Applications of Tensor Models in Wireless Communications and Mobile Computing
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Marcos T. de Oliveira, Ricardo K. Miranda, João Paulo C. L. da Costa, André L. F. de Almeida, Rafael T. de Sousa, "Low Cost Antenna Array Based Drone Tracking Device for Outdoor Environments", Wireless Communications and Mobile Computing, vol. 2019, Article ID 5437908, 14 pages, 2019. https://doi.org/10.1155/2019/5437908
Low Cost Antenna Array Based Drone Tracking Device for Outdoor Environments
Abstract
Applications of direction of arrival (DoA) techniques have dramatically increased in various areas ranging from the traditional wireless communication systems and rescue operations to GNSS systems and drone tracking. Particularly, police forces and security companies have drawn their attention to drone tracking devices, in order to provide the safeness of citizens and of clients, respectively. In this paper, we propose a low cost antenna array based drone tracking device for outdoor environments. The proposed solution is divided into hardware and software parts. The hardware part of the proposed device is based on offtheshelf components such as an omnidirectional antenna array, a 4channel software defined radio (SDR) platform with carrier frequency ranging from 70 MHz to 6 GHz, a FPGA motherboard, and a laptop. The software part includes algorithms for calibration, model order selection (MOS), and DoA estimation, including specific preprocessing steps and a tensorbased estimator to increase the DoA accuracy. We evaluate the performance of our proposed low cost solution in outdoor scenarios. According to our measurement campaigns, we show that when the array is in the front fire position, i.e., with a DoA ranging from to , the maximum and the average DoA errors are and 1,9°, respectively.
1. Introduction
Applications of direction of arrival (DoA) techniques have dramatically increased in various areas ranging from the traditional wireless communication systems [1, 2] and rescue operations [3] to GNSS systems [4–7] and drone tracking in public and private events. In the last years Unmanned Aerial Vehicles (UAVs) have been a major concern of airspace control bodies and military due to possible terrorist attacks and illegal activities. In 2015, there were more than nine hundred incidents involving drones and aircrafts in the United States [8], whereas, in April 2016, a UAV reached an aircraft landing at the Heathrow airport in London [9]. In 2016 in Dubai, four drones invaded the airport interrupting the landings and takeoffs, causing an estimated loss of one million dollars [10]. In October 2017 in Canada, the first reported collision of a drone and a commercial airplane has occurred [11]. Recently police forces and security companies have drawn their attention to drone tracking devices in order to provide the safeness of citizens and of clients, respectively. In this sense, the development of low cost devices for drone tracking is fundamental to fit such demands.
In general, the DoA estimation techniques can be broadly classified into conventional beamforming techniques, maximum likelihood techniques, and subspacebased techniques [12–14]. In [15] the authors proposed to estimate the DoA of a signal impinging the Electronically Steerable Parasitic Array Radiator (ESPAR) antenna with twelve parasitic elements by using a support vector machine (SVM) technique. In the anechoic chamber the result of the experiments reaches estimation error. No hardware details were provided.
To overcome the effects of multipath propagation on the performance of DoA estimation, the authors of [16] proposed a frequency domain multipath resolution subspacebased approach, which makes RSSbased DoA estimation applicable in multipath scenarios for smallsize and lowpower sensor networks. This approach was verified with Monte Carlo simulations with high SNR.
Aiming at the growth of connected cars systems, the authors in [17] developed a 44 MIMO antenna system and proposed the DoA function for a circular phased array antenna. Since the focus of this paper is on the methodology and basic characteristics of DoA function, no measurement or hardware information was provided.
In [18], DoA estimation using an ESPAR with 12 parasitic elements and one active monopole is carried out for wireless sensor network (WSN) applications. The authors calibrated the ESPAR array using an anechoic chamber. Since the focus of [18] is on the calibration, no outdoor or indoor measurement campaigns were performed by the authors.
An improvement of ESPAR antennas is proposed in [19]. The authors developed a Multiple Beam Parasitic Array Radiator (MBPAR) antenna that can realize six beams at the same time without the use of diodes, which increases the communication capacity. To validate the proposed design, a prototype was fabricated at 2.45 GHz. The antenna has the efficiency from 94.2 % to 95.7 % over the 2.4GWLAN bands. No DoA measurements were provided.
In [20], a square shaped 16 element antenna array is connected to switches so that a fourchannel SDR can select four antennas at each side of the square, allowing a DoA estimation in outdoor environments. Each side of the square performs a azimuth estimation. According to the authors, three Yagi antennas were used as sources at specific points, and a maximum DoA error of is achieved. No information is provided about the real distance between the sources and the receive array.
In [21], a fourelement quasiYagi antenna array system is applied for DoA estimation using the MUSIC algorithm, whereas the Minimum Description Length (MDL) criterion is used to estimate the number of dominant multipath components. Only two measurements were performed for two specific positions, showing an error of . However, no information is provided about the experimental scenario.
In [22], several DoA estimation techniques are compared considering a horizontal uniform linear array (ULA) with 12 elements inside an anechoic chamber. The measurements were conducted varying the DoA from to in steps of . The DoA estimation errors were smaller than . The MinNorm approach [12] outperformed MUSIC [23] although it has a higher standard deviation.
Finally, in [24], the authors developed system using fiveport reflectometers that allow simultaneously measuring the DoA and Time of Arrival (ToA) of coherent and incoherent signals, connected to seven quasiYagi antennas, with one reflectometer for each antenna. The MUSIC algorithm is applied for the DoA estimation, providing an error of for one source and 0,5° for two sources. The measurements were performed in a nonreflective environment.
In order to detect the presence of drones and to track them, there is a variety of mechanical, optical, or antenna array based solutions in the market. For instance, the mechanical solution in [25] detects a drone within 3 km for targets up to 55 cm in diameter and classifies the model of the drone within 1.1 km. The position accuracy (azimuth) in [25] is . In [26], a rechargeable portable drone tracking device can detect and indicate the direction of a drone in a plane even with weak line of sight (LoS) component. The device in [26] allows the communication with other devices by using an Application Programming Interface (API) framework. No technical information and patent about the principles behind the device in [26] and its DoA accuracy were provided. In [27], an antenna array based system is shown to detect with a 1 km range and with accuracy or with a 7 km range and with accuracy.
In [28], a mobile application (app) is proposed for drone detection. According to the developers, the app has an average range of 106 meters. The system allows the detection of almost 95 % of all types of drones. However, the solution in [28] does not indicate the position or the direction of the drone.
In terms of drone tracking, there are recent works on received signal strength (RSS)based DoA estimation. For example, [29, 30] propose to estimate the DoA using arrays of YagiUda directional antennas for the localization of drones exploiting by their incoming NTSC signal in a measurement campaign. The work [29] proposes a complete hardware and software framework using arrays of directional antennas and formulates a novel DoA estimation correction procedure. In [30], a novel DoA estimation algorithm for the localization of drones is validated by using an ADFMCOMMS5EBZ software defined radio (SDR). Finally, [31] implemented tests to detect and locate UAVs at 900MHz. The authors used MUSIC combined with spatial smoothing and MDL. The work [30] has a similar objective to our proposal; however the authors did not concern in to present the accuracy of the system and in use of low cost equipment.
In this paper, we propose a low cost antenna array based drone tracking device for outdoor environments. To the best of our knowledge, there are no stateoftheart low cost offtheshelf antenna array based devices applied to drone tracking. The problem of drone tracking is challenging due to the several possible modulation schemes for the data transmission, multipath propagation, and the possible long operational distances. The proposed framework can also exploit tensorbased techniques, such as the Parallel Factor Analysis (PARAFAC). In contrast to the subspacebased methods, the tensorbased approach shows to be robust in real scenarios.
The remainder of this paper is divided as follows. In Section 2, the data model is presented. Next, in Section 3, we propose a low cost antenna array based drone tracking device for outdoor environments, including a complete description of the hardware and software, and the steps involved for assembling, calibration, and signal processing. In Section 4, we validate our proposed solution by means of measurement campaigns in an outdoor scenario. In Section 5, conclusions are drawn.
2. Data Model
We assume be farfield sources transmitting narrowband signals. These planar wavefront signals impinge over a receive antenna array with omnidirectional elements that are uniformly and linearly disposed. The space between two adjacent antennas is equal to , where is the wavelength of the carrier signal. The received signals at the antenna array can be written in a matrix form as follows:where A is the steering matrix and its th steering vector is given bywhere is the spatial frequency that can be mapped into the direction of arrival of the th source, , by the following expression: . S is the symbol matrix with N being the number of snapshots. N stands for the noise matrix whose elements are assumed to be ComplexValued Circularly Symmetric Gaussian and identically and independently distributed (i.i.d.) random variables.
Given (1) and assuming that the noise and the signal are uncorrelated, the covariance matrix can be computed bywhere is one column vector from X, is the Hermitian operator, and is the expected value operator. In practice, the sample covariance matrix is calculated as follows:
The DoA techniques used along this paper exploit the sample covariance matrix in (4). As shown in Section 3, the matrix X is preprocessed before we compute the sample covariance matrix .
The goal of our proposed drone tracking device is to estimate the direction of arrival (DoA) of the line of sight (LoS) component from a drone in an outdoor scenario. We assume that there is no obstruction of the LoS component. Therefore, the LoS component is assumed to have the greatest power in comparison with the nonLoS components. Mathematically, we can express it asfor . The operator stands for the Frobenius norm.
3. Proposed Low Cost Antenna Array Based Drone Tracking Device
In this section, we detail the proposed low cost antenna array based drone tracking device. In Section 3.1, we describe the steps for the hardware calibration. The calibration ensures phase alignment for all the four channels of the SDR, allowing the DoA estimation. In Section 3.2, we present the assembling of the hardware components of the proposed drone tracking device. In Section 3.3, we propose a signal processing framework for DoA estimation.
3.1. Hardware Assembling for the Calibration
In order to perform the measurements, the SDR should be calibrated, such that all the receive channels become in phase. The phase imbalance may be caused by different time initialization of the oscillators and by hardware imperfections. The hardware vendor provides a software [32] for clock calibration of the local oscillator. However, this software does not perform phase calibration.
Therefore, in order to perform the phase calibration, the hardware components are first assembled according to Figure 1. Note that the SDR transmits the signal from one channel and receives it in four channels that should be calibrated.
As shown in Figure 1, the SDR is a 44 MIMO platform named ADFMCOMMS5 [33], with two AD9361 [34] Integrated Circuits (ICs) that contain 2 transmitters and two receivers each, ranging from 70 MHz to 6.0 GHz, and have a channel bandwidth ranging from 200 kHz to 56 MHz. The platform is connected to a microprocessor and a FPGA motherboard [35] that configures the SDR and transmits the SDR data to the PC. As shown in Figure 1, the cables for calibration should have the same length. Moreover, a power division component is included in order to lead the signal to the four receive channels at the same time and to reduce the power of the transmitted signal to avoid damaging.
To compensate the phase errors previously explained, the first step is to extract the phase of the elements of the matrix . The phase is defined as follows:where is the element in position of the measured matrix . The operators , and stand for the phase operator, the imaginary part of a complex number and the real part of a complex number, respectively.
Since each channel is related to each line of the matrix , in order to compute the phase shift between two channels, we compute the phase difference of two consecutive antennaswhere stands for the matrix containing the calculated phases by (6), indicates the reference channel, and varies from 1 to . This reference channel can be randomly selected from 1 to and is the input of the SDR that is used as a reference to compensate the phase imbalance from the other inputs. Since the vector is the th row of matrix , in case , the th row of is filled with zeros. Finally, since the phase difference may slightly vary for different samples in the same row of due to the thermal noise, we compute the arithmetic mean of the elements of each row of , obtaining the vector and its th element is given by
Hence, in order to compensate the phase shift between two different channels, we compute the vector c . The th element is given byNote that the compensation vector c is computed only once for the system initialization. The calibrated outputs of the antenna array are given by the following expression:where the operator diag transforms its argument vector into the main diagonal of a diagonal matrix.
3.2. Hardware Assembling for the Drone Tracking Measurement Campaign
After the hardware has been calibrated, the next step is to assembly it in order to perform the measurements.
The fourelement omnidirectional antenna array is connected to the calibrated hardware composed of the FPGA motherboard and SDR daughterboard according to Figure 2. Each antenna is dual band (from MHz to MHz and from 4900 MHz to 5875 MHz ) [36] and has linear polarization with dBi of gain. The space between two consecutive antennas is equal to cm. The operational frequency GHz is the maximum frequency that avoids aliasing.
3.3. Framework for DoA Estimation
Here, we first propose a sample selection approach for DoA estimation by automatic phase deviation detection. Then, we formulate a DoA estimation framework exploiting preprocessing techniques and model order selection schemes.
Figure 3 depicts the flowchart of the proposed signal processing solution for DoA estimation.
As shown in Box 2 of the Figure 3, the phase deviation correction proposed in Section 3.1 returns a matrix that is used in the sample selection scheme in Section 3.3.1.
3.3.1. Sample Selection for DoA Estimation by Automatic Phase Deviation Detection
As exemplified in Figure 4, empirically we observed that the hardware causes phase deviations on the samples in random time instants. Therefore, we propose an approach to select the samples with phase deviations for the DoA estimation.
Note that the phase compensation proposed in Section 3.1 has been applied to the samples, whose matrix containing the phases are depicted in Figure 4. Furthermore, note that there are significant deviations that can degrade the DoA estimation process. The main objective here is to remove these phase deviations.
As shown in Figure 5, such ripples can be better visualized by computing the phase difference in the time dimension according to the following expression:where is the value containing the quadratic difference of two consecutive time samples and of the th channel. The , from (6), is the element in position of the matrix .
Figure 5 draws the that contains the result of (11). By detecting the peaks, we can identify which samples should be removed. For this task, we apply the approach proposed in [37], which returns the green curve with the value of the threshold. Therefore, the samples whose phase differences are greater than the threshold are removed. The result after the samples removed is presented in the following equation:where is the matrix with the selected samples. Note that and the values of , for , are found by comparing the phase difference values with the threshold in Figure 5.
3.3.2. DoA Estimation Framework
According to Figure 3, the matrix given in (12) is used to improve the DoA estimation with preprocessing schemes. There are several DoA estimation schemes in the literature, such as beamforming approaches and subspacebased approaches. Examples of beamforming are Delay and Sum [38] and Capon [39], whereas examples of classical subspacebased approaches are MUSIC [23] and ESPRIT [40].
The DoA estimation schemes assume that the model order is known. In practice, model order selection techniques should be applied to estimate the model order , as depicted in Figure 3.
In the flowchart of Figure 3, we adopted the Exponential Fitting Test (EFT) [43, 44] as the model order selection scheme. The EFT has the deflation property that allows us to find suitable thresholds as a function of the Probability of False Alarm (). By exploiting the deflation property and by finding suitable thresholds, the EFT has been the only scheme in the literature to estimate in the presence of a strong LoS signal and in the only noise (no signal) measurements. We have compared several schemes in the literature such as Akaike Information Criterion (AIC) [45], Efficient Detection Criterion (EDC) [46], Minimum Description Length (MDL) [47], Stein’s unbiased risk estimate (SURE) [48], RADOI [49], ESTimation ERror (ESTER) [50], and Subspacebased Automatic Model Order Selection (SAMOS) [51]. The MEFT [43, 44, 52] has also been suitable, but an even smaller was required to find the thresholds. The computation of the thresholds of the EFT requires an extremely low . The complexity of such a computation is prohibitive. Therefore, we propose in Appendix A an extrapolation algorithm to compute such thresholds. Note that our proposed extrapolation algorithm has been applied in [53–56], although no details are provided. The reason for extremely low may be related to the noise behaviour as shown in Appendix B. Note that the Ilmenau Package for Model Order Selection (IPM) [57] with MATLAB and Java implementation of the model order selection schemes can be found at the LASP homepage (https://lasp.unb.br/index.php/publications/softwares/).
In order to further improve the accuracy of DoA estimation schemes, preprocessing schemes can be applied beforehand. We consider in this work the Vandermonde Invariation Technique (VIT) [58], Spatial smoothing (SPS) [59], and Forward Backward Averaging (FBA) [60, 61] as a preprocessing schemes. As depicted in Figure 3, after the preprocessing step, a matrix Z is returned and used by the DoA methods summarized in Table 1.
In Table 1, the vector w() in , , and vary according to the candidate values of . The value of that maximizes the expression in , , and is the , since, in Section 2, the data model assumes that the LoS component faces no obstacles. Therefore, the component corresponding to the greatest power should be the same component with DoA equal to . is signal subspace, which is equal to the eigenvectors corresponding to the greatest eigenvalues, whereas is a basis for the noise subspace, composed by the eigenvectors associated with to the smallest eigenvalues. In and , is the diagonal matrix that has the eigenvalues of . We compute all the spatial frequencies and the one whose component has the greatest power is the that can be mapped to .
3.4. TensorBased DoA Estimation
In this subsection, a tensor factorization, namely, the PARAFAC decomposition, is applied. The PARAFAC decomposition generates three factor matrices from a received tensor , whose structure is detailed in this subsection. One factor matrix corresponds to the estimate of the steering matrix containing DoA information. In this subsection is extracted from tensor to estimate the DoA of the impinging signal.
We first consider an unchanging sequence of symbols transmitted periodically. Such symbols can be found in a header or footer or even in the payload of a message. Alternatively, repeating sequences of symbols can be extracted from time periods when no data is being transmitted, but the carrier of the transmitter is active. An example is given as follows. Since the oscillators at the transmitter and at the receiver are never exactly the same, a small frequency deviation or constant phase change is observed at the receiver. At an MSK receiver, if the deviation is positive, a sequence of ones can be extracted and if the deviation is negative a sequence of zeros is observed.
The symbols corresponding to the sequence from the th source is represented by the vector . Accordingly, we can build a received signal matrixwhere is the period corresponding to the sequence transmission. Generalizing for signals we findwhere and .
For transmitted sequences, the symbols can be concatenated along the third dimension to form the received signal tensorwhere represents concatenation along the third dimension. Since the slices of can be written as , has a PARAFAC structure and can be decomposed into three factor matrices , and , where contains the diagonals of along its rows. To factorize , we first rewrite it in three different matrix representations or unfoldingswhere is the KhatriRao (columnwise Kroenecker) product.
It is known that minimizing , , and in the least squares sense [62] leads to the following solutions:
We consider the wellknown Alternating Least Squares (ALS) algorithm to solve , , and in an iterative way. Since is known, the estimation step is skipped, and the ALS algorithm alternates between the estimations of and in a twostep approach [63].
Once convergence is achieved, we use to extract the DoA of the th source as follows:
4. Experiments
In this section, we validate our proposed drone tracking device with measurement campaigns in an outdoor scenario. In Section 4.1, the setup for the measurement campaign is described, whereas, in Section 4.2, we present the obtained results.
4.1. Experimental Setup
In Figure 6, we depict the outdoor scenario used for the measurement campaigns. On the righthand, we placed our drone tracking device, proposed in Section 3, as the receiver, whereas, on the lefthand, the transmitter is placed. The transmitter is a 2x2 MIMO SDR platform ASPR4 [64], with frequencies ranging from 50 MHz to 6.0 GHz, a channel bandwidth varying from 200 kHz to 56 MHz and a maximum power of 10 dBm at the output port.
As shown in Figure 7, the distance between the transmitter and the receiver is 48 m. Both transmitter and receiver are placed on tripods 115 cm above the ground. Note that the red “X” in Figure 7 is the location from where the photo in Figure 6 has been taken.
Both the transmitter and the receiver were set up using a MSK message signal, at GHz carrier frequency and 250 kbps of data rate. Before starting the experiment, the GHz frequency was scanned and no noise source was detected. In order to verify that the receiver properly works and measures the Bit Error Rate (BER), we have to decode the signal. To this purpose, the transmitted package must be known and consists of pseudo random sequences of length 1024 bits and a header and footer with 16 bits each defined as 0xFFFF and 0x0000, respectively. Therefore, the total size of the package is 1056 bits. The transmitter uses both sampling frequency and bandwidth of 2MHz. At the receiver side, a 4 MHz sampling frequency and 4 MHz bandwidth are used. Each captured frame at the receiver has 5120 samples.
As shown in Figure 7, the transmitter is fixed and the receiver rotates from to in steps of . According to Table 2, which presents the number of captured frame per DoA, at each step, about 31 frames of size 5120 samples are captured.

4.2. Experimental Results
This subsection shows the performance of the DoA estimation schemes shown in Table 1. During the measurement campaign, the achieved Bit Error Rate (BER) was .
Figure 8 shows the DoA calculated by rotating the base array over the time. Our proposed device works for a DoA ranging from to . The DS, Capon, MUSIC, ESPRIT, and Tensor combined with preprocessing schemes are shown in Figure 9(b).
(a) Without preprocessing
(b) With preprocessing
In Tables 3 and 4, we present the Root Mean Square Error RMSE for the schemes in Table 1 with and without preprocessing schemes using the measurements from Figure 6, respectively. The equation for RMSE is given bywhere is one realization of a total of realizations for each stage of the measurement campaign. In other words, the acquired data at each degree step is reshaped into small matrices. The computation of the DoA is individually performed for each of these matrices. In our experiment, we chose empirically matrices of size 4 1000. Since, as shown in Table 2, at each step 31 frames of size 5120 samples are captured, in average there are approximately matrices depending on the data reduction performed in Section 3.3.1. The variables and stand for the actual and the estimated DoA, respectively.


Comparing Tables 3 and 4 we can note that, except for the Tensor, the algorithms presented improvement in terms of RMSE after incorporating the preprocessing. CAPON and the Tensor increased the variance with the preprocessing. The smallest RMSE was achieved by the DS approach after preprocessing. Note that the ESPRIT assumes the shift invariance property, while both MUSIC and ESPRIT exploit the property of the orthogonality between signal and noise subspaces. Note that both assumptions are approximations and, therefore, their exploitation may cause additional errors.
In Table 5, we compare the DoA estimation results obtained with our framework and with the stateoftheart approaches. Furthermore, we show that, even without involving anechoic chamber in our tests, precise results are obtained in comparison to references that implemented its tests in a nonreflexive environment. The ‘x’ means that the commercial solutions did not provide such information. Finally, we provide the information about which of the solutions has low cost.
As shown in Table 6, the total cost of the proposed drone tracking solution is US$ 2.222, whereas the solutions in [25] and [27] cost US$ 226.000 and US$ 120.000, respectively. Therefore, our proposed offtheshelf solution costs less than 2 % of the commercial solutions in [25, 27].
5. Conclusions
In this paper, we have proposed a low cost antenna array based drone tracking device for outdoor environments. The proposed solution is divided into hardware and software parts. The hardware part of the proposed device is based on offtheshelf components such as an omnidirectional antenna array, a 4channel SDR platform with carrier frequency ranging from 70 MHz to 6 GHz, a FPGA motherboard, and a laptop. The software part includes algorithms for calibration, model order selection (MOS), and DoA estimation, including specific preprocessing steps to increase the DoA accuracy. We have evaluated the performance of our proposed low cost solution in outdoor scenarios. Our measurement campaigns have shown that when the array is in the front fire position, i.e., with a DoA ranging from to , the maximum and the average DoA errors are and 1,9°, respectively. Our proposed offtheshelf solution costs less than 2 % of commercial solutions in [25, 27]. In order to further improve our analysis of the proposed system and our results, experiments in an anechoic chamber can be performed. Moreover, the performance of the proposed framework can be improved by incorporating interpolation schemes. Perspectives also include the adoption of a more realistic noise model to simplify the computation of the thresholds of the Exponential Fitting Test (EFT).
Appendix
A. Proposed Extrapolation Algorithm to Find the EFT Thresholds for Extremely Low Probability of False Alarm
In this appendix we propose an extrapolation algorithm to estimate the thresholds of the EFT algorithm in cases that the Probability of False Alarm () is extremely low.
The EFT is based on the approximation that the profile of the ordered noise eigenvalues has an exponential behaviour. The profile can be expressed as
Given that , our goal is to vary such that we find that , where and stand for predicted noise eigenvalue and stands for actual eigenvalue, respectively. Note that the EFT assumes that smallest eigenvalue is a noise eigenvalue. Therefore, varies from 1 to . Using (A.1), [44] has derived the following expression:where is the estimated noise power.
In order to improve further the performance of the EFT approach, thresholds coefficients are computed using noiseonly simulated data following ComplexValued Circularly Symmetric Gaussian and identically and independently distributed (i.i.d.) as indicated in Section 2. Depending on the , we have two hypothesis:where is a noise eigenvalue and is a signal eigenvalue. In order to have all depending of the , we can define the asNote that the thresholds are obtained by Monte Carlo simulations carried out in the onlynoise scenario following the steps in [44] and by choosing the following amount of realizations:
Depending on the noise behaviour and the parameters of the scenario [54, 56], the thresholds can be extremely low. Therefore, the computational complexity of (A.7) can be prohibitive. In order to overcome such limitation, we propose an extrapolation approach to compute the thresholds for extremely low values of .
Since we wish to estimate values outside the known limits, we can use an extrapolation method and approximate the descending side of the curve as a decreasing exponential. In order to simplify the approximation, we adopt a logarithmic scale as exemplified in Figure 10.
Given the two known points in Figure 10 obtained by Monte Carlo simulations and given the linear extrapolation in (A.8), we can compute the two unknown constants a and b.
The constants a and b are given by (A.9) and (A.10) by using the two known points and .
By replacing a and b in (A.8), we obtain the expression for the in (A.11).
Following the framework of Figure 3, we set up the EFT with a = and we obtained the following values for the thresholds: = , = , and = .
Note that there are only three thresholds, since the smallest eigenvalue is assumed as a noise eigenvalue in the EFT approach. In Figures 11(a), 11(b), and 11(c), we depict the extrapolation curves for the thresholds , , and , respectively.
(a)
(b)
(c)
B. Noise Analysis
In Section 2, the noise is assumed to be ComplexValued Circularly Symmetric Gaussian and identically and independently distributed (i.i.d.). The EFT relies on these properties of the noise. Due to extremely low values of the in Appendix A, we analyze the noise behaviour.
According to Figure 12, the Power Spectrum Density (PSD) is not flat, indicating that the noise is time correlated.
In Figure 13, we depict the normalized histogram for antenna 3. Note that the Gaussian approximation has errors that can be reduced with an improved model.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the colleagues Prof. Dr. Sébastien Roland Marie Joseph Rondineau, Dr.Ing. Christopher Schirmer, and Mr. Mario Lorenz for their support. This work was partially supported by Research Support Foundation of the Brazilian Federal District (FAPDF), by the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES), and by the Brazilian National Council for Scientific and Technological Development (CNPq).
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Copyright © 2019 Marcos T. de Oliveira et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.