Читать книгу Spatial Multidimensional Cooperative Transmission Theories And Key Technologies - Lin Bai - Страница 34

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.