Cramer-Rao Bound Investigation of the Double-Nested Arc Array Virtual Extension and Butterfly Antenna Configuration

Abstract

This paper exploits a sparse array representation, a novel double-nested arc array to generate a multiple virtual nested arc array that forms the virtual non-uniform circular array which has been presented and investigated. The nested arc array has been developed using higher-order statistics, namely using the fourth-order cumulants (FOC), to increase the array aperture. The extended array contains a hybrid of physical and virtual sensors which is distributed in a butterfly shape. The enhanced degrees of freedom (DOFs) in the resultant virtual array make it appropriate for over-and under-determined direction-of-arrival (DOA) estimation. The Cramer-Rao bound (CRB), the number of nested arc arrays, the structure of the double-nested arc related to each other, and the overlap between sensors for such an array are developed and investigated for various arrays. The proposed array performance is superior to that of the uniform one. Further, the proposed array is appropriate for two-dimensional (2D) DOA estimation problems.

Appendix

1.1 Direct and Fast Virtual Sensor Localization

The physical sensor ${\mathcal{M}}3_{{{\text{arc}}_{1} \ell_{2} }}$ in Fig. 13 shapes the identified position data of the generated virtual sensor ${\text{VS}}_{23}$ and it provides an element of the generated virtual arc and circle.

Computation Related to the Central Physical Sensors to Configure the Virtual Circle

$$b_{23} = \sqrt {2R^{2} (1 - \cos (\beta_{23} ))}$$

(36)

$$\rho_{23} = \cos^{ - 1} \frac{{b_{23} }}{2R}$$

(37)

(38)

Computation Related to the Generated Virtual Arc

As shown in Fig. 13

$$\beta_{23} = \beta$$

(39)

virtual arc angles sequence $\left( {\alpha ,\beta ,\gamma } \right)$, which is counterclockwise direction as the physical arc. Applying the use of the sine rule:

$$\frac{{a_{23} }}{{\sin \rho_{23} }} = \frac{R}{{\sin \rho_{23} }} = \frac{{b_{23} }}{{\sin \beta_{23} }}$$

(40)

$${ }\frac{{a_{23} }}{{\sin \rho_{23} }} = \frac{R}{{\sin \rho_{23} }} = \frac{{b_{23} }}{\sin \beta }$$

(41)

The generated virtual arc has a radius R exactly like the physical one. So

$$a_{23} = R$$

(42)

Similarly, the same derivation for ${\text{VS}}_{13} ,{\text{VS}}_{43}$ when the original central physical sensor is shared (overlapped) with a virtual sensor on the generated virtual arc. It takes the same sequence number three to be ${\text{VS}}_{3}$ as the physical sensors ${\mathcal{M}}3_{{{\text{arc}}_{1} l_{2} }}$ which is responsible to generate the virtual arc.

These parameters related to the main central physical sensors are used for configuring a virtual circle by the equiradius virtual sensors of the same $b_{MM}$ as shown in Fig. 5. Also, the generated virtual nested arc array ${\text{VS}}_{13} { },{\text{VS}}_{23} { },{\text{VS}}_{3} { },{\text{VS}}_{43}$ lays with a ${\text{radius}} = R$ for all the virtual senores from the central origin physical sensor $\mathcal{M}3_{{{\text{arc}}_{1} l_{2} }}$ and with the same angles sequence values and direction $\left( {\alpha ,\beta ,\gamma } \right)$ as the physical actual arc $\mathcal{M}1_{{{\text{arc}}_{1} \ell_{1} }}$, $\mathcal{M}2_{{{\text{arc}}_{1} \ell_{1} }}$, $\mathcal{M}3_{{{\text{arc}}_{1} \ell_{2} }}$, $\mathcal{M}4_{{{\text{arc}}_{1} \ell_{2} }}$ as shown in Fig. 5.

Similarly applying the same calculations on each physical sensor to generate its virtual arc and circle.

1.2 Extended Cumulant-Based Virtual Sensors Localization for the Physical Sensor ${\mathbf{\mathcal{M}}}5_{{{\mathbf{origin}}}}$

Each of the two physical sensors, one on the arc, and one at the origin can generate a virtual sensor [4], it is configured with the same angles sequence values as the physical actual arc and with a ${\text{radius}} = R$ from the origin sensor $\mathcal{M}5_{{{\text{origin}}}}$ as shown in Fig. 5.

Fig. 13

Arc array’s extended virtual sensor ${\text{VS}}_{23}$ localization related to the physical sensor $\mathcal{M}3_{{{\text{arc}}_{1} \ell_{2} }}$