Due to the introduction of frequency offsets, the pattern synthesis problem of the FDA becomes more complicated than traditional array pattern synthesis. In the currently reported works, global optimization or convex optimization techniques have been introduced to deal with this problem [
23,
24,
25]. However, these methods require a time-consuming search or iterative process. To alleviate this problem, an efficient non-iterative sparse FDA beampattern synthesis method is presented in this section.
3.1. Proposed Synthesis Method for Sparse SFDA
Consider a sparse SFDA with
elements, by defining
and
respectively, and according to Equation (
5), the array factor of sparse SFDA can be formulated as
where
and
represent the position and frequency offset corresponding to the
mth element respectively.
denotes the number of waves, and
is the wavelength.
For a given reference beampattern
, which can be generated by uniform SFDA with
M array elements, the sparse MCFDA pattern synthesis problem can be equivalent to finding the minimum value
to approximate the reference pattern.
where
is the reconstruction error. Based on the fact that the pattern expression is a sum of complex exponentials, the MP method is introduced to solve this question [
26].
First, we need to sample the reference pattern
in
plane uniformly. Let the number of sampling points along
u and
v axes be
and
respectively, then the sample points
and
can be written as
where
and
are the sampling spacing across
u and
v axes respectively, and
. According to the Nyquist sampling theorem,
and
should meet the conditions that
and
with
and
. After the sampling, we have
where
and
. With the sampled data, the block Hankel matrix
can be obtained as follows [
27]:
in which
K and
L are pencil parameters, which should be chosen such that
and
with
M being the element number of reference pattern [
18]. Then the singular value decomposition of the matrix
is performed as
where
and
are unitary matrices.
with
being the ordered singular values of
, and
. Based on the observation that, for many designed antenna arrays, the number of principal singularities is less than the number of antenna elements [
17], we can discard the non-principal values to obtain a low rank approximation of
. It is common practice to set these small singular values to zero [
17,
18]. That is,
where
. The minimum value of
can be roughly estimated as
Once the low-rank matrix
is acquired, the parameters
corresponding to the positions of the new sparse array with
elements can be calculated by solving the following generalized eigenvalue problem [
28],
where
and
are obtained by deleting the first
L rows and the last
L rows of
. Besides this, we can obtain eigenvalues more computationally efficient by solving the following equation
where
and
are obtained by removing the first
L rows and the last
L rows of
which contain only principal right singular vectors of
. Similarly, another set of eigenvalues
corresponding to the frequency offsets also can be obtained. Next, utilize the pairing algorithm in [
28] to get the correct pairing of
and
. Finally the locations and frequency offsets of the resulting sparse SFDA can be given by
where
Once all the frequency offsets and element positions are available, the elements’ excitations can be calculated using the following equation
and the diagonal elements of matrix
are
.
and
are shown as follows:
where
wherein
The steps are summarized in Algorithm 1.
Algorithm 1: Proposed synthesis method for sparse FDA. |
Input: |
1: Sample reference pattern in plane uniformly according to Equations (13) and (14), and construct the block Hankel matrix using Equations (15) and (16). |
2: Perform the singular value decomposition (SVD) of according to Equation (17) and calculate the singular values . |
3: According to Equation (19), determine the minimum number of elements . |
4: According to Equation (21), extract the eigenvalues and , then to pair them utilizing pairing algorithm in [28]. |
5: Detemine frequency offsets and locations of the new sparse array with using Equations (22) and (23) |
6: Calculate the excitations using Equations (24)–(30). |
Output: |
In Algorithm 1, the most computationally intensive operations mainly include the SVD of the block Hankel matrix with
in step 2 and the eigenvalue decomposition (ED) with
in step 4. Therefore, the total computational complexity is
. For the pattern synthesis problem of sparse SFDA, the typical method is to adopt global optimization algorithms, e.g., [
23]. Since the number of iterations guaranteed to converge is hard to know, it is difficult to compare in terms of computational complexity. However, according to the authors’ observations, the average execution time of the proposed method is much lower than that of [
23].
3.2. Proposed Synthesis Method for Sparse MCFDA
Suppose a sparse MCFDA consists of
antenna elements with
carriers, and according to Equation (
10), the array factor of sparse MCFDA can be expressed as
where
and
represent the position corresponding to the
mth element and frequency offset corresponding to the
nth carrier, respectively. According to Equation (
12), the pattern synthesis problem for sparse MCFDA can also be described mathematically as
Besides this, it can be seen from Equation (
31) that the array factor of MCFDA can be decomposed as the product of two individual exponential summations corresponding to
u and
v. Accordingly, Equation (
32) can further be recast as [
25]
where
and
are the cross section of reference beampattern
along the
u and
v axes, respectively.
and
are the reconstructed error threshold, and
and
are given by
It is observed that the synthesis of sparse MCFDA can be translated into two independent sparse array pattern synthesis problems which depend on the array positions and frequency offsets, respectively. Therefore, we only need to consider two simple one-dimensional parameter estimation problems, which can be easily solved by MPM.
Similarly, we sample the reference pattern
and
respectively, constructe the Hanke matrix
and
by the sampling points as follows, and perform the singular value decomposition of
and
.
where
and
are the sampled points obtained from
and
, and
and
are the pencil parameters. Generally,
and
should satisfy the conditions
and
with
M and
N being the number of elements of the reference pattern [
17]. Then, the minimum value
and the reconstructed low-rank matrix
and
can be acquired using Equations (
18) and (
19). The parameters
and
can be obtained by solving the following equations [
17].
where
(resp.,
) and
(resp.,
) are obtained by deleting the top (resp.,) row of
and
, which consist of
and
principal left-singular vectors. Next, the locations
and frequency offsets
of the new sparse MCFDA can be obtained by Equation (
22). It should be noted that the pairing operation is not required. Finally, the excitations
and
can be calculated by the least squares (LS) method. The implementation steps of sparse MCFDA are listed in Algorithm 2.
Algorithm 2: Proposed synthesis method for sparse MCFDA. |
Input: |
1: Sample two desired patterns and respectively, and construct the Hankel matrix and using Equations (35) and (36). |
2: Perform the SVD of and and determine the minimum number value and . |
3: According to Equation (37), extract the eigenvalues and . |
4: Detemine locations and carriers of the new sparse MCFDA using Equations (22) and (23) |
5: Calculate the excitations and using the LS method. |
Output: |
Similar to the analysis of Algorithm 1, the computational complexity of Algorithm 2 is close to
. For the synthesis problem of sparse MCFDA, we adopt the method presented in [
25] as comparison. The computational complexity of the method in [
25] is
, where
is the number of the initial dense array. In general, the conditions
and
are satisfied. Consequently, the complexity of the proposed algorithm is lower than that of [
25].