An Improved CACIS Configuration for DOA Estimation with Enhanced Degrees of Freedom

Abstract

In this work, we propose a novel coprime array with compressed inter-element spacing (CACIS) configuration, which achieves the more consecutive and unique degrees of freedom (DOF) over the existing CACIS configuration. We know that CACIS is a popular configuration of the coprime class of arrays for direction-of-arrival estimation, where the inter-element spacing in one of the subarrays is compressed by an integer factor. If M and N are the parameters of a coprime array, it will be proved that the increase in the number of both consecutive and unique DOF of the proposed CACIS will be $O(\breve{M}N)$ over the existing CACIS, where $\breve{M}$ is the compressed inter-element spacing. We provide the closed-form expressions of the unique and consecutive DOF of the proposed CACIS. Spatial smoothing MUltiple SIgnal Classification is applied to estimate the directions of the sources. Simulation results demonstrate the superiority of the proposed CACIS over the existing CACIS and the existing coprime arrays.

Appendices

Appendix A

Proof of Proposition 1

In the proposed I-CACIS, let us denote the forward cross-difference lags and the backward cross-difference lags by $L_{\mathrm{cd}_{I}}$ and $L^-_{\mathrm{cd}_{I}}$, respectively. Hence, $L_{\mathrm{cd}_{I}}=\frac{\breve{M}N}{2}-\breve{M}+ Nm-\breve{M}n, 0\le m \le \textit{M}-1, 0\le n \le \textit{N}-1$. From Proposition 1b of [12], $L_\mathrm{cd}$ of the original CACIS is hole-free up to $MN-\breve{M}(N-1)-1$. Hence, from $\frac{\breve{M}N}{2}-\breve{M}+1$ to $MN-\frac{\breve{M}N}{2}-1$, $L_{\mathrm{cd}_{I}}$ of the I-CACIS will have uniform lags.

Let us investigate the positive holes of $L_{\mathrm{cd}_{I}}$ up to $\frac{\breve{M}N}{2}-\breve{M}$. From Proposition 1e of [12], the negative holes of $L_\mathrm{cd}$ of the original CACIS are located at $-(a\breve{M}+bN),a\ge 0,b>0$. Hence, the positive holes of $L_{\mathrm{cd}_{I}}$ in the proposed I-CACIS up to $\frac{\breve{M}N}{2}-\breve{M}$ will be located at $\breve{M}(\frac{N}{2}-1)-(a_1\breve{M}+b_1N),a_1\ge 0,b_1>0,a_1\breve{M}+b_1N<\breve{M}(\frac{N}{2}-1)$. Now, let us look at the positive holes of $L^-_{\mathrm{cd}_{I}}$ up to $\frac{\breve{M}N}{2}-\breve{M}$. In a similar way, we can find the holes of $L^-_{\mathrm{cd}_{I}}$ up to $\frac{\breve{M}N}{2}-\breve{M}$, which will be located at $(a_2\breve{M}+b_2N)-\breve{M}(\frac{N}{2}-1),a_2\ge 0,b_2>0, a_2\breve{M}+b_2N>\breve{M}(\frac{N}{2}-1)$. For the common holes of $L_{\mathrm{cd}_{I}}$ and $L^-_{\mathrm{cd}_{I}}$ up to $\breve{M}(\frac{N}{2}-1)$, we can write

$$\begin{aligned}&\breve{M} \left( \frac{N}{2}-1\right) -(a_1\breve{M}+b_1N)= (a_2\breve{M}+b_2N) -\breve{M} \left( \frac{N}{2}-1\right) , \nonumber \\&\quad a_1\ge 0,b_1>0,a_2\ge 0,b_2>0. \end{aligned}$$

(9)

From (9), we have

$$\begin{aligned} \frac{\breve{M}}{N}=\frac{b_1+ b_2}{N-(a_1+a_2+2)} . \end{aligned}$$

(10)

here $a_1\ge 0,b_1>0,a_2\ge 0,b_2>0$. Now, $\breve{M}$ and N are coprime because M and N are coprimes and $\breve{M}$ is a factor of M. As, $a_1\ge 0, a_2\ge 0$, the denominator of the right side of (10) will always be less than N. Thus, it can be inferred that (10) cannot be satisfied. So, no common hole will be there between $L_{\mathrm{cd}_{I}}$ and $L^-_{\mathrm{cd}_{I}}$ up to $\breve{M}(\frac{N}{2}-1)$. Hence, the DC of the proposed I-CACIS is hole-free up to $\frac{\breve{M}N}{2}-\breve{M}$. It is already proven that $L_{\mathrm{cd}_{I}}$ of the proposed I-CACIS has consecutive lags from $\frac{\breve{M}N}{2}-\breve{M}+1$ to $MN-\frac{\breve{M}N}{2}-1$. Hence, we can conclude that, in this case, the DC of the proposed I-CACIS has consecutive positive lags up to $MN-\frac{\breve{M}N}{2}-1$.

Appendix B

Proof of Proposition 2

The DC of the original CACIS has $MN-\breve{M}(N-1)-1$ consecutive positive lags and $MN-\frac{\breve{M}N}{2}+\frac{\breve{M}}{2}-\frac{N}{2}-\frac{1}{2}$ unique positive lags [12]. Hence, the DC of the original CACIS has $\frac{\breve{M}N}{2}-\frac{N}{2}-\frac{\breve{M}}{2}+\frac{1}{2}$ unique lags in the range $\left[ MN-\breve{M}(N-1)+1,N(M-1)\right]$. Obviously, these lags will come from $L_\mathrm{cd}$ as the maximum positive lag of $L^-_\mathrm{cd}$ of original CACIS is $\breve{M}(N-1)$ (it can be verified that whatever value of p we choose, $\breve{M}(N-1)$ will always be less than $MN-\breve{M}(N-1)+1)$.

Now, in the proposed I-CACIS, in this case, all the forward cross lags will shift in the positive direction by $\frac{\breve{M}N}{2}-\breve{M}$. Hence, in the proposed I-CACIS, from $MN-\breve{M}(N-1)+1+\frac{\breve{M}N}{2}-\breve{M}$ to $N(M-1)+\frac{\breve{M}N}{2}-\breve{M}$, $L_{\mathrm{cd}_{I}}$ will have $\frac{\breve{M}N}{2}-\frac{N}{2}-\frac{\breve{M}}{2}+\frac{1}{2}$ unique lags. Again, it can be checked that $L^-_{\mathrm{cd}_{I}}$ will not have any lag in that range. It is already proved that the proposed I-CACIS has $MN-\frac{\breve{M}N}{2}-1$ consecutive positive lags. Hence, in this case, the number of unique positive lags in the DC of the proposed I-CACIS is $MN-\frac{\breve{M}N}{2}-1+\frac{\breve{M}N}{2}-\frac{N}{2}-\frac{\breve{M}}{2}+\frac{1}{2}=MN-\frac{\breve{M}}{2}-\frac{N}{2}-\frac{1}{2}$.