A novel method for doa estimation of wideband signals based on total forward-backward matrix pencil algorithm

In addition, the performance and the efficiency of the proposed method are not affected significantly by the correlation between the received signals, which permits us to estimate not only wideband incoherent signals but also wideband coherent signals. The simulation results for DOA estimation of wideband signals using the proposed method will be shown and analyzed to verify its performance.

A NOVEL METHOD FOR DOA ESTIMATION OF WIDEBAND SIGNALS BASED ON TOTAL FORWARD-BACKWARD

MATRIX PENCIL ALGORITHM

Han Trong Thanh1*, Vu Van Yem1

Abstract: Direction Of Arrival (DOA) estimation for wideband signals has been

drawing a lot of research interest recently In this paper, a novel method based on Total Forward-Backward Matrix Pencil (TFBMP) algorithm to estimate the direction of arrival of wideband signals using a Uniform Linear Antenna Array (ULA) is proposed TFBMP is an extension of Matrix Pencil algorithm that works directly on signal samples received by an M – element ULA One of the most remarkable advantages of the method is that it can extract the DOA information with one snapshot, which means that the sampling frequency in real time receivers as well as implementation cost of the system can be considerably reduced In addition, the performance and the efficiency of the proposed method are not affected significantly by the correlation between the received signals, which permits us to estimate not only wideband incoherent signals but also wideband coherent signals The simulation results for DOA estimation of wideband signals using the proposed method will be shown and analyzed to verify its performance

Keywords: Direction of arrival (DOA); Wideband signal; Total Forward-Backward

Matrix Pencil (TFBMP), Uniform Linear Antenna Array (ULA)

1 INTRODUCTION

Wideband signals are widely used in wireless localization system such as Radar, Sonar or Direction Finding (DF) systems Several techniques to estimate the Direction Of Arrival of wideband signals have been investigated recently [1-9] In order to do that most of them decompose the incident wideband signals into narrowband bins, and then extract the DOA information by using high resolution DOA techniques such as MUSIC, ESPRIT [1-4]

In [10-14], the Matrix Pencil algorithm was proposed as a high – resolution technique to estimate the DOA of the incoming radio signals Matrix Pencil (MP) has some advantages in comparison to the other super – resolution methods for DOA estimation such as MUSIC, ESPRIT, which generally have to calculate the signal covariance matrix Unlike other algorithms, Matrix Pencil works directly on signal samples and does not require independent data samples Furthermore, the Matrix Pencil algorithm offers some benefits such as less processing power and faster executing than some other ones [12] One of the most remarkable advantages

of this technique is that it can extract the DOA information with one snapshot

The Total Forward – Backward Matrix Pencil (TFBMP) algorithm is an

extension of the Matrix Pencil Method [15] The Total Forward – Backward is the

preprocessing technique to break the correlative property of the received signals

[16-17] Therefore, this fact helps the Matrix Pencil method to estimate the DOA

information of the incoming signals with the higher accuracy [18], especially in a

high correlative environment TFBMP was utilized for the high – resolution

frequency estimator In some scenarios, it provides better estimation results than

the other methods such as Fourier technique [15]

In [18], the DOA information of the narrowband coherent incoming signals can

be successfully determined However, the DOA of wideband signals had not been

mentioned yet In this paper, we propose a method based on the TFBMP to

estimate the DOA information of wideband incoming signals received by an M –

element Uniform Linear Antenna (ULA) array In order to do that we have to

divide the received wideband signals into several narrowband bins by using the

Discrette Fourier Transform (DFT) method [4,9] and then estimate the DOA

information of incoming signals in each bin The performance of this method will

be assessed in many cases that depend on the characteristics of incoming signals as

well as number of snapshots of data

The paper is organized as follows Section II describes the model of the

incoming signals In section III, we present in detail the TFBMP technique for

DOAs estimation of wideband signals The simulation results are shown in the

section IV The conclusion is given in the section V

2 SIGNAL MODEL

Fig 1 Antenna array in the coordinate system

In this research, an element Uniform Linear Antenna Array (ULA) is used

This is one of the most convenient mathematical models for array processing due

to its simplicity and regularity The ULA model can be described as a set of

isotropic antenna elements spaced at a uniform interval along some line in space

The reference point has been defined as the origin of the three – Dimensional –

Cartesian – coordinate system shown in Fig.1 In Fig.1, we assume that the incoming signals at the far field of the array impinging on the ULA are wideband and have DOA information in both elevation and azimuth ( ) However, this work concerns about the signals in the same plane with antenna array This means that we estimate the DOA of signals of interest in azimuth, , and

In practice, there are several radio signals crossing the antenna array simultaneously The received signal at each antenna element will be the sum of all arriving radio signals In case of signals approaching the array from some azimuth directions , according to [4-6], the wideband signal received at the antenna element can be modeled as

where is the incoming signal; is noise at the antenna element, which is assumed to be uncorrelated with the signal sources and is white noise in both temporal and spatial domain; , in which is the distance between the element and the reference point, and is the speed of the signal propagation

Assuming the array manifolds of different DOAs are independent In other words, array manifolds with different DOAs should span a dimensional subspace Moreover, considering the number of signal sources is either known or can be estimated The bandwidths of the wideband sources need not be identical,

and maximum angular frequency of wideband signal spectrum, respectively In order to ensure the Fourier transform of the output signal at each antenna element has a good resolution, we suppose the observation time is long enough Then the DFT of the element output is

Equation (2) describes the received wideband signals at each antenna element in frequency domain In order to estimate the DOA information, this signal is splitted into several narrowband bins using filter banks or the DFT technique If the intersection of the frequency bands of all incoming signals is , then the output of the filter bank or DFT module can be written in frequency vector form as follows:

(3) where is number of bins and

(4)

(5)

is the steering matrix:

(6) The columns of the matrix are the array manifolds at

frequency The array manifold is defined as

(7)

3 DOA ESTIMATION BASED ON TFBMP

According to [15,18], the DOA information could be extracted by using

TFBMP with the following steps

Step 1 – Hankel Matrix Establishment: The Hankel matrix of could be

written as:





L

L m

Y

(9)

where are the Pencil parameter Because of the efficient noise filtering issue

described in [10], is chosen with the conditions as:

, if is even

Step 2 – Complex conjugate Matrix Establishment: Based on Equation (9), the

matrix is established as:







b

Y

(11)

where symbol denotes the complex conjugate

Step 3 – All Data Matrix Establishment: The “All data” matrix can be

defined as follow:

The matrix will be processed to get the DOA information For noisy data, based on [10], the best way is to perform a Singular Value Decomposition (SVD)

on The SVD of is given as:

(13) where the characters denotes conjugate transpose and are unitary matrices and is the diagonal matrix which is composed by the singular values (SVs) of :

The matrix can be divided into two subspace and is a

diagonal matrix This matrix has largest SVs in main diagonal and displays the

noise of relative SVs This matrix characterizes the additive noise subspace Similar to the above analysis, can also be separated into two spaces: noise subspace and signal subspace as:

(14)

Step 4 – All Data Matrix Decomposition: Based on the properties of SVD

operation can be rewritten as follow:

are obtained from In matlab, and can be presented as

(16) (17)

Step 5 – DOA Estimation: As above analysis, the DOA information could be

gathered from It can be seen that the can be extracted from matrix In order to do that, two matrices and are created by deleting the last and the first columns from After that, the matrix will be established, in which is Moore – Penrose pseudo – inverse of as

(18)

is a matrix This matrix has the eigenvalues which is the value of Therefore, by using the values of the generalized eigenvalues of , angles

of arrival can be estimated as

4 SIMULATION RESULTS

The proposed method is simulated using Matlab to examine its performance for

DOA estimation of far field wideband incoming signals which are assumed as a

sum of complex exponentials as follow:

where the amplitude is a Rayleigh random variable; the phase is uniformly

distributed in and is the number of frequency components of

wideband incoming signal

In our research, we suppose that the wideband signals impinge on an 8 –

element antenna array These signals are based on IEEE 802.15.4a

standard [19] and they can be divided into 11 bins ( ), in which

of wideband signal spectrum, respectively In our simulation, we choose the Pencil

parameter and inter – spacing of antenna array , where

with is the frequency center of the selected narrowband signal bin

Table 1 The DOAs (Degrees) estimated in each narrow bins.

Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin

10

Bin

11 9.56 10.09 9.96 10.28 10.38 9.91 10.04 10.06 10.05 9.71 10.25

-29.71

-30.28

-29.73

-30.33

-30.01

-29.91

-29.49

-29.80

-30.11

-29.83

-29.88 59.85 60.58 58.96 60.43 60.13 60.16 60.67 59.96 60.05 60.18 59.86

In the first simulation, the proposed method is executed to estimate the DOAs of

The simulation results in each bin are presented in the table 1 In this table, we can

see that the DOAs are estimated accurately for all bins However, in order to get

the best result, the average value in each row is calculated and choosed as the final

estimated DOA information The final result is plotted in Figure 2

Fig 2 DOA estimation with 1

snapshot

Fig 3 Estimation accuracy of DOA

estimation with one snapshot

However, we need to note that the estimated DOAs in the simulation are the numerical values as in Eq.19 Therefore, in order to demonstrate visually the results, we illustrate the DOA in XOY plane, in which the X – Axis is the DOA of incoming signal and the Y – Axis is the indicating factor This factor is set to 1 corresponding to the estimated DOA in X - Axis

In Figure 2, it can be seen that the DOAs are estimated accurately by the proposed method with very small error Figure 3 presents the performance of the proposed method in the AWGN channel with the variable SNRs from −10dB to 20dB The simulation result also indicates that the proposed method works well in white noise environment although with one snapshot Figure 4 illustrates the DOA estimation accuracy of the TFBMP and the original MP method with the above incoming signals From this figure, it is easy to see that TFBMP is more efficient than MP Obviously, DOA estimation with only one snapshot is the remarkable advantage in comparison with other super high resolution methods In similar situation, the DOA information could not be estimated by the MUSIC algorithm as shown in Fig.5 The fact is due to the inaccurate estimate of the correlation matrix

Fig 4 DOA estimation accuracy of

TFBMP and MP

Fig.5 DOA estimation using MUSIC

with one snapshot

The angle resolution of this method is also a factor to be assessed In order to do that this algorithm is performed to estimate the DOA of some pairs of incoming wideband signals whose SNRs are set to 10dB with one snapshot The simulation results are stored in Table 2 If we consider the RMSE of desired results less than 1 degrees, from Table 2, it can be seen that this method will work well with the resolution approximately 3 degrees This result is similar to the MP However, the

resolution of this method is less than other super high resolution algorithms such as

MUSIC Obviously, this is the drawback of the method

Table 2 Angle resolution for wideband signals

Assumed DOA (Degrees) Estimated DOA (Degrees) RMSE (Degrees)

Moreover, in order to evaluate the influence of the number of snapshot on the

performance of this algorithm, this method is executed with the same inputs as

above with 1000 snapshots The simulation results plotted in Figure 6 and Figure 7

show that when we increase the number of snapshots, the accuracy of this method

increases significantly However, it can be observed that when the number of

snapshots is more than 300, the accuracy of the algorithm seems to be invariable

On the other hand, when the number of snapshot is increased, the computation time

also significantly increases Therefore, the trade – off between the computation

time and the accuracy of the algorithm should be taken into account

Fig 6 DOA estimation with 1000

snapshots

Fig 7 Estimation accuracy of DOA

estimation with the different number of

snapshot

5 CONCLUSIONS

We propose an efficient method for DOA estimation of wideband signals by

using Total Forward – Backward Matrix Pencil algorithm The remarkable

advantage of the proposed method is that it can extract accurately the DOA

information with only one snapshot This fact reduces significantly the

computation time in comparison with other high resolution algorithms as well as

the sampling frequency and the size of buffer in real time receivers Therefore, the

proposed method is one of the practical ways to estimate the DOAs information of wideband signals in real application

Acknowledgment: This research is carried out in the framework of the project

titled “Design and Implementation of an Earth Station based on Software Defined Radio in the satellite communication system” in the national program on space technology under the grant number VT/CN-02/14-15 This research is funded by the Vietnam Academy of Science and Technology The authors would like to thank the Vietnam Academy of Science and Technology and the Ministry of Science and Technology, Vietnam for their financial support

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Nhận bài ngày 21 tháng 07 năm 2015 Hoàn thiện ngày 10 tháng 08 năm 2015 Chấp nhận đăng ngày 07 tháng 09 năm 2015

Address: 1 School of Electronics and Telecommunications,

Hanoi University of Science and Technology, Vietnam.

*

Email: Thanh.hantrong@hust.edu.vn ;