2. System Model
Considering an SAA with radius
r having
V isotropic and mutually independent elements with the assumption that no mutual coupling exists among the elements. For an SAA configuration, with the
v-th element localization expressed as
$r{\left[sin{\mathsf{\Psi}}_{v}cos{\psi}_{v},sin{\mathsf{\Psi}}_{v}sin{\psi}_{v},sin{\mathsf{\Psi}}_{v}\right]}^{T}$ in spherical domain,
${\mathsf{\Psi}}_{v}$ and
${\psi}_{v}$ represent the elevation and azimuth of
v-th element, respectively.
${\left[\cdot \right]}^{T}$ is transpose operation. Assuming
Q number of EM waves arrive on SAA from far field. Let
${\mathsf{\Phi}}_{d}=\left({\mathsf{\vartheta}}_{d},{\phi}_{d}\right)$ for
$d=1,\dots ,D,$ where
$\mathsf{\vartheta}$ and
$\phi $ represent azimuth and elevation angles of arriving signal. Then the SAA output is expressed in form of matrix as
where
$\overline{G}\left(\mathsf{\Phi}\right)=\left[\overline{g}\left({\mathsf{\Phi}}_{1}\right),\dots ,\overline{g}\left({\mathsf{\Phi}}_{D}\right)\right]$ represents the steering matrix, which is the system-function from origin to the SAA, and it carries the data of the DoA.
t is index of snapshot.
$\overline{x}\left(t\right)={\left[\overline{{x}_{1}}\left(t\right),\dots ,\overline{{x}_{V}}\left(t\right)\right]}^{T}$ defines the vector of the output,
$\overline{s}\left(t\right)={\left[\overline{{s}_{1}}\left(t\right),\dots ,\overline{{s}_{D}}\left(t\right)\right]}^{T}$ is the signal source vector.
$\overline{n}\left(t\right)={\left[\overline{{n}_{1}}\left(t\right),\dots ,\overline{{n}_{V}}\left(t\right)\right]}^{T}$ is the noise that is Gaussian white in vector form with zero mean and
${\sigma}_{n}^{2}$ variance.
After SH decomposition, the output of SAA is expressed as [
21]
where
$B\left(kr\right)=diag\{{b}_{0}\left(kr\right),{b}_{1}\left(kr\right),{b}_{2}\left(kr\right),{b}_{3}\left(kr\right),\dots ,{b}_{N}\left(kr\right)\}$ is an
$A\times A$ strength matrix in far field mode and
$A={\left(N+1\right)}^{2}.N$ represents the highest order of SAA and
diag {⋅} is a diagonal matrix and
${b}_{n}\left(kr\right)$ is described as
where
${j}_{n}\left(kr\right)$ and
${h}_{n}\left(kr\right)$ represent the
nth order spherical Bessel function and spherical Hankel function of first kind, respectively. The
${j}_{n}^{\prime}\left(kr\right)$,
${h}_{{n}^{\prime}}\left(kr\right)$ are the defined corresponding derivatives. In open sphere, elements are positioned in the open with no capturing impairment, while in rigid sphere, the elements are placed in rigid sphere. Here, the received EM wave gets scattered by the interface with the rigid sphere [
22].
${\left[\cdot \right]}^{H}$ is the Hermitian transpose operator.
$k=2\pi f/c$ denotes the signal wave number,
f is the operation frequency, and
c is the speed of EM propagation.
${\mathit{Y}}^{H}\left(\mathsf{\Phi}\right)$ is a
$A\times D$ SH matrix having the
d-th column as
and
$Y\left(\mathsf{\Omega}\right)={\left[y\left({\mathsf{\Omega}}_{1}\right),\dots ,y\left({\mathsf{\Omega}}_{V}\right)\right]}^{H}\u03f5{\u2102}^{V\times A},$where
${Y}_{n}^{m}\left(\cdot \right)$ denotes the SH function with
m, n representing degree and order [
22].
${\mathsf{\Omega}}_{v}=\left({\mathsf{\Psi}}_{v},\text{}{\psi}_{v}\right)$ is the spherical angle of the
v-th position of the element.
${Y}_{n}^{m}\left(\cdot \right)$ can be expressed as
where
$n\in \left[0,N\right],m\in \left[-n,n\right]$ and
${P}_{n}^{m}\left(\cdot \right)$ is the Legendre polynomial.
Furthermore, applying the SFT (spherical Fourier transform) on the output with the consideration of orthogonality relation
${Y}^{H}\left(\mathsf{\Omega}\right)\alpha Y\left(\mathsf{\Omega}\right)={I}_{A}$ [
23] and
$\alpha =diag\left\{{\alpha}_{1},\dots ,{\alpha}_{V}\right\},$ where
${\alpha}_{v}$ denotes the sampling weight of the sampling scheme. From the left hand side of Equation (2), multiply it by
${Y}^{H}\left(\mathsf{\Omega}\right)\alpha $, then the model can be obtained in SH domain as
where
$\widehat{x}\left(t\right)={Y}^{H}\left(\mathsf{\Omega}\right)\alpha \overline{x}\left(\mathrm{t}\right)$ and
$\widehat{n}\left(t\right)={Y}^{H}\left(\Omega \right)\mathit{\alpha}\overline{n}\left(\mathrm{t}\right)$. Multiplying Equation (6) from the left with
${B}^{-1}\left(kr\right)$ to ensure frequency independent steering matrix, the system model can be expressed as
where
$x\left(t\right)={B}^{-1}\left(kr\right)\widehat{x}\left(t\right),G\left(\mathsf{\Phi}\right)=\left[y\left({\mathsf{\Phi}}_{1}\right),\dots ,y\left({\mathsf{\Phi}}_{D}\right)\right]$ and
$n\left(t\right)$ is defined as
$n\left(t\right)={B}^{-1}\left(kr\right)\widehat{n}\left(t\right).$ The covariance matrix is expressed as
where
${R}_{s}=E\left[s\left(t\right){s}^{H}\left(t\right)\right],$ ${R}_{n}=E\left[n\left(t\right){n}^{H}\left(t\right)\right].$ Nonetheless, the covariance matrix
${R}_{c}$ is unseen in practical sense, an approach is to employ sample covariance matrix
${\widehat{R}}_{c}$ in place of the
${R}_{c}$
where
T denotes the number of snapshots and
$X=\left[x\left(1\right),\dots ,x\left(T\right)\right]$ represents system matrix. If we take
${\widehat{R}}_{c}$ eigenvalue decomposition, the noise subspace
${U}_{n}=\left[{u}_{D+1},\dots ,{u}_{A}\right]$ and signal subspace
${U}_{s}=\left[{u}_{1},\dots ,{u}_{D}\right]$can be obtained.
${\left\{{u}_{i}\right\}}_{i=1}$ are regarded as the eigenvectors of the signal corresponding to the
D sizeable
${\lambda}_{1},\dots ,{\lambda}_{D}$ eigenvalues.
${\left\{{u}_{i}\right\}}_{i=1}{}_{D+1}^{A}$ represents eigenvectors of the noise, corresponding to the other eigenvalues
${\lambda}_{D+1},\dots ,{\lambda}_{A}$.
On this note, the output of the SAA has been transformed from spatial to SH domain. The steering vector is only made up of the SH function. If we use the properties of SH function, a mapping matrix can easily be constructed to design a connection between Fourier series and SH function. The SH matrix is decomposed into the multiplication of two matrixes, consisting of the azimuth and elevation information, respectively.
Considering the defined SH function in Equation (5), the SH vector
$\mathit{y}\left(\mathsf{\vartheta},\text{}\mathsf{\phi}\right)$ can be described as
$M\u03f5{\mathbb{R}}^{A\times A}$ gives a diagonal matrix which consists the coefficient of normalization as
where
${f}_{n,m}=\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}$,
$n\in \left[0,N\right],m\in \left[-n,n\right].$ $D\left(\mathsf{\phi}\right)\in {\u2102}^{A\times A}$ is diagonal with the form
$q\left(\mathsf{\vartheta}\right)$ is expressed as
If
${L}_{e}\in {\u2102}^{A\times \left(2N+1\right)}$ and
${L}_{a}\in {\mathbb{R}}^{A\times \left(2N+1\right)}$, then
${L}_{a}$ matrix will satisfy
$f\left(\mathsf{\phi}\right)={\left[{e}^{jN\phi},\dots ,1,\dots ,{e}^{-jN\phi}\right]}^{T}$ and
${L}_{e}$ is expressed as
where
$f\left(\mathsf{\vartheta}\right)=[{\left[{e}^{-jN\mathsf{\vartheta}},\dots ,1,\dots ,{e}^{jN\mathsf{\vartheta}}\right]}^{T}$ [
24]. We can establish a relationship between Fourier series and SH function as
The matrix
$\Re $ is expressed as
where
$\tilde{I}={I}_{{A}^{2}}\left(:,\left[1:A:{A}^{2}\right]\right)\in {\mathbb{R}}^{{A}^{2}\times A}$, this implies
$\tilde{I}$ can be modeled via extraction from columns
${I}_{{A}^{2}}$ and the interval of extraction is
A. $\u2a02$ is Kronecker product operator.
$\Re $ is the mapping matrix that establish the connection between SH vector and Fourier basis function (2D). The reason for
$\Re $ calculation is to determine
${L}_{a}$ and
${L}_{e}$ because the
${L}_{a}$ and
${L}_{e}$ are not known in Equation (18). Hence, the next event is to the best method for
${L}_{a}$ and
${L}_{e}$ representation. Based on the features of
$d\left(\mathsf{\vartheta}\right)$ and
$f\left(\mathsf{\vartheta}\right)$ vectors, it becomes easier to verify that the
${L}_{a}$ matrix is of the form
where
${Q}_{z}={\left[{0}_{\left(2z-1\right)\times {A}_{2}}{I}_{2z-1}{0}_{\left(2z-1\right)\times {A}_{2}}\right]}^{T},$ $z=1,\dots ,N+1$ and
${A}_{2}=N-z+1.$ ${0}_{p\times p}$ is zero matrix of
$p\times p$ size.
${L}_{e}$ is calculated using two independent recurrence Legendre function relationships that is complex and requires initializing the Legendre function [
24]. To ensure accuracy of
${L}_{e}$, it is important to consider an easier method to calculate
${L}_{e}$.
${L}_{e}$ is described as the result or solution to Least Square problem
${||\cdot ||}_{F}$is the Frobenius norm [
25]. The optimization problem of Equation (20) is solvable if we choose the Q direction
${\mathsf{\vartheta}}_{1},\dots ,{\mathsf{\vartheta}}_{Q}$distributed uniformly over (0,
$\pi $]. The
$A\times Q$ matrix
$P=\left[q({\mathsf{\vartheta}}_{1}\right),\dots ,q({\mathsf{\vartheta}}_{Q})]$ can be designed for various angles
${\mathsf{\vartheta}}_{1},\dots ,{\mathsf{\vartheta}}_{Q}$. In a similar way,
${A}_{1}\times Q$ matrix can be formed
$P=\left[f({\mathsf{\vartheta}}_{1}\right),\dots ,f({\mathsf{\vartheta}}_{Q})]$ and
${A}_{1}=\left(2N+1\right).$ Then
${L}_{e}$ matrix is expressed below
The angle size satisfies $Q\ge 2N+1$. Therefore, inversion of Equation (21) exists. Expressing Equation (18) as a function of Equations (19) and (21), the mapping matrix $\Re $ is obtainable. We can obtain it offline before the estimation of DoA.
Following the derivation of
$\Re $, Equation (7) can be described as
The information of DoA can be decomposed into matrixes with Vandermonde structure. Therefore, the root polynomial methods can be developed to compute the azimuth and elevation.
4. Evaluation
This section presents the computational complexity evaluation and analysis of 1D-MUSIC in comparison with FFT-MUSIC and SH-MUSIC. Multiplication operation is the major computational factor in the estimators that causes complexity. The azimuth is estimated at first and then used as initialization. In a one-dimensional spectrum search, the time taken to initialize with the azimuth is double that when initializing with the elevation. Hence, elevation is considered for initialization.
In 1D-MUSIC, the computational complexity majorly occurs when elevation is used as initialization. Employing the fast subspace decomposition in [
25], the complexity when finding D-dimensional noise subspace
${\mathit{U}}_{n}$ is
$o\left({A}^{2}D\right)$. Computing
$\mathit{H}\left(\mathsf{\vartheta}\right)$ and corresponding inverse costs
$o\left({A}_{1}A+{A}_{1}^{2}\right)\left(A-D\right)+{A}_{1}^{3}+A{A}_{1}^{2}),$ then the one-dimensional spectral search for elevations requires
$o\left({N}_{e}\left({A}_{1}A+{A}_{1}^{2}\right)\left(A-D\right)+{A}_{1}^{3}+A{A}_{1}^{2}\right),$ where
${N}_{e}$ is the value of spectral points searched along the path of elevation. 1D-MUSIC computes both the azimuth and elevation using the two root polynomials when initialization ends. The nature of the complexity involved while solving the roots of the polynomial will not be more than
$o\left({A}^{3}\right)$; hence, the complexity of 1D-MUSIC equals
$o\left({A}^{2}D+{A}^{3}+{N}_{e}\left(\left({A}_{1}A+{A}_{1}^{2}\right)\left(A-D\right)+{A}_{1}^{3}+A{A}_{1}^{2}\right)\right).$ The conventional SH-MUSIC algorithm is
$o\left({N}_{a}{N}_{e}\left(A+1\right)\left(A-D\right)+{A}^{2}D\right),$ where
${N}_{a}$ is the spectral points search across the azimuth. In FFT-MUSIC, the coefficient matrix computational complexity and two-dimensional fast Fourier transform is
$o\left({V}^{2}D+{M}_{t}^{2}{V}^{2}+{N}_{a}{N}_{e}+{N}_{a}{N}_{e}\mathrm{log}\left({N}_{a}{N}_{e}\right)\right)$ [
26], for
${M}_{t}\approx 2.5\mathrm{kR}$, where
R denotes the biggest range between the elements of the SAA. An overview of the comparison between SH-MUSIC, FFT-MUSIC, and the new 1D-MUSIC is as given in
Table 2. For
${N}_{a}=4096,{N}_{e}=256,512,1024,\mathrm{and}2048;$ search step in path of the elevation is
${0.6}^{0},{0.34}^{0},{0.16}^{0},\mathrm{and}{0.07}^{0}$. EM signal considered is 2. For SAA,
${M}_{t}=16$.
Table 3 shows the operation of multiplication. Therefore, it is right to say that the proposed method exhibits a lower complexity in computation compared with FFT-MUSIC and SH-MUSIC.
5. Numerical Experiments, Results, and Discussion
This section presents the results of simulations for the purpose of illustration and performance evaluation of the proposed 1D-MUSIC method in comparison with the MUSIC, TSDM, ESPRIT, and SH-MUSIC methods. An SAA of 1.8 cm radius
r, and 32 elements operating at 8.4 GHz, which is uniformly distributed on a rigid sphere, was considered for the simulations. The range between consecutive samples within the uniform sampling framework was constant. The uniform sampling framework causes reduced platonic solids. This only happens for a specific number of elements [
23].The SAA has the highest order of
$N=4$. We conducted simulations with 700 independent Monte Carlo events utilizing MATLAB software, 2018b edition. A
$0.1$ degree search step was used for SH-MUSIC, MUSIC, and the proposed 1D-MUSIC. The number of the employed iterations was two. The
kr was to be less than
N to avoid an aliasing problem. In addition, narrowband AM signals (amplitude modulation) in the far field were considered in all the simulated cases.
Case 1: Here, we conducted a study on the fitting error (Equation (20)). We calculated the matrix
${\mathit{L}}_{e}$ using Equation (21) for a given value of Q. Assuming the elevation is from
0 to
180 degree to verify the validity of Equation (20). The equation for the error is as given below
where
$\mathsf{\vartheta}\text{}\in \text{}\left(0,\mathsf{\pi}\right]$ [
27]. Equation (21) is solvable when
$Q\ge 2N+1.$ We tested the impacts of various
Q on the error.
Q is divided into two sections using 9 as a benchmark. Hence,
Q is put at 7, 9, 10, and 180, the consequent results of the simulation are depicted in
Figure 2. At
Q above 9, the estimated error falls below
${10}^{-8}$; this implies that
${\mathit{L}}_{e}$ matrix is seen as independent of the elevation. If
Q is below 9, the error in fitting is big, which renders Equation (21) invalid due to the fact that the inverse of Equation (21) does not always exist. The curve slightly varies with elevation due to the Legendre function that takes
$\mathrm{cos}\mathsf{\vartheta}$ as the input. For other cases considered in this paper,
Q is set at 180.
Case 2: In this case, for various initialization, we evaluated the performance of both the elevation and azimuth. The estimation accuracy of DoA was conducted using RMSEs. Two uncorrelated waves arriving on SAA from
$\left(45,34\right)$ and
$\left(76,56\right)$ degree. The SNR ranges were from −2 dB to 16 dB, with 200 snapshots. Equation (27) and TS-SH-ESPRIT were utilized for the estimation of elevation to end the initialization process. As shown in
Figure 3a, based on a comparison with TS-SH-ESPRIT, higher estimation accuracy of elevation is observed according to Equation (27). Based on the computed elevation, the azimuth is obtained using SH-MUSIC. As depicted in
Figure 3b, this indicates a higher accuracy of azimuth estimation based on the initialization via Equation (27). Furthermore, Equation (31) is employed to estimate elevation as a function of the estimated azimuth that is known to be an iteration for the elevation.
In conclusion, the elevation accuracy of the estimation could be enhanced further with an iteration, which shows that the proposed algorithm runs iteratively.
Case 3: This example studies the impact of the number of iterations on the accuracy of estimation of both the azimuth and elevation. We considered 400 snapshots and 3 iterations. Two uncorrelated waves arrived on the SAA from
$\left(45,34\right)$ and
$\left(76,56\right)$ degrees. The SNR ranged from −2 dB to 16 dB. Based on the results obtained in
Figure 4b, there is an improvement in azimuth accuracy of estimation after the second iteration. The impact of iteration on the elevation is depicted in
Figure 3a. Since an initialization for elevation has been taken with Equation (31) used for the estimation of elevation that is taken as an iteration, then the elevation estimated is used for initialization and for commencement of the other iteration. The RMSEs result after two iterations is as shown in
Figure 4a. It reveals the accuracy level of 1D-MUSIC remains constant after the second iteration.
Case 4: This case presents the simulation of the maximum number of signals. For
$\mathit{H}\left(\mathsf{\vartheta}\right)$ of size
${A}_{1}\times {A}_{1}$ in Equation (26), the highest number of signal is
${A}_{1}$. The results of 9 uncorrelated signals, with 15 dB SNR and 200 snapshots from (100, 35), (60, 140), (140, 60), (169, 100), (45, 67), (67, 78), (53, 24), (78, 45) and (106, 89) degrees is as shown in
Figure 5. We can infer that the proposed 1D-MUSIC detects the whole of the sources effectively.
Case 5: This scenario presents the performance at 10 dB SNR with 2 uncorrelated EM waves impinging SAA from (45, 34) and (34, 66) degrees sources. Snapshots range between 100 and 600. The RMSEs decay as we increase the snapshots as indicated in
Figure 6, together with CRB (Cramer–Rao bound). This demonstrates how good the proposed 1D-MUSIC performance is.
Case 6: This section verifies the performance of the proposed method by comparing the 1D-MUSIC, MUSIC, SH-MUSIC, TSDM, and ESPRIT algorithms to CRB using the RMSEs. We computed the RMSEs of the methods against the SNR with 100 snapshots. Two uncorrelated signals were impinging on the SAA from (45, 34) and (34, 66) degrees. The computed RMSEs of the azimuth and elevation estimate are as shown in
Figure 7. The proposed 1D-MUSIC method shows better performance than TSDM, TS-SH-ESPRIT, and DoA
_{v}-SH-ESPRIT methods.
Furthermore, the 1D-MUSIC method shares close performance with MUSIC and SH-MUSIC, but reduced complexity in computation. This can be attributed to the fact that both MUSIC and SH-MUSIC require a 2D angle search, but 1D-MUSIC requires a 1D angle search for the computation of the elevation and uses the estimated elevation for initialization. The application of root polynomials to compute the elevation and azimuth reduces the complexity in the computation of SH-MUSIC. Zhuang et al. [
19] reports that FFT-MUSIC shares close performance with the traditional MUSIC; this result exhibits identical behavior with the result reported in this paper; however, the proposed algorithm exhibits reduced complexity in computation than the FFT-MUSIC.
Case 7: In addition, using two EM signals the performance of the proposed 1D-MUSIC method is evaluated. The signal length employed is 10 s and sampled at 48 MHz. A short-time Fourier transform is performed of SAA output. The parameters considered for the short-time Fourier transform are: discrete Fourier transform of 128 size, 64 samples hopsize, and the Hamming window. The two signals impinging on the SAA are from (45, 54) and (65, 76) degree. The signal is transformed into a short-time frequency domain via short-time Fourier transform. Each signal frame in the short-time frequency domain is transformed into an SH domain utilizing the SFT. The proposed 1D-MUSIC is then used to compute the azimuth and elevation for all signal frames in the SH domain. The average of the results for all of the frames is considered the final azimuth and elevation. The comparison of performance between DoA
_{V}-ESPRIT, 1D-MUSIC, and SH-MUSIC is as shown in
Figure 8. 1D-MUSIC exhibits greater performance than DoA
_{V}-ESPRIT and SH-MUSIC. Therefore, the proposed 1D-MUSIC algorithm has been demonstrated to be a better candidate in DoA estimation.
Case 8: Finally, fulfilling the growing application requirements, a specific number of elements are situated on the systems. The inter-element distance becomes shorter causing strong mutual coupling with poor radiation performance and impedance matching. In order to incorporate the mutual coupling effect that exists between elements, experimental measured data, which are the ground truth to systematically evaluate any procedure, are used. Therefore, experimental measurement data are further used for performance evaluation and analysis. The SAA is positioned in the middle of the chamber, and the source is situated at 74 DoAs, which are obtained from different combinations of 4 different elevations and 18 different azimuths. We selected the azimuths from 5 degrees to 365 degrees with 20 degrees as a step size. For detailed information on the measurement architecture/setup using SAA, readers are referred to the previous paper [
4] where the measured data were first published. Gross error (GE) performance evaluation metrics is equally employed to evaluate the methods. The comparison of performance between DoA
_{V}-ESPRIT, 1D-MUSIC, and SH-MUSIC is as shown in
Figure 9. 1D-MUSIC exhibits greater performance than DoA
_{V}-ESPRIT and closer performance to SH-MUSIC, even with unknown mutual coupling. Hence, 1D-MUSIC is a good candidate for estimation of the DoA, particularly in practical cases.