Fundamental Limit of Angular Resolution in Partly Calibrated Arrays with Position Errors

To meet the challenge of analyzing the resolution limit of partly calibrated arrays, we propose a new resolution criterion using the turning point of CRB. The proposed criterion states that: two signals are said to be resolvable w.r.t. the frequencies if the difference between the frequencies, $\Delta f=f_{2}-f_{1}$, is greater than the turning point of CRB w.r.t. $\Delta f$ denoted by $\Omega_{f}\equiv\mathcal{T}\left({\rm CRB}(\bm{f})\right)$. The definition of the CRB turning point is detailed in Section V-D.

The basic principle of this criterion is based on a significant phenomenon of ${\rm CRB}(\bm{f})$: Fix the source of $f_{1}$ as a reference and gradually increase the frequency difference $\Delta f>0$. The ${\rm CRB}(\bm{f})$ first declines rapidly w.r.t. $\Delta f$, and then it almost remains constant (with a small fluctuation). The turning point is located close to the angular resolution limit. We show this phenomenon in fully and partly calibrated arrays as an example. Consider using $10$ half-wavelength, uniform linear subarrays to estimate the directions of $2$ sources with the angles $\bm{\theta}=[1.2^{\circ},1.2^{\circ}+\Delta\theta]^{T}$, where $\Delta\theta>0$ denotes the angle difference. There are $10$ elements in each subarray and the adjacent subarrays are spaced at half-wavelength apart. Denote by $\Omega_{f}\equiv 2\pi/D$ the angular resolution limit, where $D$ is the whole array aperture. The CRBs of fully and partly calibrated arrays, ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$, are shown in Fig. 2.

From Fig. 2, we find that the two CRBs decline when $\Delta f/\Omega_{f}\leqslant 1$, and tend to be stable when $\Delta f/\Omega_{f}>1$. We provide an intuitive explanation of this phenomenon: when the frequency difference $\Delta f$ is within the resolution limit $\Omega_{f}$, the sources cannot be distinguished, disrupting the estimation performance and greatly increasing the CRB; in this case, increasing $\Delta f$ increases the resolvability and hence decreases the CRB, which corresponds to the declining phase of the CRB. When $\Delta f$ exceeds $\Omega_{f}$, the estimation accuracy is close to the counterpart of estimating the sources separately [25]; consequently, the accuracy is less relevant to $\Delta f$, keeping the CRB constant, which corresponds to the plateau phase of the CRB. Here, ’plateaus phase’ refers to a situation where, after some change, a trend or curve stabilizes and no longer shows significant variation or fluctuation, which is consistent with the CRB when $\Delta f\geqslant\Omega_{f}$. Therefore, the turning point between the declining and the plateau phases indicates angular resolution.

For another reason, we use the turning point of CRB as the angular resolution limit since it can reflect the influence on each other for parameter estimation, which is precisely the meaning of resolution. Particularly, when two sources are not distinguished, they significantly influence each other, resulting in poor estimation performance as reflected in the decline part of CRB. When the two sources can be separated, their mutual influence is minimal, and the CRB approaches the performance of separate estimation. Thus, the transition point between these two states can be used to indicate the resolution, which is the turning point of CRB.

The reason that the turning point is not exactly located at $\Delta f=\Omega_{f}$ is that on one hand, the theoretical proof in the paper represents the result in a statistical average sense (see the A2 assumption in Section V), which cannot ensure that the turning point of the CRB in every specific scenario precisely locates on the resolution cell; on the other hand, resolution criterion, either the Rayleigh resolution limit or 3dB beam width, is an empirical concept, which is not an absolute indication of separability or non-separability. Overall, it can ensure that the magnitude of the resolution is correct.

In the proposed resolution criterion, angular resolution depends not only on the angular difference but also on the absolute angle values. This is because that we use the frequency separation, $\Delta f$, to reflect source separation instead of angle separation, $\Delta\theta$. Particularly in Fig. 2, we have $\Delta f=2\pi(\sin\theta_{2}-\sin\theta_{1})/\lambda$ and $\Delta\theta=\theta_{2}-\theta_{1}$. Through trigonometric transformation, we have

In the above equation, the sin part reflects the angular resolution and the cos part reflects the angle values themselves. Therefore, the CRB is not only related to the angle separation $\Delta\theta$, but also angle values $(\theta_{2}+\theta_{1})/2$. It can be found that large angle values have worse resolution.

Note that the proposed resolution criterion using the CRB turning point is almost unaffected by SNR. This is because in the CRB expression (see (35) and (36) in Appendix A), the component of white noise $\sigma^{2}$ can be isolated, which means that it only affects the absolute value of the CRB without altering its relative variation with respect to the angle difference. Although this property contradicts the commonly held conclusion that resolution is related to SNR, it is rational in analyzing the effect of position errors in distributed arrays. This is because that inter-subarray position errors fundamentally differ from white noise errors: the former are multiplicative errors, while the latter are additive errors. When the SNR is extremely low, it becomes impossible to distinguish and estimate the angles, making it ineffective to analyze the impact of multiplicative errors on resolution. Therefore, we aim to conduct an analysis method that is unaffected or minimally affected by additive noise. The proposed resolution criterion using the CRB turning point can effectively address this challenge.

In the sequel, we apply the proposed criterion to analyze the angular resolution of fully and partly calibrated arrays. A main conclusion is that partly calibrated arrays achieve high resolution similar to that of fully calibrated arrays. Note that the above phenomenon of CRB w.r.t. $\Delta f$ was also mentioned in previous works [25, 26], but they did not use it to indicate resolution and provide the corresponding theoretical guarantees. The key challenge is to analyze the partial derivative of CRB to $\Delta f$, which is hard to analytically calculate. In Section V, we provide an approximate method to solve this problem.

Here, we analyze the declining phase, plateau phase, and turning point of ${\rm CRB}_{\rm PC}$, and use the turning point to indicate angular resolution of partly calibrated arrays.

For comparison, we analyze the declining phase, plateau phase, and turning point of ${\rm CRB}_{\rm FC}$, and use the turning point to indicate angular resolution of fully calibrated arrays.

To analyze the angular resolution in fully and partly calibrated arrays, we impose the following assumptions (A3 is not necessary for fully calibrated arrays.). A diagram is shown in Fig. 3.

• A1:

  Consider $L=2$ sources with spatial frequencies denoted by $\omega_{1}=\omega_{0}$ and $\omega_{2}=\omega_{0}+\Delta\omega$.

• A2:

  The average of the element positions of each subarray is uniformly distributed in $[0,D]$, i.e., $\bar{\varphi}_{k}\sim\mathcal{U}[0,D]$, where

  $\displaystyle\bar{\varphi}_{k}=\frac{\sum_{n\in\mathcal{N}_{k}}\varphi_{n}}{|\mathcal{N}_{k}|}.$

  (15)

  Each subarray has the same number of elements, i.e., $|\mathcal{N}_{k}|=\frac{N}{K}$, and the number of subarrays, $K$, is large such that $1/K\approx 0$.

• A3:

  The interval between the array elements within a subarray is small relative to the whole distributed array, i.e., a subarray can be approximated as a point in the geometry. Particularly, assume $\varphi_{n}\approx\bar{\varphi}_{k}$ for $n\in\mathcal{N}_{k}$, such that

  $\displaystyle\frac{1}{|\mathcal{N}_{k}|}\sum_{n\in\mathcal{N}_{k}}f(\varphi_{n})\cdot g(e^{j\Delta\omega\varphi_{n}})\approx f(\bar{\varphi}_{k})\cdot g(Q_{0}^{k}),$

  (16)

  where

  $\displaystyle Q_{0}^{k}=\frac{\sum_{n\in\mathcal{N}_{k}}e^{j\Delta\omega\varphi_{n}}}{|\mathcal{N}_{k}|},$

  (17)

  and $f(\cdot),g(\cdot)$ are any general polynomial functions.

In Assumption A1, we mainly consider the case $L=2$ to simplify the expressions. In Assumption A2, uniformly distributed subarrays are common in practice. Note that Assumption 2 can be extended to more general cases, such as the case where the inter-subarray position errors are bounded instead of being distributed on the whole array aperture. These proofs are similar and we take Assumption A2 for example in this paper. In Assumption A3, we consider that the apertures of the subarrays, $d_{k}\equiv\varphi_{\bar{n}_{k+1}}-\varphi_{\bar{n}_{k}+1},k=1,\dots,K$, are far less than the whole distributed array aperture $D$, such that a subarray can be viewed as a point geometrically, i.e., $\varphi_{n}\approx\bar{\varphi}_{k}$ for $n\in\mathcal{N}_{k}$. In an extreme case, substituting $\varphi_{n}=\bar{\varphi}_{k}$ for $n\in\mathcal{N}_{k}$ to (16) yields the corresponding equation. Therefore, closer intra-subarray displacements between $\varphi_{n},n\in\mathcal{N}_{k}$ implies smaller approximation errors in (16). This means that assumption A3 actually corresponds to a large, sparsely distributed array sensor network, which is usually used for high angular resolution.

The proof of Proposition 2 is not direct due to the complex form of ${\rm CRB}_{\rm PC}$ w.r.t. $\Delta\omega$. We then introduce intermediate variables w.r.t. $\Delta\omega$, denoted by $Q(\Delta\omega)$. We divide the proof of Proposition 2 into several tractable lemmas. In the sequel, we first introduce the intermediate variables $Q$ and the lemmas, and then show how Proposition 2 is proved with these lemmas.

The intermediate variables $Q$ are defined as

where $i=0,1,2$ and $k=1,\dots,K$. In the sequel, we abbreviate $Q_{i}(\Delta\omega)$ and $Q_{i}^{k}(\Delta\omega)$ as $Q_{i}$ and $Q_{i}^{k}$, respectively. The intermediate variables $Q_{i}$ correspond to the whole distributed array, and $Q_{i}^{k}$ correspond to the $k$-th subarray. Our theoretical results are mainly about $Q_{i}$, while $Q_{i}^{k}$ are used for intermediate derivations. These intermediate variables are all bounded by 1. Abandoning extreme scenarios ($e^{j\Delta\omega\varphi_{n}}=1,\ n=1,\dots,N$), we assume

With the help of $Q$, we rewrite the CRB w.r.t. $Q$ instead of $\Delta\omega$, facilitating the analysis of the plateau phase. This is feasible because the proof of Lemma 2 and 3 is equivalent to that of Proposition 2:

Here we give an intuitive explanation of how $|Q(\Delta\omega)|$ is close to 0 in Lemma 2. From Lemma 2. 1 in Appendix C, when $t=0.25$, $\Delta\omega D=2\pi$ and $N=200$, we have

The above conclusion can be extended to the cases of $\Delta\omega D\geqslant 2\pi$. Therefore, when $N$ is large enough, we have $|Q(\Delta\omega)|\approx 0$ with high probability, yielding that Proposition 2 also holds with high probability.

Based on the chain rule of partial derivatives, we have

Substituting ${\rm CRB}_{\rm PC}(\Delta\omega)\approx{\rm CRB}_{\rm PC}(Q(p\Delta\omega))$ in Lemma 3 to (24) yields that

Under assumptions A1-A3, the Fisher information matrix (FIM) of CRB is not singular, yielding $\left|\frac{\partial{\rm CRB}_{\rm PC}(Q(p\Delta\omega))}{\partial Q(p\Delta\omega)}\right|\leqslant C$, where $C$ is some constant. Based on Lemma 2, we have $\left|\partial Q(p\Delta\omega)/\partial p\Delta\omega\right|\approx 0$ when $\Delta\omega\geqslant\Omega/p$, which is also satisfied for $\Delta\omega\geqslant\Omega$ since $|p|\geqslant 1$. Therefore, when $\Delta\omega\geqslant\Omega$ we have

completing the proof.

The proof of Proposition 3 is similar to the counterpart of Proposition 2, except that Lemma 3 is replaced by Lemma 4, given by:

Combining Lemma 2 and 4 yields Proposition 3. The proof is the same as the counterpart of Proposition 2 and is thus omitted here.

We explain how to determine the turning of CRB. From the above propositions, we know that when $\Delta\omega\ll\Omega$, ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ decline rapidly as $\Delta\omega$ increases, and plateau out when $\Delta\omega\geqslant\Omega$. A criterion is in demand to distinguish the plateau phase and the declining phase in a strict sense, which implies the turning point of CRB.

As the CRB curve has the identical trend with the intermediate variables $Q(\Delta\omega)$, by observing the structure of $Q(\Delta\omega)$ defined in (18), we use the criterion

to determine the turning point of CRB. Consequently, the turning point is located at

This criterion is inspired by [35], which considers a similar problem that distinguishes the correlated and uncorrelated signals, detailed as follows: Denote the correlation of two signals by $E(\Delta f)$,

where $\Delta f$ is the frequency difference between two signals. The signals are regarded as correlated if $E(\Delta f)$ is close to 1 or regarded as uncorrelated if $E(\Delta f)$ approaches 0. In [35], it is explained that a sufficient condition for $|E(\Delta f)|$ to be much less than one is that the integrand completes at least one cycle or equivalently

We use the similarity between $Q(\Delta\omega)$ in (18) and $E(\Delta f)$ in (30) and determine the turning point of CRB as (29).

Finally, we give an intuitive explanation of this criterion applied in $Q(\Delta\omega)$. Take $Q_{0}=\sum_{n=1}^{N}e^{j\Delta\omega\varphi_{n}}/N$ as an example. When $\Delta\omega\ll\Omega$, we have $\Delta\omega\varphi_{N}\ll 2\pi$, which means that the phases of $e^{j\Delta\omega\varphi_{n}}$ are centralized in a small range of $[0,2\pi)$, yielding large $|Q_{0}|$. When $\Delta\omega\geqslant\Omega$, we have $\Delta\omega\varphi_{N}\geqslant 2\pi$, which means that the phases of $e^{j\Delta\omega\varphi_{n}}$ are distributed in $[0,2\pi)$ and vectors in different directions cancel each other, yielding small $|Q_{0}|$. A diagram w.r.t. the phases of $e^{j\Delta\omega\varphi_{n}}$ and $Q_{0}$ is shown in Fig. 4 Since $Q_{1},Q_{2}$ are the weighted extension of $Q_{0}$, they also approximately have the above characteristics.

We show the declining and plateau phases of ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ w.r.t. $\Delta\omega$ by simulations to verify the theoretical analysis of CRB in Section IV.

Consider $K=10$ half-wavelength uniform linear subarrays, each composed of 10 elements, yielding $N=100$. These subarrays are uniformly spaced on a straight line with $\xi_{k}=I(k-1)\lambda$ for $k=1,\dots,K$, where $I>0$ reflects the size of interval between subarrays and is set as $I=50$. The wavelength of the received signals is $\lambda=1m$, and the resolution limit of $\bm{\omega}$ is thus $\Omega=0.014{\rm m}^{-1}$. The geometry of the distributed array is shown in Fig. 5.

The directions $\bm{\theta}$ are uniformly in the range of $[\theta_{\min},\theta_{\max}]$, i.e., $\theta_{l}=\theta_{\min}+(l-1)(\theta_{\max}-\theta_{\min})/(L-1)$ for $l=1,\dots,L$. In this case, we have $\Delta\omega=2\pi(\sin(\theta_{\max})-\sin(\theta_{\min}))/\lambda$, and $\Delta\omega/(L-1)$ denotes the minimum separation between sources. We set $\theta_{\min}=1.2^{\circ}$. The complex coefficients $\bm{s}$ are set as $s_{l}=e^{j\pi/5}$ for $l=1,\dots,L$. The SNR is defined as $1/\sigma^{2}$ being 20dB. We consider $L=2,3,4$, and show ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ w.r.t. $(\Delta\omega/(L-1))/\Omega$ with a logarithmic coordinate in Fig. 6.

From Fig. 6, when $\Delta\omega/(L-1)\ll\Omega$, we find that the slopes of ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ w.r.t. $\Delta\omega$ in the $L=2,3,4$ cases are close to $-2,-4,-6$ in the logarithmic coordinate, respectively. This verifies the conclusions in Lemma 1 and Proposition 1 that ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ are mainly proportional to $(\Delta\omega)^{-2(L-1)}$ when $\Delta\omega\ll\Omega$, which correspond to the declining phase of CRB. When $\Delta\omega/(L-1)\geqslant\Omega$, we find that ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ both begin to plateau out w.r.t. $\Delta\omega$. This verifies the conclusions in Proposition 2 and Proposition 3, which correspond to the plateau phase of CRB.

Note that there is a fluctuation of CRB near the turning point, which is a common phenomenon in the CRB based resolution criterion. An intuitive reason is that the CRB involving the inverse of matrix usually has a complex form w.r.t. $\Delta\omega$, particularly when $\Delta\omega$ is close to the resolution limit $\Omega$. Based on the theoretical analysis method proposed in this paper, we can also give a more convincing explanation on this phenomenon. Particularly, from the discussion in Subsection V-B and Subsection V-C, we transform the analysis of $\partial{\rm CRB}(\Delta\omega)/\partial\Delta\omega$ into that of $\partial Q(\Delta\omega)/\partial\Delta\omega$. We take $Q_{0}$ as an example and statistically analyze the how $Q_{0}(\Delta\omega)$ varies with $\Delta\omega$. Based on the A1-A3 assumptions, we calculate the expectations of $Q_{0}$ as follows.

We show the turning points of ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ in different SNRs to explain that the proposed resolution criterion is not sensitive to noise.

Consider the same simulation setting as Subsection VI-A except $L=3$. We plot the ${\rm CRB}_{\rm FC}$ and ${\rm CRB}_{\rm PC}$ curves for SNR being $10$, $20$, and $30$ dB, shown as Fig. 7. From Fig. 7, we find that the curves of ${\rm CRB}_{\rm PC}$ in different SNRs have the same shapes, as well as the turning points. How the noise affects the CRB reflects on the absolute values of CRB, instead of the relative relationship between CRB and $\Delta\omega$. This verifies that the proposed resolution criterion using the CRB turning point is not sensitive to SNR in Subsection IV-A.

Since $\bm{M}\bm{G}^{-1}\bm{M}^{T}$ is the main component of ${\rm CRB}_{\rm PC}$ different from ${\rm CRB}_{\rm FC}$, we show its approximation errors in the proof of Lemma 3. Particularly, we compare the approximate $\bm{M}\bm{G}^{-1}\bm{M}^{T}$ with the true counterpart.

Consider the same simulation setting as Subsection VI-A except $L=2$. We take the (1,1) entries of the true and approximate $\bm{M}\bm{G}^{-1}\bm{M}^{T}$ as an example, and the comparison is shown in Fig. 8. From Fig. 8, we find that the approximation errors are large when $\Delta\omega<\Omega$, and become small when $\Delta\omega\geqslant\Omega$. This is because in the proof of Lemma 3, we use $|Q(\Delta\omega)|\approx 0$ in Lemma 2 to simplify the proof. This conclusion is feasible for $\Delta\omega\geqslant\Omega$, but not for $\Delta\omega<\Omega$. When $\Delta\omega\geqslant\Omega$, the approximation results are close to the true values, yielding the feasibility of the approximation.

We show that the angular resolution of fully and partly calibrated arrays is similar, which is achieved by comparing the upper resolution limit of the corresponding direction-finding algorithms. Particularly, we provide the estimation accuracy of these algorithms varying with the source separation, and regard the turning point where the estimation accuracy initially tends to plateau out as the resolution limit. For fair comparison, the directions are estimated using Multiple Signal Classification (MUSIC) [38] for fully calibrated arrays, and root-RARE [4], spectral-RARE [5], and ESPRIT-GP [16] for partly calibrated arrays, since these algorithms are all based on subspace separation techniques with the difference being whether errors exist.

Consider the same simulation settings as Subsection VI-A except $L=2$, the complex coefficients $\bm{s}$ being standard Gaussian variables and the number of snapshots being $T=50$. We use the root mean square error (RMSE) of $\Delta\omega$ to indicate the estimation performance of directions. We carry out $T_{m}=300$ Monte Carlo trials and denote the RMSE of $\bm{\omega}$ by

where $\hat{\bm{\omega}}_{t}$ is the estimate in the $t$-th trial and $\bm{\omega}^{*}$ is the true value. The estimation results of MUSIC for fully calibrated arrays and root-RARE, spectral-RARE, and ESPRIT-GP for partly calibrated arrays w.r.t. $\Delta\omega$ are shown in Fig. 9.

From Fig. 9(a), we find that when $\Delta\omega\leqslant\Omega$, the RMSEs of root-RARE, spectral-RARE, and ESPRIT-GP decline as $\Delta\omega$ increases; when $\Delta\omega>\Omega$, the RMSEs tend to be stable. Similar phenomenon is found in the fully calibrated case in Fig. 9(b), yielding that the resolution limit of these algorithms is close to $\Omega$. This also heuristically indicates that the resolution limit of fully and partly calibrated arrays is similar, both close to $\Omega$, corresponding to our main conclusion. We note that the turning points of the RMSEs w.r.t. $\Delta\omega$ in Fig. 9 are not exactly located at $\Omega$ since $\Omega$ is an empirical bound and the subspace based algorithms have super-resolution ability.

Then, we construct the dependence of the resolution probability for the RARE/ESPRIT methods in partly calibrated arrays and MUSIC method in fully calibrated arrays. Particularly, we define the resolution probability $P_{t}$ as the probability that the estimation error is less than a threshold, given by

where $T_{p}$ is the number of trials in which the estimation error is less than $-13$dB for partly calibrated case and $-30$dB for fully calibrated case, respectively, and the thresholds are empirically chosen based on the corresponding CRBs. The resolution probability of RARE/ESPRIT and MUSIC is shown as Fig. 10, which also verifies that the angular resolution between fully and partly calibrated arrays is similar.