HongqiYu,2,*,David Day-UeiLi2,Kun Zhang3,Jietao Diao,and Haijun Liu

1.Schoolof Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China;

2.Centre for Biophotonics,Strathclyde Institute of Pharmacy&BiomedicalSciences,University of Strathclyde,Glasgow G4 0RE,UK;

3.College of Mechanical and Electrical Engineering,Henan Agricultural University,Zhengzhou 450002,China

Wideband direction-of-arrivalestimation based on cubic spline function

HongqiYu1,2,*,David Day-UeiLi2,Kun Zhang3,Jietao Diao1,and Haijun Liu1

1.Schoolof Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China;

2.Centre for Biophotonics,Strathclyde Institute of Pharmacy&BiomedicalSciences,University of Strathclyde,Glasgow G4 0RE,UK;

3.College of Mechanical and Electrical Engineering,Henan Agricultural University,Zhengzhou 450002,China

A new direction-of-arrival(DOA)estimation algorithm for wideband sources is introduced.The new method obtains the output of the virtual arrays in the signal bandwidth using cubic spline function interpolation techniques.The narrowband highresolution algorithm is then used to get the DOA estimation.This technique does not require any preliminary knowledge of DOA angles.Simulation results demonstrate the effectiveness of the method.

array signal processing,wideband sources,cubic spline function,direction-of-arrival(DOA)estimation.

1.Introduction

The direction-of-arrival(DOA)estimation is widely used in reconnaissance and communications systems[1-4].At fi rst,narrowband subspace algorithms are widely studied. Based on the eigenvalue decomposition techniques[5-7],it is easy to obtain the signal and noise subspaces. The main idea of subspace based methods for DOA estimation is using the orthogonality of the two subspaces [8,9].However,narrowband methods are not suitable for wideband signals,the reason is that for wideband signals the phase difference not only depends on the DOA but also on the frequency[10].When the signal is wideband,the frequency is a relatively wide domain.The variations of the phase difference with the frequency cannot be neglected.At some special conditions,such as easily resolved sources,large number of arrays,and high signalto-noise ratio(SNR),the DOA can be measured by a certain frequency in the domain.Nevertheless atmostcircumstances,all frequency bandwidth should be considered to getthe necessary high processing gain.The study of wideband DOA estimation algorithms is,therefore,ofgreatimportance.

One intuitive way of processing a wideband signal is dividing the signal into several narrowband ones through discrete Fourier transform(DFT)or non-overlapping narrowband bandpass fi lters[11].Then the narrowband signals are processed by narrowband DOA estimation methods to getthe fi nalresults.One research pointof wideband DOA estimations is combining the covariance matrix of these narrowband signals to get a better performance instead of simply using one or more certain frequencies.The incoherentsignal-subspace method(ISSM)[12]is the simplest method to process wideband array signals.The outputof the array is passed through a number of narrowband fi lters to get a group of narrowband signals.The narrowband estimation algorithm is used to obtain the DOA of each narrowband signal,and all the estimations are averaged to obtain the fi nalresults.The testof orthogonality of projected subspaces(TOPS)method[13,14]constructs a new matrix based on the orthogonality of the signal noise subspaces.The DOAs are obtained by searching the degree of the rand de fi ciency of the new matrix.The ISSM and TOPS methods are ef fi cient when the SNR is high. However,when the SNR is low,the performance is poor. To lower the threshold of the SNR,the coherent signalsubspace method(CSSM)[15]was proposed.The CSSM focused the manifold matrices of pre-estimation DOAs at every frequency point to the reference frequency.It has been shown that CSSM does not produce consistent estimations[16].The bias increases as the bandwidth increases.Based on unitary matrices,the rotational signalsubspace(RSS)method was proposed[16],which is a special case of the signal subspace transformation(SST) method[17].To further decrease the focusing fi tting error, Valaee and Kabalproposed two-sided correlation transformation(TCT)method[18].CSSM,RSS,SST and TCT require pre-estimation of DOA to constructthe focusing matrix.The error of the pre-estimation DOA can signi fi cantly affectthe performance.Recently,Mahata proposed a new low-rank signal model[19].The new model has no rela-tions with the narrowband signal model.In this way,the DOAcan be estimated withoutresorting to narrowband fi ltering.Zeng and Liproposed a new DOA estimation algorithm via short-time Fourier transform for wideband nonstationary sources[20].Other researchers used the sparse reconstruction techniques for wideband signal DOA estimations[21,22].

In this paper,we introduce a simple wideband DOA estimation algorithm.The method constructs a new virtual array atevery frequency point.The manifolds of each virtualarray are independentofthe frequency by adjusting the position of the elements.The outputof the virtualarray is obtained by using cubic spline data interpolation.Then,the narrowband algorithm can be used to obtain the wideband DOA estimation.The proposed method does not require any pre-estimation of the DOA,and therefore it provides the more robust performance than the methods mentioned above.

2.Theory

2.1 Signalmodel

As shown in Fig.1,suppose the array has M isotropic elements,the spacing between the two adjacentelements is d,and the number of wideband signalis P.

Fig.1 Uniform linear array diagram

The outputof theμth elementis

For uniform linear array(ULA),vm=(m?1)d/c.And nm(t)is the Gaussian white noise of the m th element.

The DFT ofthe outputof the m th elementis as follows:

For mostsubspace based wideband DOA estimation methods,the array outputis divided into many narrowband signals by using fi lters or DFT as follows:

Equation(1)is the signalmodelof the wideband array.

2.2 Wideband focusing method

As the rank of A(fj)(j=1,...,J)is P for any j orθk(k=1,...,P),we can fi nd an M×M nonsingularmatrix T(fj)(j=1,...,J),which can satisfy

where T(fj)is called the focusing matrix.T(fj)can be used to transform the outputof the array X(fj)as

The weighted summation ofthe spectrum density spectrum is

Wjcan be used to normalize the SNR.For simplicity,let Wj=1.

When allthe DOAs are in the same beamwidth,a simple and effective method introduced in[23]can be used to constructthe focusing matrix as follows:

where ai(θ,fj)is the i th element of a(θ,fj)andθis the pre-estimation DOA.

Hung and Kaveh showed that when the focusing matrix is a unitary matrix,there is no loss after focusing[24]. They used the following equation to constructthe focusing matrix

where‖·‖F(xiàn)is 2-norm.The RSS focusing matrix thatsatis fi es the above equation is given as

where U and V are the left and right singular vectors of AH(fj,θ)A(f0,θ).

Doron and Weiss testi fi ed thatthe following SST focusing matrix can also have no focusing loss[25]

Valaee and Kabalproposed the TCT method[26].They used the following equation to construct the focusing matrix.

where P(fj)is the signal covariance matrix at the frequency fj.

One solution to(3)is given as

where X0and Xjare the eigenvectors of P(f0)and P(fj),respectively.

2.3 The proposed method

When the array is ULA,A(fj,θ)can be expressed as

Suppose the reference frequency is f0,the corresponding manifold is

If we replace d with df0/fj,then(5)is the same as(6). Thus,we can eliminate the difference in the manifold caused by the frequency.Suppose the corresponding distance between two adjacent elements is djfor the frequency fj,the realdistance d is corresponding to the reference frequency f0.The manifold is a function of the position of the elements.

The snapshotof the array at the reference frequency f0is

The snapshotof the array atthe frequency fjis

For a virtual array at the frequency fjwith the distance between two adjacent elements being dj=df0/fj,the snapshotof the array is

and the covariance matrix is

All the covariance matrix of the entire virtualarray can be added as

then we can apply the following theorem.

Theorem 1Letλiand ei(i=1,2,...,M)be the eigenvalues and eigenvectors of the matrix pencil (RU,RN),andλiare in descending order,then the column span of En=[ep+1,ep+2,...,eM]is orthogonalto the column span of{A(f0)},which means A(f0)HEn= 0.The proof of this theorem is similar to thatof[23].

We can estimate the covariance matrix as follows[26]:

According to(6),we can multiply RS(fj)with the manifold of the virtual array to obtain the covariance matrix of the virtual array.Based on(7),RUand RNcan be calculated,and the noise subspace or the signalsubspace can be obtained.The narrowband algorithms can then be usedto obtain the wideband DOA estimation.For example,the narrowband multiple signal classi fi cation(MUSIC)algorithm can be used with the spatialspectrum is

Searching the peaks of the spatialspectrum,we can obtain the DOA estimations.

The performance of this focusing algorithm is excellent,but it requires pre-estimation of the DOA to obtain the manifold in(10)and(13).To avoid this shortcoming, this paper uses the cubic spline function to obtain the covariance matrix of the virtualarray.

Based on(8)and the position of the elements,we can fi nd the cubic spline function that fi ts them.According to (9),the covariance matrix of the virtualarray atevery frequency point can be obtained.RUand RNcan be calculated accordingly.Based on Theorem 1,the signalsubspace and noise subspaces can be found.Then,the narrowband DOA estimation algorithms can be used to obtain the fi nalestimation.

In summary,the calculate process of the proposed method is described as follows.

Step 1DFT the array output;

Step 2Based on the element position and sampling date,get the cubic spline interpolation polynomial.The spline()function in Matlab can be used;

Step 3Calculate the output of the virtual array using(9)by using cubic spline interpolation polynomials and setting the intervalof the virtualarray to df0/fj;

Step 4Add allthe outputs of the virtualarray based on (11)and obtain RUand RN;

Step 5Getthe noise subspace?G;

Step 6Getthe spatial spectrum based on(14),search for the peak,and obtain the DOA estimation.

3.Simulation

3.1 DOAs in a beamwidth

Suppose the uniform linear array has 15 omnidirectional elements,the distance between two adjacent elements is the half wavelength of the mid-band frequency.The sources are temporally zero-mean bandpass white Gaussian processes.The central frequency of the sources is 100 Hz.The bandwidth of the sources is 40 Hz.The noise is independent zero-mean bandpass white Gaussian process.Suppose the DOAs are 3°and 0°.The total observation time T0is 51.2 s.T0is divided into 64 segments. Suppose the pre-estimation angle is 2,the focusing angles are-1°,2°and 5°for the TCT method.The spatial spectrum of the TCT method is shown in Fig.2.When the preestimation angle is 6°,and the focusing angles are 3°,6°and 9°,the spatial spectrum of the TCT method is shown in Fig.3.Letthe rightestsensoras the reference sensor,the lowest frequency as the reference frequency,the times of Monte-Carlo is 5,and the spatialspectrum of the proposed method is shown in Fig.4.From these fi gures,we can see thatpeaks of the proposed method are more evident.

Fig.2 TCT spatial spectrum with focusing angles being-1°,2°and 5°

Fig.3 TCT spatial spectrum with focusing angles being 3°,6°and 9°

Fig.4 Spatialspectrum of the proposed method when DOAs are 0°and 3°

3.2 DOAs allin multi-beamwidth

The DOAs are 0°,3°,33°and 40°.The spatial spectrum of the proposed method is shown in Fig.5.Simulations show that the proposed method can correctly identify all the DOAs existing in differentbandwidth.

4.Conclusions

The proposed method fi rst obtains the distance between two adjacent elements of the virtual array according to the frequency.The output of every virtual array is independentof the frequencies.The outputof the virtualarray is obtained by cubic spline interpolation polynomial.Then the narrowband DOA algorithms can be used to obtain the wideband DOA estimation.Simulations demonstrate the effectiveness of the method.The proposed method does notrequire any pre-estimation of the DOAs,and is therefore free from pre-estimation errors.Simulations also show that the spectrum peaks of the new method are more evidentthan the TCT method.

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Biographies

HongqiYuwas born in 1978.He received his B.S., M.S.,and Ph.D.degrees from National University of Defense Technology(NUDT)respectively in 2000,2003 and 2007.Currently,he is an associate professor with the Department of Electrical Science and Technology,in the School of Electrical Science and Engineering,NUDT.Now,he is an academic visitor at University of Strathclyde.His research interests are array signalprocessing and embedded system. E-mail:yhq@nudt.edu.cn

David Day-Uei Lireceived his Ph.D.degree in optical waveguide devices from National Taiwan University in 2001.In 2013,he joined the Centre for Biophotonics,Strathclyde Institute of Pharmacy &Biomedical Sciences,University of Strathclyde as a lecturer.His current research interests include mixed-signal integrated circuits design and CMOS SPAD-based FLIMcameras.

E-mail:david.li@strath.ac.uk

Kun Zhangwas born in 1994.From 2012,she became a bachelor majored at electrical information and engineering.She has published a book on MCU design at 2015.Her current research interests include embedded system design and signal processing.

E-mail:zhangkun zkn@163.com

Jietao Diaowas born in 1965.He received his B.S.and M.S.degree from National University of Defense Technology(NUDT)respectively in 1988 and 1996.Currently,he is a professor with the Department of Electrical Science and Technology,in the School of Electrical Science and Engineering, NUDT.His main research interests include embedded system,circuitand system.

E-mail:diaojietao@vip.sina.com

Haijun Liureceived his B.S.degree in electrical information engineering from Shandong University of Technology,in 2004,and his Ph.D.degree in information and telecommunication systems from National University of Defense Technology(NUDT), in 2010.Currently,he is a lecturer with the Department of Electrical Science and Technology,in the School of Electrical Science and Engineering, NUDT.His main research interests include theory study of memristor and memristive system,and applications based on two-terminal resistive switches.

E-mail:liuhaijun@nudt.edu.cn

10.1109/JSEE.2015.00076

Manuscriptreceived December 20,2014.

*Corresponding author.