The basic idea of self-adapting root-MUSIC algorithm is: firstly weight summation for three-way signal of vector sensor, select the self-adaptive lead orientation, then construct polynomial by noise subspace, and finally estimate DOA of signals by finding the roots of polynomial.

### Selection of lead orientation vector

Weight 1, cos *ϕ*, sin *ϕ* to the output signal *p*_{
i
}(*t*),*v*_{
ix
}(*t*),*v*_{
iy
}(*t*) of *i* th vector sensor respectively, and make sum

{y}_{i}\left(t\right)={p}_{i}\left(t\right)+{v}_{\mathit{ix}}\left(t\right)cos\varphi +{v}_{\mathit{iy}}\left(t\right)sin\varphi \text{,}

(6)

then the average power is P _{
i
}(*ϕ*) = E[|*y*_{
i
}(*t*)|^{2}].

The function of weight corresponds to make electronic rotary for the output of the vector sensor, the direction *ϕ* which reflects the maximum energy is the signal direction[13].

P _{
i
}(*ϕ*) is the output of spatial spectrum of *i* th vector sensor with *ϕ* relevant, reflects the energy distribution in space. It is the equivalent of a spatial filter, and can implement the signal and noise separation based on the orientation difference of the signal and interference.

The vector form of (6) for vector sensor array can be written as

\mathbf{Y}=\mathbf{W}\cdot \mathbf{Z}\text{,}

(7)

where **W** = diag[1, cos *ϕ*, sin *ϕ*, ⋯, 1, cos *ϕ*, sin *ϕ*].

Take

P\left(\varphi \right)=\frac{1}{M}{\displaystyle \sum _{i=1}^{M}{P}_{i}\left(\varphi \right)}\text{,}

(8)

where P(*ϕ*) is spatial spectrum of array. The lead orientation *ϕ*_{0} can be obtained from the maximum of P(*ϕ*) for *ϕ* ∈ [0, 2*π*].

The lead orientation vector can be received as

\mathbf{u}={\left[1,cos{\varphi}_{0},sin{\varphi}_{0}\right]}^{T}\text{,}

(9)

where **u** is also known as self-adaptive lead vector.

### Construction of the polynomial

Define the polynomial

f\left(z\right)={z}^{M-1}{\mathbf{F}}^{T}\left(1/z\right){\mathbf{U}}_{N}{\mathbf{U}}_{N}^{H}\mathbf{F}\left(z\right)\text{,}

(10)

where **F**(*z*) = [1, *z*, ⋯, *z*^{M − 1}]^{T} ⊗ **u**, *z* = exp(*jβ*), *β* = (2*π*/*λ*)*d* sin *θ*, and *θ* is the azimuth angle of the signals to be estimated.

Let

\mathbf{B}=\left(\begin{array}{cccc}\hfill {\mathbf{b}}_{11}\hfill & \hfill {\mathbf{b}}_{12}\hfill & \hfill \cdots \hfill & \hfill {\mathbf{b}}_{1M}\hfill \\ \hfill {\mathbf{b}}_{21}\hfill & \hfill {\mathbf{b}}_{22}\hfill & \hfill \cdots \hfill & \hfill {\mathbf{b}}_{2M}\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {\mathbf{b}}_{M1}\hfill & \hfill {\mathbf{b}}_{M2}\hfill & \hfill \cdots \hfill & \hfill {\mathbf{b}}_{\mathit{MM}}\hfill \end{array}\right)={\mathbf{U}}_{N}{\mathbf{U}}_{N}^{H}\text{,}

(11)

where **b**_{
ij
} (*i*, *j* = 1, 2, ⋯, *M*) are 3 × 3 symmetry sub-matrix.

Then

\begin{array}{c}f\left(z\right)={z}^{M-1}{\mathbf{F}}^{T}(1/z)\mathbf{B}\mathbf{F}\left(z\right)=\mathbf{u}{\mathbf{b}}_{M1}{\mathbf{u}}^{H}\\ \phantom{\rule{7em}{0ex}}+z\mathbf{u}{\displaystyle \sum _{i=1}^{2}{\mathbf{b}}_{i+M-2,i}}{\mathbf{u}}^{T}+\cdots +{z}^{M-1}\mathbf{u}{\displaystyle \sum _{i=1}^{M}{\mathbf{b}}_{i,i}}{\mathbf{u}}^{T}\\ \phantom{\rule{7em}{0ex}}+{z}^{M}\mathbf{u}{\displaystyle \sum _{i=1}^{M-1}{\mathbf{b}}_{i,i+1}}{\mathbf{u}}^{T}+\cdots +{z}^{2M-3}\mathbf{u}{\displaystyle \sum _{i=1}^{2}{\mathbf{b}}_{i,i+M-2}}{\mathbf{u}}^{T}\\ \phantom{\rule{7em}{0ex}}+{z}^{2M-2}\mathbf{u}{\mathbf{b}}_{1M}{\mathbf{u}}^{T}={\displaystyle \sum _{k=1}^{M}\left(\mathbf{u}{\displaystyle \sum _{i=1}^{k}{\mathbf{b}}_{i+M-k,i}}{\mathbf{u}}^{T}\right){z}^{k-1}}\\ \phantom{\rule{4em}{0ex}}+{\displaystyle \sum _{k=1}^{M-1}\left(\mathbf{u}{\displaystyle \sum _{i=1}^{M-k}{\mathbf{b}}_{i,i+k}}{\mathbf{u}}^{T}\right){z}^{M+k-1}}\text{,}\end{array}

(12)

So the order of the polynomial *f*(*z*)is 2(*M* − 1), it has (*M* − 1) pair roots which every two conjugate with each another. and there are *N* roots which lie on the unit circle,

{z}_{i}=exp\left(j{\beta}_{i}\right),i=1,2,\cdots ,N\text{.}

(13)

In practical calculation, considering the error of covariance matrix, the *N* roots{\widehat{z}}_{i} nearest to the unit circle can be estimated as the DOAs of the signals.

{\widehat{\theta}}_{i}=arcsin\left(\frac{\lambda}{2\pi d}arg\left\{{\widehat{z}}_{i}\right\}\right),i=1,2,\cdots ,N\text{.}

(14)

To sum up, the self-adapting root-MUSIC algorithm can be formulated as the following six-step procedure:

Step 1: Compute **R** by (3), and the estimate is given by (5).

Step 2: Obtain **U**_{
N
} from the eigendecomposition of **R** by (4).

Step 3: Compute the lead vector **u** by (8).

Step 4: Construct the polynomial *f*(*z*) by (11).

Step 5: Find the root of the polynomial *f*(*z*), and select the roots{\widehat{z}}_{i} that are nearest to the unit circle as being the roots corresponding to the DOA estimates.

Step 6: Receive{\widehat{\theta}}_{i} to the DOA estimates by (12).