Study of Smart Antenas on Mobile Communications

1.1.2 Dolph-Chebyshev Method

Comparing the Uniform, Dolph-Chebyshev and Binomial distribution arrays, the uniform amplitude arrays yields the smallest half-power beamwidth while the binomial arrays usually possess the smallest side lobes. On the other hand, Dolph- Chebyshev array is mainly a compromise between uniform and binomial arrays.

Its excitation coefficients are affiliated to the Chebyshev polynomials and a Dolph-

Chebyshev array with zero side lobes (or side lobes of -8dB) is simply a binomial design. Thus, the excitation coefficients for this case would be the same if both methods were used for calculation.

In [6], the array factor of an array of odd and even number of elements with symmetric excitation is given by

(1.6)

(1.7)

M is an integer, an is the excitation coefficients and

(1.8)

The array factor is merely a summation of M or M+1 cosine terms. The largest harmonic of the cosine terms is one less than the total number of elements in the array. Each cosine term, whose argument is an integer times a frequency, can be rewritten as a series of cosine functions with the fundamental frequency as the argument [5], which is,

m = 0; cos(mu) = 1

m = 1; cos(mu) = cos u

m = 2; cos(mu) = cos (2u) = 2cos²u -1

m = 3; cos(mu) = cos (3u) = 4cos³u - 3cos u

m = 4; cos(mu) = cos (4u) = 8cos⁴u - 8cos²u + 1 (1.9)

The above are achieved by using the Euler's formula

(1.10)

Where m= number of antennas on x-plane and the trigonometric identity

sin²u = 1 - cos²u.

Assuming the elements of the array is placed along the z-axis, and thus, replacing cos u

with z in (1.8), will relate each of the expression to a Chebyshev polynomial Tm(z).

m = 0; cos(mu) = 1 = T0(z)

m = 1; cos(mu) = z = T1(z)

m = 2; cos(mu) = 2z² -1 = T2(z)

m = 3; cos(mu) = 4z³ - 3z = T3(z)

m = 4; cos(mu) = 8z⁴ - 8z² + 1 = T4(z) (1.11)

These relations between the cosine functions and the Chebyshev polynomials are valid

only in the range of -1=Z=+1. Because |cos(mu)| =?1, each Chebyshev polynomial is

|Tm(z)| =1 for -1 =Z =+1. For |z| > 1, the Chebyshev polynomials are related too the

hyperbolic cosine function [5].

The recursive formula can be used to determine the Chebyshev polynomial if the polynomials of the previous two orders are known. This is given by

T_m(z) = 2zT_m-1(z) - T_m-2(z) (1.12)

It can be seen that the array factor of an odd and even number of elements is a summation of cosine terms whose form is similar with the Chebyshev polynomials. Therefore, by equating the series representing the cosine terms of the array to the appropriate Chebyshev polynomial, the unknown coefficients of the array factor can be determined. Note that the order of the polynomial should be one less than the total number of elements of the array.