Uniform Non-Exhaustive Search on Sparse Reconstruction for Direction of Arrival Estimation

  

Uniform Non-Exhaustive Search on Sparse

Reconstruction for Direction of Arrival Estimation

Koredianto Usman

  I. I

  By using this approach, angle sparsity scheme has ad- vantage over time sparsity and angle sparsity, which are lesser samples. In addition, angle sparsity performs the CS construction and DoA estimation in single step. This technique was introduced by Gorotnitsky and Rao [12]. In the paper, they showed that one sample is sufficient to perform DoA estimation.

  A · s = x (1)

  Angle sparsity has a difference approach as the previous schemes. It does not compressed the received signal in time or space direction. However it assumes that the received signal is coming from a limited number of directions. Based on this, CS construction is performed using sensing matrix A taken from a set of antenna steering vectors. The CS formulation is composed from sensing matrix A, a sparse matrix s, and snapshots of received signal x. This CS construction can be represented by Eq.1.

  Space sparsity scheme is basically similar to the time sparsity. It is based on assumption that the signal received by each sensors (i.e. antennas) are similar to each others, thus can be assumed sparse as well. Signal reduction is performed by selecting sensing matrix A that reduces dimension on sensors direction (i.e. a k times N matrix). The signal needs to be reconstructed before DoA estimation is performed. Compared to time sparsity, spase sparsity has less compression capability. Space sparsity, however, has advantage on robustness against noise. In addition, the space sparsty can be implemented directly by modifying hardware at the receiver side ([10] and [11]). Thus, no extra computation is needed.

  Time sparsity scheme based on an assumption that trans- mitted signal is sparse in time. The signal x is collected by M antennas of N time snapshots to produce a block of M times N size of input signal. Using block processing, each block of received signal of size is pre-multiplied by sensing matrix A (a M times k matrix, with k << N) to produce a smaller set of sensed signal. In the DoA estimation side, the compressed then DoA estimation is performed upon the reconstructed signal.

  In the radar area, especially in direction of arrival (DoA) estimation of incoming object, various CS schemes had been applied as alternative of classical DoA estimation algorithms, such as MVDR [5], MUSIC [6], and ESPRIT [7]. As far as CS is concerned, there are three main CS schemes that applied to the DoA estimation, namely: time sparsity ([8], [9]), space sparsity ([10], [11]), and angle sparsity ([12], [13]).

  The development of compressive sensing (CS) has attracted considerable attention in recent decade. CS gives an alternative to classical Shannon-Nyquist sampling criteria for perfect construction. It was shown that sparse signal can be sampled at much lower sampling rate that the minimum Shannon-Nyquist sampling theorem. Sparse signal is defined as signal with few non-zero values while majority of the samples have zero values. Pioneers work in the field of CS are, among others, Donoho [1], Candes and Wakin [2], and Baraniuk [3]. With its advantage on small amount of samples, CS has been applied in various engineering applications, such as wireless sensors network, channel estimation, biomedical engineering, sonar, and radar applications ([4]).

  NTRODUCTION

  Abstract—Direction of arrival estimation (DoA) using sparse reconstruction gives advantage on minimizing the number of re- quired samples. Among available sparse reconstruction schemes, angle sparsity has shown a favorable advantage as it requires fewer samples compare to other schemes. Previous researches on angle sparsity utilized an exhaustive scanning on every possible arrival angles. This technique leads to a problem of large sensing matrix A. The result presented in this paper proves that partial scanning (i.e. non-exhaustive search) also gives similar accurate result. The advantage of this scheme is smaller sensing matrix. In simulation, this scheme requires sensing matrix six times less than the exhaustive search with similar accuracy. Thus, this scheme is potential for practical application of DoA estimation based on sparse reconstruction.

  Faculty of Electical Engineering Telkom University;

  Email : hgunawan@math.itb.ac.id

  Faculty of Mathematics and Science Institut Teknologi Bandung

  Hendra Gunawan

  Email: suksmono@stei.itb.ac.id

  School of Electical and Informatics Institut Teknologi Bandung

  Andriyan Bayu Suksmono

  Email: korediantousman@telkomuniversity.ac.id

  School of Electical and Informatics Institut Teknologi Bandung

  Given this advantage, angle sparsity is, however, less robust to the noise. Usman et al. [5] verified this condition and proposed improvement by using multi-snaps signal. In this multi-snaps scheme, the estimated DoA is determined by averaging over each samples. Stoica et al. [13] independently proposed covariance-based estimation technique (SPICE) in angle sparsity DoA. Their scheme also accommodates multi- snaps samples to improve robustness. As researchers mitigated the noise problem, angle sparsity has another weakness, which is large sensing matrix A. Usman et al. [14] and Stoica et al. C. CS solver by convex programming There are several CS solver to solve Eq.7. Some of these solvers are, for examples, cvx-programming and l ¹ -

  < ǫ (7)

  1p a

  (5)

  B. Compressive Sensing Formulation The angle sparsity can be represented by Eq.1. Fig.?? clarifies the scheme into a detail structure.

  a

  11 a

  21 a M

  1 a

  12 a

  22 a M

  2 a

  2p a _{M p} A s

  1 e ^−jψ e ^{−j(M −1)ψ}

  1 s

  2 s

  3 s p x 1i x 2i x

  3i x M i a (θ ₁ ) a(θ ₂ ) a (θ _M ) steering vector at angle θ i

  Fig. 2. The CS construction of angle sparsity DoA

  The sparse reconstruction is achieved by solving Eq.1 for sparse matrix s. Solving the equation for s is on the same time solving the DoA estimation problem, since location of non-zero element of s indicates the angle of arrival ([12]). However, since the number of equations is less then the number of unknowns, then Eq.1 leads to an ill-posed matrix problem. In other word, there are many set of s that fulfilled 1.

  Back to the assumption the received signal is sparse in angle of arrival (i.e. in steering vector a), then the optimal solution is obtained by selecting the solution that minimize l-norm of s ([2]). For sparse reconstruction, the appropriate norm is either l or l ¹ -norm. Norm l ¹ is normally chosen for computation simplicity ([15]). The formal problem of CS reconstruction is, therefore, described by Eq.6 min ksk

  1

  subject to A · s = x.

  (6) In the case of noisy environment, which is common in telecommunication, the formulation of CS problem can be generalized by minimizing l ¹ -norm on s, while constraining the l ² -norm of difference between the estimation and the actual signal less than a certain threshold (ǫ). This formulation is written in Eq.7. This equation is called l ¹ − l ² optimization problem (detailed discussion in [4] and references therein). min ksk ₁ subject to kA · s − xk ₂

  T

  In Eq.4, s denotes the received signal at antennas’ input (p times N snaps matrix; p is number of objects), x denotes the received signal after antennas (M times N snaps matrix), n is white gaussian noise, and a is array steering vector or array manifold. The steering vector a is expressed as a =

  Assuming angle scanning resolution as 0.5 , this produces sensing matrix A of size M times 360. If the number of antennas M is large, then this leads to CS reconstruction problem deal with large size matrix. Computation complexity increases correspondingly.

  ON

  This paper proposes a solution to the exhaustive scanning problem by using a non-exhaustive scanning technique. The technique in principle is to reduce the scanning window into a narrower range. A pre-scanning technique is used to initially located the objects. After location of objects are pre-located, the next scanning is performed on narrow range around these objects. In the case of moving objects, the scheme can easily adapt the position by updating the searching window. We also proposed a tail scanning to further improve the accuracy of this non-exhaustive technique in case of a very narrow range of scanning window.

  The presentation in this paper is arranged as follow: Section

  II discuss mathematical model and detail of the proposed scheme, Section III shows the simulation result and discussion, and finally section IV concludes this paper.

  II. N

• EXHAUSTIVE SEARCH

  Fig. 1. Antennas arrangement in ULA with distance d between element

  d d θ l

  · d · sin(θ) (3)

  2π λ

  ψ =

  ∆ = d · sin(θ). (2) This amount of distance corresponding to phase delay of

  Assuming that the source is located at distance much larger than the size of the array, then the beam of signals arrived at each antennas can be considered parallel to each others. The difference of traveled distance of neighboring antenna is given by

  A. Mathematical Model We consider the set of M antennas arranged linearly with constant distance betwen antennas. This arrangement is called uniform linear array (ULA). Assumed that a source signal coming at angle θ relative to a reference line (Fig.1).

  1 l

  M (t)

  2 l

  3 l

  M ∆ 2∆

  Collecting each received signal at each antennas, and write it in matrix form, we obtain x = a

  1 (t) x

  2 (t) x

  3 (t) x

  (M − 1)∆ R x while l ^{1 -magic is developed by Candes and Romberg. CVX-} programming is more generic to solve various convex program- ming problem. Thus, it is flexible. As a convex programming tool, cvx-programming utilize some solver engine such as SDPT3 and SeDumi. To solve the CS problem as depicted in Eq.7, in CVX we write, for example: begin_cvx variable s(n) complex; minimize(norm(s,1)); subject to norm(A*s-x,2) < epsilon; end_cvx .

  D. Uniform non-exhaustive searching In order to reduce the computation complexity, we propose a uniform non-exhaustive searching on the DoA. At the beginning of operation, classical method such as MVDR is used to coarsely estimate the DoA. After the coarse location is determined, the scanning window is narrowed at next iteration. The center of scanning window is at the coarse DoA. The steering vectors of narrowed scanning window is updated as sensing matrix A. As the location of object is changed from time to time, it is also important to update the scanning window. The DoA of object is directly obtained from non- zero element of vector s in each iteration.

  IMULATION

  1 n · ^{n X} _i=1 θ ai − ˆ θ i (10)

  M AE =

  90 ), and then non-exhaustive search by reducing scanning window from −30 to 90 and to 90 . A actual position of object is used as reference. As performance parameter, we use mean absolute error (MAE) of each scheme. MAE is defined as:

  After coarse DoA is determined, the non-exhaustive al- gorithm is performed by limited the scanned signal around this maximum power spectrum. We performed simulation on exhaustive scanning (scanning window −90 to

  Here P (θ) is DoA power spectrum, a(θ) is steering vector at angle θ, and R ^{− 1} _xx is inverse of covariance matrix of received signal x. Coarse DoA is taken at an angle whose DoA power spectrum is maximum.

  1 a ^H (θ) · R ^{− 1 xx · a(θ)} (9)

  P (θ) =

  To evaluate the performance of the proposed scheme, we performed a computer simulation. In this simulation, a single source of object is moving from 30 to 60 with angular speed of 1.4 per second. This angular speed is equivalent to commercial airplane moving at linear speed of 1,000 kmph at altitude 10 km (about 30,000 feet) above the ground. The object is assumed to move circularly at a constant distance to the receiver. As the receiver, we use ULA of 12 antennas. The distance between antenna elements is a half of the signal wavelength. We assume that the signal strength as compare to noise (SNR) is 10 dB. At the initial operation of the DoA estimation, a coarse DoA is performed using MVDR algorithm. The signal spectrum as function of scanning angle θ is given by:

  ESULTS

  R

  III. S

  Acqusition Coarse Pre-Scanning Angle Framing Sample

  In Eq.8, θ P max is the estimated DoA, θ min denotes lower border of scanning angle, while θ ^max denotes the upper border.

  θ P _{max = med([θ} ^{min , θ max} ]) (8)

  aligning the center of the window to the object’s DoA. The higher and lower border of scanning range is adjust so that center aligning process is given by

  Fig. 4. Schemes illustration: a). exhaustive search b). uniform non-exhaustive search without tail scan, c). uniform non-exhaustive search with tail scan

  The are several techniques to update the scanning range. ^{object object object tail scan scanning direction (a) (b) (c)}

  In this paper, we propose two schemes of non-exhaustive scanning, i.e., scanning without tail scan and scanning with tail scan. Tail scan is an additional scanning at outside of the main scanning window. In the without tail scan, the scheme is suffered from the situation where cvx-programming is failed to converge, especially if main scanning window is too narrow. Thus, additional tail scan is useful to ensure the convergence of the cvx-programming. Fig.4 shows illustration of the exhaustive search, the uniform non-exhaustive search without tail scan, and the uniform non-exhaustive search with tail scan.

  It is worth to note that the original scheme as proposed by Gorotnitsky and Rao utilizes a single snapshot ([12]). However we can extend the scheme by using multiple snapshots by extending column of vector x and column of vector s corre- spondingly (as also done in [14]). If the time duration of each snapshot is short enough, then we can assume that the object is still on its location, therefore we can do averaging on the value of every column of matrix s to obtain a robust estimate of the DoA.

  Fig. 3. Block diagram of proposed scheme

  Estimated DoA received signal

  CS CS Solver Snapshot Construction

  Actual angle of arrival at time i is denoted by θ ^ai , while θ i indicates a better estimate. Fig. 5 shows the simulation result. As observed in the figure, the actual position of the object is ⁸⁰ linearly changed by time. The exhaustive search performs well ₆₀ in tracking the object’s DoA with MAE is 0.6. The narrower ₄₀ scanning of −30 to

  90 performs slightly worse (with MAE _UniTail value of 1.6). This is because narrower scanning window has ₂₀ ^RNDTail _Actual less information for cvx-programming to iterate. _{70 (d e ngl a eg ree ) -20 -40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 degr ee)}

  60 _{50 40 w 0 to 90 -100 -60 -80 time an g le} ( _{30 20 w -90 to 90 w -30 to 90 Actual 10} Fig. 6. Comparison of uniform and random tail scan exhaustive search _{1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22} also included as a reference. In the depicted figure, we observe _time(second) that non-exhaustive search with uniform tail scan and random tail scan performed as good as exhaustive search. Uniform

  and random tail scan gave MAE of 0.81 and 0.56 respectively. Exhaustive search, on the other hand gave MAE of 0.61.

  Fig. 5. Comparison of exhaustive search (-90 to 90 ) and non-exhaustive

  If we compare the computation complexity, non-exhaustive

  searches

  search has smaller sensing matrix A, which is M times

  30 for non-exhaustive compared to M times 180 for exhaustive

  The situation is different in the case of searching range search. In other word, in this simulation, non-exhaustive search to 90 . The estimated angle is incorrectly in off positions. The has about 150M less computation compared to the exhaustive estimated angles are bounced back between 5 and 65 . This search. Here, M is number of antennas in the array. phenomena indicates failures of cvx-programming to give a converge solution. The bounce back values, indicates that the cvx iteration is trapped within these values. ₇₀ To improve this situation, we add a few additional scanning outside the window scanning range (i.e. the tail scanning). ⁶⁰ There are several methods to add these tail scanning. In ₅₀ the simulation, we tried two methods, which are, uniform _{exhaustive search} tail scan and random tail scan. In the uniform tail scan, we _{ee) 40 Actual} scan in additional 12 scanning directions which are uniformly _{( uniform tail scan degr} ^{random tail scan} separated at 20 to each others. Random tail scanning, on the le _{g an 30} other hand, scattered these 12 scanning directions randomly.

  Fig. 6 shows the simulation result of these tail scanning. In ²⁰ this simulation, window scanning range is to .

  90 ₁₀ As shown in Fig. 6, the performance of non-exhaustive searches, both uniform tail scan and random tail scan, improve _{1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22} accuracy of estimation. There is, however, an off-position _time(second) which took place at the time 11 (uniform scan tail) and 15 (random scan tail). This off-position statistically can take place at any time. This indicate that cvx-programming failed to give a convergence solution. This off-position is can be easily

  Fig. 7. Comparison of exhaustive search and non-exhaustive search using

  detected and removed. Interpolation of neighbors value can be

  uniform and random tail scan. Off-position is removed by interpolation used to substitute these off-positions.

  As the tail scan has successfully improve the non-

  IV. C ONCLUSION exhaustive search, we further narrowing the scanning window from 30 to 60 . This scanning window is exactly the same A non-exhaustive searching on DoA estimation based on to the track of the object. Here again, we simulate both sparse reconstruction has been presented in this paper. Non- uniform tail scanning and random tail scanning. Fig. 6 show exhaustive scanning without tail scanning simulation result has window. The main limitation of this scheme, especially as combined with the cvx-programming, is that it does not converge if the window range is narrow. We show also that adding tail scan can mitigate this problem. Two methods of tail scans were tested in this paper, which are uniform tail scan and random tail scan. These two methods perform very closed to the exhaustive search in term of accuracy. Main advantage of this schemes is smaller sensing matrix A, which means a faster computation. This advantage may add practicality of this scheme for actual implementation. Mathematical background for tail scan, however, is still necessary to be explored in future research.

  A CKNOWLEDGMENT The authors would like to thank to Yayasan Pendidikan

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