.

a = (a₀, a₁) = (1, – 1), a = (a₀, a₁) = (1, –j) with excitation values of d = 0.25λ, d = 0.5λ, and d = λ, respectively.

      The main beam-pointing direction of the pattern changes with the relative phases of the excitation values a₀ and a₁. When the main beam points to ϕ = 0° or ϕ = 180°, the antenna array is called an end-fire array.

      As shown in Fig. 2.7, the main lobe width gradually increases with the main lobe of the pattern moving from ϕ = 90° to ϕ = 0°.

      In addition, when d ≥ λ, there will be multiple main lobes in the pattern, which is called the grating lobe as shown in Fig. 2.8.

      Consider a two-dimensional array and three half-wave oscillators placed along the z-axis, among which one is at the origin on the x-axis and the other is on the y-axis with a spacing d = λ/2, as shown in Figs. 2.9 and 2.10.

      The array element excitation values are a₀, a₁, and a₂, and the corresponding position vectors are d₁ = d and d₂ = d. And then

      The array factor of the antenna array is

       Fig. 2.7. Patterns of array antenna (Fig. 2.6).

      Therefore, the normalized gain of the array is

      where g(θ, ϕ) is the pattern function of the half-wave oscillator.

      In the xoy plane (θ = 90°), the gain pattern is given as

       Fig. 2.8. Antenna patterns when d ≥ λ.

       Fig. 2.9. Two-dimensional array.

2.2.2Array antenna freedom

      A binary array with two weighting coefficients can maximize the response of the antenna in a desired signal direction or produce a zero in an interference direction by adjusting the weighting coefficients, which is defined as a degree of freedom. When M array elements are used, the degree of freedom of the antenna array is M – 1. This property has important applications in the pattern synthesis of array antennas.

      Assume that the radiation pattern of the array is

      where is the array steering vector and W is the array element weight vector. By expanding the above equation, we can obtain

       Fig. 2.10. Patterns of the two-dimensional antenna array (Fig. 2.9).

      which refers to

      In Eq. (2.74), when L ≤ M – 1, the equations have a non-zero solution.

      And it also needs to establish a constraint equation when it is required by the pattern to produce a maximum in a certain direction.

      

      This is also a homogeneous linear equation for w_m. Therefore, it also requires the degrees of freedom of an array when generating a beam maximum in a certain direction.

      In a word, there are M weighted M-ary arrays with (M – 1) degrees of freedom, and at most L₁ independent beam maxima and L₂ = M – 1 – L₁ beam zeros can be achieved.

2.2.3Array antenna pattern synthesis

      The analytical methods of array antenna pattern synthesis are mainly for uniform linear arrays and uniform planar arrays, while numerical methods are generally used for non-uniform arrays. For more than half a century, many analytical methods for array antenna pattern synthesis have been studied. The two most basic methods, namely the Dolph–Chebyshev pattern synthesis method and the Taylor single parameter pattern synthesis method, are introduced here.

      2.2.3.1Dolph–Chebyshev pattern synthesis method

      For a uniform linear array, the first sidelobe is approximately 13.5 dB lower than the main lobe when the same excitation is used for each array element. For many practical applications, lower sidelobe levels are often required. In 1946, C.L. Dolphy proposed a method for obtaining lower sidelobe patterns in a classic paper. This method considers the properties of the Chebyshev polynomial and establishes the relationship from polynomial to array sidelobe level.

      The Chebyshev polynomial T_2N(u) has the characteristics of an undamped oscillation function when –1 ≤ u ≤ 1, and the monotonic increase is characteristic of the absolute value outside this oscillation interval. The undamped oscillation characteristics correspond to equal sidelobe levels, while the monotonic characteristics correspond to the main lobe. The Chebyshev polynomial whose order is 2N and number of elements is 2N + 1 can be expressed as

      

      The relationship between the Chebyshev polynomial and the array antenna parameters is

      where θ denotes the angle between the spatial orientation and the array.

      The sidelobe level of the array antenna is expressed as 201gη in dB, where η = T_2N(u₀).

      The above polynomial can also be expressed in the product form of the polynomial root.