A versatile approach to take into account the above discussed requirements consists in joining them into a cost function that contains selectable weights, which can be suitably modified in adherence to the specifications of a given problem [

5]. However, a further basic aspect that should be considered when multibeam patterns must be synthesized is the equalization of the gains corresponding to the

P directions of maxima. Thus, beside the terms deriving from (

3a)–(

3c), the cost function should include a further term having the objective of reducing the difference between the gains achieved in different desired directions. In agreement with these observations, we hence consider the minimization of the following weighted cost function:

in which the real parameters

${w}_{1},\dots ,{w}_{P},{w}_{P+1},{w}_{P+2},{w}_{P+3}$ are proper non-negative weights. In particular, in (

5), the minimization of the first term has the purpose of forming the

P required maxima, while the minimization of the second term aims to guarantee that the difference between the amplitudes corresponding to these maxima is sufficiently low. Hence, the first two terms jointly impose condition (

3a), in order to obtain an identical radiation level for all the

P desired directions. The minimization of the third term aims to form the

Q required nulls, thus imposing condition (

3b), and finally, the minimization of the last term, identified by the integral, allows one to maintain the generated pattern as low as possible in the other directions, thus imposing condition (

3c). By inserting (

2) in (

5) and using (

4), after some algebra,

$\mathcal{F}\left(\mathbf{a}\right)$ can be expressed in compact form as

where

in which

Each of the four terms in (8) may be viewed as the generic component of an

$N\times N$ matrix. Thus, we can define the matrices

$\mathbf{A}=\left[{A}_{mn}\right]$,

$\mathbf{B}=\left[{B}_{mn}\right]$,

$\mathbf{C}=\left[{C}_{mn}\right]$, and

$\mathbf{D}=\left[{D}_{mn}\right]$, which are all Hermitian, since it can be easily proved that

${A}_{mn}={A}_{nm}^{*}$,

${B}_{mn}={B}_{nm}^{*}$,

${C}_{mn}={C}_{nm}^{*}$, and

${D}_{mn}={D}_{nm}^{*}$. This, in turn, implies that the matrix

$\mathbf{T}=\left[{T}_{mn}\right]=\mathbf{A}+\mathbf{B}+\mathbf{C}+\mathbf{D}$ is also Hermitian.

Let us now impose condition (

3d) according to which

$\mathcal{F}\left(\mathbf{a}\right)$ becomes a function of the sole set of phases

$\mathsf{\Psi}={[{\psi}_{1},\dots ,{\psi}_{N}]}^{T}$. Therefore, recalling that

${T}_{mn}={T}_{nm}^{*}$, (

6) can be rewritten considering its upper triangular elements as

the minimization of this function cannot be accomplished in closed-form, thus an iterative procedure is required.

This procedure can be inferred from the single co-ordinate method [

21], which performs the minimization process through the calculation of the unknown phases one at a time, following a sequential order. Accordingly, (

9) is minimized by individually evaluating each of the

N unknown phases, assuming the other

$N-1$ phases as constants. To allow this operation, the cost function is rearranged to put into evidence its dependence on the generic phase

${\psi}_{k}$, thus obtaining

where

is a term independent of

${\psi}_{k}$. A further manipulation of (

10) leads to

where

It can be immediately observed that the final expression in (

12) is minimized for

Thus, the generic unknown phase that minimizes

$\mathcal{F}\left(\mathsf{\Psi}\right)$ can be evaluated in closed form. This advantage can be exploited to design an iterative algorithm to solve the original 3D synthesis problem described in

Section 2. The proposed algorithm develops as follows (

Figure 1).

First, given the formulated problem (i.e., the antenna array, the sets of desired and undesired directions, and the weights of the cost function), the elements of the matrix

$\mathbf{T}$ are calculated by (

7) and (8). The starting point

${\mathsf{\Psi}}^{0}={[{\psi}_{1}^{0},\dots ,{\psi}_{N}^{0}]}^{T}$ is then selected. Its first component, that is,

${\psi}_{1}^{0}$, is taken as the reference phase, thus its value is maintained equal to zero for the entire evolution of the algorithm. Subsequently, the second element of

${\mathsf{\Psi}}^{0}$ is modified by using (13) and (

14), while keeping constant all the remaining phases of

${\mathsf{\Psi}}^{0}$. This leads to the updated value

${\psi}_{2}^{1}$. By using the novel set

${[{\psi}_{1}^{0},{\psi}_{2}^{1},{\psi}_{3}^{0},\dots ,{\psi}_{N}^{0}]}^{T}$, the same operation is carried out to update the third phase

${\psi}_{3}^{0}$. This procedure is sequentially repeated until the last element

${\psi}_{N}^{0}$ is updated, thus concluding the first iteration of the algorithm, which provides an updated set

${\mathsf{\Psi}}^{1}={[{\psi}_{1}^{0},{\psi}_{2}^{1},\dots ,{\psi}_{N}^{1}]}^{T}$. Proceeding in this way, at the generic

i-th iteration, the last

$N-1$ elements of

${\mathsf{\Psi}}^{i-1}={[{\psi}_{1}^{0},{\psi}_{2}^{i-1},\dots ,{\psi}_{N}^{i-1}]}^{T}$ are updated to obtain

${\mathsf{\Psi}}^{i}={[{\psi}_{1}^{0},{\psi}_{2}^{i},\dots ,{\psi}_{N}^{i}]}^{T}$. Therefore, each iteration consists of

$N-1$ subiterations necessary to update the

$N-1$ phases different from the reference one. Since the cost function

$\mathcal{F}\left(\mathsf{\Psi}\right)$ is minimized at each step, the algorithm generates a non-increasing, and hence convergent, sequence

$\left\{{\mathcal{F}}_{i}\right\}$, where

${\mathcal{F}}_{i}=\mathcal{F}\left({\mathsf{\Psi}}^{i}\right)$. The iterative procedure is terminated when the maximum allowed number of iterations

${I}_{max}$ is reached.