Two-dimensional Off-Grid DOA Estimation with Improved Three-Parallel Coprime Arrays on Moving Platform

Abstract

In this paper, two-dimensional (2-D) direction-of-arrival (DOA) estimation issue is investigated by constructing array model on moving platform. Based on the moving array model, we propose two improved three-parallel coprime arrays (TPCPAs), which utilize the redundancy in the physical structure of moving TPCPA (MTPCPA) and are able to generate the same virtual array as MTPCPA using much fewer sensors. Accordingly, higher sensor utilization is achieved by the proposed arrays, which can contribute to the increase in the number of degrees of freedom. Besides, with the proposed arrays, we also present a 2-D off-grid DOA estimation algorithm, in which the estimated 2-D angles are automatically paired without extra pairing procedure. Particularly, by utilizing the ${\ell }_{p}(0<p<1)$ norm and majorization–minimization method jointly, the proposed algorithm solves the grid mismatch problem effectively and accordingly enhances the 2-D DOA estimation performance. Finally, numerical simulations demonstrate the effectiveness and superiority of the proposed arrays and 2-D off-grid DOA estimation algorithm.

Appendix: Proof of Proposition 2

1) First, we will prove that the physical sensors located at $\{mN\left| {\left\lfloor {{M /2}} \right\rfloor + 1 \le m} \right. \le M\}$ with $M \ge 2$ in subarray 1 of MTPCPA can be removed without changing the DOFs of resulting virtual array.

Let $\bar{{\mathbb {D}}}={{\bar{{\mathbb {D}}}}_{1}}\cup {{\bar{{\mathbb {D}}}}_{2}}$, where ${{\bar{{\mathbb {D}}}}_{1}}=\{Mn-Nm\}$, ${{\bar{{\mathbb {D}}}}_{2}}=\{Nm-Mn\}$ for $0 \le n \le N - 1$, and $0 \le m \le 2M - 1$. For convenience of expression, we use the index (n, m) to denote the position of virtual sensor located at $Mn - Nm$. Similarly, the index (m, n) denotes the position of virtual sensor located at $Nm - Mn$. Then, in the covariance matrix, if two elements with indices $({n_1},{m_1})$ and $({n_2},{m_2})$ are complex conjugate, then the following relationship will be satisfied:

$$\begin{aligned} M{n_1} - N{m_1} = - (M{n_2} - N{m_2}). \end{aligned}$$

(54)

Obviously, we can modify (54) as

$$\begin{aligned} M({n_1} + {n_2}) = N({m_1} + {m_2}). \end{aligned}$$

(55)

Since M and N are coprime integers, the condition that equation (55) holds is ${n_1} + {n_2} = N$ and ${m_1} + {m_2} = M$. Then, for a given index $({n_1},{m_1})$ with $\left\lfloor {{M/2}} \right\rfloor + 1 \le {m_1} \le M$ and $1 \le {n_1} \le N - 1$, there will be a corresponding index $({n_2},{m_2})$ with ${m_2} = M - {m_1}$ in the range $\left[ {0,M - \left\lfloor {{M/2}} \right\rfloor - 1} \right]$ and ${n_2} = N - {n_1}$ in the range $[1,N - 1]$ satisfying (55). Therefore, the elements in ${\bar{{\mathbb {D}}}_1}$ and ${\bar{{\mathbb {D}}}_2}$ will have the following relationship:

$$\begin{aligned} \left\{ {\begin{array}{c} {\bar{{\mathbb {D}}}_1^{({n_2},{m_2})} = - \bar{{\mathbb {D}}}_1^{({n_1},{m_1})}} \\ {\bar{{\mathbb {D}}}_1^{({n_2},{m_2})} = - \bar{{\mathbb {D}}}_2^{({m_2},{n_2})}} \end{array}} \right. . \end{aligned}$$

(56)

From (56), we can know that $\bar{{\mathbb {D}}}_1^{({n_1},{m_1})} =\bar{{\mathbb {D}}}_2^{({m_2},{n_2})}$, which indicates that the virtual sensors corresponding to index range $({n_1},{m_1})$ of ${\bar{{\mathbb {D}}}_1}$ can be found in virtual sensors corresponding to index range $({m_2},{n_2})$ of ${\bar{{\mathbb {D}}}_2}$. Similarly, the virtual sensors corresponding to index range $({m_1},{n_1})$ of ${\bar{{\mathbb {D}}}_2}$ can also be found in virtual sensors corresponding to index range $({n_2},{m_2})$ of ${\bar{{\mathbb {D}}}_1}$. Therefore, physical sensors in subarray 1 and located at $\{ mN\left| {\left\lfloor {{M/2}} \right\rfloor + 1 \le m} \right. \le M\}$ are redundant, whose removal will not reduce the number of virtual sensors extended under the case that $n \in [1,N - 1]$.

Now, we will discuss the case that $n = 0$. If the physical sensors located at $\{mN|\left\lfloor M/2 \right\rfloor +1\le m\le M\}$ are removed, then the corresponding reduced virtual sensors with $n = 0$ in the positive range can be represented as ${{\mathbb {L}}^ + } =\{mN\left| {\left\lfloor {{M /2}} \right\rfloor + 1 \le m} \right. \le M\}$. ${{\mathbb {L}}^ - } = \{ - l\left| {l \in {{\mathbb {L}}^ + }} \right. \}$ is the corresponding reduced virtual sensors in the negative range. Similar to the 1-D CPA with $2M + N -1$ physical sensors, the consecutive part of $\bar{{\mathbb {D}}}$ lies in the range ${\mathbb {C}} = [ - (MN + M - 1),MN + M - 1]$. Since M and N are coprime integers, and $M < N$, there is the relationship that $M \ge 2$. Thereby, the inequality $MN = \max ({{\mathbb {L}}^ + }) \le MN + M - 1$ will hold. Likewise, $\min ({{\mathbb {L}}^ - }) = - MN \ge - (MN + M - 1)$ also holds. Therefore, the reduced virtual sensors ${\mathbb {L}} = {{\mathbb {L}}^ + } \cup {{\mathbb {L}}^ - }$ are actually some discontinuous discrete points in the consecutive range of $\bar{{\mathbb {D}}}$. Besides, according to the analysis in Sect. 3, we can know that the discontinuous holes in the virtual array of the original array can be filled with the virtual sensors increased by array motion. Accordingly, the holes in ${\mathbb {L}}$ caused by the removal of sensors located at $\{mN\left| {\left\lfloor {{M /2}} \right\rfloor + 1 \le m} \right. \le M\}$ will be filled with the increased virtual sensors due to array motion.

As a conclusion, the physical sensors located at $\{ mN\left| {\left\lfloor {{M /2}} \right\rfloor + 1 \le m} \right. \le M\}$ with $M \ge 2$ in subarray 1 of MTPCPA can be removed without changing the resulting virtual array structure.

2) Next, we will prove that the removal of physical sensors located at $\{ mN\left| {0 \le m} \right. \le \left\lfloor {{M/2}} \right\rfloor \}$ with $M > 2$ in subarray 1 of MTPCPA will not change the resulting virtual array. And, the proof will be given as follows according to the parity of M.

C1: Consider the case that M is even, where $\left\lfloor {{M/2}} \right\rfloor$ is thus equal to M/2. For index $({n_1},{m_1})$ with $0 \le {m_1} \le {M /2} - 1$ and $1 \le {n_1} \le N - 1$, there will be a corresponding index $({n_2},{m_2})$, i.e., $(N -{n_1},M - {m_1})$ with ${m_2}$ in the range ${M/2} + 1 \le {m_2} \le M$ (for even M, ${M /2} + 1$ is equivalent to $\left\lfloor {{M /2}} \right\rfloor + 1$) and ${n_2}$ in the range $1 \le {n_2} \le N - 1$ to make (55) be true. Similarly, the entries corresponding to index $({n_1},{m_1})$ and $({n_2},{m_2})$ will satisfy (56). Then, for $1 \le n \le N - 1$, physical sensors located at $\{ mN\left| {0 \le m} \right. \le \left\lfloor {{M /2}} \right\rfloor - 1\}$ can be removed without reducing the virtual sensors. Next, we will consider the removal of physical sensor located at $\{ mN\left| m \right. = \left\lfloor {{M/2}} \right\rfloor = {M /2}\}$ under the case that $1 \le n \le N -1$, which will result in the generation of holes located at ${{\mathbb {H}}_1} = \{ \pm (Mn - {{MN} /2})\left| {1 \le n} \right. \le N - 1\}$. While under the case that $n = 0$, the removal of sensors located at $\{ mN\left| {0 \le m} \right. \le \left\lfloor {{M /2}} \right\rfloor \}$ will lead to the generation of holes located at ${{\mathbb {H}}_2} = \{ \pm Nm\left| {0 \le m} \right. \le {M /2}\}$. Then, all holes can be represented as ${\mathbb {H}} = {{\mathbb {H}}_1} \cup {{\mathbb {H}}_2}$. Since $\max ({{\mathbb {H}}_1}) = {{MN} /2} - M < MN + M - 1$ and $\min ({{\mathbb {H}}_1}) = M - {{MN} /2} > - (MN + M - 1)$, ${{\mathbb {H}}_1} \subset {\mathbb {C}}$ holds. Similarly, $\max ({{\mathbb {H}}_2}) = {{MN} /2} < MN + M - 1$ and $\min ({{\mathbb {H}}_2}) = {{ - MN} /2} > - (MN + M - 1)$ hold, then ${{\mathbb {H}}_2} \subset {\mathbb {C}}$ is true. Accordingly, ${\mathbb {H}} \subset {\mathbb {C}}$ holds true, which indicates that the holes in ${\mathbb {H}}$ are some discrete points in the consecutive range of $\bar{{\mathbb {D}}}$. As mentioned before, when the number of consecutive holes does not exceed 2, all of them can be filled with the virtual sensors increased by array motion. Therefore, we will first analyze whether there are three consecutive holes in ${\mathbb {H}}$. For convenience, we use circles and crosses to represent the holes in ${{\mathbb {H}}_1}$ and ${{\mathbb {H}}_2}$, respectively. Then, the relative positions of holes in ${{\mathbb {H}}_1}$ and ${{\mathbb {H}}_2}$ will have the following four cases in Fig. 13.

Fig. 13

Four cases for the relative positions of holes in ${{\mathbb {H}}_1}$ and ${{\mathbb {H}}_2}$, where circles and crosses represent the holes in ${{\mathbb {H}}_1}$ and ${{\mathbb {H}}_2}$, respectively

For Fig. 13a, the spacing between holes 2 and 3 is N unit length, which means that the three holes are discontinuous. Similarly, the distance between holes 2 and 3 in Fig. 13b is M unit length, which indicates that the three holes are also discontinuous. Since the holes in ${\mathbb {H}}$ are mirror-symmetrical with respect to the origin, we just need to analyze the holes of ${\mathbb {H}}$ in the positive range here. For Fig. 13c, we can set the positions of holes 1, 2, and 3 as Nx, $My - {{MN} /2}$, and $(x + 1)N$, respectively, according to the expressions of ${{\mathbb {H}}_1}$ and ${{\mathbb {H}}_2}$, where x and y are integers in the range $0 \le x \le {M /2}$ and $1 \le y \le N - 1$, respectively. Accordingly, if the three holes in Fig. 13c are continuous, then the following equation should be satisfied, i.e.,

$$\begin{aligned} \left\{ \begin{array}{l} Nx + 1 = My - {{MN} /2} \\ My - {{MN}/2} + 1 = (x + 1)N \\ \end{array}\right. . \end{aligned}$$

(57)

Simplifying (57), we can get $N = 2$, which contradicts the given condition $N> M > 2$. Thus, the three holes in Fig. 13c are also discontinuous.

Finally, we will analyze the positional relationship of the three holes in Fig. 13d, whose positions can be similarly set as $My - {{MN} / 2}$, Nx, and $M(y + 1) - {{MN} /2}$, respectively, where the range of x and y is the same as that in (57). Then, the condition that the three holes in Fig. 13d are continuous is as follows.

$$\begin{aligned} \left\{ \begin{array}{l} My - {{MN} / 2} + 1 = Nx \\ Nx + 1 = M(y + 1) - {{MN} /2} \\ \end{array}\right. . \end{aligned}$$

(58)

Simplifying (58), we can get $M = 2$, which contradicts the given condition $M > 2$. Thus, the three holes in Fig. 13d are not continuous.

Based on the above analysis, we can see that ${\mathbb {H}}$ does not contain three or more continuous holes, so it can be fully filled by the virtual sensors increased by array motion. As a conclusion, physical sensors located at $\{ mN\left| {0 \le m} \right. \le \left\lfloor {{M /2}} \right\rfloor \}$ with $M>2$ can be removed without changing the resulting virtual array under the case that M is even.

C2: For the case that M is odd, we consider the index $({n_1},{m_1})$ with ${m_1}$ in the range $0 \le {m_1} \le \left\lfloor {{M /2}} \right\rfloor = {{(M - 1)} /2}$ and ${n_1}$ in the range $1 \le {n_1} \le N - 1$. Then, there will be a corresponding index $({n_2},{m_2})$, i.e., $(N - {n_1},M - {m_1})$ with ${{(M + 1)} /2} \le {m_2} \le M$ (for odd M, ${{(M + 1)} /2}$ is equal to $\left\lfloor {{M/2}} \right\rfloor + 1$) and $1 \le {n_2} \le N - 1$ satisfying (55). The corresponding entries of index $({n_1},{m_1})$ and $({n_2},{m_2})$ will satisfy (56), which indicates that when $1 \le n \le N - 1$, the reduced virtual sensors in ${\bar{{\mathbb {D}}}_1}$ due to the removal of physical sensors located at $\{ mN\left| {0 \le m} \right. \le \left\lfloor {{M /2}} \right\rfloor \}$ with $M > 2$ can be found in virtual sensors corresponding to index range $({m_2},{n_2})$ of ${\bar{{\mathbb {D}}}_2}$. Similarly, the reduced virtual sensors in ${\bar{{\mathbb {D}}}_2}$ can also be found in that corresponding to index range $({n_2},{m_2})$ of ${\bar{{\mathbb {D}}}_1}$.

As for $n = 0$, the removal of sensors located at $\{ mN\left| {0 \le m} \right. \le {{(M - 1)} /2}\}$ will cause the generation of holes located at $\{ \pm Nm\left| {0 \le m} \right. \le {{(M - 1)} /2}\}$. Obviously, these holes are discrete points scattered in ${\mathbb {C}}$ at intervals of N unit length. Therefore, they can be fully filled by virtual sensors increased by array motion. As a conclusion, physical sensors located at $\{ mN\left| {0 \le m} \right. \le \left\lfloor {{M /2}} \right\rfloor \}$ with $M > 2$ can be removed without changing the resulting virtual array under the case that M is odd.

Synthesizing the above analysis, we can conclude that the redundant sensors located at $\{ mN\left| {\left\lfloor {{M /2}} \right\rfloor + 1 \le m} \right. \le M\}$ with $M \ge 2$ or $\{ mN\left| {0 \le m} \right. \le \left\lfloor {{M /2}} \right\rfloor \}$ with $M > 2$ in subarray 1 of MTPCPA can be removed without affecting the DOFs of the resulting virtual array.