ratios 1:2:3:2:1 (as in Example 11–9).
e) Based on the locations of ψ0m for the two arrays in parts (c) and (d), explain why the pattern for the array in part (d) has lower sidelobes but a wider beamwidth.
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