2.7 Array response vector/matrix

Commonly used array forms include uniform linear array, uniform circular array, L-shaped array, planar array and arbitrary array.

1. Uniform line array

Assuming that the received signal meets the narrowband condition, that is, the time required for the signal to pass through the array length should be much smaller than the coherence time of the signal, and the signal envelope does not change much during the propagation time of the antenna array. For simplicity, it is assumed that the source and the antenna array are in the same plane, and that the wave incident on the array is a plane wave.

The incoming wave direction is $\theta_K(k=1,2,...,K)$

There are a total of $M$ elements

Then $d$ the array response vector of a uniform linear array with an element spacing of :

$\vec a(\theta)=\begin{bmatrix} 1 &exp(-j2\pi \frac{d}{\lambda}sin\theta_k) &... & exp(-j2\pi(M-1)\ frac{d}{\lambda}sin\theta_k) \end{bmatrix}^T$

Define the orientation matrix as: $A=[\vec a(\theta_1),\vec a(\theta_2),...,\vec a(\theta_k)]=\begin{bmatrix} 1 &1 &1 &1 \\e^{-j\frac {2\pi d}{\lambda}sin\theta_1}& e^{-j\frac{2\pi d}{\lambda}sin\theta_2} & ... & e^{-j\frac{2 \pi d}{\lambda}sin\theta_k}\\ ... & ... & ... &... \\ e^{-j\frac{2\pi d}{\lambda}(M -1)sin\theta_1} &e^{-j\frac{2\pi d}{\lambda}(M-1)sin\theta_2} &... & e^{-j\frac{2\pi d }{\lambda}(M-1)sin\theta_k}\end{bmatrix}$

2. Uniform circular array

M identical omnidirectional arrays of a uniform circle are evenly distributed on a circle of radius R on the plane xy , as shown in the figure.

A spherical coordinate system is used to represent the direction of arrival of the incident plane wave, and the origin O of the coordinate system is at the center of the array. The elevation angle $\theta$ of the source is the angle between the line from the origin to the source and the z-axis, and the azimuth $\phi$ is the angle between the projection of the line from the origin to the source on the plane xy and the x-axis.

The direction vector $\vec a(\theta,\phi)$ is $(\theta,\phi)$ the array response of DOA, which $\vec a(\theta,\phi)$ can be expressed as:

$\vec a(\theta,\phi)=\begin{bmatrix} exp(j2\pi Rsin\theta cos(\phi-\gamma_0)/ \lambda\\ exp(j2\pi Rsin\theta cos(\phi- \gamma_1)/ \lambda\\ ...\\ exp(j2\pi Rsin\theta cos(\phi-\gamma_{M-1})/ \lambda\end{bmatrix}$

Among them, $\gamma_m = 2\pi m/M,m=0,1,...,M-1$ , $R$ is the radius

3. L-shaped array

The L-shaped array is composed of a uniform linear array with N array elements on the x-axis and a uniform linear array with M array elements on the y-axis, and one has M+N-1 array elements. The array element spacing is d.

Assuming that there are K sources in the space that irradiate the array, the two-dimensional direction of arrival is $(\theta_k,\phi_k),k=1,2,...,K$

where $\theta_k$ and $\phi_k$ represent the elevation and azimuth angles of the kth source, respectively.

Assuming that the number of sources incident on this array is K, the direction matrix corresponding to N array elements on the x-axis is

$A_x=\begin{bmatrix} 1 &1 &1 &1 \\ e^{j2\pi dcos\phi_1 sin\theta_1/\lambda}& e^{j2\pi dcos\phi_2 sin\theta_2/\lambda}& ... & e^{j2\pi dcos\phi_k sin\theta_k/\lambda}\\ ...& ... & ... &... \\ e^{j2\pi d(N-1)cos\ phi_1 sin\theta_1/\lambda} & e^{j2\pi d(N-1)cos\phi_2 sin\theta_2/\lambda} & ...& e^{j2\pi d(N-1)cos\ phi_k sin\theta_k/\lambda}\end{bmatrix}$

The direction matrix corresponding to the M array elements on the y-axis is:

$A_y=\begin{bmatrix} 1 &1 &1 &1 \\ e^{j2\pi dsin\phi_1 sin\theta_1/\lambda}& e^{j2\pi dsin\phi_2 sin\theta_2/\lambda}& ... & e^{j2\pi dsin\phi_k sin\theta_k/\lambda}\\ ...& ... & ... &... \\ e^{j2\pi d(M-1)sin\ phi_1 sin\theta_1/\lambda} & e^{j2\pi d(M-1)sin\phi_2 sin\theta_2/\lambda} & ...& e^{j2\pi d(M-1)sin\ phi_k sin\theta_k/\lambda}\end{bmatrix}$

where $A_x$ and $Oh$ are Vandermonde matrices.

4. Planar array

Let the number of array elements of the planar array be M*N, and the number of information sources be K.

where $\theta_k$ and $\phi_k$ represent the elevation and azimuth angles of the kth source, respectively.

Then the wave path difference between the ith array element in the space and the reference array element is:

$\beta = 2\pi(x_icos\phi sin\theta+y_isin\phi cos\theta+z_icos\theta)/\lambda$

In the formula, $(x_i,y_i)$ is the coordinate of the i-th array element, and the surface array is generally in the xy plane, so it $z_i$ is generally 0

From the above analysis of the L-shaped array, it can be seen that the direction of the N array elements on the x-axis is $A_x$ , and the direction of the M array elements on the y-axis is $Oh$ . Therefore, the direction matrix of subarray 1 as shown in the figure above is $A_x$ , while the direction matrix of subarray 2 needs to consider the offset along the y-axis, and the wave length difference of each array element relative to the reference array element is equal to that of subarray 1 The wave path difference of the array element is added $2\pi dsin\phi sin\theta/\lambda$ , so we can get:

Subarray 1: $A_1 = A_xD_1(A_y)$

Subarray 2: $A_2 = A_xD_2(A_y)$

......

Subarray M: $A_M = A_xD_M(A_y)$

where $D_m(\cdot )$ is a diagonal matrix constructed from m rows of the matrix.

5. Arbitrary array

Assume that the M-element array is located in any three-dimensional space, as shown in the figure. Define the mth sensor in the array as $r_m=(x_m,y_m,z_m)$ . The orientation matrix is:

$A=[\vec a(\theta_1,\phi_1),\vec a(\theta_2,\phi_2),...,\vec a(\theta_k,\phi_k)]\in\mathbb{C}^{M\times K}$

Among them, $\vec a(\theta_k,\phi_k)$ is the direction vector of the kth information source, which can be expressed as:

$\vec a(\theta_k,\phi_k)=\begin{bmatrix} 1\\ e^{j2\pi(x_2sin\theta_kcos\phi_k+y_2sin\theta_ksin\phi_k+z_2cos\theta_k)/\lambda}\\ .. . . \\ e^{j2\pi(x_Msin\theta_kcos\phi_k+y_Msin\theta_ksin\phi_k+z_Mcos\theta_k)/\lambda} \end{bmatrix}\in\mathbb{C}^{M\times1} .$

where $\lambda$ is the wavelength.

"Array Signal Processing and MATLAB Realization" Array Response Matrix (Uniform Linear Array, Uniform Circular Array, L-shaped Array, Planar Array and Arbitrary Array)