Manifold Separation-Based DOA Estimation for Nonlinear Arrays via Compressed Super-Resolution of Positive Sources

Abstract

Manifold separation technique plays an important role in array modeling and signal processing for arbitrary arrays. By utilizing this technique, the atomic norm minimization (ANM) methods can be extended to the nonlinear arrays for DOA estimation via the generalized line spectral estimation approach. However, such an approach results in a high computational burden for the large-scale arrays. In this paper, a low-dimensional semidefinite programming (SDP) implementation to the coarray manifold separation atomic norm minimization (CMS-ANM) is proposed for a class of nonlinear arrays based on the compressed super-resolution of positive sources. The theoretical guarantee of the proposed SDP implementation for the exact recovery is presented, and the CMS-ANM-based low-complexity DOA estimation method is developed. The simulation results validate the theoretical analysis and demonstrate the satisfying trade-off for the performance and complexity.

Appendices

Appendix A Proof of Condition 1

Provided that there exists another atomic decomposition

$$\begin{aligned} \mathbf {r}=\sum _{k=1}^K{\mathbf {a}(\theta _k )}{\mathbf {s}'_{k}}+\sum _{l=1}^L{\mathbf {a}(\hat{\theta }_l )}{\hat{\mathbf {s}}_{l}} \end{aligned}$$

(A1)

such that it is also optimal to (10), where $\mathbf {a}(\theta )\in \mathcal {M}$, $\mathbf {s}'_{k} \ge 0$, ${\hat{\mathbf {s}}}_{l} \ge 0$ and $\{{{\theta }_{k}}\} \bigcap \{{{\hat{\theta }}_{l}}\}=\emptyset$, which gives

$$\begin{aligned} \begin{aligned} \sum _{k=1}^K{\mathbf {a}(\theta _k )}({\mathbf {s}'_{k}}-\mathbf {s}_{k})+\sum _{l=1}^L{\mathbf {a}(\hat{\theta }_l )}{\hat{\mathbf {s}}_{l}}=0 \end{aligned} \end{aligned}$$

(A2)

followed from the definition of (10), where $\left\{ \theta _k,{\mathbf {s}_{k}}\right\}$ is the truth atomic decomposition.

According to the assumption in Condition 1, there exists the nonnegative polynomial $q(\theta )=\mathbf {Q}^{\text {H}}_{\star }\bar{\mathbf {G}d}_{2M}(\theta )$ such that it vanishes only on $\left\{ \theta _k \right\} _{k=1,\ldots ,K}$, and thus there is the polynomial

$$\begin{aligned} \begin{aligned} q'(\theta )&=q(\theta )/\left\| \mathbf {Gd}_M(\theta ) \right\| _F^2\\&=\mathbf {Q}^{\text {H}}_{\star }\mathbf {a}(\theta ),\mathbf {a}(\theta )\in \mathcal {M}\\ \end{aligned} \end{aligned}$$

(A3)

satisfying

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&q'({\theta _{k}})=0,\forall k=1,2,\ldots ,K \\&q'({\theta }) > 0,\forall \theta \ne {\theta _{k}}, \\ \end{aligned} \right. \end{aligned} \end{aligned}$$

(A4)

which implies

$$\begin{aligned} \begin{aligned}&\mathbf {Q}^{\text {H}}_{\star }\left\{ \sum _{k=1}^K{\mathbf {a}(\theta _k )}({\mathbf {s}'_{k}}-\mathbf {s}_{k})+\sum _{l=1}^L{\mathbf {a}(\hat{\theta }_l )}{\hat{\mathbf { s}}_{l}}\right\} \\&\quad =\sum _{k=1}^K q'(\theta _k)({\mathbf {s}'_{k}}-\mathbf {s}_{k})+\sum _{l=1}^L q'(\hat{\theta }_l){{\hat{\mathbf {s}}}_{l}}\\&\quad =\sum _{l=1}^L q'(\hat{\theta }_l){{\hat{\mathbf {s}}}_{l}}\\&\quad =0.\\ \end{aligned} \end{aligned}$$

(A5)

Since $q'({\theta }) > 0$ on $\left\{ \hat{\theta }_{l}\right\}$, it can be concluded that ${{\hat{\mathbf {s}}}_{l}}=0$ for any $l \in [1,L]$, followed from (A5), and thus, the "another" atomic optimal decomposition can only have the form

$$\begin{aligned} \mathbf {r}=\sum _{k=1}^K{\mathbf {a}(\theta _k )}{\mathbf {s}'_{k}}. \end{aligned}$$

(A6)

Let $\mathbf {A}(\varvec{\Theta })=[\mathbf {a}(\theta _1),\ldots ,\mathbf {a}(\theta _K)]$ such that

(A7)

it implies that $\mathbf {A}(\varvec{\Theta })$ is full rank, followed from the assumption in Condition 1, i.e., $\bar{\mathbf {G}D}_{2M}(\varvec{\Theta })$ is full rank.

Note that combining (A2) and (A6) gives

$$\begin{aligned} \begin{aligned} \sum _{k=1}^K{\mathbf {a}(\theta _k )}({\mathbf {s}'_{k}}-\mathbf {s}_{k})=0, \end{aligned} \end{aligned}$$

(A8)

which implies ${\mathbf {s}'_{k}}=\mathbf {s}_{k}$, because $\mathbf {A}(\varvec{\Theta })$ is full rank by assumption. The atomic decomposition in (4) can be retrieved uniquely by solving (10) if Condition 1 is to be satisfied.

Appendix B Proof of Theorem 1

From the constraint in (11), it gives

$$\begin{aligned} \begin{aligned}&{{\mathbf {a}^{\text {H}}(\theta )}}\varvec{\Omega }\le 1, \forall \mathbf {a}(\theta )\in \mathcal {M}\\&\quad \Leftrightarrow \mathbf {d}^\text {H}_M({{\theta } })\mathbf {G}^\text {H}\mathbf {Gd}_M({{\theta } })-\mathbf {d}^\text {H}_{2M}(\theta )\bar{\mathbf {G}}^\text {H}\varvec{\Omega } \ge 0. \end{aligned} \end{aligned}$$

(B9)

According to Fejer–Riesz theorem, the nonnegative univariate polynomial is always sum-of-squares, i.e., there exists a positive-semidefinite matrix $\mathbf {P}\succeq 0$ such that

$$\begin{aligned} \mathbf {d}^\text {H}_M({{\theta } })\mathbf {G}^\text {H}\mathbf {Gd}_M({{\theta } })-\mathbf {d}^\text {H}_{2M}(\theta )\bar{\mathbf {G}}^\text {H}\varvec{\Omega } =\mathbf {d}^\text {H}_M({{\theta } })\mathbf {P}\mathbf {d}_M({{\theta } }), \end{aligned}$$

(B10)

which implies

$$\begin{aligned} \begin{aligned}&\sum \limits _{m}\text {Tr}({{\Theta }_{m}}\mathbf {P})z^m+\sum \limits _{m}\bar{\mathbf {g}}^{\text {H}}_{m}\varvec{\Omega }z^m=\sum \limits _{m}\text {Tr}({{\Theta }_{m}}{{\mathbf {G}^{\text {H}}}}\mathbf {G})z^m\\&\quad \Leftrightarrow \text {Tr}({{\Theta }_{m}}\mathbf {P})+\bar{\mathbf {g}}^{\text {H}}_{m}\varvec{\Omega }=\text {Tr}({{\Theta }_{m}}{{\mathbf {G}^{\text {H}}}}\mathbf {G}), \quad m\in \mathcal {H}\\ \end{aligned} \end{aligned}$$

(B11)

with $z=e^{-j\theta }$, i.e., the SDP implementation in (14) shares the identical feasible solution set with (11), and thus results the same solution since the same objective.

Appendix C Proof of Theorem 3

Note that the line spectral atomic set $\mathcal {A}$ in (5) can be described as a semi-algebraic set $\mathbf {Z}_M=\frac{1}{\sqrt{2M+1}}[z_1, \ldots , z_{2M+1}]$ that

(C12)

According to the Tarski–Seidenberg principle (Corollary 2.4 in [11]) and (C12), the array manifold $\mathcal {B}=\left\{ \mathbf {Td}_M(\theta ) \mid \mathbf {d}_M(\theta ) \in \mathcal {A} \right\}$ can be described as a compact semi-algebraic set

$$\begin{aligned} \begin{aligned} \mathcal {E}(g)&=\left\{ \mathbf {b} \in \mathbb {C}^{N} \mid g_v(\mathbf {b}) = 0, v=1,2, \ldots , \chi \right\} \end{aligned} \end{aligned}$$

(C13)

as well, where $\left\{ g_v(\mathbf {b}) \right\}$ are the polynomials on $\mathbf {b}$. And thus, any polynomial $f(\mathbf {b}) \ge 0$ with $\mathbf {b} \in \mathcal {B}$ is the sum-of-squares (SOS) polynomial by means of SOS relaxation [12], i.e., there exits $\mathbf {P }\succeq 0$ and the integer $\alpha \ge 1$ such that

$$\begin{aligned} \begin{aligned} f(\mathbf {b})={\mathbf {b}_{\alpha }^\text {H}}\mathbf {P}\mathbf {b}_{\alpha }, \end{aligned} \end{aligned}$$

(C14)

where $\mathbf {b}=[b_1,\ldots ,b_N]$, $\mathbf {b}_{\alpha }$ is the vector of monomials

(C15)

from the assumptions in Theorem 3. Let the polynomial $f(\mathbf {b})=\mathbf {d}^\text {H}_M({{\theta } })\mathbf {G}^\text {H}\mathbf {Gd}_M({{\theta } })-\mathbf {d}^\text {H}_{2M}(\theta )\bar{\mathbf {G}}^\text {H}\varvec{\Omega }$ with $\mathbf {b} \in \mathcal {B}$, it gives

$$\begin{aligned} \begin{aligned}&\mathbf {a}^{\text {H}}(\theta )\varvec{\Omega } \le 1, \forall \mathbf {a}(\theta )\in \mathcal {M}\\&\quad \Leftrightarrow f(\mathbf {b}) \ge 0\\&\quad \Rightarrow f(\mathbf {b})=\mathbf {d}^\text {H}_{M\alpha }({{\theta } })\mathbf {T}_{(\alpha )}^\text {H}\mathbf {P}\mathbf {T}_{(\alpha )}\mathbf {d}_{M\alpha }({{\theta } }),\\ \end{aligned} \end{aligned}$$

(C16)

according to (C14), which implies

$$\begin{aligned} \begin{aligned}&{{\mathbf {a}^{\text {H}}(\theta )}}\varvec{\Omega } \le 1, \forall \mathbf {a}(\theta )\in \mathcal {M}\\&\Leftrightarrow \left\{ \begin{aligned}&\text {Tr}({{\Theta }_{m}}{{\mathbf {T}}_{(\alpha )}^{\text {H}}}\mathbf {P}\mathbf {T}_{(\alpha )})+\hat{\mathbf {g}}^{\text {H}}_{m}\varvec{\Omega }=\text {Tr}({{\Theta }_{m}}{{\mathbf {G}^{\text {H}}_{(\alpha )}}}{\mathbf {G}}_{(\alpha )}) \quad m\in \mathcal {H}'\\&\mathbf {P}\succeq 0\\ \end{aligned} \right. \end{aligned} \end{aligned}$$

(C17)

where

$$\begin{aligned} \begin{aligned} \hat{\mathbf {g}}_m=\left\{ \begin{array}{ll} \bar{\mathbf {g}}_m,&{}\quad \vert m \vert \le 2M \\ \mathbf {0}_{N^2 \times 1},&{}\quad \text {Otherwise}\\ \end{array} \right. \end{aligned} \end{aligned}$$

(C18)

${\mathbf {G}}_{(\alpha )}=[\mathbf {0}_{N \times (\alpha -1)M}\quad \mathbf {G}\quad \mathbf {0}_{N \times (\alpha -1)M}]$, and $\mathbf {0}_{m \times n}$ is a $m \times n$ matrix with zeros. Then the following SDP

$$\begin{aligned} \begin{aligned}&\underset{\mathbf {P},{\varvec{\Omega }}}{\mathop {\max }}\,{\text {Re}}\left\{ \text {Tr}\left[ {{\mathbf {r}}^{\text {H}}}{\varvec{\Omega }} \right] \right\} \\&\text {s.t.}\quad \text {Tr}({{\Theta }_{m}}{{\mathbf {T}}_{(\alpha )}^{\text {H}}}\mathbf {P}\mathbf {T}_{(\alpha )})+\hat{\mathbf {g}}^{\text {H}}_{m}\varvec{\Omega }=\text {Tr}({{\Theta }_{m}}{{\mathbf {G}^{\text {H}}_{(\alpha )}}}{\mathbf {G}}_{(\alpha )}) \quad m\in \mathcal {H}' \\&\quad \quad \mathbf {P} \succeq 0\\ \end{aligned} \end{aligned}$$

(C19)

yields the identical solution to (11).