Mathematical model of the antenna array

Let's consider an M-element array, which during the time T receives a signal with a complex envelope u ₀(t) from the direction (θ ₀, ϕ ₀).

Let the radiation patterns of the elements in the composition of the antenna array be described by complex functions f _m(θ, ϕ)(m = 1, 3, …, M), then the radiation patterns of the antenna array can be represented as:

(1)$$F( \theta , \;\phi ) = \sum\limits_{m = 1}^M {A_m} f_m( \theta , \;\phi ) , \;$$

where A_m is the complex excitation amplitude of the m-th element.

Obviously, the power of the received signal, taking into account expression (1), is determined by an expression of the form:

(2)$$\eqalign{equP_0 &= \vert {F( \theta_0, \;\phi_0) } \vert ^2\int\limits_0^T {\vert {u_0( t) } \vert } dt \cr &= p_0\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^M {A_mA_nf_m( \theta _0, \;\phi _0) } } f_n^\ast ( \theta _0, \;\phi _0) = p_0AS_0A^H, \;}$$

where the symbol “*” denotes the operation of complex conjugation; H denotes the Hermitian conjugation of a vector; $p_0 = \int_0^T {\vert {u_0( t) } \vert } dt-$signal power at the input of the antenna array.

Each channel of the antenna array contains noise with a complex envelope of n _m(t). The total noise power can be represented using the relation

(3)$$N = \sum\limits_{m = 1}^M {A_m} \sum\limits_{n = 1}^M {A_n^\ast } \int\limits_0^T {n_m( t) } n_n^\ast ( t) dt = \sum\limits_{m = 1}^M {A_m\sum\limits_{n = 1}^M {A_n^\ast I_{m, n}} } = AIA^H.$$

The thermal noise in the channels of the antenna array can be considered statistically independent, with zero mean and equal noise variances σ ² in each channel. In this case, the off-diagonal elements of the matrix I vanish, and the general expression for the elements of the matrix I can be represented as:

(4)$$I_{m, n} = \left\{\matrix{\sigma^2, \;m = n \hfill \cr 0, \;m\ne n. \hfill} \right.$$

The distribution of the power of external interference and noise will be set using function ρ(θ, ϕ). In this case, the resulting noise power at the output of the antenna array can be represented as:

(5)$$P = \sum\limits_{m = 1}^M {\sum\limits_{n = 1}^M {A_m} } A_n^\ast \iint\limits_\Omega {\rho ( \theta , \;\phi ) f_m( \theta , \;\phi ) f_n^\ast ( \theta , \;\phi ) } \sin \theta d\theta d\phi = ASA^H.$$

Taking into account the introduced notation, the signal/(interference + noise) ratio at the output of the antenna array can be represented in matrix form:

(6)$$q^2 = \displaystyle{{P_0} \over {P + N}} = p_0\displaystyle{{AS_0A^H} \over {A( S + I) A^H}}.$$

Note that in expression (6) the elements of the matrix S₀ depend on the direction of maximum radiation patterns. The elements of the matrix S for a given geometry of the antenna array are determined by the spatial distribution of interference and their power. These parameters depend on the form of the function ρ(θ, ϕ). Depending on the emerging interference situation, the matrix S is not always positively definite. On the contrary, matrix I is diagonal, which can always be assigned an inverse matrix. If the noise power is commensurate with the interference power, then the sum S + I is a positive-definite Hermitian matrix.

In matrix theory, there are theorems defining the condition maximum ratio of Hermitian forms [Reference Liu, Zhang, Ren, Wang and Sim12–Reference Khalilov, Islamov, Hunbataliyev, Shukurov and Abdullayev15]. Based on these theorems, for a positive-definite Hermitian matrix S + I, the maximum expression (8) is achieved for the vector

(7)$$A_{opt} = A_0( A + I) ^{{-}1}, \;$$

where, as applied to the methods of matrix synthesis of the antenna array, the elements of the vector A₀ are described by the expression:

(8)$$A_{0m} = exp( {-}ik( x_m\sin \theta _0\cos \phi _0 + y_m\sin \theta _0\sin \phi _0 + z_m\cos \theta _0) ) , \;$$

where (x _m, y _m, z _m) − are coordinates of phase centers of antenna array elements; k is the wave number.

Substantiation of the synthesis method

Let us consider the application of the matrix model of the antenna array to solve the problem of forming the required envelope of the side lobes.

Let r(θ, ϕ) be a real function describing a given side lobe envelope, which is given in the Ω_SL region. It is required to find a vector A _opt that provides the maximum functional

(9)$$q^2( A) = \displaystyle{{AS_0A^H} \over {A( S + I) A^H}}$$

and allows one to form a normalized radiation pattern of the antenna array, which in the Ω_SL region satisfies the condition

(10)$$\vert {F( \theta , \;\phi ) } \vert \le r( \theta , \;\phi ).$$

Comparison of functional (9) with expression (6) shows that they coincide up to a constant factor equal to p ₀. In this regard, to solve the formulated synthesis problem, it is possible to use expression (7).

Based on expression (7), it can be argued that in the absence of interference, when all elements of the matrix S are equal to zero, the maximum of expression (9) is achieved at A_opt = A₀. This solution also corresponds to the maximum value of the directivity of the antenna array. From the theory of matrix synthesis, it is also known that in the presence of one or more interferences, as a result of applying expression (7), the maximum value of the signal/(interference + noise) ratio is achieved due to the simultaneous formation of “nulls” of the radiation patterns in the directions of interference and maximizing the power P ₀ toward the signal source. In this case, the depth of the formed zeros depends on the specified interference power. Expression (7) is also used to form extended nulls of the radiation patterns. In this case, the depth of interference suppression also depends on the distribution of interference power within the area Ω_SL. This means that the choice of function ρ(θ, ϕ) affects the nature of the distribution and the envelope of the side lobes of the generated radiation patterns |F(θ, ϕ)|. At the same time, the relationship between function ρ(θ, ϕ) and the envelope of the side lobe is not unambiguous. For example, if the direction of arrival of the interference coincides with the zero of the radiation patterns, then this interference will not affect the amplitude-phase distribution. Therefore, to solve the formulated problem, it is necessary to find a weight function ρ(θ, ϕ) that, after substitution into expression (5) and determination of the amplitude-phase distribution by formula (7), will provide the maximum value of the expression (6) and fulfillment of requirement (9).

Let us represent the solution of the formulated synthesis problem in the form of an iterative process, at each step of which, with the number t, the form of the distribution function of external noise sources ρ ^t(θ, ϕ) is specified. Here and below, the value in triangular brackets will denote the step number of the iterative process.

When initializing the iterative process, we set ρ ^t=0(θ, ϕ). In accordance with expression (7), with such an initialization of the iterative process, we obtain $A_{opt}^0 = ( 1/\sigma ^2) A_0, \;$ from which, using formula (1), we can calculate the radiation patterns of the antenna array F ⁰(θ, ϕ).

Taking into account the introduced notation, at an arbitrary step t ≥ 0 of the iterative process, we will define a new weight function

(11)$$\rho ^t( \theta , \;\phi ) = \left\{\matrix{0, \;( \theta , \;\phi ) \notin \Omega_{SL}; \;\hfill \cr \rho^{t-1}( \theta , \;\phi ) + \alpha , \;\vert {F^{t-1}( \theta , \;\phi ) } \vert \ge r( \theta , \;\phi ) ; \;\hfill \cr \rho^{t-1}( \theta , \;\phi ) /2, \;\vert {F^{t-1}( \theta , \;\phi ) } \vert \le 0, \;7r( \theta , \;\phi ) ; \;\hfill \cr \rho^{t-1}( \theta , \;\phi ) , \;0, \;7( \theta , \;\phi ) \le \vert {F^{t-1}( \theta , \;\phi ) } \vert \le r( \theta , \;\phi ). \hfill} \right.$$

Consider expression (10) in more detail. Condition ρ ^t(θ, ϕ) at $( \theta , \;\phi ) \notin \Omega _{SL}$ determines that there are no external interference sources in the main beam region. This is part of the expression (11) not dependent on iteration number t.

If in some direction $( \theta , \;\phi ) \notin \Omega _{SL}$ at the previous step the value of the amplitude pattern |F ^t−1(θ, ϕ)| exceeded the specified level of the side lobe envelope, then this means that the weight function ρ ^t(θ, ϕ) needs to be increased. In this regard, this situation corresponds to the second line of the expression (10) and

(12)$$\rho ^t( \theta , \;\phi ) = \rho ^{t-1}( \theta , \;\phi ) + \alpha , \;$$

where the coefficient α ≥ 0 determines the rate of change of the function ρ ^t(θ, ϕ).

The reverse situation is also possible, in which the level of the side lobes in the direction (θ, ϕ) is noticeably lower than the given envelope of the side lobes. If in this direction the grating factor ensures the formation of zero radiation patterns, then the value of the function ρ ^t(θ, ϕ) will not affect the solution of the synthesis problem according to formula (7). However, much more often the situation arises in which the requirements for the envelope of the side lobe are met at the expense of the loss of antenna gain when the level of side lobes is reduced in the entire region (θ, ϕ) ∈ Ω_SL. In this case, it is necessary to decrease the values of function ρ ^t(θ, ϕ). This is achieved using the third line of expression (11). It should be noted that the numerical coefficient of 0.7 in the third and fourth lines of expression (11) determines the accuracy of the approximation given by the side lobe envelope. In fact, with its help, a little lower than the specified side lobe envelope, another auxiliary side lobe envelope is formed, with the help of which the sensitivity of the selected weight function to the level of the side lobes of the generated radiation patterns is regulated. The fourth line in the formation of the function ρ ^t(θ, ϕ) corresponds to the case in which the level of side lobes, on the one hand, does not exceed the specified level and is in close proximity to the required level. In this case, as can be seen from expression (10),

(13)$$\rho ^t( \theta , \;\phi ) = \rho ^{t-1}( \theta , \;\phi ).$$

After the formation of the function ρ ^t(θ, ϕ) formula (5), the elements of the matrix S^t can be determined, and then, by formula (7), the amplitude-phase distribution $A_{opt}^t$ can be determined.

Exit rules from the iterative process can be set when one or more conditions are met:

1. (1) reaching the limited number of iterations;

2. (2) meeting the requirements for the level of side lobes;

3. (3) achievement of the minimum allowable value of the directivity of the antenna array.

Cyclic repetition of the process of refining function ρ ^t(θ, ϕ) by formula (10) is equivalent to the selection of such a spatial distribution of sources of external interference, in which, according to formula (7), the maximum value of the expression (6), which determines the value of the signal/(interference + noise) ratio, is provided, and the requirements for the level of side lobe are met. As a result, the main result of the proposed method is achieved: the formation of directivity diagrams with the best energy when meeting the requirements for the level of side lobe.