In this section, we introduce our proposed method. In Section 3.1, we firstly derive the virtual array signal mode in beamspace domain by using beamspace transforming. And in Section 3.2, we introduce the covariance matrix representation by KR-product. In Section 3.3, we introduce the sparse beamspace DOA estimation method via single measurement vector.

### 3.1 Virtual array signal model in beamspace

Consider an array composed of *M* sensors located along the circumference of a uniform circular array with radius *r*. There are *P*(*P*<*M*) narrowband uncorrelated signals impinging on the array in the far-field. *ϕ*={*ϕ*_{1},*ϕ*_{2},⋯,*ϕ*_{P}} is the set of the incident angles of the signals. The observation data is formulated as

$$ {\mathbf{x}}(t) = {\mathbf{A(\boldsymbol{\phi})s}}(t) + {\mathbf{n}}(t), \quad {t = 1,\cdots,N}, $$

(5)

where **x**(*t*)=[*x*_{1}(*t*),*x*_{2}(*t*),⋯,*x*_{M}(*t*)]^{T} is an *M*×1 noise-corrupted snapshot vector. **s**(*t*)=[*s*_{1}(*t*),*s*_{2}(*t*),⋯,*s*_{P}(*t*)]^{T} is a *P*×1 signal vector, and \({\mathbf {n}}(t) \in {\mathbb {C}^{M}}\) is the assumed zero-mean Gaussian white noise. *N* is the number of snapshots. \({\mathbf {A(\boldsymbol {\phi })}} = [{\mathbf {a}}({\phi _{\mathrm {1}}}),{\mathbf {a}}({\phi _{\mathrm {2}}}),\cdots, {\mathbf {a}}({\phi _{P}})]\in {\mathbb {C}^{M \times P}}\) is the array manifold matrix of the UCA, here **a**(*ϕ*_{p}),*p*=1,⋯,*P*, are the *M*×1 steering vectors. It can be expressed as

$$ {\mathbf{a}}({\phi_{p}}) = \left[ {\begin{array}{*{20}{c}} {{e^{j\zeta\cos ({\phi_{p}} - {\gamma_{\mathrm{1}}})}}}\\ {{e^{j\zeta\cos ({\phi_{p}} - {\gamma_{\mathrm{2}}})}}}\\ { \cdot \cdot \cdot }\\ {{e^{j\zeta\cos ({\phi_{p}} - {\gamma_{M}})}}} \end{array}} \right], $$

(6)

where \(j=\sqrt {-1}, \zeta =kr\), and the wavenumber *k*=2*π*/*λ* with *λ* being the wavelength of the incident signals. *γ*_{m}=2*π*(*m*−1)/*M,m*=1,2,⋯,*M*, are sensors’ locations along the circumference of the UCA.

Assume that the signals \(\{s_{p}(t)\}_{p=1}^{P}\) are uncorrelated for different sources, and also independent of **n**(*t*). The covariance matrix of the observation data **x**(*t*) is given by

$$ {\mathbf{R}} = {\mathrm{E}}[{\mathbf{x}}(t){{\mathbf{x}}^{H}}(t)] = {\mathbf{A}}(\boldsymbol{\phi}){{\mathbf{R}}_{\mathrm{s}}}{{\mathbf{A}}^{H}}(\boldsymbol{\phi}) + {{\sigma}_{n}^{\mathrm{2}}}{{\mathbf{I}}_{M}}, $$

(7)

where **R**_{s}=E[**s**(*t*)**s**^{H}(*t*)] is the signal covariance matrix, whose diagonal elements are \(\{{\sigma _{{\mathrm {s}}p}^{2}}\}_{p=1}^{P}\). \({\sigma }_{n}^{\mathrm {2}}\) is the noise power, E(∙) and (∙)^{H} are the expectation and the conjugate transpose operator respectively. The signal mode of the UCA can be transformed to that of a ULA-type array by synthesizing the beamspace manifold, which is similar to that of ULA using phase mode excitation of continuous circular aperture[20]. The signal model of the virtual ULA essentially takes discrete spatial sampling of far-field pattern resulting from all harmonics of array excitation (each harmonic means one phase mode, theoretically it ranges from −*∞* to +*∞*. Actually the magnitude of harmonic decays super-exponentially with increasing harmonic order *h*, i.e. *h*- th phase mode. If *h* is large enough and reach a certain number *H*_{e}, the magnitude is asymptotically approaching zero) by incoming signals over continuous aperture of UCA[27]. The beamspace manifold synthesized by a beamformer \({\mathbf {F}}_{e}^{H}={{\mathbf {C}}_{v}}{{\mathbf {V}}^{H}}\)[20] is given by

$$ {{\mathbf{a}}_{e}}({\phi_{p}}) = {\mathbf{F}}_{e}^{H}{\mathbf{a}}({\phi_{p}}) = {{\mathbf{C}}_{v}}{{\mathbf{V}}^{H}}{\mathbf{a}}({\phi_{p}}) \approx \sqrt M {{\mathbf{J}}_{\zeta}}{\mathbf{d}}({\phi_{p}}), $$

(8)

where

$$ {{\mathbf{C}}_{v}} = {\text{diag}} \left\{ {{j^{- {H_{e}}}}, \ldots,{j^{- 1}},{j^{0}},{j^{- 1}}, \ldots,{j^{- {H_{e}}}}} \right\}, $$

(9)

$$ {\mathbf{V}} = \sqrt M \left[ {{{\mathbf{w}}_{- {H_{e}}}}: \cdots :{{\mathbf{w}}_{0}}: \cdots :{{\mathbf{w}}_{{H_{e}}}}} \right], $$

(10)

$$ {{\mathbf{J}}_{\zeta}} = {\text{diag}} \left[ {{J_{H_{e}}}(\zeta), \ldots,{J_{1}}(\zeta),{J_{0}}(\zeta),{J_{1}}(\zeta), \ldots,{J_{H_{e}}}(\zeta)} \right], $$

(11)

where **C**_{v} and **J**_{ζ} are (2*H*_{e}+1)×(2*H*_{e}+1) diagonal matrices. **J**_{ζ} is a matrix of Bessel functions. The amplitudes of Bessel-function coefficients on the master diagonal taper symmetrically, and the coefficients act as such a function, that is, linking the amplitude of each phase-mode excitation with that of the corresponding far-field phase-mode pattern. The matrix **V** is a normalized beamforming weight matrix that excites the array with a finite number of excitation modes. *h*∈[−*H*_{e},*H*_{e}] are phase modes that can be excited. Here a rule of thumb for determining *H*_{e} is given as *H*_{e}≈*ζ* and *H*_{e} should satisfy *H*_{e}<*M*/2. The vectors of \({\{\mathbf {w}}_{h}\}_{h=-{H_{e}}}^{H_{e}}\) are regarded as the spatial discrete sampling corresponding to the far-field pattern, which are caused by the *h*-th phase mode excitation along the continuous circular aperture. It is defined by

$$ {\mathbf{w}}_{h}^{H} = \frac{1}{M}\left[ {{e^{jh{\gamma_{\mathrm{1}}}}},{e^{jh{\gamma_{\mathrm{2}}}}}, \ldots,{e^{jh{\gamma_{M }}}}} \right]. $$

(12)

From (8), we know that the steering vectors **a**_{e}(*ϕ*_{p}) in beamspace domain can be represented by the vector **d**(*ϕ*_{p}). It is expressed as

$$ {\mathbf{d}}({\phi_{p}}) = {\left[ {{e^{- j{H_{e}}{\phi_{p}}}}, \ldots,{e^{- j\phi }},1,{e^{j\phi }}, \ldots,{e^{j{H_{e}}{\phi_{p}}}}} \right]^{T}}. $$

(13)

Here \( {\sqrt {M}}{{\mathbf {J}}_{\zeta }}{{\mathbf {d}}(\phi _{p})}, p=1,2,...,P,\) are the ideal steering vectors of the virtual ULA with Vandermonde structure.

The methodology of phase mode excitation-based beamformer offers the operation on transforming observation data in element-space to that of beamspace. For the observation data illustrated in (5), using the methodology, we have \({\mathbf {y}}(t)\in {\mathbb {C}}^{{M_{e}} \times P}\), which is given by

$$ {\mathbf{y}}(t)={{\mathbf{F}}_{e}^{H}}{\mathbf{x}}(t)={\sqrt{M}}{\mathbf{J}_{\zeta}}{\mathbf{D(\boldsymbol{\phi })}}{\mathbf{s}}(t) + {\mathbf{F}}_{e}^{H}{\mathbf{n}}(t), $$

(14)

here *M*_{e}=2*H*_{e}+1 is the total number of excited modes. From (14), we know that the observation data **x**(*t*) of *M*×*P* dimensions in element-space domain is mapped to a dimension-reduction matrix **y**(*t*) of *M*_{e}×*P* dimensions in beamspace domain. And the term \({\sqrt {M}}{\mathbf {J}_{\zeta }}{\mathbf {D(\boldsymbol {\phi })}}{\mathbf {s}}(t)\) is a noise-free beamspace data matrix, which is expressed as a product of virtual array manifold **D****(***ϕ***)**, the source vector **s**(*t*) and Bessel functions. Here **D****(***ϕ***)**=[**d**(*ϕ*_{1}),**d**(*ϕ*_{2}),...,**d**(*ϕ*_{P})] has centro-Hermitian columns with Vandermonde structure, \({{\mathbf {F}}_{e}^{H}}\) is a unitary matrix that satisfies \({{\mathbf {F}}_{e}^{H}}{{\mathbf {F}}_{e}}={\mathbf {I}_{M_{e}}}\). The by-product \({{\mathbf {F}}_{e}^{H}}{\mathbf {n}(t)}\) of the transformation still remains the white Gaussian process. Thus, we have the covariance matrix of the observation data **y**(*t*). It is given by

$$ {\mathbf{R}}_{\mathbf{y}}=M{\mathbf{J}}_{\mathrm{\zeta}}{\mathbf{D}}({\mathrm{\phi}}){\mathbf{R_{s}}}{\mathbf{D}}^{H}({\mathrm{\phi}}){\mathbf{J}}_{\mathrm{\zeta}}+ {\sigma_{n}^{2}}{\mathbf{I}_{M_{e}}}. $$

(15)

### 3.2 KR-based covariance matrix representation

In this subsection, we apply the KR subspace approach to DOA estimation[24] of the virtual ULA. For the signal representation formulated in the above section, applying the vectorization operator on (15), we have a new array model expressed as

$$ {\mathbf{Y}}:={\text{vec}}({{\mathbf{R}}_{\mathbf{y}}}) = M\left[({\mathbf{J}}_{\zeta}^{\ast}{{\mathbf{D}}^{\ast}}(\boldsymbol{\phi})) \odot ({{\mathbf{J}}_{\zeta} }{\mathbf{D}}(\boldsymbol{\phi}))\right]{\vec{\boldsymbol{\sigma}}}_{\mathbf{s}}^{\mathrm{2}} + \sigma_{n}^{\mathrm{2}}{\mathbf{1}}. $$

(16)

Here \({\mathbf {1}}=[{\mathbf {e}}_{1}^{T},{\mathbf {e}}_{2}^{T},...{\mathbf {e}}_{M_{e}}^{T}]^{T}\), where \({\{\mathbf {e}}_{p}\}_{l=1}^{M_{e}}, l=1, 2, \cdots, {M_{e}}\), are the *M*_{e}×1 vectors with one at the *p*-th position and nought otherwise. \(\vec {\boldsymbol {\sigma }}_{\mathbf {s}}^{\mathrm {2}}\) is a column vector composed of nonzero elements on the diagonal of **R**_{s}. The virtual array response matrix \((\mathbf {J}_{\zeta }^{\ast }{\mathbf {D}^{\ast }}(\boldsymbol {\phi })) \odot ({\mathbf {J}_{\zeta } }\mathbf {D}(\boldsymbol {\phi })) \in {\mathbb {C}^{{M_{e}^{2}}\times {P}}}\) can be formulated as

$$ \left(\mathbf{J}_{\zeta}^{\ast}{\mathbf{D}^{\ast}}(\boldsymbol{\phi})\right) \odot \left({\mathbf{J}_{\zeta} }\mathbf{D}(\boldsymbol{\phi})\right)={\mathbf{G}}{\mathbf{B}(\boldsymbol{\phi})}, $$

(17)

here \({\mathbf {B}}(\boldsymbol {\phi })\in {\mathbb {C}^{{(2M_{e}-1)}\times {P}}}\) is a dimension-reduced virtual array response matrix that expressed as

$$ {\mathbf{B(\boldsymbol{\phi})}}=\left[{\mathbf{b}}(\phi_{1}),{\mathbf{b}}(\phi_{2}),...,{\mathbf{b}}(\phi_{p}),...{\mathbf{b}}(\phi_{P})\right], $$

(18)

where

$$ {{\mathbf{b}}(\phi_{p})}=\left[e^{-j(M_{e}-1){\phi_{p}}},...,e^{-j{\phi_{p}}},1,e^{j{\phi_{p}}},...e^{j(M_{e}-1){\phi_{p}}}\right]^{T}, $$

(19)

and \({\mathbf {G}}\in {\mathbb {C}^{{M_{e}^{2}}\times {(2M_{e}-1)}}} \) is given by

$$ {\mathbf{G}}=({{\mathbf{J}}_{\zeta} }\otimes{{\mathbf{J}}_{\zeta} }){\mathbf{H}}, $$

(20)

where ⊗ symbolises Kronecker product. Here **H** is the selection matrix [9] of ULA given by

$$ {\mathbf{H}}=\left[{\text{vec}}\left({\mathbf{H}}_{M_{e}-1}\right),\cdots, {\text{vec}}\left(\mathbf{H}_{1}\right),{\text{vec}}\left(\mathbf{H}_{0}\right), {\text{vec}}\left(\mathbf{H}^{T}_{1}\right),\cdots, {\text{vec}}\left({\mathbf{H}}^{T}_{M_{e}-1}\right)\right] $$

(21)

with

$$ {\mathbf{H}_{i}} = \left[ {\begin{array}{*{20}{c}} {{{\mathbf{0}}_{M_{e}- i}}}&{{\mathbf{I}_{M_{e} - i}}}\\ {{{\mathbf{0}}_{i,i}}}&{{{\mathbf{0}}_{i,M_{e} - i}}} \end{array}} \right],i = {\mathrm{0}},{\mathrm{1}},...M_{e} - {\mathrm{1}}. $$

(22)

As is known from (20), the selection matrix **G** of the virtual ULA is just a derivation of **H**, that is, an inner product of **J**_{ζ}⊗**J**_{ζ} and the selection matrix of ULA. (16) can be reformulated as below,

$$ {\mathbf{Y}}=M{\mathbf{G}}{\mathbf{B}}{(\boldsymbol{\phi})} {\vec{\boldsymbol{\sigma}}}_{\mathrm{s}}^{2}+ \sigma_{n}^{\mathrm{2}}{\mathbf{1}}. $$

(23)

From (23), we know that the observation data **Y**, vectorization of **R**_{y}, behaves like a new signal model. \({\vec {\boldsymbol {\sigma }}}_{\mathrm {s}}^{2}\) is the equivalent signal vector, which describes the power of each signal. The virtual array response matrix **G****B**(*ϕ*) is a new observation matrix, which has a larger aperture than the array which is not vectorized. \({\sigma _{n}^{2}}\) represents the power of noise. When no knowledge of noise is available, \({\sigma _{n}^{2}}\) is estimated and given by the minimum of the eigenvalue of **R**_{y}. From [28], we know that Rank(**G****B**(*ϕ*)) is *P*, which satisfies *P*<2*H*_{e}+1. So if any complete basis of *P*-dimension vectors is given, (23) can be expressed as a linear combination of the signal powers in the *P*-dimension vectors space.

###
**Remark 1**

In [9] and [28], the dimension of the array manifold matrix is reduced from *M*^{2}×*P* to (2*M*−1)×*P* by Khatri-Rao product. It works for ULA, but not for UCA. For an arbitrary array, the array response matrix is generally expressed as \(({\mathbf {A}^{\ast }}\odot {\mathbf {A}})\in \mathbb {C}^{{M^{2}}\times {P}}\). But for UCA, having vectorized the observation data by BT technique, the response matrix is reduced to \(\phantom {\dot {i}\!}{\mathbf {B}}\in \mathbb {C}^{(4H_{e}+1)\times {P}}\). According to the spatial sampling criterion of *H*_{e}<*M*/2, we know that if *M*>10, then *M*^{2}≃(5∼6)(4*H*_{e}+1), that is, the array response matrix is reduced from *M*^{2}×*P* to (4*H*_{e}+1)×*P*.

### 3.3 Sparse beamspace DOA estimation via single measurement vector

Assuming that the overcomplete basis \(\left \{{\mathbf {b}}\left ({\widetilde {\phi }}_{q}\right)\right \}_{q=1}^{Q}\left ({Q}\gg {M_{e}^{2}}\right)\), where \({\left \{\widetilde {\phi }_{q}\right \}_{q=1}^{Q}}, q=1, 2, \cdots, Q\), are the discrete samples from the potential incident directions of signals in beamspace domain. Here denote \({\{\widetilde {\phi }_{q}\}_{q=1}^{Q} }\) by the vector \({\boldsymbol {\widetilde {\phi }}}\). Therefore, (23) can be reformulated as the SMV form

$$ {\mathbf{Y}}=M{\mathbf{G}}{\mathbf{B}}({\widetilde{\boldsymbol{\phi}}}){\mathbf{u}}+\sigma_{n}^{\mathrm{2}}{\mathbf{1}}, $$

(24)

which is essentially an underdetermined signal reconstruction problem. We can estimate the DOAs of the signals by recovering the sparse vector **u** of the single measurement vector **Y**. At this point, if the grid resolution of \({\widetilde {\boldsymbol {\phi }}}\) is dense enough, then some *P* column vectors of \({\mathbf {B}}({\widetilde {\boldsymbol {\phi }}})\) are approaching to or equal to \({\{\mathbf {b}}(\phi _{p})\}_{p=1}^{P}\). Correspondingly, a *P*-sparse vector \(\tilde {{\mathbf {u}}}\) is estimated, whose nonzero elements are close to or equal to \({\{\sigma _{{\mathrm {s}}{p}}^{2}}\}_{p=1}^{P}\). In theory, (24) can be solved by the following constraint *ℓ*_{1} optimization that expressed as [29]

$$ \underset{\tilde{\mathbf{u}}}{\min}||{\tilde{\mathbf{u}}}||_{1}\ {\mathrm{subject\ to}}\ \widetilde{\mathbf{Y}}=M{\mathbf{G}}{\mathbf{B}}({\widetilde{\boldsymbol{\phi}}}) \tilde{\mathbf{u}}+\sigma_{n}^{\mathrm{2}}{\mathbf{1}}, $$

(25)

here \(\tilde {{\mathbf {u}}}\) and \(\widetilde {\mathbf {Y}}\) are the estimates of **u** and **Y** respectively. From (25), we know that if \(\tilde {{\mathbf {u}}}\) is approaching to \({\vec {\boldsymbol {\sigma }}}_{\mathrm {s}}^{2}\), then \({\widetilde {\mathbf {Y}}}\) approximates to **Y**. And some \({\{{\widetilde {\phi }\}_{p=1}^{P}}}\) are very close to the DOAs of the incident signals. We know that the estimate error of \({\mathbf {\widetilde {\mathbf {Y}}} - {{\mathbf {Y}}}}\) with the weighted matrix of \({\mathbf {W}={\frac {1}{N}}{\mathbf {R}_{\mathbf {y}}^{T}}} \otimes {{\mathbf {R}}_{\mathbf {y}}}\) follows asymptotically normal (AsN) distribution[29], which is given by

$$ {\mathbf{W}}^{-\frac{1}{2}}\lbrack{\widetilde{\mathbf{Y}}}-M\mathbf{G}{\mathbf{B}} ({\widetilde{\boldsymbol{\phi}}}){\tilde{\mathbf{u}}}-{\sigma_{n}^{2}}{\mathbf{1}} \rbrack\sim{{\text{AsN}}({\mathbf{0}},{\mathbf{I}_{M_{e}^{2}}})}. $$

(26)

Using least-squares criterion, the weighted estimate error follows asymptotic chi-square distribution with \({M_{e}^{2}}\) degree-of-freedoms. It is formulated as

$$ {\parallel {\mathbf{W}}^{-\frac{1}{2}}\lbrack{\widetilde{\mathbf{Y}}}-M\mathbf{G}{\mathbf{B}} ({\widetilde{\boldsymbol{\phi}}}){\tilde{\mathbf{u}}}-{\sigma_{n}^{2}}{\mathbf{1}}\rbrack \parallel}_{2}^{2}\sim{As}{\chi}^{2}({M_{e}^{2}}), $$

(27)

here \({\widetilde {\mathbf {Y}}}={\text {vec}(\widetilde {{\mathbf {R}}}_{\mathbf {y}})}\), where \({\widetilde {{\mathbf {R}}}_{\mathbf {y}}}=\frac {1}{N}\sum _{t=1}^{N}{\mathbf {y}(t)}{\mathbf {y}^{H}(t)}\). Thus, a modified DOA estimation mode is derived from (25) by introducing the parameter of *β*, which makes the inequality \(\parallel {{\mathbf {W}}^{-\frac {1}{2}}({\widetilde {\mathbf {Y}}}-{\mathbf {Y}})\parallel }_{2}^{2}\leqslant {\beta }^{2}\) hold with a high probability \(\tilde {p}\). It specifies how much estimate error we wish to allow. It is expressed as follows

$$ {\mathrm{P}}\left({{\chi}^{2}({M_{e}^{2}})\leqslant{\beta}^{2}}\right)=\tilde{p}, $$

(28)

where P(∙) denotes the probability distribution function. For the probability value of \(\tilde {p}\), by looking up the probability table of chi-square distribution, we have the regularization parameter *β*, that is, \(\beta =\sqrt {{{\chi }_{\tilde {p}}^{2}({M_{e}^{2}})}}\). Then, (25) can be expressed as

$$ \underset{\tilde{\mathbf{u}}}{\min}{\parallel}{\tilde{\mathbf{u}}}{\parallel}_{1} \ {\mathrm{subject\ to}}\ {\parallel{\mathbf{W}}^{-\frac{1}{2}}({\widetilde{\mathbf{Y}}}- M\mathbf{G}\mathbf{B}({\widetilde{\boldsymbol{\phi}}}){{\tilde{{\mathbf{u}}}}-{\sigma}_{n}^{2}} {\mathbf{1}})\parallel}_{2}^{2} \leqslant {\beta}^{2}. $$

(29)

Using the Matlab convex optimization toolbox, the *P*-sparse vector \({\tilde {\mathbf {u}}}\) can be obtained. We can plot the peaks versus the directions and determine the DOAs of the incoming signals. The proposed method is summarized in Algorithm 1.

###
**Remark 2**

From (16), we know that krank(**G****B**(*ϕ*))≥min{*P*,2×krank(**J**_{ζ}**D**(*ϕ*))−1}. Here krank(**J**_{ζ}**D**(*ϕ*))=*M*_{e}, where krank(∙) denotes the *Kruskal* *rank* (see definition in [28] for details). It means that every collection of 2*M*_{e}−1 column vectors of **G****B**(*ϕ*) is linearly independent and there exits a set of 2*M*_{e} column vectors linearly dependent. That is Spark(**G****B**(*ϕ*))=2*M*_{e}. The constraint condition of *ℓ*_{1} optimization for a unique *P*-sparse vector **u** is Spark(**G****B**(*ϕ*))>2*P*. i.e. 2*H*_{e}+1>*P*, which means that the DOA estimator for virtual ULA with *H*_{e} modes can handle 2*H*_{e} signals at most.