When emitting plane waves with a linear array, all elements are triggered simultaneously, while the time delay of each element will be designed individually when emitting with a ring array. In this section, the PW emitting and receiving methods based on ring arrays are introduced. The MV and ESBMV beamforming methods are then mentioned briefly.

### Plane wave transmitting method

Considering a ring array with \(N\) elements in total, it is obvious that not all \(N\) elements will be triggered in a single emission event to form a plane wave. Here, \(N^{\prime}\) (\(N^{\prime} < N\)) elements are used in each emission event to produce the plane wave and the time delay \(t_{delay} \left( i \right)\) of the \(i th\) element of each transmitting aperture can be calculated as follows.

$$t_{delay} \left( i \right) = \left[ {r - r*\cos \left( {\left| {\frac{{N^{\prime} + 1}}{2} - i} \right|*\Delta \theta } \right)} \right]/c \quad i = 1,2, \ldots ,N^{\prime}$$

(1)

where \(r\) is the radius of the ring array and \(c\) is the speed of sound, which is considered to be a constant throughout the imaging area. \(\Delta \theta\) is the central angle between two adjacent elements, as the elements of the ring array are uniformly distributed.

In each emission event, the plane wave can be formed in front of \(N^{\prime}\) elements, and the area in front of the \(N^{\prime}\) elements can be scanned and reconstructed. The transmitting aperture can be shifted around the target to yield a whole view. For a shift of \(\Delta N\) elements between two adjacent emission events, the total number of plane wave emission events \(M\) will be calculated as follows:

$$M = floor\left( {\frac{N}{\Delta N}} \right) + 1$$

(2)

where \(floor\left( {\frac{N}{\Delta N}} \right)\) rounds the \(\frac{N}{\Delta N}\) to the nearest integers less than or equal to \(\frac{N}{\Delta N}\). Obviously, \(M\) is much smaller than \(N\), which means that the scanning time with the plane wave method can be \(\frac{M}{N}\) of the time with the SA method.

### Receiving method

In the \(j th\) emission event, \(N^{\prime}\) elements are triggered by order according to the time delay mentioned in Eq. (1). The element in the centre of the transmitting aperture is triggered first (\(t = 0\)), and the plane wave is formed completely after the time \(t = \mathop {\hbox{max} }\nolimits_{i} \left\{ {t_{delay} \left( i \right)} \right\}\), which means that the receiving signal before time \(t = \mathop {\hbox{max} }\nolimits_{i} \left\{ {t_{delay} \left( i \right)} \right\}\) cannot be used and may cause a dead zone. However, the dead zone is relatively small and can be avoided in application by placing the target in the centre of the ring array.

For a certain position \(\varvec{p}\) in front of the \(j th\) transmitting aperture, the plane wave will reach \(\varvec{p}\) when the time is

$$t_{j} = \frac{{r + \varvec{p} \cdot \varvec{d}_{j} }}{c}$$

(3)

The echo wave from the position \(\varvec{p}\) can be received by the \(i th\) element at time

$$t_{i,j} = \frac{{r + \varvec{p} \cdot \varvec{d}_{j} }}{c} + \frac{{\left| {\varvec{p} - \varvec{e}_{i} } \right|}}{c}$$

(4)

where \(\varvec{d}_{j}\) is a unit vector that represents the spreading direction of the plane wave in the \(j th\) emission event, as shown in Fig. 1, and \(\varvec{e}_{i}\) is the position of the \(i th\) element on the transmitting aperture. The origin of the two vectors \(\varvec{d}_{j}\) and \(\varvec{e}_{i}\) is the same as the geometry centre of the ring array.

Thus, the area in front of the \(j th\) transmitting aperture can be reconstructed. The whole result can also be calculated after all \(M\) emission events are finished. The total \(M\) emission processes are shown in Fig. 2.

### Eigenspace-based minimum variance beamforming for a ring array

Minimum variance is the most commonly used adaptive beamforming method. Instead of being added up easily as in the delay-and-sum (DAS) method, the signals from different elements have different weights applied before being added. The weight is calculated according to the received signal. Here, the MV beamformer is briefly introduced according to [14,15,16,17,18,19]. In a certain emission event, \(N^{\prime}\) elements of the ring array are used as both transmitters and receivers, so the signal received from each element can be arranged as a vector \(\varvec{X}\) after the time delay at position \(\varvec{p}\).

$$\varvec{X}\left( \varvec{p} \right) = \left[ {x_{1} ,x_{2} , \ldots ,x_{{N^{\prime}}} } \right]^{T}$$

(5)

Then, the final output for a certain position reconstructed by this emission is

$$z\left( \varvec{p} \right) = \varvec{W}^{H} \cdot \varvec{X} = \mathop \sum \limits_{i = 1}^{{N^{\prime}}} w_{i}^{*} \cdot x_{i}$$

(6)

where \(x_{i}\) is the sample received at the \(i th\) element and \(w_{i}\) is the adaptive weight on the \(i th\) element in this emission event. The \(N^{\prime}\) different weights can also be arranged as a vector.

$$\varvec{W} = \left[ {w_{1} ,w_{2} , \ldots ,w_{{N^{\prime}}} } \right]^{T}$$

(7)

\(\cdot^{H}\) is the conjugate transpose, and \(\cdot^{ *}\) represents the complex conjugate.

In the MV method, the optimization problem can be expressed as

$$\begin{aligned} {\text{min }}E\left[ {z^{2} } \right] =\, & \hbox{min} \varvec{W}^{H} \varvec{RW} \\ {\text{s}}.{\text{t}}. \varvec{W}^{H} \cdot \varvec{d} =\, & 1 \\ \end{aligned}$$

(8)

where \(\varvec{d}\) is the steering vector and will become \(\varvec{d} = \left[ {1,1, \ldots ,1} \right]^{T}\) after the time delay. The covariance matrix \(\varvec{R}\) is defined as \(\varvec{R} = E\left[ {\varvec{X} \cdot \varvec{X}^{H} } \right]\). The solution to the MV problem is calculated by

$$\varvec{W}_{MV} = \frac{{\varvec{R}^{ - 1} \varvec{d}}}{{\varvec{d}^{H} \varvec{R}^{ - 1} \varvec{d}}}$$

(9)

The covariance matrix \(R\) is not given directly. In practice, \(R\) is estimated with the receiving signal as in Eq. (10).

$$\hat{\varvec{R}} = \frac{1}{{N^{\prime} - L + 1}}\mathop \sum \limits_{q = 1}^{{N^{\prime} - L + 1}} {\text{x}}_{q} {\text{x}}_{q}^{H}$$

(10)

where \(L\) is the length of the subarray and \({\text{x}}_{q} = \left[ {x_{q} ,x_{q + 1} , \ldots ,x_{q + L - 1} } \right]^{T}\) is the signal vector on the \(q th\) subarray. To make the result more robust, diagonal loading is also applied in estimating \(\hat{R}\), in the form of \(\varepsilon \varvec{I}\).

$$\varepsilon = \frac{1}{{100*N^{\prime}}}*tr\left\{ {\hat{\varvec{R}}} \right\}$$

(11)

where, \(tr\left\{ \cdot \right\}\) is the trace of matrix \(\hat{\varvec{R}}\). Then, the estimated covariance matrix \(\hat{\varvec{R}}\) is applied to calculate the adaptive weight \(\varvec{W}_{MV}\).

To further improve the reconstruction result, the covariance matrix \(\varvec{R}\) can be eigen-decomposed into two orthogonal subspaces, called the signal subspace \(\varvec{R}_{s}\) and the noise subspace \(\varvec{R}_{n}\), in the ESBMV beamformer [16].

$$\varvec{R} = \varvec{U\varLambda U}^{H} = \mathop \sum \limits_{i = 1}^{L} \lambda_{i} \varvec{v}_{i} \varvec{v}_{i}^{H} = \mathop \sum \limits_{i = 1}^{{N_{sig} }} \lambda_{i} \varvec{v}_{i} \varvec{v}_{i}^{H} + \mathop \sum \limits_{{i = N_{sig} + 1}}^{L} \lambda_{i} \varvec{v}_{i} \varvec{v}_{i}^{H} = \varvec{U}_{s}\varvec{\varLambda}_{s} \varvec{U}_{s}^{H} + \varvec{U}_{p}\varvec{\varLambda}_{p} \varvec{U}_{p}^{H} = \varvec{R}_{s} + \varvec{R}_{n}$$

(12)

where \(\varvec{\varLambda}= {\text{diag}}\left[ {\lambda_{1} ,\lambda_{2} , \ldots ,\lambda_{L} } \right]\) are the eigenvalues arranged in descending order and \(\varvec{U} = \left[ {\varvec{v}_{1} ,\varvec{v}_{2} , \ldots ,\varvec{v}_{L} } \right]\) is composed of the \(L\) orthonormal eigenvectors \(\varvec{v}_{i}\) corresponding to the eigenvalues \(\lambda_{i}\), while the signal subspace \(\varvec{U}_{s}\) is composed of \(N_{sig}\) orthonormal eigenvectors \(\varvec{v}_{i}\) corresponding to the \(N_{sig}\) largest eigenvalues \(\lambda_{i}\). Then, the adaptive weights in the ESBMV beamformer are calculated as

$$\varvec{W}_{ESBMV} = \varvec{U}_{s} \varvec{U}_{s}^{H} \varvec{W}_{MV}$$

(13)

The final imaging result can be reconstructed by adding the \(M\) results of all emission events.