A Novel Translated Coprime Array Configuration for Moving Platform in Direction-of-Arrival Estimation

Abstract

This paper proposes a novel coprime array configuration, which achieves more degrees of freedom (DOFs) than the existing coprime arrays for half wavelength array motion in direction-of-arrival estimation. Generally, in a moving coprime array, there exists a large number of redundant lags in its difference coarray, which limits the number of achievable DOFs. In the proposed configuration, we place the sensors of the array strategically so that the redundant lags decrease. The mathematical expression for the number of consecutive lags of the proposed moving coprime array is provided in the paper. There will be a substantial improvement in the number of consecutive DOFs of the proposed moving coprime array over the existing moving coprime arrays. Simulations are carried out to demonstrate the effectiveness of the proposed coprime array over the existing coprime arrays in moving platform.

Appendices

Appendix A

1.1 Proof of Proposition 1

Let us represent the forward cross-difference set and backward cross-difference set of the new translated configuration by $L_{cd_{n}}$ and $L^-_{cd_{n}}$, respectively. So, we have $L_{cd_{n}}=\frac{M}{2}p_1+ Nm-\frac{M}{2}n, 0\le m \le \textit{M}-1,0\le n \le \textit{N}-1$. $L_{cd}$ of Configuration-I has positive consecutive lags up to $\frac{MN}{2}+\frac{M}{2}-1$. Hence, $L_{cd_{n}}$ of the new translated configuration will be hole-free from $\frac{M}{2}p_1+1$ to $\frac{MN}{2}+\frac{M}{2}-1+\frac{M}{2}p_1$.

Now, let us look into the positive holes of $L_{cd_{n}}$ up to $\frac{M}{2}p_1$. The negative holes of $L_{cd}$ in Configuration-I are located at -$(a_1\frac{M}{2}+b_1N), a_1\ge 0,b_1>0$ [14]. So, in the new configuration, the locations of the positive holes of $L_{cd_{n}}$ up to $\frac{M}{2}p_1$ will be on $\frac{M}{2}p_1-(a_1\frac{M}{2}+b_1N),a_1\ge 0,b_1>0,a_1\frac{M}{2}+b_1N<\frac{M}{2}p_1$. By similar logic, we can get the locations of positive holes of $L^-_{cd_{n}}$ up to $\frac{M}{2}p_1$. The locations of these holes will be on $(c_1\frac{M}{2}+d_1N)-\frac{M}{2}p_1,c_1\ge 0,d_1>0,c_1\frac{M}{2}+d_1N>\frac{M}{2}p_1$.

Let us investigate the common holes between $L_{cd_{n}}$ and $L^-_{cd_{n}}$ up to $\frac{M}{2}p_1$. For common holes, we have

$$\begin{aligned} \frac{M}{2}p_1-(a_1\frac{M}{2}+b_1N)&= (c_1\frac{M}{2}+d_1N)-\frac{M}{2}p_1,\\ a_1&\ge 0,b_1>0,c_1\ge 0,d_1>0\nonumber . \end{aligned}$$

(6)

Simplifying (6), we get

$$\begin{aligned} \frac{M/2}{N}=\frac{b_1+ d_1}{2p_1-(a_1+c_1)}. \end{aligned}$$

(7)

Now, as $2p_1\le (N-1)$ and $a_1\ge 0,c_1\ge 0$, $2p_1-(a_1+c_1)$ will always be less than N. So, (7) cannot be satisfied. Hence, no common hole is there between $L_{cd_{n}}$ and $L^-_{cd_{n}}$ in that specified range. Thus, the difference coarray of the new configuration has no holes up to $\frac{M}{2}p_1$. Hence, the difference coarray of the new translated configuration is hole-free up to $\frac{MN}{2}+\frac{M}{2}-1+\frac{M}{2}p_1$, as we have already proved that $L_{cd_{n}}$ of the new configuration is hole-free from $\frac{M}{2}p_1+1$ to $\frac{MN}{2}+\frac{M}{2}-1+\frac{M}{2}p_1$.

Appendix B

1.1 Proof of Proposition 2

If we proceed the same way as the proof of Proposition 1, to find the common holes between $L_{cd_{n}}$ and $L^-_{cd_{n}}$ up to $\frac{M}{2}(\frac{N+1}{2})$ in the new configuration, the corresponding equation of (7) will be

$$\begin{aligned} \frac{M/2}{N}=\frac{b_2+ d_2}{N+1-(a_2+c_2)}, a_2\ge 0,b_2>0,c_2\ge 0,d_2>0. \end{aligned}$$

(8)

Here, (8) will be satisfied whenever $N+1-(a_2+c_2)=N$ (as $a_2\ge 0,c_2\ge 0$). Hence, there will be at least one hole in the new configuration up to $\frac{M}{2}(\frac{N+1}{2})$.