Direction Finding in Partly Calibrated Arrays Exploiting the Whole Array Aperture

Consider $K$ isotropic linear antenna subarrays, each composed of $M_{k}$ sensors for $k=1,\dots,K$, where $M\equiv\sum_{k=1}^{K}M_{k}$. We construct a planar Cartesian coordinate system, where the array is on the x-axis. For these subarrays, the partly calibrated model assumes the precisely known intra-subarray displacement and unknown inter-subarray displacement. Particularly, denote by $\xi_{k}\in\mathbb{R}$ the unknown inter-subarray displacement of the first (reference) sensor in the $k$-th subarray relative to the the first sensor in the 1-st subarray for $k=1,\dots,K$, thus, $\xi_{1}=0$. We use $\bm{\xi}=[\xi_{2},\dots,\xi_{K}]^{T}\in\mathbb{R}^{(K-1)\times 1}$ to represent the inter-subarray displacement vector. Denote by $\eta_{k,m}\in\mathbb{R}$ the known intra-subarray displacement of the $m$-th sensor relative to the 1-st sensor in the $k$-th subarray for $m=1,\dots,M_{k},k=1,\dots,K$, thus, $\eta_{k,1}=0$. Assume that the $M$ sensors share a common sampling clock, or the clocks have been synchronized exactly. We illustrate the partly calibrated model in Fig. 1.

There are $L$ far-field [27], closely spaced emitters impinging narrow-band signals onto the whole array from different directions $\theta_{1},\dots,\theta_{L}$. Denote the direction vector by $\bm{\theta}=[\theta_{1},\dots,\theta_{L}]^{T}\in\mathbb{R}^{L\times 1}$. In general cases, we assume $\bm{\theta}\geqslant\bm{0}$ or $\bm{\theta}\leqslant\bm{0}$. The steering vector of the $k$-th subarray corresponding to an emitter at direction $\theta$ is given by

where $\phi_{k}(\theta,\xi)={\rm exp}\left(\jmath\frac{2\pi}{\lambda}\xi_{k}\sin\theta\right)$ is an unknown phase offset between the $k$-th and the 1-st subarray, $\lambda$ denotes the wavelength of the transmitted signals by emitters and the vector $\bm{b}_{k}(\theta)\in\mathbb{C}^{M_{k}\times 1}$ is defined by

Here $\bm{b}_{k}(\theta)$ is a known function of $\theta$ in contrast to the unknown phase offset $\phi_{k}(\theta,\xi)$ for $k=1,\dots,K$.

In this paper, we assume that only a single snapshot is available and all the subarrays sample at the same time. Therefore, the signal received by the $k$-th subarray is a summation of signals transmitted by the $L$ emitters, given by

where $s_{l}$ represents an unknown complex coefficient and $\bm{n}_{k}$ represents noise for $k=1,\dots,K$. By substituting (1) into (3), we rewrite the received signals (3) as

The fully calibrated arrays assume that $\bm{\xi}$ in (4) is exactly known or well calibrated. By substituting (2) and $\phi_{k}(\theta,\xi)={\rm exp}\left(\jmath\frac{2\pi}{\lambda}\xi_{k}\sin\theta\right)$ to (4), we have

where $[\bm{C}_{k}(\bm{\theta})]_{m,l}={\rm exp}\left(\jmath\frac{2\pi}{\lambda}(\eta_{k,m}+\xi_{k})\sin\theta_{l}\right)$ for $m=1,\dots,M_{k}$ and $\bm{s}=[s_{1},\dots,s_{L}]^{T}\in\mathbb{C}^{L\times 1}$. Denote $\tilde{\bm{x}}_{c}=[\bm{x}_{1}^{T},\dots,\bm{x}_{K}^{T}]^{T}\in\mathbb{C}^{M\times 1}$, $\bm{C}(\bm{\theta})=[\bm{C}_{1}^{T}(\bm{\theta}),\dots,\bm{C}_{K}^{T}(\bm{\theta})]^{T}\in\mathbb{C}^{M\times L}$ and $\tilde{\bm{n}}_{c}=[\bm{n}_{1}^{T},\dots,\bm{n}_{K}^{T}]^{T}\in\mathbb{C}^{M\times 1}$. We then stack the $K$ signals in (5) together as

which is a typical single-snapshot signal model. In (6), $\xi_{k}$ and $\eta_{k,m}$ are known and the unknowns in $\bm{C}(\bm{\theta})$ are only $\bm{\theta}$. Direction finding is to estimate $\bm{\theta}$ given $\tilde{\bm{x}}_{c}$, which can be achieved by existing classical algorithms such as Multiple Signal Classification (MUSIC) [28] and compressed sensing (CS) [29].

Direction finding by fully calibrated arrays can achieve high angular resolution, inversely proportional to the whole array aperture. This is because the fully calibrated model assumes completely known $\bm{\xi}$ and the received signals are recast as (6). In this case, the subarrays can be regarded as sparsely distributed array elements in a single array with a large aperture, where the array manifold is denoted by $\bm{C}(\bm{\theta})$ in (6). In this single array, the positions of array elements are in the range from $0$ to $\eta_{K,M_{K}}+\xi_{K}$, yielding the whole array aperture. Therefore, the angular resolution of (6) is inversely proportional to the whole array aperture [2].

However, for the partly calibrated model, an accurate array manifold $\bm{C}(\bm{\theta})$ is not available due to the unknown $\bm{\xi}$, which makes it hard to synthesize the subarrays into a single large-aperture array. The unknown $\bm{\xi}$ introduces unknown phase errors $\phi_{k}(\theta_{l},\xi)$, which are related to both subarrays and sources, to the signal model. Particularly, define $\bm{\Phi}_{k}(\bm{\theta},\bm{\xi})={\rm diag}(\phi_{k}(\theta_{1},\xi),\dots,\phi_{k}(\theta_{L},\xi))\in\mathbb{C}^{L\times L}$ and $\bm{B}_{k}(\bm{\theta})=[\bm{b}_{k}(\theta_{1}),\dots,\bm{b}_{k}(\theta_{L})]\in\mathbb{C}^{M_{k}\times L}$ for $k=1,\dots,K$. We then rewrite (4) as

where $\bm{\Phi}_{k}(\bm{\theta},\bm{\xi})$ denotes the unknown phase error w.r.t. $\bm{\xi}$. Compared with (5), the unknown $\bm{\Phi}_{k}(\bm{\theta},\bm{\xi})$ in (7) makes high-resolution direction finding a challenging problem in single-snapshot cases [11]. For brevity, we denote $\bm{\Phi}_{k}(\bm{\theta},\bm{\xi})$ and $\bm{B}_{k}(\bm{\theta})$ by $\bm{\Phi}_{k}$ and $\bm{B}_{k}$, respectively.

There is a key question in partly calibrated arrays:

Can partly calibrated subarrays achieve angular resolution comparable to that of fully calibrated subarrays?

To illustrate our approach, we consider a typical case in partly calibrated arrays, where the intra-subarray displacement of each subarray is the same. In this case, we transform the partly calibrated model into a multiple-measurement model [23, 24]. Particularly, $\{\eta_{k,\cdot}\}$ and $\{M_{k}\}$ reduce to the same values for different $k$, and are denoted by $\bar{\eta}_{\cdot}=\eta_{k,\cdot}$ and $\bar{M}=M_{k}$, respectively. We use $\bm{\eta}=[\bar{\eta}_{2},\dots,\bar{\eta}_{\bar{M}}]^{T}\in\mathbb{R}^{(\bar{M}-1)\times 1}$ to represent the intra-subarray displacement vector. The steering matrices $\bm{B}_{k}$ become the same for all the subarrays, thus we denote $\bar{\bm{B}}=\bm{B}_{k}\in\mathbb{C}^{\bar{M}\times L}$ for $k=1,\dots,K$. We assume that for each subarray, $\bar{M}>L$, such that emitters are identifiable [30]. Under these assumptions, $\bar{\bm{B}}$ is a full column rank matrix. Then, the received signals (7) are reorganized as

where $\bar{\bm{s}}_{l}=s_{l}\left[\phi_{1}(\theta_{l},\xi),\dots,\phi_{K}(\theta_{l},\xi)\right]^{T}\in\mathbb{C}^{K\times 1}$ are viewed as the received signals from the $l$-th emitter, sampled $K$ times by different subarrays. Denote $\bm{X}=[\bm{x}_{1},\dots,\bm{x}_{K}]\in\mathbb{C}^{\bar{M}\times K}$, $\bm{S}=\left[\bar{\bm{s}}_{1},\dots,\bar{\bm{s}}_{L}\right]^{T}\in\mathbb{C}^{L\times K}$ and $\bm{N}=[\bm{n}_{1},\dots,\bm{n}_{K}]\in\mathbb{C}^{\bar{M}\times K}$. Then (II-C) is recast as

where $\bar{\bm{B}}$ is related to $\{\bm{\theta},\bm{\eta}\}$, and $\bm{S}$ is related to $\{\bm{\theta},\bm{\xi},\bm{s}\}$. In (9), $\{\bm{X},\bm{\eta}\}$ are known, and $\{\bm{\theta},\bm{\xi},\bm{s},\bm{N}\}$ are unknown. In addition, $L$ is assumed to be known by existing algorithms [31, 32, 33]. The recast signal model (9) is a multiple-measurement model, where each row of $\bm{S}$ represents the multiple measurements of the $K$ subarrays. Direction finding is to estimate $\bm{\theta}$ given $\bm{X}$.

Note that $\bm{S}$ in (9) is constructed by multiple measurements of subarrays instead of multiple time samples as typical multiple-measurement models, and has special signal structure. Particularly, the entries of the $l$-th row of $\bm{S}$ have the same modulus $|s_{l}|$ and different phase offsets $\phi_{k}(\theta_{l},\xi)$. In (9), high-resolution capability of partly calibrated arrays is encoded in this special signal structure of $\bm{S}$, mainly in $\phi_{k}(\theta_{l},\xi)$. The key issue of enhancing angular resolution lies in how to exactly recover $\phi_{k}(\theta_{l},\xi)$ by exploiting the signal structure of $\bm{S}$.

Existing algorithms encounter difficulties in achieving high angular resolution based on (9). This is because $\bm{S}$ is a complex function of $\{\bm{\theta},\bm{\xi},\bm{s}\}$ and exploiting the signal structure of $\bm{S}$ to recover $\phi_{k}(\theta_{l},\xi)$ is not a trivial problem. Particularly, MUSIC algorithms [34, 28] suffer from imprecise subspace structure of signals due to the unknown $\bm{\xi}$. Typical sparse recovery algorithms [35] divide the parameter space of $\bm{\theta}$ into finite grids, construct a over-complete, grid-based dictionary of $\bar{\bm{B}}$, and solve a convex lasso [36] problem. However, these algorithms ignore the phase relationship between subarrays $\phi_{k}(\theta_{l},\xi)$ and assume $\bm{S}$ to be completely unknown, which corresponds to non-coherent processing which cannot achieve high angular resolution. Therefore, we do not use gridding on (9). To enhance angular resolution, more efficient constraints on $\bm{S}$ are required in the problem formulation. However, it is hard to propose a proper constraint that fully characterizes the signal structure, and the introduction of such constraints increase the difficulty of the solution.

Our goal is to find a way that both makes full use of the signal structure and is tractable to solve. As mentioned above, the key of high angular resolution lies in the exact recovery of $\phi_{k}(\theta_{l},\xi)$, which are the phases of $\bm{S}$ in (9). Therefore, we propose to exploit a specific characteristic of $\bm{S}$ to recover $\phi_{k}(\theta_{l},\xi)$ first, and then find the directions based on the phase estimates. The characteristic we consider is approximate orthogonality between the source signals of partly calibrated arrays, which is detailed next.

First, we calculate the elements of $\hat{\bm{R}}_{\bar{\bm{s}}}\equiv\frac{1}{K}\bm{S}\bm{S}^{H}$. By substituting the definitions of $\bm{S}$, $\bar{\bm{s}}_{l}$ and $\phi_{k}(\theta,\xi)$ into $\hat{\bm{R}}_{\bar{\bm{s}}}$, the $(i,j)$-th element of $\hat{\bm{R}}_{\bar{\bm{s}}}$, denoted by $R_{i,j}$ for abbreviation, is

where $R_{i,j}=R_{j,i}^{*}$ for $i,j=1,\dots,L$. The diagonal elements of $\hat{\bm{R}}_{\bar{\bm{s}}}$ are $R_{i,i}=|s_{i}|^{2}$. We assume that the source intensities are all equal, $|s_{i}|=1$, $i=1,\dots,L$ in the sequel. Under this assumption, the intensities of the off-diagonal elements rely on the inter-subarray displacements $\bm{\xi}$ and the directions $\bm{\theta}$.

Denote the whole array aperture and the intra-subarray element spacing by $D$ and $d$, respectively. To analyze the statistics of $\hat{\bm{R}}_{\bar{\bm{s}}}$, we consider a common practical scenario that the subarrays are randomly, uniformly placed in $[0,D]$, i.e., $\xi_{k}\sim\mathcal{U}\left[0,D\right]$ for $k=1,\dots,K$. The corresponding geometry is shown in Fig. 2.

This typical case leads to the following proposition, a similar analysis can be found in [37].

The term $\left|\frac{\sin\rho_{i,j}}{\rho_{i,j}}\right|$ plays an important role in (11) and (12). We draw the curve of $\left|\frac{\sin\rho_{i,j}}{\rho_{i,j}}\right|$ w.r.t. $(\sin\theta_{i}-\sin\theta_{j})/\Delta$ in Fig. 3, where $\Delta\equiv\lambda/D$ denotes the empirical angular resolution of the whole array.

From Fig. 3, we see that the curve is composed of a main lobe and several side lobes. When the interval $\sin\theta_{i}-\sin\theta_{j}$ equals $\Delta$, the intensity $\left|R_{i,j}\right|$ reaches its first zero point, which is related to the angular resolution $\Delta$. When the interval is in the region $[0,\Delta)$, the distributed array cannot distinguish sources at such close interval and $\left|R_{i,j}\right|$ has large values. When the interval belongs to the region $[\Delta,\infty)$, $\left|R_{i,j}\right|$ is typically small. In this region, the zero points of $\left|R_{i,j}\right|$ occur in a period of $\Delta$ and there are side lobes between these zero points.

Based on the statistics of $\hat{\bm{R}}_{\bar{\bm{s}}}$, there is approximate orthogonality in partly calibrated arrays. Here, orthogonality is defined as the orthogonality between the rows of $\bm{S}$ in (9). Particularly, we say there is orthogonality in partly calibrated arrays if the off-diagonal elements of $\hat{\bm{R}}_{\bar{\bm{s}}}$ are 0, i.e.,

In Proposition 1 and Fig. 3, we show that when the source separation exceeds one resolution cell, the off-diagonal elements of $\hat{\bm{R}}_{\bar{\bm{s}}}$ are small relative to the diagonal elements in a statistical sense. In other words, there is approximate orthogonality between the rows of $\bm{S}$ in (9), and

We may also understand the orthogonality in partly calibrated arrays using the concept of coherence [38], given by

where $0\leqslant\mu\leqslant 1$. In partly calibrated arrays, the coherence $\mu$ corresponds to the largest modulus of the off-diagonal elements of $\hat{\bm{R}}_{\bar{\bm{s}}}$. Orthogonality means that $\mu=0$ and approximate orthogonality means that $0<\mu\ll 1$.

In summary, when the source separation is within a resolution cell, there is no obvious orthogonality between the source signals. When the source separation exceeds a resolution cell, the modulus of the off-diagonal $R_{i,j}$ is small in a statistical sense, yielding approximate orthogonality. This phenomenon not only illustrates the orthogonality in partly calibrated arrays, but also builds a connection between orthogonality and angular resolution, which will be discussed in Section V.

In (9), when $\bar{\bm{B}}$ is unknown, how to exploit the orthogonality between the rows of $\bm{S}$ is a challenging problem. Here, we propose a solution to this problem. Particularly, we introduce BSS techniques to the self-calibration problems in distributed arrays. To this end, we first review basic BSS ideas. Then, we show how a specific BSS algorithm exploits the orthogonality of $\bm{S}$ in (9). Finally, we show how we recover the phase offsets between subarrays.

When the phase offsets between subarrays is recovered by (23), direction finding reduces to a coherent array signal processing problem: Recover $\bm{\theta}$ from the following signals,

This problem can be solved by many classical algorithms. In this subsection, we introduce two representative techniques for direction finding.

Here, we show the performance of phase offset estimation with JADE by comparing the true $|R_{i,j}|$ with the estimated one by JADE. This is because more accurate phase offset estimation yields better recovery of $|R_{i,j}|$. We also verify that higher level of orthogonality likely yields better performance of JADE.

Particularly, we set $L=2$, $s_{1}=s_{2}=1$, $\bar{M}=10$, $K=10$, $D=450\lambda$, $d=\frac{\lambda}{2}$ and fix $\theta_{1}=1.2^{\circ}$. With $\sin\theta_{2}-\sin\theta_{1}$ as the variable, we first calculate $\left|R_{2,1}\right|$ of (III-A) in the equidistant and random distributed arrays. Then, we substitute the phase offset estimation as (23) by JADE to (III-A) and calculate the corresponding $\left|R_{2,1}\right|$ as $\left|\hat{R}_{2,1}\right|=\left|\frac{1}{K}\sum_{k=1}^{K}\hat{\phi}_{2,k}\hat{\phi}_{1,k}^{*}\right|$. The simulation results are shown in Fig. 4.

From Fig. 4, first we find that the estimation $|\hat{R}_{2,1}|$ is close to the truth $|R_{2,1}|$ when $|R_{2,1}|$ is small, yielding the feasibility of phase recovery in partly calibrated arrays using JADE. Second, larger $|R_{2,1}|$ usually corresponds to a bigger gap between $|\hat{R}_{2,1}|$ and $|R_{2,1}|$ and vice versus, which verifies that high level of orthogonality (smaller $|R_{2,1}|$) likely means better performance of JADE (smaller gap between $|\hat{R}_{2,1}|$ and $|R_{2,1}|$). Third, $|R_{2,1}|$ in the equidistant case has apparent grating lobes at $(\sin\theta_{2}-\sin\theta_{1})/\Delta=10$, which do not appear in the random case due to the randomness of the displacement.

In this subsection, we compare our proposed algorithms, BSS-MF and BSS-NLS, with the root-RARE [18], spectral-RARE [19], COBRAS [11], and group sparse [43] methods in the scenarios with only a single snapshot. We compare both the angular resolution and direction estimation accuracy of our proposed techniques with other methods and the corresponding CRBs. The simulations are carried out under various signal-to-noise rate (SNR)s and angle differences.

The CRBs for partly calibrated arrays, denoted by PC-CRB [18], is both given as the benchmarks. We use the root mean square error (RMSE) of $\bm{\theta}$ to indicate the estimation performance of directions. We carry out $T_{m}$ Monte Carlo trials and denote the RMSE of $\bm{\theta}$ by

where $\hat{\bm{\theta}}_{t}$ is the direction estimate in the $t$-th trial and $\bm{\theta}^{*}$ is the true value.

We consider that there are $L=2$ emitters in the directions of $\bm{\theta}$, transmitting signals with wavelength $\lambda$. There are $K=10$ uniform linear subarrays receiving the signals and each subarray has $\bar{M}$ array elements. The intra-subarray and inter-subarray displacements are respectively set as $\bar{\eta}_{m}=(m-1)\lambda/2$ for $m=1,\dots,\bar{M}$ and $\xi_{k}=\frac{(k-1)D}{K-1}$ for $k=1,\dots,K$, where the whole array aperture is $D=450\lambda$. In this case, the angular resolution of a single subarray and the whole array are $\frac{\lambda}{(\bar{M}-1)d\cos\theta}\approx 12.73^{\circ}$ and $\frac{\lambda}{D\cos\theta}\approx 0.13^{\circ}$, respectively. The received signal amplitudes are set as $\bm{s}=[e^{\jmath\frac{\pi}{5}},3e^{\jmath\frac{3\pi}{5}}]^{T}$. In the scenario, there are additive noises $\bm{n}_{k}$ for $k=1,\dots,K$, which are i.i.d. white Gaussian with mean $\bm{0}$ and variance $\sigma^{2}\bm{I}$. Here SNR is defined as ${1}/{\sigma^{2}}$.