Snapshot averaged Matrix Pencil Method (SAM) for direction of arrival estimation

Abstract

The estimation of the direction of electromagnetic (EM) waves from a radio source using electrically short antennas is one of the challenging problems in the field of radio astronomy. In this paper we have developed an algorithm which performs better in direction and polarization estimations than the existing algorithms. Our proposed algorithm Snapshot Averaged Matrix Pencil Method (SAM) is a modification to the existing Matrix Pencil Method (MPM) based Direction of Arrival (DoA) algorithm. In general, MPM estimates DoA of the incoherent EM waves in the spectra using unitary transformations and least square method (LSM). Our proposed SAM modification is made in context to the proposed Space Electric and Magnetic Sensor (SEAMS) mission to study the radio universe below 16 MHz. SAM introduces a snapshot averaging method to improve the incoherent frequency estimation thereby improving the accuracy of DoA estimation. It can also detect polarization to differentiate between Right Hand Circular Polarlization (RHCP), Right Hand Elliptical Polarlization (RHEP), Left Hand Circular Polarlization (LHCP), Left Hand Elliptical Polarlization (LHEP) and Linear Polarlization (LP). This paper discusses the formalism of SAM and shows the initial results of a scaled version of a DoA experiment at a resonant frequency of $\sim$72 MHz.

Appendices

Appendix A: Phase contamination by electronic components

In circuit theory any receiving antenna can be viewed as a independent voltage source with a source impedance called antenna impedance or radiation resistance [4]. Figure 15 is the circuit equivalent diagram of an receiving antenna with a load resistance of 50 $\Omega$.

Fig. 15

Receiving antenna circuit equivalent. $V_{RX}$ is the voltage received by the antenna, $Z_{ANT}$ is the intrinsic impedance or radiation resistance of the antenna, R is the load resistance of 50 $\Omega$, and $V_{out}$ voltage across the load

Since, the voltage received ($V_{RX}$) by the antenna is due to electric field of EM wave then, $V_{RX}\,=\,h_{eff}\textbf{E}$ where $h_{eff}$ is the effective height of the antenna and $\textbf{E}$ is the electric field present in the EM wave. Using the plane wave consideration the electric field component can be written as $\textbf{E}\,=\,E_0 e^{j(\textbf{k}\cdot \textbf{r} - \omega t)}$ here, $\omega \,=\,2\pi f$. Thus, the voltage received can be written as following:

$$\begin{aligned} V_{RX} = h_{eff}E_0 e^{j\textbf{k}\cdot \textbf{r}} e^{-j \omega t} = A e^{-j \omega t} \end{aligned}$$

(A1)

Using Eq. (A1) and circuit in Fig. 15 the received signal $V_{out}$ can be written as

$$\begin{aligned} V_{out} = \frac{V_{RX}\times 50}{50+Z_{ANT}} \end{aligned}$$

(A2)

As impedance comprises of resistive (R) and reactive component thus antenna impedance can be written as $Z_{ANT}\,=\,R_{ANT}+jX_{ANT}$. Considering the antenna impedance and Eq. (A2) on can observe analytically how phase is being modified due to the impedance in Eq. (A3).

$$\begin{aligned} V_{out} = \frac{50 A}{\sqrt{R_{ANT}^2+50^2}} e^{-j[wt + tan^{-1}({X_{ANT}/(R_{ANT}+50)})]} \end{aligned}$$

(A3)

In case of addition of several circuit components either in series or in parallel, the antenna impedance $Z_{ANT}$ in Eq. (A3) has to be replaced by the effective impedance of the circuit also known as the Thevenin’s equivalent.

Appendix B: Matrix pencil method

The Matrix Pencil method is used to obtain the best estimates since it interacts directly with the data instead of generating a co-variance matrix, reducing computer complexity [48]. Eq. (6) is used to generate a Hankel matrix in order to estimate N and $\omega ^n$.

$$\begin{aligned} \Lambda = \begin{bmatrix} S(0) &{} S(1) &{} \cdots &{} S(L)\\ S(1) &{} S(2) &{} \cdots &{} S(L+1)\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ S(M-L-1) &{} S(M-L) &{} \cdots &{} S(M-1) \end{bmatrix}_{(M-L) \times (L+1)} \end{aligned}$$

where, L is selected between (M/3, M/2] for optimum performance and is known as the pencil parameter [37]; M is the total sample length. The real matrix($\Lambda _R$) is computed using a Unitary matrix transformation [37] ( $\Lambda _R = U^\dagger [\Lambda \mid \Pi _{M-L} \Lambda ^* \Pi _{L+1}]U$; where, $^\dagger$ represents hermitian conjugate and U is the unitary matrix [23]) and the complex number matrix $\Lambda$. Later, an estimate of the singular values of $\Lambda _R$ is generated using SVD formulation. Matrix $A_s$ consisting of N largest singular vectors of $\Lambda _R$ is estimated by performing a thresholding operation on the normalized Eigen value i.e., $\sigma _i/\sigma _{\max }$. N generalized singular values are then calculated using an unitary transformation ($-[Re(U^\dagger J_1 U)A_s]^{-1}\cdot Im(U^\dagger J_1 U)A_s$) which are, $\psi _1, \psi _2,\cdots ,\psi _N$. Also, N incoherent frequencies are calculated by $\omega ^n = 2arctan(\psi _n)/\delta$ for $n = 1,2,\cdots ,N$ [21, 23, 40].