Joint 2-D angles and time delay estimation of coherent wideband signals

Abstract. Joint direction of arrival (DOA) and time delay (TD) estimation of wideband signals such as WIFI and mobile communication signals has to face the challenges aroused by multipath effect in indoor environments. New methods that can deal with coherent wideband signals with high estimation accuracy are necessary. To achieve localization when signals are coherent, this paper proposes a new joint estimation method which fully utilizes the subcarriers of wideband signals. This method could greatly extend the array aperture and accomplish the joint estimation through UCA-ESPRIT and one-dimensional (1-D) search. Simulation experiments are conducted to show the validity and efficiency of this method.


Introduction
WIFI and mobile communication signals are typical wideband signals that have been widely applied in recent decades. As location-based services are becoming more and more popular, localization based on WIFI and mobile communication signals in indoor environments has a promising prospect [1]. These wideband signals utilize multi-carrier modulation (MCM) technology which contains more frequency points than traditional narrowband signals [2]. To achieve accurate localization, joint two-dimensional (2-D) angles and time delay (TD) estimation is considered by most of previous researchers. Especially in indoor environments where multipath effect is common, joint estimation methods that can deal with coherent signals are needed [3,4].
A number of methods have been proposed for joint estimation of DOA and TD. However, some of them cannot be used when signals are coherent [5]. While some others have too high computational complexity and poor estimation accuracy [6]. In this way, we propose a novel method for joint estimation in multipath environments based on UCA that can achieve both low computation and high accuracy. We first expand the virtual array aperture and obtain the extended array output. Then virtual spatial smoothing is deployed to eliminate the influence of multipath effect. Finally, UCA-ESPRIT and 1-D search are used to get the estimation results of DOA and TD. The estimation results show the performance of our proposed method.

Signal model
Consider a uniform circular array (UCA) consisting of sensors with the radius . There are reflections of a far-field wideband signal impinging on the array from directions , where is the angle between the direction of the -th path and the x axis, while is the angle between the direction of the -th path and the z axis, as is shown in Fig.1.
is the time delay of the -th path from the emission source to the center of the array. The array spatial responding vector is represented as where is the speed of light and is the frequency.
where , , (7) (8) Since the frequency-domain incident signal could be got, Eq. 5 could be expressed as where is the statistical average value of , written as Then the extended array output which contains all subcarriers could be expressed as , where is the reconstructed manifold matrix which contains DOA and TD information of all the paths; is the noise matrix. The uniformly distributed subcarriers are used to expand the dimension of the manifold matrix in frequency domain, functioning the same as the spatially distributed sensors. As can be seen from Eq. 11, is a matrix, which means that the number of virtual array elements is , so the array aperture is greatly expanded.

Joint estimation method
For this section, we'll introduce the proposed estimation method. Generally, the covariance matrix of the extended array output is required in joint estimation, which could be expressed as .
In Eq. 12, is the covariance matrix of complex attenuation matrix .
When the paths are independent, it would be clear that . As is of full column rank, we can get . In this way, the MUSIC algorithm, one of subspace-based methods, could be used to estimate the parameters. However, when the channels are coherent, the rank of the covariance matrix is 1. In this case, the accuracy of methods based on eigen decomposition cannot be ensured and there will be interference in the estimation results of different paths. In this way, the multipath effect needs to be solved and spatial smoothing is one typical solution.

Virtual spatial smoothing
Smoothing processing is adopted when the rank of covariance matrix is smaller than the number of paths to estimate. Generally, the array will be divided into several subarrays that have the same structure. By making full use of the relationship between adjacent sub-array manifold matrices, the negative impact of multipath effect on the estimation could be greatly eliminated.
In this paper, the extended manifold matrix is divided into sub-array matrices, which are . That is to say, the -th virtual sub-array contains the information of all sensors received from the -th subcarrier.
We first calculate the covariance matrix of each sub-array. Assume the covariance matrix of the -th subcarrier is , then we have .
According to Eq. 6, it could be got that , where , so could be expressed as The key step of virtual spatial smoothing is calculating the mean of to get a full-rank covariance matrix, which is given as , where .

UCA-ESPRIT for DOA estimation
The next step of the proposed joint estimation method is DOA estimation using UCA-ESPRIT. UCA-ESPRIT is an improved ESPRIT method designed for uniform circular arrays because the array manifold of UCAs is not a Vandermonde matrix, leading to the lack of rotational invariance. In this way, the technology of beam-space transformation is deployed to process the data received by UCAs. Assume is the beam-space transform matrix, and is signal space after the beamspace transformation. Then we have , where is a full rank matrix. Next, define , where , , and . could be expressed as , and . (20) Then take the eigenvalue decomposition of and get its eigenvalue . The closed-from solutions of DOA are . (21)

1-D search for time delay estimation
Based on the estimation results of DOA obtained through UCA-ESPRIT, we then use onedimensional (1-D) research method to estimate time delays, the spatial spectrum function could be expressed as .
We can get corresponding estimation result for each . In this way, joint estimation of DOA and time delay is accomplished.

Simulation results
In this section, we conduct two simulation experiments to examine the validity and accuracy of the proposed method. We consider a wideband signal with , subcarriers, guard duration 1.6µs, fast Fourier transform period 3.2µs, and bandwidth 80MHz. In the first simulation, the performance of this method under low signal-to-noise ratio (SNR) condition is examined. With 200 Monte Carlo simulations, the estimation results distribution is shown in Fig.2 and Fig.3.  It can be seen from the above figures that the estimation results are all concentrated in the true values, which indicates the effectiveness and validity of our method when dealing with coherent signals under low SNR.
The second simulation is conducted to evaluate the performance of the method proposed in this paper. We choose root mean square error (RMSE) as the measure index, which is defined as , where is the number of Monte Carlo simulations, and are the true value and the -th estimation value. SNR in this simulation ranges from -15dB to 20dB at 5dB intervals. The estimation performance of our method and the approximate maximun likelihood (AML) algorithm in [6] is shown in Fig.4 and Fig.5.  It is evident that the proposed method is more accurate than the AML method and is more close to the CRB. Consequently, our method is more efficient when dealing with coherent signals.

Conclusion
For wideband signals in indoor environments, this paper proposes a novel joint DOA and TD estimation method that can deal with coherent signals. The virtual array aperture is greatly expanded by dividing wideband signals into several subcarriers. After virtual spatial smoothing processing, the joint estimation is accomplished by using UCA-ESPRIT and 1-D search method. Simulation results show that the proposed method has good performance under low SNR conditions and brings better estimation accuracy than typical AML algorithm.
This research was financially supported by Grant No.61401513 from the National Science Foundation.