Direction of Arrival Estimation in the presence of Scatterer in noisy environment

We p resent a n a lgorithm to e stimate d irection o f a rrival (DOA) of an incoming wave received at an array antenna in the scenario where the incoming wave is contaminated by the additive white Gaussian noise and scattered by arbitrary shaped 3 D s catterer(s). We p resent d ifferent s imulation examples to show the validity of the proposed method. It is observed that the proposed algorithm is capable of closely estimating the DOA of an incoming wave irrespective of the shape o f t he s catterer p rovided t he d ecision i s made o ver multiple iterations. Moreover, presence of noise affects the estimate especially in the case of low signal-to-noise ratio (SNR) that gi ves a r elatively l arge e stimation er ror. However, for larger SNR the DOA estimation is primarily dependent on the scatterer only.


Introduction
Array signal processing emerged in the last few decades as an active area of research. It is an important area in the field of signal processing, which uses antenna array to detect the useful signals while rejecting the interference and noise [1]. Direction-of-arrival e stimation ( DOA) p lays a n i mportant role i n ar ray s ignal p rocessing. T he main p urpose o f t he DOA a lgorithm is to e stimate th e d irection o f in coming signals while r estraining t he i nterference an d n oise. The accuracy of the estimate depends on the number of received signal samples. T he benefit of using an array antenna is to enhance the resolution of multiple signals DOAs and has a better performance in s ignal detection and estimation t han using a s ingle an tenna [ 2]. DOA es timation h as s everal potential ap plications such as s earch a nd r escue, l aw enforcement and wireless emergency call locating etc. DOA estimation h as c onsiderable a ttention in wireless communication, r adar s ystem o f co mmercial a nd military application and sonar system. The prime advantage of using DOA estimation algorithm is to improve the performance of an a ntenna b y c ontrolling t he directivity o f a ntenna to reduce t he e ffects l ike i nterference, d elay s pread an d multipath fading [ 3]. D OA e stimation is a lso u sed t o increase t he cap acity a nd t hroughput o f a network i n wireless communication [4].
Several D OA e stimation a lgorithms o f n arrowband signals are presented in the literature targeting the problem of DOA estimation i n t he p resence o f e ither noise [5][6][7][8] o r in the presence of scatterer [9][10][11][12]. For the case of noise the DOA e stimation is a chieved b y d irectly a pplying t he algorithm o n Uniform Linear Array without pre-processing techniques such as forward-backward averaging of the cross correlation o f a rray o utput d ata o r s patial s moothing. F or the cas e o f scatterer, s pherical h armonics are u sed t o remove the effects of scattered field. It has better realization of scattered field because the number of harmonics used is less a nd it also r educes t he number o f a ntenna elements in comparison of using cylindrical harmonics [10].
In t his pa per we a ddress t he problem of e stimating t he DOA in the situation where Additive White Gaussian Noise and 3D near zone scatterer are simultaneously present. The noise is independent of a signal and present at each antenna elements.

DOA Estimation
DOA estimation is a process for determining the signal of interest while rejecting the signal not o f interest [13] using antenna ar rays [ 14]. T he p resence o f s cattered f ield an d noise in the received signals generate unintended copies of the s ignal t hat n eed t o b e rejected. W e b egin with t he description of the considered environment and then present the proposed solution.

Environment Description
Consider u niform l inear a rray (U LA) g eometry with N identical x-directed dipole elements numbered from 1 t o N as s hown i n F ig. 1 . T he ar ray el ements have a uniform spacing ' d' between them. P lane wave ar e u sed b ecause source o f i ncident wave i s l ocated s ufficiently far a way from the antenna elements [15]. Consider TM plane wave incident o n a ntenna a rray i n x direction. Near f ield scatterers are also present, whose locations are known b ut geometries a re unk nown a s shown i n Fig. 1. The p lane wave i s s cattered b y t he n ear-zone s catterer at l ocation = ( , , ) producing spherical waves/ harmonics. The plane wave and the scattered waves are incident on the ℎ antenna element located at = ( , , ) . Therefore plane wave f rom f ar field region is desired s ignal a nd spherical waves d ue t o n ear zo ne s catterer f ield are interfering s ignals. T he to tal e lectric f ield a t ℎ antenna element is given by An antenna ar ray ca n be designed t o es timate t he direction of incoming signals based on samples of received signals. The accuracy of estimation method depends on the number of received signal samples K. It is also assumed that antenna an d en vironment i s s tationary d uring K number of samples. The receiver is capable of measuring total voltage at ℎ antenna terminal that can be expressed as Where, is th e v oltage d ue to in cident f ield ( ) at ℎ terminal, is the voltage due to scattered field ar ising from n ear zo ne s catterer ( ) , is th e voltage at the ℎ antenna terminal. If is additive white Gaussian noise (AWGN), the output of receiver can be expressed as:

Classical Method
Classical method for d irection o f a rrival ( DOA) e stimation is based on the concept of beam forming. A commonly used classical method is D elay-and-sum method [ 16,17]. An array can steer beams through space and measure the output power. T he di rection from which maximum a mount o f power is o btained yields direction o f a rrival ( DOA) estimation [ 18,19]. F ig. 2, s hows t hat t he o utput s ignals ( [ ]) is c omputed by us ing l inear w eights ( ) combined with received data ( ). The received data can be expressed as: Where, [ ] is the k-th r eceived s ample for to tal L incident waves, [ ] is the -th incident wave, ( ) is a column of array manifold matrix relating the -th incident wave to the receiver terminal, and [ ] represents sample form AWGN. For known number of signal samples K, covariance matrix (R uu ) can be expressed as where [•] represents expectation operator. In this case, the t otal o utput po wer of de lay a nd s um method c an be expressed as: In classical beam forming, the signal power is measured over a ngular r egion o f i nterest b y setting b eam forming weights equal to s teering weights = ( ) corresponding to the particular direction. The output power is obtained as a function of angle of arrival as [20].
The direction of arrival of the incident wave is taken as the di rection c orresponding t o t he maximum r eceived power.

Proposed Solution
The total voltage at the output of the receiver is measured or known. In the absence of noise and scatterer the received signal is same as . However, actually the signal is co rrupted b y n oise a nd s cattered field. To r emove the effect o f AWGN f rom t he t otal r eceiver v oltage , we assumed that the total voltages received by incident field and scatterer field can be expressed as This total voltage is known or measured as mention earlier. T he n oise at eac h an tenna terminal i s i ndependent from s napshot to s napshot a nd it is u ncorrelated. B ut th e signal r emains s ame d uring each s napshot. T he o utput o f signal is given as Where, is the received output signal which is corrupted by Noise V. DOA estimation method uses sampled version of array output at k-th snapshot (k = 1, 2, …, K) is given by [21].
The key factor for this e valuation is Signal to Noise (SNR) of the environment surrounding the antenna arrays and incident sources, while the numbers of snapshots (K) is kept constant.
Next step is to remove the effect of scattering b y using spherical harmonics. I t i s assumed t hat s catterers ar e exterior to array elements. It is to be noted that the incorrect assumption of letting = not only causes errors in DOA estimate but may also give rise to false peaks in DOA spectrum. The linear equation for an array of N elements is given in [22].
Classical D OA es timation t echniques ar e ap plied f or ( ) number of sources an d e stimate t heir el evation = The in cident v oltage is used to f ind th e e levation o f desired incident sources and as iterative index is increases and algorithm is repeated until plot of convergence of DOA estimation is achieved.

Numerical Examples and Results
The electromagnetic simulations are carried ou t by u sing COMSOL multiphysics e nvironment. I n t he c onsidered scenarios we assumed (x-directed) horizontal half wave dipoles an tenna el ements o f a u niform l inear ar ray. T he radius of half wave di pole i s = 0.001 . T he ope rating frequency is 2.4GHz. The first element center is (0,0,0) and its axis is along z direction as shown in Fig. 1. Two to three spherical h armonics will b e s ufficient to r epresent t he field due t o s catterer. Here we as sumed t hat a ntenna a nd environment is s tationary d uring a single s ample. In r eal environment 3 D s catterer can b e ap proximated t o a s phere, therefore s pherical harmonics i s u sed t o r emove t he e ffects of scattered field. It has better realization of scattered field because t he n umber o f h armonics u sed i s l ess an d i t al so reduces t he n umber o f an tenna el ements i n co mparison o f using cylindrical h armonics. I n classical m ethod, w hen the amplitude of D OA a ngle e quals or e xceeds t o 4 0% of t he maximum a mplitude in s pectrum the i ncident s ource i s detected.

Case 1: Single Scatterer, Single Wave
The assumed geometry for case 1 i s s hown i n F ig. 3, h ere number o f s catterer S= 1, s catterer in th e f orm o f e llipsoid (semi axis a= 0.5λ, b=0.5λ and c= 0.8λ) and is located at (0.1,-0.6,3)λ. The number o f ar ray el ements N= 10, t he spacing between the elements is d= 0.5λ. The incident wave L= 1 a nd th e e levation o f in cident wave is = 75°. Gaussian noise i s added at each a ntenna element a nd it is assumed that noise is complex and uncorrelated.  The effect of noise in the DOA estimation is elaborated in Fig. 6 that shows the plot of DOA estimation with respect to SNR. Relatively large estimation error is observed in the low SNR regime. As SNR increases the error tend to reduce and eventually the DOA estimation is converged to the case of scatterer only (i.e. no noise). The result is quite intuitive as in case of low SNR the effect of noise is dominating the signal thereby p roducing larger er ror. As SNR is increased the ef fect o f noise r educes in co mparison t o t he s ignal a nd the decision is mainly depend on the scattering effect.

Case 2(a): Single Scatterer, Two Waves
In case 2, we use two different scatterer geometries (a) cube (b) e llipsoid with a pproximately s ame size a nd s ame location to show that the proposed algorithm is applicable to any 3D geometry. The simulation environment for case 2(a) is s hown in Fig. 7

Case 2(b): Single Scatterer, Two Waves
The case 2(a) is r epeated with different shape of scatterer. Here the scatterer is in t he form o f e llipsoid (semi a xis a= 0.5λ, b=0.5λ and c= 0.8λ) and it is located at same location as in previous case at (0.2,-0.6, 2.5) λ as shown in Fig. 11. Here the number of incident wave L=2 and the elevation of incident w ave i s 1 = 80° and 2 = 120°. The number o f array elements N=10 and the spacing between the element is d= 0.5λ. Here Gaussian noise is added and it is assumed that noise is uncorrelated.

Case 3: Two Scatterers, Single Wave
The setup o f c ase 3 i s s hown i n F ig. 15. This cas e i s more complex because the number of array elements are increased to N= 20 and the spacing between the element is d= 0.25λ. Here number of scatterer S=2. Both scatterer are in the form of sphere (radius = 0.5 λ) and are located at (−0.2,−0.6, 4) λ and ( −0.2,−0.6, 1) λ. There i s o ne incident wave (L=1) and the elevation of incident wave is 1 = 95°. Gaussian noise is added and it is assumed that noise is uncorrelated. The convergence of decided DOA is ( ) to ( ) = 95.1° as s hown i n F ig. 1 7 using Q = 2 S pherical h armonics. The Fig. 18 shows the plot of decided DOA with respect to SNR. In t his case when t he s catterer i s p resent, d ecided D OA is detected at 95.1°.But in the presence of noise and scatterer, the algorithm gives the same results in high SNR.

Case 4: Two Scatterers, Two Waves
The geometry of case 4 is shown in Fig. 19. Here number of scatterer S=2. One scatterer is in the form of ellipsoid (semi axis a=0.4λ, b=0.5λ and c=0.7λ) and is located at (0.1,-0.6, 4) λ. Another scatterer is in the form of sphere (radius= 0.5 λ) and is located at (-0.1,-0.6, 1.5) λ. The number of array elements N= 20 a nd s pacing b etween t he el ement i s d= 0.25λ. The incident wave L= 2 and the elevation of incident wave i s 1 = 65° and 2 = 120°. Gaussian N oise i s a dded and it is assumed that noise is uncorrelated.   Fig. 22 s hows t he plot of decided DOA with respect to SNR. In this case when the scatterer is present, the first decided DOA is detected at 64.2° and the second decided DOA is detected at 119.7°. But i n t he p resence o f noise an d s catterer, t he al gorithm gives the same results at high SNR for both decided DOAs.

Conclusions
An iterative algorithm for D OA e stimation is p resented in the cas e where A dditive W hite G aussian N oise ( AWGN) and 3D scatterer(s) are simultaneously present. Although all the s imulations a re p erformed with th e c ubic, s pherical, or ellipsoidal scatterer, the algorithm imposes no condition on the s hape o f t he s catterer. H owever t he l ocation o f t he scatterer must b e k nown. T he co nvergence o f D OA i s achieved iteratively an d al gorithm i s r epeated until t he correct (converged) DOA i s a chieved. A num ber of numerical e xperiments were co nducted where multiple incident sources and multiple scatterers ar e present. W here noise i s a ssumed t o b e i ndependent an d p resent at each antenna terminal. It is also assumed that signal remain same at each s ample. S NR d irectly a ffect t he p erformance o f DOA e stimation especially i n t he l ow S NR regime. I t i s observed that the algorithm is capable of closely estimating the DOA in the presence of noise and scatterers.