Joint mean angle of arrival, angular and Doppler spreads estimation in macrocell environments

In wireless systems, the propagation environment has a great impact on smart antenna performance. Indeed, the multipath phenomenon produces the fading of signal strength due to constructive and destructive interferences. It also induces multiple angles of arrival (AoA) and angular spreads (AS), which reduces signal quality and hence degrades the performance of smart antennas. The estimation of those parameters is crucial and would improve the potential of smart antennas.

The mean AoA and the AS are critical parameters. Their estimates are used in several applications like source localization and detection [1]. The maximum Doppler spread (DS) is also a key parameter. Indeed, it provides information about the fading severity, and its knowledge at the base station can be used for hand-off purposes [2]. It is also needed in dynamic channel assignment [2], so that it can improve link quality. In this paper, we propose a new joint estimator for the mean AoA, the AS, and the maximum DS with a low-complexity approach. This work is motivated by the need to develop a new simple and accurate approach to jointly estimate the desired parameters since they are all required by several applications in mobile communication systems. To the best of our knowledge, there is no estimator which jointly estimates the three parameters. One can argue that two estimators among the literature could be easily combined. But implementing a different method for each parameter increases the overall computational complexity. Besides, there is room for performance improvement by recurring to a joint estimation approach while keeping overall complexity at a modest order, thereby resulting in a significantly improved performance vs. complexity trade-off. It is from this perspective that we propose in this work a low-complexity algorithm to jointly estimate the desired parameters with high accuracy.

Mean AoA and AS estimation has been studied in recent works. The maximum likelihood (ML) method is used in [3, 4]. The Gaussian-Newton algorithm used in [4] shows high computational complexity, whereas in [3], a new derivative of the ML method is developed. The latter considers the problem of localizing a source by means of a sensor array for a noisy received signal. This estimator is based on two solutions, considering both high and low signal-to-noise ratio (SNR) cases. In [5], the Gaussian-Newton algorithm is applied using the estimated covariance matrix of a single-input multiple-output (SIMO) configuration. This approach provides accurate estimates over a high computational complexity. In [6], a simple and accurate mean AoA estimator in the case of imperfect spatial coherence is proposed. In [1] and [7], low-complexity estimators are developed. The idea of replicating the transmitting source into two virtual sources is used. Then, the mean AoA and the AS are estimated by, respectively, averaging and differentiating the two virtual AoAs. In [7], the spread root-MUSIC algorithm is used, while in [1], the two-stage (TS) approach is developed using closed-form expressions.

Several methods were developed in the context of mobile communications to estimate the maximum DS. The auto-correlation function (ACF) was exploited in [3] and [8] to offer accurate estimates. In [3], a ML method based on a polynomial approximation of the ACF is used. While the estimator developed in [8] uses the ACF derivatives and takes into account the incoming wave distribution. Unlike the method described in [3], it presents a low computational complexity. In [9], a level crossing rate (LCR) approach is proposed. It considers a novel Doppler adaptive noise suppression process in the frequency domain to reduce the effect of the additive noise. In [10–12], Azemi et al. investigate the maximum DS estimation using three different techniques. The first approach is based on the reduced interference time-frequency distribution of the received signals [10]. The second one considers the ambiguity function [11], while the proposed algorithm in [12] uses the instantaneous frequency of the received signal. The two-ray (TR) approximation proposed in [13] offers a robust maximum DS estimation. It offers a closed-form expression and considers the presence of residual carrier frequency offset (CFO), which is closer to real-life scenarios.

In this work, we consider as benchmarks the TS approach [1] and the SRM algorithm [7] for the mean AoA and AS estimation, and the TR approach [13] and the ACF-based algorithm [8] for the maximum DS estimation. These recent works were chosen because they currently offer best trade-off between estimation accuracy and computational complexity.

This paper is organized as follows: In Section 2, we describe the considered signal model then define the space-time correlation matrix. We consider here the Gaussian and the Laplacian angular distributions for the incoming AoAs [14], the most popular ones in the literature. Nonetheless, the algorithm can be applied for other distributions like the uniform one. Next, we propose our joint estimator for the mean AoA, the AS, and the maximum DS. In Section 3, we evaluate the performance of the proposed approach before drawing out our conclusions in Section 4.

Notation: We use (.)^∗ for conjugate operator, |.| for absolute value, E[.] for mathematical expectation, and ∠ for phase. ℜ(.) represents the real parts of a complex number. We also use (.) ^H for trans-conjugate operator. The bold uppercase and lowercase letters represent, respectively, the matrices and vectors, while the non-bold lowercase letters represent scalars.

so that ${\sigma }_{s_i}^2=E\left[{\left|x_i\left(t\right)\right|}^2\right]-{\sigma }_{n_i}^2$ where ${\sigma }_{n_i}^2$ is the power of the additive white Gaussian noise (AWGN), n _i (t), at the i th antenna. The AoAs θ_p of the received signals follow an angular distribution with a mean and a standard deviation defined by the mean AoA, θ_m, and the AS, σ _θ , respectively. The phases ϕ_p are uniformly distributed over (-π π]. ω_D denotes the normalized maximum DS and is given by ω_D= 2 π F_DT_s where F_D is the Doppler frequency [15] and T_s is the sampling interval.

where P(θ_p|θ_m,σ _θ ) is the probability density function (PDF) of the incoming AoAs.

where P(θ_D|θ_m,σ _θ ) is the PDF of the Doppler angles θ_D. The latter is given by (θ_p-α), where α is the direction of travel (DoT) defined as the angle between the direction of the mobile and the antenna axis as shown in Figure 1. In this work, instead of combining two methods from the ones developed in the literature, we propose a unique algorithm to jointly estimate the three parameters. To this end, we jointly exploit both the spatial and the temporal correlations. The cross-correlation matrix of the received signals is then given by

$\begin{array}{lcr}{\mathbf{\text{R}}}_{\mathit{\text{kl}}}(\tau )& =& \frac{E\left[x_k\left(t\right)x_l^{\ast }(t+\tau )\right]}{\sqrt{E\left[{\left|x_k\left(t\right)\right|}^2\right]E\left[{\left|x_l\left(t\right)\right|}^2\right]}}.\end{array}$

where N_s is the number of the received signal samples.

In this paper, we consider both Gaussian and Laplacian angular distributions for the incoming AoAs [14]. Other angular distribution for the incoming AoAs can be applied like the uniform one, but this would yield to different closed-form expressions. The von Mises distribution approximate all these angular distributions over κ parameter value, but in our approach, it does not yield to closed-form expressions of the auto-correlation and cross-correlations functions.

We illustrate the performance of the new joint estimator (JE) in macrocell environments by means of N_b= 1,000 Monte-Carlo simulations. We consider N_s = 1,024 samples and a ULA with N_a= 5 elements spaced by a half wavelength. We also use the non-selective frequency Rayleigh channel model described in [19]. The simulations are CFO free and run at SNR = 20 dB and the sampling interval is set to $T_{\text{s}}=\frac{1}{1,500}$ s. Two time lags τ are needed to ensure high accuracy. This is why we consider in this section τ = 1 for the mean AoA and the AS estimation and τ = 100 for the maximum DS estimation. However, if the targeted application does not require accurate estimates, one time lag could be used then. The sampling rate 1/T_s is sufficiently small to guarantee τ T_s≪ 1. Exhaustive simulations were performed and showed that averaging over all antenna pairs induces several possible AoAs that give the same phase difference. This is why only the closest antenna elements (d = λ/2) are considered for the three-parameter estimation to avoid the ambiguity problem. In that case, only the first subdiagonal of the cross-correlation matrix is used. The subdiagonal cross-correlation coefficients are nominally equal; this is why considering more antenna elements would not improve the estimation accuracy of the joint estimation.

We evaluate our approach by comparing it to the TS approach [1] and the SRM [7] for the mean AoA and the AS estimation. For the maximum DS, we take as benchmark the TR approach [13] and the ACF-based algorithm [8]. We also compare these estimators to the Cramér-Rao lower bounds (CRLBs). For the mean AoA and AS, we consider the CRLB developed in [20]. For the maximum DS, we use the one developed in [21]. The used CRLBs for each given parameter assume the two others to be perfectly known and hence very likely overestimate the true joint CRLB.

Figures 2 and 3 show the NRMSE of the mean AoA estimates using the JE, TS, and SRM approaches at σ _θ =6° for both Doppler frequencies F_D=50 and 100 Hz, respectively, for the Gaussian and the Laplacian angular distributions. We notice that for both high and low F_D values, the SRM algorithm offers a lower error rate than the TS approach. The TS approach offers the same error rate than the JE estimator for high mean AoA values, while for low values, the JE provides a higher accuracy. For low θ_m values, the JE and the SRM estimators have almost the same performance, while for high mean AoA values, the JE outperforms the SRM algorithm.

For the AS estimation, as shown in Figures 4 and 5, the TS and JE approaches have similar error rate for low AS values. The JE estimator achieves a lower NRMSE than the TS approach [1] for high AS region where accuracy is precisely more beneficial and is the most encountered in practice. The inaccuracy shown by the TS approach for mean AoA and AS estimation is due to the approximation of the AS angular distribution while our algorithm considers an exact expression. Indeed, the proposed JE offers accurate estimates even for small ASs, while the SRM algorithm offers higher NRMSE’s. For high AS values, the SRM and JE estimators have similar performances. We note that for both mean AoA and AS estimates, we obtain a difference less than 1 dB between the NRMSEs given by the JE estimator and the CRLB. We note that, for low AS values, the JE NRMSEs are almost optimal, since they coincide with the CRLB.

For the maximum DS estimation, we notice in Figures 6 and 7 that the JE and TR approaches [13] perform nearly the same in terms of NRMSEs, while the ACF algorithm provides higher error rate than the TR and JE ones for both low and high ω_D values. At high ω_D values, TR estimator outperforms the JE one. While at low ω_D, the JE yields to more accurate estimates. Since T_s is extremely small, ω_D is as well, even if operating at relatively high Doppler or mobility. Hence, accuracy is precisely more needed in the low normalized DS region, the most encountered in practice in today’s high data rate transmissions characterizing 3G/4G technologies and beyond. We notice that the TR approach considers p time lags with p ∈ [ 0 … 19], while our new method uses only two time lags. The first is (τ = 1) for the mean AoA and AS estimation. For the maximum DS estimation, we determine empirically the second time lag (τ = 100). Moreover, our approach allows the joint estimation of the three parameters.

In order to study the robustness of our algorithm against the additive noise effect, we illustrate in Figures 8 and 9 the estimation performance of the three parameters vs. the SNR for both the Gaussian and the Laplacian angular distributions. We take as benchmark the TS and TR algorithms in dashed lines, because these approaches use the eigenvalue decomposition of the cross-correlation matrix to estimate the additive noise power and to reduce its effect. The proposed JE does not require such procedure since the cross-correlation coefficients are nominally noise free, i.e., $\left(E\left[n_k\left(t\right)n_l^{\ast }(t-\tau )\right]=0\right)$.As one can notice, both methods seem unaffected by the SNR values variation. Figures 8 and 9 show that JE offers better estimation accuracy than the TS approach even at low SNR values for mean AoA. For AS estimation, the two approaches present close NRMSEs. For the maximum DS estimation, the TR approach outperforms our JE at the expense of a noticeably higher complexity.

In this paper, we proposed a new low-complexity approach to jointly estimate the mean AoA, the AS, and the maximum DS in macrocell environments. The magnitudes and the phases of the cross-correlation matrix were used to estimate the three parameters. We developed the joint estimation algorithm for both the Gaussian and the Laplacian angular distributions in a NLOS scenario. Simulation results showed that our method provides more accurate estimates of the mean AoA and the AS than the TS and SRM approaches. For the maximum DS estimation, the new joint estimator outperforms the TR and ACF approaches at small DSs with lower computational complexity.