Direct One-Bit DOA Estimation Robust in Presence of Unequal Power Signals

Direction-of-arrival (DOA) estimation is a crucial task in wireless communication and radar systems, with applications spanning beamforming, localization, and target tracking. Conventional methods often require high-resolution quantization, imposing challenges and complexities, particularly in large-scale antenna arrays. One-bit DOA estimation has emerged as a groundbreaking alternative, aiming to achieve accurate results without the need for high-resolution measurements. However, state-of-the-art approaches either require reconstruction of an unquantized covariance matrix or sparse signal recovery, or are based on restrictive assumptions such as the equality of power of signal sources. In this paper, a novel approach for direct one-bit DOA estimation is presented, overcoming the limitations of previous methods by introducing a generalized one-bit covariance matrix and smoothing it. Through analytical and numerical analyses, we reveal the shortcomings of the direct application of the one-bit covariance matrix, particularly in scenarios with unequal signal powers. Comparative simulations demonstrate the superiority of the proposed approach, especially in scenarios with significant signal-to-noise ratio differences and a limited number of snapshots.


I. INTRODUCTION
In the ever-evolving landscape of wireless communication and radar systems, direction-of-arrival (DOA) estimation stands as a critical and fundamental task [1], [2], [3].Accurate DOA estimation plays a pivotal role in a wide range of applications, including beamforming, localization, and target tracking [4], [5], [6].Over the years, extensive research efforts have been directed towards enhancing the performance of DOA estimation methods.However, one of the persistent challenges in DOA estimation is the need for high-resolution quantization, which often, and especially in emerging large-scale antenna array systems, requires costly and complex hardware setups [7].The power consumption of analog-to-digital converters (ADCs) increases exponentially with quantization bit number [8].Therefore, one-bit ADCs, composed of simple comparators, are of great interest in The associate editor coordinating the review of this manuscript and approving it for publication was Hassan Tariq Chattha .massive multiple-input multiple-output (MIMO) systems due to their minimal circuit power consumption [9], [10].
In response to the aforementioned challenge, a breakthrough has emerged in the form of one-bit DOA estimation, a paradigm-shifting approach that seeks to provide accurate DOA estimation without relying on high-resolution measurements [11], [12], [13].In [14], an estimator is provided to obtain the normalized scatter matrix of the unquantized data from one-bit samples, which is robust against outliers.In [15], a framework is presented to solve the problem of DOA estimation from one-bit measurements received by a sparse linear array.The performance of such a framework is studied in detail in [11].Chen et al. [16] modeled the DOA estimation problem of incoherent signals as a binary classification problem and reduced the physical complexity by using sparse arrays.In [17], a gridless approach robust against off-grid errors and sign inconsistency is provided.
All the works mentioned above rely either on the reconstruction of the unquantized covariance matrix or the sparse signal recovery.Huang and Liao [18] showed that the covariance matrix of one-bit array measurements can be used without mediation for DOA estimation based on subspaces decomposition.However, in modeling and implementation, they have considered the limiting assumption of equal power of signal sources.Such an assumption is not necessarily true in practical applications, because the sources in the environment can be located at different distances from the array or radiate different powers [19].
To overcome the above limitation, in this paper, we first derive the one-bit covariance matrix without restrictions on the power of the sources, generalized one-bit covariance (GOBC) matrix, and express it in terms of the unquantized covariance matrix.Then, with analytical and numerical analyses, we will show how the covariance matrix obtained directly from one-bit measurements may not be reliable against unbalanced signal-to-noise ratios (SNRs).We will also show that such a matrix suffers from the problem of swapping subspaces in conditions of imbalance between sources' powers and limits its use for DOA estimation based on subspaces decomposition techniques.Finally, a solution based on smoothing the GOBC matrix is presented to significantly improve the final results.
The rest of this paper is organized as follows.Section II establishes the data model, providing the main assumptions for subsequent developments.Section III introduces the proposed generalized direct one-bit DOA estimation approach, deriving the GOBC matrix and addressing the limitation of equal power assumptions.Finally, Section IV presents the performance evaluation through computer simulations, comparing the proposed method with existing one-bit approaches.
signal and noise vectors, respectively, which are assumed to be uncorrelated and modeled as zero-mean circular complex Gaussian random processes.The power of the k-th signal source and the noise power are denoted by σ 2 k and σ 2 w , respectively.The steering vector of the steering matrix A = [a (θ 1 ) , a (θ 2 ) , . . ., a (θ where τ m, k = −2π (m − 1) d sin θ k λ denotes the propagation time delay between the first sensor (reference) and the m-th one, and m = 1, 2, . . ., M .In (1), Q 1 {z} is defined as [22] Q

III. GENERALIZED DIRECT ONE-BIT DOA ESTIMATION
The arcsine law [14], [23] states that if x (t) is a circularly-symmetric complex Gaussian vector with zero mean and covariance R x = E x (t) x H (t) (consistent with the assumptions of the data model in Section II), then the covariance matrix R y = E y (t) y H (t) corresponding to the signal y (t) (see (1)) is related to R x as follows: where ) and arcsin (z) = arcsin (ℜ {z})+j arcsin (ℑ {z}).
According to (1) and ( 2), and the uncorrelation of signals and noise, the m, m ′ -th entry of the matrix R x is obtained from the following equation: where , and as a result we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. = where χ k ≜ σ 2 k σ 2 w represents the SNR of the k-th signal.According to ( 2) and ( 6), we have By using the generalized triangle inequality [24], it can be easily proved that Therefore, considering (7) and ( 8), we can conclude that Similarly, for the imaginary part, we can write Since in practice, always for m ̸ = m ′ , occurs only when a noise-free scenario is considered, or the signal power is infinite.Therefore, in practice, it can be said that for m ̸ = m ′ , ℜ Rx m, m ′ < 1 and ℑ Rx m, m ′ < 1.Also, ( 9) and (10) indicate that the lower the sum of SNRs, the smaller the upper bounds of  According to the derived bounds in the previous paragraph, the convergence of Taylor series expansions arcsin ℜ Rx m, m ′ and arcsin ℑ Rx m, m ′ are guaranteed [25].In addition, if ℜ Rx m, m ′ and ℑ Rx m, m ′ are small enough, or equivalently, the sum of the SNRs is sufficiently low, the high-order terms of the arcsin expansion can be omitted and according to (4) we write: For m = m ′ , since according to ( 6) and ( 4), Rx m, m ′ = R y m, m ′ = 1, therefore, the above approximation cannot be applied.However, by rewriting (11) into the form ( 12) the following closed form (GOBC matrix) can be obtained: Let v ∈ C M ×1 be an eigenvector of R x corresponding to an eigenvalue λ , then R x v = λ v [26].Therefore, we can write From ( 14), we can conclude that although the eigenvalues of R y are obtained by scaling the eigenvalues of R x by 2 (πP) and adding 0.36, the eigenvectors of R y and R x are almost the same.As a result, for DOA estimation methods based on eigenvectors, such as subspace techniques, the GOBC matrix can be used directly, considering applied approximations.However, considering that measurements y (t), unlike measurements x (t), are limited to only four values (±1 ± j) √ 2 (according to (1) and ( 3)), the data of R y , which in practice are estimated as Ry = L t=1 y (t) y H (t) L, experience faster (highfrequency) changes, especially when the number of snapshots is small.The results of the simulations that will be presented in Section IV show that the direct application of the GOBC matrix may lead to the phenomenon of swapping subspaces under conditions where the powers of the received signals are too unbalanced.In particular, the greater the differences of signals' powers, the greater the difference between the eigenvalues corresponding to the signals (i.e.λ 1 , λ 2 , . . ., λ K ); and the eigenvalues of the weaker signals are closer to the λ K +1 , λ K +2 , . . ., λ M , where λ 1 ≥ λ 2 ≥ . . .≥ λ M .This leads to inter-subspace leakage [27], in which a share of the true signal subspace resides in the estimated noise subspace.Therefore, some steering vectors may not be completely orthogonal to the noise subspace.Ultimately, this may cause DOAs to be incorrectly estimated.Note that this becomes more acute when the estimation relies on low-resolution measurements (quantization).To overcome this problem, we apply to the data Ry , before subspace decomposition, a smoothing operation that acts as a low-frequency filtering.This preprocessing step reduces the impact of high-frequency changes in the data, making the subsequent subspace decomposition more robust and accurate.After applying the filtering, an estimate of the smoothed GOBC matrix Ry_Smoothed is obtained.For this purpose, various smoothing filters such as moving mean, moving median, Gaussian, etc. [28] can be employed.
DOAs can be estimated by the following spectral search function: where where Q s_Smoothed , spanning the signal subspace of Ry_Smoothed , consists of the eigenvectors related to the diagonal elements of s_Smoothed .Similarly, Q n_Smoothed consists of the eigenvectors related to the diagonal elements of n_Smoothed , which spans the noise subspace of Ry_Smoothed .

IV. PERFORMANCE EVALUATION AND DISCUSSION
In this section, the results of computer simulations to verify the performance of the proposed approach (generalized direct one-bit DOA estimation) are presented along with the discussion.Also, comparisons are made with one-bit multiple signal classification (MUSIC) with covariance matrix reconstruction [23], MUSIC with unquantized measurements [29] and one-bit MUSIC algorithm [18].A 10-element ULA is considered.The signals and noise are independent and identically distributed complex Gaussian processes with zero mean.In all simulations, σ 2 w = 1.Unless otherwise specified, L = 1000, and two narrowband signals impinge on the array from θ 1 = −10 • and θ 2 = 3.5 • .In all methods, the step size of the angle search is 0.1 • .To smooth the GOBC matrix, we have used the Gaussian filter [30], which provided the best performance in our experiments.All results are obtained from averaging N T = 1000 Monte Carlo runs.To evaluate the accuracy of the estimates, the root-meansquare error (RMSE) criterion with the definition of is used, where θk, i represents an estimate of θ k in the i-th trial [31].
Fig. 2(a) shows the normalized estimated spatial spectra of various methods when the SNRs of the signals are different (the first is 10 dB and the second is −10 dB).As can be seen, the output of the proposed approach (green diagram) has a better resolution than other one-bit approaches, although logically the best resolution belongs to unquantized MUSIC (black diagram).More noteworthy is the performance of the one-bit MUSIC method (red diagram).It can be seen that the estimated DOA of the second signal with lower SNR has a significant error (see right peak).Fig. 2(b) shows the simulation outputs when the second source is moved to 5.5 • (keeping other parameters constant).Again, the peaks of all the methods are around the correct DOAs, except for the one-bit MUSIC method.The estimated DOA of the second source, like Fig. 2(a), is about 31.5 • .In other words,  in the case of the one-bit MUSIC method, although the original DOA of the second source is changed, there is no change in the estimated DOA.Now let us consider Fig. 3. Fig. 3(a) and Fig. 3(b) show the outputs of the mode where the SNRs of both signals are χ 1 = χ 2 = −10 dB and χ 1 = χ 2 = 10 dB, respectively.This time, the one-bit MUSIC method works correctly (whether the SNR is low or the SNR is high).Considering the above, it can be concluded that the one-bit MUSIC method [18], which is a direct estimation approach from one-bit data, fails when the SNRs are different.In Section III, it was stated that the reason for this issue is the problem of swapping subspaces.To confirm this, we have demonstrated three different cases in Fig. 4. In all three cases, χ 1 = 10 dB and χ 2 = −10 dB.Fig. 5 shows the allocation of signal and noise subspaces in the corresponding cases.Case 1 represents the normal state of decomposition of signal and noise subspaces, where the two eigenvectors corresponding to the largest eigenvalues of the covariance matrix in [18] constitute the signal subspace, and the remaining eigenvectors form the noise subspace [18].In this case, only one of the peaks (corresponding to the stronger signal) is correct (see Fig. 4).In Case 2, although there are only two sources, three eigenvectors corresponding to three larger eigenvalues are considered as the signal subspace.The result is that three peaks can be identified in Fig. 4, the two on the left side are correct and the right peak does not correspond to any signal.Finally, considering Case 3, we find that the problem is in the eigendecomposition of the covariance matrix in the approach [18] so that since the SNR of the second signal is significantly different from the SNR of the first one, the eigenvector corresponding to the weaker signal is swapped with the strongest component of the noise subspace.As we showed in Fig. 1, the proposed approach does not suffer from such a practical problem.Now, the estimation accuracy in various methods is compared.In this experiment, the SNR of the first source is assumed to be fixed and equal to 10 dB.By changing the SNR of the second source from −10 to 10 dB, the RMSE values for different numbers of snapshots are calculated in Fig. 6.
From the results, it can be seen that when the SNR difference between two sources is more than about 10 dB, the proposed one-bit approach performs much better than other one-bit methods, especially when the number of snapshots is less.This is consistent with the explanation in Section III.Also, when the SNRs of the sources are close to each other, the proposed approach exhibits competitive performance.In addition, according to the theoretical analyzes related to correlation coefficient and power in Section III, when the sum of SNRs is lower (compared to the case where the sum of SNRs is higher), we expect the performance of the proposed approach to be closer to the outputs of unquantized data.The results of Fig. 6 confirm this.It is quite natural that the lowest RMSEs always belong to the results obtained by unquantized data (with infinite precision).For further investigation, in Fig. 7, a 3-D surface plot of the performance of each method is extracted.In Fig. 7, in addition to changing  the SNR of the second signal from −10 dB to 10 dB, the DOA of the second source is also variable (between 2.5 and 4.5 degrees).DOA and SNR of the first signal are still fixed.It is still observed that, in general, the proposed approach, among the one-bit methods, has the closest performance to the unquantized data output.It can also be seen that generally increasing the spatial distance (increasing the angular gap between the sources) decreases the RMSE, which is reasonable.
In the last experiment, the resolution probabilities (RPs) [18] of the algorithms are compared.For this purpose, it is assumed that the true DOAs of the two sources are θ 1 = −10 • and θ 2 = −10 • + , and have SNRs of 10 and −10 dB, respectively, where ∈ [1 • , 15 • ] represents the angular gap between the two sources.In each independent run, it is assumed that the test is successful if the estimation errors of both DOAs are less than 0.5 (the middle angle of the two sources).Therefore, in the simulation, we have defined the RP as the ratio of the number of successful trials (N S ), to the total number of runs, i.e.RP ≜ N S N T .The results are shown in Fig. 8.For both the number of snapshots equal to 100 and 1000, the proposed approach provided the best performance using one-bit measurements.Similar to the previous experiment, it can be seen that in fewer snapshots, the performance of the proposed approach is closer to the output of unquantized data.The more important point is that the RP values corresponding to the one-bit MUSIC algorithm, regardless of the number of snapshots, are always equal to zero, which means that the method [18], in a more practical scenario, where the powers of the signals are not necessarily equal, is absolutely not reliable.

V. CONCLUSION
In this paper, we first derived the GOBC matrix.Then, we highlighted the limitations of directly applying the one-bit covariance matrix, particularly in the presence of unbalanced SNRs, leading to the swapping of subspaces.Furthermore, we introduced a solution involving the smoothing of the GOBC matrix, significantly improving the final DOA estimation results.The presented analytical findings and numerical simulations demonstrated the efficacy of the proposed approach in overcoming practical challenges associated with unequal signal powers for one-bit DOA estimation.Comparative simulations with existing methods underscored the superiority of the proposed method, especially in scenarios with significant SNR differences and a limited number of snapshots.The research contributes a robust solution to enhance the reliability of one-bit DOA estimation in practical applications.

ℜ
Rx m, m ′ and ℑ Rx m, m ′ (go towards zero).For example, suppose two signals with DOAs of −40 • and 10 • and with SNRs of χ 1 and χ 1 − 5 dB, respectively, impinge onto a 10-element array.The values of R m ′ ≜ ℜ Rx 1, m ′ and I m ′ ≜ ℑ Rx 1, m ′ for different χ 1 s are illustrated in Fig. 1.The curves in Fig. 1 confirm the theoretical findings derived above.

FIGURE 1 .
FIGURE 1.The magnitude of the real and imaginary parts of the correlation coefficient versus different SNRs.

FIGURE 3 .
FIGURE 3. Comparison of the estimated spatial spectra of various methods when the SNRs of the signals are the same; (a) χ 1 = χ 2 = −10 dB.Note that the same performance of the blue and red spectra in the case where the SNRs of the signals are equal and low is consistent with the analysis of [18], (b) χ 1 = χ 2 = 10 dB.

FIGURE 5 .
FIGURE 5. Allocation of signal and noise subspaces in one-bit MUSIC algorithm [18] corresponding to Fig. 4. The eigenvectors 1 to 10 corresponding to the largest to smallest eigenvalues are sorted; (a) Case 1, (b) Case 2, (c) Case 3.

FIGURE 7 .
FIGURE 7. 3-D surface plot of the performance of various methods.L = 100.Different methods are distinguished by different edge line colors.Faces and colorbar are color-coded according to the RMSE range.

FIGURE 8 .
FIGURE 8. Comparison of RPs of various methods versus angular gap .