As a useful algorithm, GS orthogonalization has been widely used in array signal processing. Specifically, the GS orthogonalization technique has been proved to be suitable for least-squares estimation because of its good robustness [26], and this technique has also shown superior performance in arithmetic efficiency, stability, and convergence times over other adaptive algorithms for adaptive cancelation [27, 28]. Moreover, the GS-based algorithm was used to synthesize desired beampattern for different arrays [29], and an optimizing approach based on the GS orthogonalization was applied in [30] to optimize loudspeaker and microphone configurations for sound reproduction systems.

It is clear that applying the GS orthogonalization in array signal processing is not novel, but this technique has not yet been used to develop a general model of superdirectivity for arbitrary sensor arrays. In this study, the solutions of superdirectivity will be accurately expressed in full closed-form based on the GS orthogonalization, and the derived Gram-Schmidt mode-beam decomposition and synthesis (GSMDS) superdirectivity model will facilitate the implementation of high-order superdirectivity in practice. See more details in the following sections.

### 3.1 Matrix form of the optimal weighting vector

The mechanism of AG optimization was explained in [31]. An optimal array processor can be implemented in two steps: pre-whitening and matching. Initially, the received data vector will be sent to a pre-whitening processor that pre-whitens its contained noise data vector so that the noise power can be largely decreased. The larger correlation between channels, the more will be the reduction of the noise power. A matching operation follows, which matches the data output modified by the pre-whitening processor to a signal steering vector, and then the received signals impinging from the desired direction are added coherently, whereas the noises are added incoherently. Therefore, the output signal-to-noise ratio can be greatly improved and the processing gain is much higher than the simple DAS method that does not pre-whiten the noises. This process is shown in Fig. 1 in which the received data vector **X** contains the signal vector **S** and noise vector **U**. The vector **V** is the pre-whitened output noise vector, and **ε** is the normalized version of **V**. The entries of both **U** and **V** are supposed to be narrow band complex analytic functions, the matrix **C** performs the orthogonal transform, and the relation **V** = **CU** holds. Therefore, *V*
_{
j
} can be expressed as a linear combination of the *j* terms of *U*
_{
h
} ( *h* = 0, 1,....., *j*), i.e.,

$$ {V}_j={c}_{j0}{U}_0+{c}_{j1}{U}_1+\cdot \cdot \cdot \cdot +{c}_{jj}{U}_j={\displaystyle \sum_{h=0}^j{c}_{jh}{U}_h}. $$

(8)

The pre-whitening process can be performed with the GS orthogonal transform, and **V** can be recursively derived from **U**:

$$ {V}_k={U}_k-{\displaystyle \sum_{j=0}^{k-1}\frac{\left\langle {U}_k,{V}_j\right\rangle }{\left\langle {V}_j,{V}_j\right\rangle }{V}_j},\kern0.62em {\varepsilon}_k=\frac{V_k}{\left|{V}_k\right|}\kern0.5em \left(k=0,1,\cdots, N-1\right), $$

(9)

where **V** is viewed as an orthogonal basis for the *N* × 1 complex space **C**
^{N} formed by the noise data vector **U**. The modulus of the vector component *V*
_{
k
} is \( \left|{V}_k\right|=\sqrt{\left\langle {V}_k,{V}_k\right\rangle } \), where 〈⋅, ⋅ 〉 indicates the inner product that expresses the cross correlation between the related vector components.

Substituting Eq. (8) into Eq. (9) and using some mathematical derivations yield

$$ {c}_{ki}=\kern1em \left\{\begin{array}{l}\kern8.5em 1\kern7em \mathrm{if}\kern0.5em i=k,\\ {}-{\displaystyle \sum_{j=i}^{k-1}\left\{\frac{c_{ji}}{\chi_j}\left({\displaystyle \sum_{h=0}^j{c}_{jh}{\rho}_{kh}}\right)\right\}}\kern1em \mathrm{if}\kern0.5em 1\le k\le N-1\kern0.5em \mathrm{and}\kern0.5em h,i,j\le \left(k-1\right),\\ {}\kern8.5em 0\kern7em \mathrm{otherwise},\end{array}\right. $$

(10)

where

$$ {\chi}_j=\left\langle \left({\displaystyle \sum_{h=0}^j{c}_{jh}{U}_h}\right),\left({\displaystyle \sum_{m=0}^j{c}_{jm}{U}_m}\right)\right\rangle ={\displaystyle \sum_{h=0}^j{\displaystyle \sum_{m=0}^j{c}_{jh}{c}_{jm}\left\langle {U}_h,{U}_m\right\rangle }}={\displaystyle \sum_{h=0}^j{\displaystyle \sum_{m=0}^j{c}_{jh}{c}_{jm}{\rho}_{hm}}} $$

(11)

and *ρ*
_{
ij
} = 〈*U*
_{
i
}, *U*
_{
j
}〉 are the correlation coefficients. These equations are similar to the results presented in [29] in which array synthesis is the main focus.

Utilizing the recursive deduction for *c*
_{
ki
} provides the matrix **C** as

$$ \mathbf{C}={\left[{\mathbf{C}}_0\kern1em {\mathbf{C}}_1\kern1em \cdots \kern1em {\mathbf{C}}_k\kern1em \cdots \kern1em {\mathbf{C}}_{N-1}\right]}^{\mathrm{T}} $$

(12)

with **C**
_{
k
} = [*c*
_{
k0}
*c*
_{
k1} ⋯ *c*
_{
kk
} 0 ⋯ 0]^{T}. Each entry of this lower triangular matrix will be completely determined by the noise correlation coefficients. For the isotropic noise field, the correlation coefficients are

$$ {\rho}_{ij}=\frac{ \sin \left(2\pi {d}_{ij}/\lambda \right)}{2\pi {d}_{ij}/\lambda }, $$

(13)

where *d*
_{
ij
} is the spacing between the *i*th and *j*th sensors.

Because the vector **V** is pre-whitened, the following is obtained:

$$ \left\langle \mathbf{V},{\mathbf{V}}^{\mathrm{T}}\right\rangle ={\mathbf{D}}^2, $$

(14)

where **D** = diag{|*V*
_{0}|, |*V*
_{1}|, ⋯, |*V*
_{
N − 1}|}, **D**
^{2} = **D** ⋅ **D**, and diag{⋅} indicate a square matrix with the elements of its arguments on the diagonal.

Substituting the relation **V** = **CU** into Eq. (14) yields

$$ \mathbf{C}\left\langle \mathbf{U},{\mathbf{U}}^{\mathrm{T}}\right\rangle {\mathbf{C}}^{\mathrm{T}}=\mathbf{C}{\mathbf{R}}_{nn}{\mathbf{C}}^{\mathrm{T}}={\mathbf{D}}^2, $$

(15)

which then obtains

$$ {\mathbf{R}}_{nn}^{-1}={\mathbf{C}}^{\mathrm{T}}{\mathbf{D}}^{-2}\mathbf{C}. $$

(16)

It is now clear that the inverse of matrix **R**
_{
nn
} can be calculated analytically based on the GS orthogonalization. Actually, the inverse of matrix **R**
_{
nn
} can also be calculated using the eigen-decomposition, as was the case for circular arrays in [21, 22], but the eigenvalues and eigenvectors in relation to other array geometries cannot be expressed in closed-form and computed analytically. Therefore, the GS-based decomposition is more advantageous than the eigen-decomposition for computing the inverse of the normalized noise covariance matrix.

The optimal weighting vector in Eq. (7) can be modified to

$$ {\mathbf{w}}_{\mathrm{opt}}={\mathbf{C}}^{\mathrm{T}}{\mathbf{D}}^{-2}\mathbf{C}\cdot \mathbf{a}\left({\theta}_0,{\phi}_0\right), $$

(17)

where the matrix **D**
^{− 2} performs the normalization of the noise input. The matching vector is

$$ {\mathbf{w}}_{\mathrm{match}}=\mathbf{C}\mathbf{a}\left({\theta}_0,{\phi}_0\right). $$

(18)

The elements of this vector are strictly marched with the data output modified by the pre-whitening processor, which can make the signals to be added coherently. The beampattern and other array performance parameters, such as AG, can be directly deduced from this solution.

The general model of superdirectivity will be developed based on the above results in the following section.

### 3.2 Mode-beam decomposition and synthesis

Substituting Eq. (17) into Eq. (3) gives the superdirective beampattern as

$$ \begin{array}{c}B\left({\phi}_0,{\theta}_0,\phi, \theta \right)={\mathbf{w}}_{\mathrm{opt}}^{\mathrm{H}}\mathbf{a}\left(\theta, \phi \right)\kern0.3em \\ {}={\left[\mathbf{C}\cdot \mathbf{a}\left({\theta}_0,{\phi}_0\right)\right]}^{\mathrm{H}}\cdot {\mathbf{D}}^{-2}\cdot \left[\mathbf{C}\cdot \mathbf{a}\left(\theta, \phi \right)\right]\\ {}={\displaystyle \sum_{k=0}^{N-1}{b}_k\left({\theta}_0,{\phi}_0,\theta, \phi \right)}.\end{array} $$

(19)

The *k*th-order mode-beam in the above equation is

$$ {b}_k\left({\theta}_0,{\phi}_0,\theta, \phi \right)=\frac{1}{\lambda_k}\cdot {E_k}^{*}\left({\theta}_0,{\phi}_0\right){E}_k\left(\theta, \phi \right), $$

(20)

where

$$ {E}_k\left(\theta, \phi \right)={\mathbf{C}}_k^{\mathrm{T}}\mathbf{a}\left(\theta, \phi \right), $$

(21)

and

$$ {\lambda}_k={\left|{V}_k\right|}^2=\left\langle {V}_k,{V}_k\right\rangle ={\mathbf{C}}_k^{\mathrm{T}}{\mathbf{R}}_{nn}{\mathbf{C}}_k\kern0.9em \left(k=0,1,\cdots, N-1\right), $$

(22)

and the superscript asterisk indicates complex conjugation. The DF (or the AG because they are equal in this case) is derived as

$$ DF={\left[\mathbf{C}\cdot \mathbf{a}\left({\theta}_0,{\phi}_0\right)\right]}^{\mathrm{H}}{\mathbf{D}}^{-2}\left[\mathbf{C}\cdot \mathbf{a}\left({\theta}_0,{\phi}_0\right)\right]={\displaystyle \sum_{k=0}^{N-1}{\left|{E}_k\left({\theta}_0,{\phi}_0\right)\right|}^2/{\lambda}_k}={\displaystyle \sum_{k=0}^{N-1}{Q}_k} $$

(23)

with

$$ {Q}_k={\left|{E}_k\left({\theta}_0,{\phi}_0\right)\right|}^2/{\lambda}_k $$

(24)

with the use of Eqs. (5), (16), and (17). The symbol *Q*
_{
k
} is the AG or DF of the *k*th-order mode-beam.

The sensitivity function (SF) is always used to measure the robustness of beamformers. A large SF corresponds to poor robustness, implying that the errors that can be tolerated are small. The SF is defined as follows [1]:

$$ T={\left\Vert \mathbf{w}\right\Vert}^2, $$

(25)

where ‖ ⋅ ‖ indicates the Euclidean norm. Substitution of Eq. (17) into Eq. (25) yields the total SF:

$$ \begin{array}{c}SF={\mathbf{E}}^{\mathrm{H}}\left({\theta}_0,{\phi}_0\right){\mathbf{D}}^{-2}\widehat{\mathbf{C}}{\mathbf{D}}^{-2}\mathbf{E}\left({\theta}_0,{\phi}_0\right)\\ {}={\displaystyle \sum_{k=0}^{N-1}{\displaystyle \sum_{k^{\prime }=0}^{N-1}\frac{1}{\lambda_k{\lambda}_{k^{\prime }}}{E}_k^{\ast}\left({\theta}_0,{\phi}_0\right){E}_{k^{\prime }}\left({\theta}_0,{\phi}_0\right){\widehat{\mathbf{C}}}_{k^{\prime}}^{\mathrm{T}}{\widehat{\mathbf{C}}}_k}}\\ {}={\displaystyle \sum_{k=0}^{N-1}\underset{T_k}{\underbrace{\frac{1}{\lambda_k^2}{\left|{E}_k\left({\theta}_0,{\phi}_0\right)\right|}^2{\widehat{\mathbf{C}}}_k^{\mathrm{T}}{\widehat{\mathbf{C}}}_k}}}+{\displaystyle \sum_{k=0}^{N-1}{\displaystyle \sum_{k^{\prime }=0,k\ne {k}^{\prime}}^{N-1}\frac{1}{\lambda_k{\lambda}_{k^{\prime }}}{E}_k^{\ast}\left({\theta}_0,{\phi}_0\right){E}_{k^{\prime }}\left({\theta}_0,{\phi}_0\right){\widehat{\mathbf{C}}}_{k^{\prime}}^{\mathrm{T}}{\widehat{\mathbf{C}}}_k}},\end{array} $$

(26)

where **E**(*θ*
_{0}, *ϕ*
_{0}) = [*E*
_{0}(*θ*
_{0}, *ϕ*
_{0}) *E*
_{1}(*θ*
_{0}, *ϕ*
_{0}) ⋯ *E*
_{
N − 1}(*θ*
_{0}, *ϕ*
_{0})]^{T}, **Ĉ** = **CC**
^{T} = [**Ĉ**
_{0}
**Ĉ**
_{1} ⋯ **Ĉ**
_{
k
} ⋯ **Ĉ**
_{
N − 1}]^{T}. The SF of the *k*th-order mode-beam is \( {T}_k={\lambda}_k^{-2}{\left|{E}_k\left({\theta}_0,{\phi}_0\right)\right|}^2\cdot {\widehat{\mathbf{C}}}_k^{\mathrm{T}}{\widehat{\mathbf{C}}}_k \), which is obtained by combining Eqs. (20) and (25). It is observed that *λ*
_{
k
} is directly related to the total SF and the SF of each mode-beam, which can be used to measure the robustness. The larger the value of *λ*
_{
k
}, the lesser is the SF of the *k*th-order mode-beam, meaning the smaller is the sensitivity to errors. However, the total SF is not equal to the sum of the SFs of mode-beams because the matrix **Ĉ** is not an orthogonal matrix, which is different from that of the EBDS model. Since the expression of total SF is somewhat complex and no superposition property similar to that of the EBDS model exists, the parameter *λ*
_{
k
} will be used in this paper to measure the robustness instead of the SF.

From Eqs. (19) and (23), it is clear that the overall beampattern and DF of an *N*-sensor superdirective array can be decomposed into *N* mode-beams and their associated DFs, respectively. These properties of the GSMDS method are similar to those of the EBDS method. The value of *λ*
_{
k
} in the low-frequency range for specific arrays (e.g., linear arrays) will be proved to decrease with an increase in its order, which means that the error sensitivity of this mode-beam increases. Therefore, a reduced-rank treatment that requires truncation of unsatisfactory high-order mode-beams and retention of robust low-order ones can be straightforwardly used to synthesize robust superdirective beampatterns. The maximum order of the required mode-beams in Eqs. (19) and (23), which will be denoted as *K* in the following, should then be smaller than or equal to *N*−1. Mode-beam extraction for a practical case of error distribution can be conducted through experimental measurements or computer simulations.

It is noteworthy that the optimal solution that provides maximum processing gain in different noise fields can also be derived using the above-mentioned procedure. However, the noise correlation coefficients between array elements cannot be always determined analytically, and sometimes, they should be estimated using in situ measurements. In other words, the general solution of the optimal array signal processing can be expressed analytically or adjusted adaptively. Because the superdirectivity is an important topic as mentioned previously, this study is only focused on the isotropic noise field, and the analytical and close-form solutions are directly derived.

For the purpose of clearly showing the performance of the GSMDS model, it is necessary to study the actual DF in the presence of errors.

In actual systems, sensor mismatches will affect the manifold and corrupt the beampattern. For simplicity, the present study takes into account just the gain and phase errors.

The gain and phase errors of the *k*th sensor are assumed to be *g*
_{
k
} and *ψ*
_{
k
}, respectively, and the actual received pressure will be

$$ {\tilde{a}}_k\left(\theta, \phi \right)={A}_k{a}_k\left(\theta, \phi \right)=\left(1+{g}_k\right){e}^{-i{\psi}_k}{a}_k\left(\theta, \phi \right). $$

(27)

Then, the beampattern in Eq. (3) changes to

$$ \tilde{B}\left(\theta, \phi \right)={\mathbf{w}}^{\mathrm{H}}\left[\mathbf{Aa}\left(\theta, \phi \right)\right]. $$

(28)

In the presence of sensor gain and phase errors, the directivity of the array will inevitably degrade. From the above discussion, the actual DF of the *k*th-order mode-beam can be defined as

$$ \begin{array}{c}{\tilde{Q}}_k=\frac{{\left|{\tilde{b}}_k\left({\theta}_0,{\phi}_0\right)\right|}^2}{\frac{1}{4\pi }{\displaystyle {\int}_0^{2\pi }{\displaystyle {\int}_0^{\pi }{\left|{\tilde{b}}_k\left(\theta, \phi \right)\right|}^2 \sin \theta d\theta d\phi}}}\\ {}=\frac{{\left|\frac{1}{\lambda_k}{E}_k^{\ast}\left({\theta}_0,{\phi}_0\right){\tilde{E}}_k\left({\theta}_0,{\phi}_0\right)\right|}^2}{\frac{1}{\lambda_k^2}{\left|{E}_k\left({\theta}_0,{\phi}_0\right)\right|}^2\left[\frac{1}{4\pi }{\displaystyle {\int}_0^{2\pi }{\displaystyle {\int}_0^{\pi }{\tilde{E}}_k\left(\theta, \phi \right){\tilde{E}}_k^{\ast}\left(\theta, \phi \right) \sin \theta d\theta d\phi}}\right]}\\ {}=\frac{{\left|{\tilde{E}}_k\left({\theta}_0,{\phi}_0\right)\right|}^2}{{\mathbf{C}}_k^{\mathrm{H}}{\tilde{\mathbf{R}}}_{nn}{\mathbf{C}}_k},\end{array} $$

(29)

where \( {\tilde{\mathbf{R}}}_{nn} \) denotes the practical noise covariance matrix, which is

$$ {\tilde{\mathbf{R}}}_{nn}=\frac{1}{4\pi }{\displaystyle {\int}_0^{2\pi }{\displaystyle {\int}_0^{\pi}\tilde{\mathbf{a}}\left(\theta, \phi \right)\cdot {\tilde{\mathbf{a}}}^{\mathrm{H}}\left(\theta, \phi \right) \sin \theta \mathrm{d}\theta \mathrm{d}\phi }}=\mathbf{A}{\mathbf{R}}_{nn}\mathbf{A}. $$

(30)

If there are no errors, Eq. (29) will be equal to Eq. (24). The average of the actual DF of the *k*th-order mode-beam is defined as

$$ {\overline{Q}}_k=E\left\{\frac{{\left|{\tilde{E}}_k\left({\theta}_0,{\phi}_0\right)\right|}^2}{{\mathbf{C}}_k^{\mathrm{H}}{\tilde{\mathbf{R}}}_{nn}{\mathbf{C}}_k}\right\}. $$

(31)

Actually, the sensor errors also increase the signal gain, which will enhance the DF. However, this additional benefit can be neglected in view of the degradation caused by errors [3]. Thus, for obtaining a conservative estimate, the average of the actual DF will be defined as

$$ {\overline{Q}}_k\approx \frac{{\left|{E}_k\left({\theta}_0,{\phi}_0\right)\right|}^2}{{\mathbf{C}}_k^{\mathrm{H}}{\overline{\mathbf{R}}}_{nn}{\mathbf{C}}_k}, $$

(32)

where the derivation of \( {\overline{\mathbf{R}}}_{nn}=E\left\{{\tilde{\mathbf{R}}}_{nn}\right\} \) and its calculation method can be found in [13]. The average DFs of other beamforming methods can be similarly defined as

$$ \overset{-}{DF}=E\left\{\frac{{\left|{\mathbf{w}}^{\mathrm{H}}\tilde{\mathbf{p}}\left({\theta}_0,{\phi}_0\right)\right|}^2}{{\mathbf{w}}^{\mathrm{H}}{\tilde{\mathbf{R}}}_{nn}\mathbf{w}}\right\}\approx \frac{{\left|{\mathbf{w}}^{\mathrm{H}}\mathbf{p}\left({\theta}_0,{\phi}_0\right)\right|}^2}{{\mathbf{w}}^{\mathrm{H}}{\overline{\mathbf{R}}}_{nn}\mathbf{w}}. $$

(33)