Iterative robust Capon beamforming

Signal Processing 118 (2016) 211–220

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Iterative robust Capon beamforming Yang Li a,n, Hong Ma a, De Yu a, Li Cheng a,b a b

School of Electric Information and Communications, Huazhong University of Science and Technology, Wuhan, China School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan, China

a r t i c l e i n f o

abstract

Article history: Received 2 December 2014 Received in revised form 21 June 2015 Accepted 6 July 2015 Available online 21 July 2015

Signal steering vector (SV) error can cause desired signal cancellation in adaptive beamforming. A covariance fitting based robust Capon beamforming (CFRCB) has been developed to solve this problem. Such a solution cannot be expressed in a closed form and its performance is highly affected by the initial value of SV error norm bound. In this paper, we propose an approximate closed-form expression of CFRCB, then develop two novel beamformers based on iterative implementation of this closed-form expression. Theoretical analysis and simulation results indicate that these beamformers improve in performance with every iterative step and converge to a stabilized solution. In addition, they perform well through a wide range of initial SV errors norm bound range, are easily implemented and computationally efficient. We also present a number of numerical examples comparing the proposed beamformers with similar classical beamformers. & 2015 Elsevier B.V. All rights reserved.

Keywords: Array signal processing Robust adaptive beamforming Steering vector error Array perturbations Signal power estimation

1. Introduction The standard Capon beamformer [1] maximizes the output signal to interference plus noise ratio (SINR) by minimizing the total beamformer output power, subject to a distortionless constraint for the signal of interest (SOI). If the training data contains the SOI component, then even small estimation error in the signal steering vector (SV) and/or array covariance matrix can lead to severe performance degradation [2]. In practice, factors such as inaccurate signal model [3], direction of arrival (DOA) estimation error [4], array perturbations (including array element position [5] and calibration errors [6]), and a moving target [7] can lead to SV estimation errors. Finite sampling sequence [8] leads to an inaccurate covariance matrix. Therefore, a robust technology [9] is required to overcome these problems.

n

Corresponding author. E-mail address: [email protected] (Y. Li).

http://dx.doi.org/10.1016/j.sigpro.2015.07.004 0165-1684/& 2015 Elsevier B.V. All rights reserved.

Quite often, SV error is dominant and easily reduced, while covariance matrix error is subordinate and more difficult to be dealt with [10]. Over the past two decades, a number of methods have been developed to improve the robustness of the Capon beamformer against SV error. They are roughly divided into two categories: the SV updating approach and robust constrained approach. The SV updating approach considers prior values in an effort to approach the actual signal SV. Algorithms such as the eigenspace projection approach [11], the Bayesian approach [12], and the Taylor series approximation approach [13] are effective. However, the actual signal SV cannot be achieved, due to inaccurate parameters used in the updating methods. The robust constrained approach does not need an actual signal SV. In general, it uses the signal SV error constraint or beampattern constraint, such as the worst-case performance optimization based beamformer [14], the doubly constrained beamformer [15], or the maximally robust Capon beamformer [16]. These robust constrained approaches generally exhibit two main defects: firstly, an optimum norm constrained bound of SV error is difficult to obtain in practice; secondly, the constraint is a second-order cone

212

Y. Li et al. / Signal Processing 118 (2016) 211–220

program problem, therefore it has no closed-form solution. A clear difference between these categories is that the main beam peak of SV from the updating approach could point closer to actual DOA than the robust constrained approach under the same given initial SV. Actually, many constrained robust adaptive beamformers turn out to be equivalent to a diagonal loading (DL) approach or belong to an extended class of the DL approach. In dealing with the aforementioned defects, we propose an approximate closed-form expression of covariance fitting based robust Capon beamforming (CFRCB) [17], and develop two novel iterative robust Capon beamformers (IRCB) based on an iterative implementation of the expression. The first proposed beamformer named IRCB1 belongs to the robust constrained approach category, and belongs to the class of DL approach. The second proposed beamformer named IRCB2 belongs to the SV updating approach category. The proposed IRCB appears to be similar to the iterative robust minimum variance beamforming (IRMVB) method [18], but there are fundamental differences as discussed in Section 3.4.7 The outline of this paper is as follows. The data model and background on adaptive beamforming are provided in Section 2. The proposed beamformers and their implementation are developed in Section 3. The simulation results are presented in Section 4. Finally, a brief conclusion appears in Section 5. In the paper, E½, ðÞH , ðÞ  1 , kk, and ? denote the expectation, Hermitian transpose, inverse, the two-norm, and orthogonal, respectively.

2. Problem formulation 2.1. Standard Capon beamformer Considering L þ 1 uncorrelated, narrowband signals impinging on a uniform linear array (ULA) with M omnidirectional sensors located along the x-axis in Cartesian coordinate system, L þ1 o M, the complex envelope representation of the received data at k-th snapshot xðkÞ can be expressed as xðkÞ ¼ aS sS ðkÞ þ

L X

ai si ðkÞ þnðkÞ

ð1Þ

the problem minwH RIN w w

subject to wH aS ¼ 1

ð3Þ

where w ¼ ½w1 ; …; wM T is the weight vector of beamforP mer, R IN ¼ Li ¼ 1 P i ai aH i þP N I is the actual interference plus noise covariance matrix. The solution is the standard Capon beamformer (SCB): wopt ¼

1 RIN aS

1 aH S R IN aS

ð4Þ

When desired signal is present in the received snapshots, R IN is replaced by R, which leads to an additional constant multiplied by weight vector, so it does not affect the output SINR. In this paper, we assume that one desired signal is always present in the received data. However, we cannot achieve the optimum weight in practice due to two inaccurate parameters. On one hand, since R is unknown in practice, it is replaced by K snapshots sample covariance matrix: K 1X R^ ¼ xðkÞxH ðkÞ Kk¼1

ð5Þ

On the other hand, factors such as inaccurate signal model, DOA estimation error, array perturbations, and moving target can lead to SV error. The array perturbations of i-th signal can be modeled as [1] 8 T jΔϕ > < ai ¼ ð1 þ ΔgÞe expfjð2π =λ0 ÞðP þ ΔPÞ Ai g ΔP ¼ ½ΔP1 ; ΔP2 ; …ΔPM  ð6Þ > : ΔP ¼ ½Δ ; Δ T ; m ¼ 1; 2; …; M m xm ym where Δg, Δϕ, and ΔP denote calibration magnitude errors, calibration phase errors, and array element position errors, respectively. This paper does not aim to study the finite sample effect but focuses on SV error with robust Capon beamforming. The actual R is used in all of the following formulas.

2.2. Robust Capon beamforming

i¼1

where ai ¼ expfjð2π =λ0 ÞPT Ai g, i ¼ S; 1; …; L is the actual SV of the i-th signal, λ0 is the wavelength. For a twodimensional problem, P ¼ ½P1 ; P2 ; …PM , Pm ¼ ½pxm ; pym T , m ¼ 1; 2; …; M, is each sensor's axis location, Ai ¼ ½ cos ðθi Þ; sin ðθi ÞT , θi is the angle between the DOA of i-th signal and the x-axis. si ðkÞ is the zero-mean stationary i-th signal and nðkÞ denotes the noise. Desired signal sS , interference signal si , i ¼ 1; 2; …; L, and noise are statistically independent. The covariance matrix of the array output is given by R ¼ E½xðkÞxH ðkÞ ¼ P S aS aH S þ

L X

P i ai aH i þP N I

ð2Þ

i¼1

where P S , P i , and P N denote the power of SOI, i-th interference, and noise, respectively. The problem of maximizing the output SINR is mathematically equivalent to

In this paper, we only consider the case of a nondegenerate ellipsoidal uncertainty set on the SV, about it is only known that aS belongs to the following uncertainty ellipsoid [17]: ðaS  a^ S ÞH C  1 ðaS  a^ S Þ r1

ð7Þ

where a^ S is the assumed SV of SOI (obtained from estimated DOA of SOI and nominal array manifold), C is a positive definite matrix. Without loss of generality, we consider C as a scaled identity matrix, that is, C ¼ εI, then we have the following uncertainty sphere:   aS  a^ S 2 r ε

ð8Þ

The prior known positive constant ε can be explained as a norm bound of the unknown mismatch/error between aS and a^ S .

Y. Li et al. / Signal Processing 118 (2016) 211–220

The CFRCB can be formulated as the following quadratic optimization problem under a spherical constraint [17]:  2 H mina~ R  1 a~ subject to a~  a^ S  r ε ð9Þ a~

where a~ is an enhanced SOI's SV obtained by the method of CFRCB. Because the solution to (9) will evidently occur on the boundary of the constraint set [17], or assuming that strong duality is achieved in (9) [18], we can reformulate the constraint of (9) with a quadratic equality constraint, and then this problem can be solved by using the Lagrange multiplier methodology, which is based on the function  2 H f ¼ a~ R  1 a~ þ λða~  a^ S   εÞ ð10Þ where λ 4 0 is the Lagrange multiplier. Differentiation of (10) with respect to a~ gives the optimal solution a~ ¼ ðR  1 =λ þ IÞ  1 a^ S ¼ a^ S  ðλR þ IÞ  1 a^ S

ð11Þ

The Lagrange multiplier λ is obtained as the solution to the constraint equation, which can be solved by Newton's method:  2   ð12Þ ðλR þIÞ  1 a^ S  ¼ ε Once

λ is obtained, the weight vector of CFRCB is given

by w¼

R  1 a~ a~ R  1 a~ H

¼

ðR þ I=λÞ  1 a^ S H a^ S ðR þ I=λÞ  1 a^ S

ð13Þ

We use wH Rw as an estimation of SOI's power P^ S . Substituting the weight vector (13) into P^ S ¼ wH Rw yields P^ S ¼

1 H a~ R  1 a~

¼

1 H a^ S ðR þ I=λÞ  1 RðR þ I=λÞ  1 a^ S

ð14Þ

3. The proposed beamformers In this section, we develop two novel beamformers based on the closed-form expression of CFRCB mentioned above. 3.1. Approximate closed-form solution of the CFRCB The proposed approximate closed-form solution of the CFRCB is extended from the covariance matrix decomposition in [19]. Using Woodbury's identity, it can be shown as

γ 1 Z ⋯ Z γ L ⪢γ L þ 1 ¼ ⋯ ¼ γ M ¼ P N , UI ¼ ½u1 ; …; uL  spans the interference subspace, UN ¼ ½uL þ 1 ; …; uM  spans the noise subspace. Similar to the eigen-decomposition properties of R, the following formulas hold: H UI U H I þUN UN ¼ I

λP S ðλRIN þIÞ  1 aS aHS ðλR IN þ IÞ  1 1 1 þ λP S aH aS S ðλR IN þ IÞ ð15Þ

spanfai g ¼ spanfui g

RIN ¼ UΓU ¼ UI Γ L X ¼ γ i ui uHi þP N i¼1

H I UI þ UN M X

Γ

ui uH i

ð16Þ

i ¼ Lþ1

where γ i and ui are the eigenvalues and corresponding eigenvectors of RIN , γ i is sorted in descending order

ð18Þ

S

H aH S UI UI aS ⪡M,

which

can

be

further

expanded

to

H H H the approximation as aH S UN UN aS ¼ aS ðI UI UI ÞaS ¼ M   2  2     H H H aH S UI UI aS  M, and UI aS  ⪡UN aS  . To derive the proposed approach, the following lemma is used:

Lemma 1. L L L  X X X  2   ¼ f ðγ~ Þ aH ui 2 aH f ðγ i Þui uH f ðγ i ÞaH S i aS ¼ S ui S i¼1

i¼1

i¼1

ð19Þ where f ðÞ is a monotonic function in this paper, and γ 1 4 γ~ 4 γ L ⪢P N always holds. We use this lemma directly because it is easy to be proved. Using Lemma 1, we have 1 aS ¼ aH aH S ðλR IN þIÞ S

M X ui uH i aS λγ i þ1 i¼1

     H 2  H 2  H 2 M a ui  UI aS  UN aS  X S þ ¼ þ ¼ λγ i þ1 i ¼ L þ 1 λP N þ1 λγ~ þ 1 λP N þ 1 i¼1   L  H 2 X a ui S

ð20Þ  2  2     H H Since γ~ 4 γ L ⪢P N , and UI aS  ⪡UN aS  , we can further obtain the approximation value of (20) as    H 2 UN aS  M 1 H aS ðλRIN þIÞ aS   9κ ð21Þ λP N þ1 λP N þ 1 Since a^ S is usually close to aS in practice, we make other H^ H^ ^H two approximations: aH S UN UN a S  M, and a S UN UN a S  M, which can be further extended to 1 ^ aS  aH S ðλR IN þIÞ

M

λP N þ1

¼κ

ð22Þ

¼κ

ð23Þ

M

λP N þ 1

Similar to (20), we have H H a^ S ðλR IN þ IÞ  2 a^ S ¼ a^ S



H N UN

for i ¼ 1; …; L

When angular separation between signal and interfer 2  ence is larger than a beam width, aH S ai =M ⪡1, i ¼ 1; …; L [20]. It is assumed that this condition always holds,  2 therefore we can make the approximation aH ui  ⪡M and

RIN can be written in eigen-decomposition form as H

ð17Þ

H a^ S ðλR IN þ IÞ  1 a^ S 

1 ðλR þ IÞ  1 ¼ ½λP S aS aH S þ ðλR IN þIÞ

¼ ðλR IN þ IÞ  1 

213

1 ðλP N þ 1Þ

H a^ 2 S

M X i¼1ð

M X i ¼ Lþ1

ui uH i

λγ i þ1Þ2

^ ui uH i aS ¼

a^ S

H ^ a^ S UN UH N aS

ðλP N þ1Þ

2



κ2 M

ð24Þ

Consequently, according to (15), (22), and (24), the following approximation holds:  2   ðλR þIÞ  1 a^ S 

214

Y. Li et al. / Signal Processing 118 (2016) 211–220

 2  λP S ðλR IN þ IÞ  1 aS aHS ðλRIN þIÞ  1 a^ S   1 ^ ¼ ðλR IN þ IÞ a S     1 þ λP S aH ðλRIN þ IÞ  1 aS S

 2    λP S κ 1 ^  a  1  λ R þ IÞ ð IN S  1 þ λP κ

ð25Þ

S

¼

H a^ S ðλRIN þ IÞ  2 a^ S

ð1 þ λP S κ Þ

2



κ2 Mð1 þ λP S κ Þ2

S

P ðkÞ S ¼

The approximate DL level can be solved by using (12) and (25) as pffiffiffi 1 εðMP S þP N Þ DL ¼  pffiffiffiffiffi pffiffiffi ð26Þ λ M ε Eq. (26) reveals that there are three factors which affect the DL level: SOI's power P S , noise power P N , and SV error norm bound ε. Once we obtain the reliable P S and P N , closed-form expression of approximate DL level is obtained. Eq. (26) is based on the approximation Eqs. (21)–(23). If a^ S ¼ aS , Eqs. (22) and (23) are strictly true, and (26) is reliable. If the error is large between a^ S and aS , the proposed iterative methods in the following sections will reduce this error step by step so as to make (26) reliable.

3.2. The proposed iterative RCB approach 1 Reliable P S cannot be obtained according to inaccurate SV a^ S . However, Eqs. (11), (14), and (26) reveal an iterative updating relationship. Using superscript number to denote the iterative step, we firstly estimate initial P ð0Þ with S ð0Þ að0Þ ¼ a^ S and λ ¼ þ1 through Eq. (14), then we obtain λð1Þ by using (26) with P ð0Þ S , update the power of SOI as ð1Þ ð1Þ P ð1Þ S ¼ P S by using (14) with λ ¼ λ . Next, we use P S as a ð2Þ new initial value to solve the next power P S . This process is repeated until the desired power of SOI is reached, which means the iterative process stops if the estimated power of SOI is stabilized. The stopping criterion is ðk  1Þ ðP ðkÞ Þ=P Sðk  1Þ o δ, where δ is a small positive numS  PS ber. At last, we use final estimated P^ S to obtain optimum DL level. Before continuing, we make a further expression of MP S þP N . Under the condition that the interferences are absent, the covariance matrix becomes R ¼ P S aS aH S þ P N I. Using Woodbury's identity, its inverse is obtained by P S aS aH 1 S R1 ¼ I P N P N ðP N þ MP S Þ

ð27Þ

1 1 aS aH SR

¼ PS þ

PN M

1 aðkÞH R  1 aðkÞ

wðkÞ ¼

;

Eq. (28) reveals that our methods have a tiny overestimation of SOI's power. However, the stopping criterion in (60) leads to a tiny underestimation. Besides, certain interference SV component included in SOI's SV leads to a tiny overestimation, as analyzed in simulation result.

aðkÞH R  1 aðkÞ

ðk  1Þ Property 1. P ðkÞ for each iteration, and P S has an S 4 PS upper bound.

Proof. We write R in eigen-decomposition form as H R ¼ Q ΣQ H ¼ Q S ΣS Q H S þQ N ΣN Q N LX þ1

¼

M X

r i qi qH i þ

i¼1

r N qi qH i

ð29Þ

i ¼ Lþ2

where r i and qi are the eigenvalues and corresponding eigenvectors of R, r i is sorted in descending order r 1 Z⋯ Zr L þ 1 ⪢r L þ 2 ¼ ⋯ ¼ r M ¼ P N , Q I ¼ ½q1 ; …; qL þ 1  spans the signal plus interference subspace, Q N ¼ ½qL þ 2 ; …; qM  spans the noise subspace. ð0Þ Defining a J 9 Q S Q H and a ? 9Q N Q N að0Þ , we have the Sa following formulas: pffiffiffiffiffi  2 Ma J ð30Þ a J ? a ? ; að0Þ  ¼ ka J k2 þ ka ? k2 ; aS ¼ ka J k The k-th iterative step SV can be expressed by a J and a ? as !1 R ðkÞ ð0Þ þI að0Þ a ¼a   1Þ τP ðk S ¼ ðτP Sðk  1Þ R  1 þIÞ  1 ða J þa ? Þ ¼

LX þ1

qi qH i

ðk  1Þ =r i þ 1 i ¼ 1 τP S

aJ þ

M X

qi qH i

a?

ðk  1Þ =P N þ 1 i ¼ L þ 2 τP S

  þ1  LX  a qi qH a?   J i þ aJ  ¼ ðk  1Þ  τP =r i þ1 ka J k τP ðk  1Þ =P N þ 1 i¼1

S

ð31Þ

S

Similar to Lemma 1, the following formula holds: aHJ

LX þ1 i¼1

f ðr i Þqi qH i aJ ¼

LX þ1 i¼1

 2 f ðr i ÞaHJ qi  LX þ1  i¼1

ð28Þ

R  1 aðkÞ

where the superscript ðkÞ denotes the estimated value of kth iterative step. The following three properties hold for the IRCB1:

¼ f ðr~ Þ

Then the power of SOI is estimated by P^ S ¼

We summarize the implementation of IRCB1 as pffiffiffi M ε 1 τ ¼ pffiffiffiffiffi pffiffiffi; að0Þ ¼ a^ S ; P ð0Þ S ¼ ð0ÞH  1 ð0Þ M ε R a a for k ¼ 1; 2; … !1 pffiffiffiffiffi ðkÞ R Ma ðkÞ ð0Þ  þI að0Þ ; aðkÞ ¼  a ¼a  ðk  1Þ aðkÞ  τP

 aH q 2 ¼ f ðr~ Þka J k2 J i

ð32Þ

where f ðÞ is a monotonic function in this paper, r~ may be different (f ðÞ is different in (33) and (38), hence r~ ¼ r a in (33) and r~ ¼ r b in (38)), but r 1 4 r~ 4 r L þ 1 ⪢P N always holds. Therefore, we have  2 þ1 LX þ1  LX  qi qH aHJ qi qH   i i aJ ¼ a   J ðk  1Þ ðk  1Þ   =r i þ 1 =r i þ1Þ2 i ¼ 1 τP S i ¼ 1 ðτ P S

Y. Li et al. / Signal Processing 118 (2016) 211–220

¼

ðτ

ka J k2 ðk  1Þ PS =r a þ 1Þ2

ð33Þ

where r 1 4r a 4 r L þ 1 ⪢P N . Substituting (33) into (31), we have aðkÞ ¼

aJ þ ðk  1Þ PS =r a þ 1 ðkÞ 1 ða J þ 1 a ? Þ

τ ¼μ

τ

a? ðk  1Þ PS =P N þ 1

η

ð34Þ  ðkÞ 2 where μ1 is a constant constraint subject to a  ¼ M, and

ηðkÞ 1 9

 1Þ τP ðk =r a þ 1 S ðk  1Þ τP S =P N þ 1

ð35Þ

Because r a 4P N , we have 0 o ηðkÞ o 1, and   ∂ηðkÞ 1 1 1 ¼ μ2  o0 ðk  1Þ ra PN ∂P

ð36Þ

where μ2 is a positive constant. Then we have  2   ðkÞ   ða J þ η1 a ? Þ ðkÞ PS ¼ R  1 a J þ ηðkÞ 1 a? ðkÞ H Mða J þ η1 a ? Þ

¼

ka J k2 þ η1ðkÞ2 ka ? k2

1

1

ðkÞ2 2 1 ka ? k

ka J k þ η  1 1 ka J k2 þ η1ðkÞ2 ka ? k2 M rb PN 

ð37Þ

where r b satisfies þ1   1  H 2 1 LX aH q 2 ¼ 1 ka J k2 a J qi ¼ J i r r rb bi¼1 i¼1 i

ð38Þ

and r 1 4 r b 4r L þ 1 ⪢P N . Therefore,   ∂P ðkÞ 1 1 s ¼ μ  o0 3 rb P N ∂ηðkÞ

ð39Þ

1 ða J þa ? ÞH R  1 ða J þ a ? Þ

¼

P ð0Þ S

as

ka J k2 þ ka ? k2  1 1 ka J k2 þ ka ? k2 M rb PN 

ka J k2 þ η1ð0Þ2 ka ? k2  ¼  1 1 M ka J k2 þ η1ð0Þ2 ka ? k2 rb PN

η

η

)⋯

ð0Þ ð2Þ ð1Þ ) P ð1Þ S 4 PS ) 1 o 1 ðkÞ ðk  1Þ ðkÞ  1Þ ) 1 o 1 ) P S 4 P ðk S

η

η

η

η

ð41Þ

ðkÞ The upper bound of P ðkÞ S is achieved if η1 ¼ 0:

P ðkÞ S o

ka J k2 r ¼ b 1 M 2 M ka J k rb

□

wðkÞH aS ¼ aðkÞH R  1 aS ffi  pffiffiffiffi Ma J 1 1 H ¼ ða J þ ηðkÞ a ? ÞH Q S ΣS Q H þ Q Σ Q N N S N ka J k pffiffiffiffiffi p ffiffiffiffi ffi 1  1 H Ma J H ¼ a J Q S ΣS Q S M ka J k ¼ ka J k rb

¼

1 1 ka J k2 þ ηðkÞ2 ka ? k2 rb PN 1

¼

1 þ SINRðkÞ

ð44Þ

 2 P S wðkÞH aS  wðkÞH RwðkÞ

ð42Þ

 2  2 Property 2. aðkÞ aS  o aðk  1Þ  aS  for each iteration.

ð45Þ

1 MP S ka J k2 r 2b ¼ 1 1 ka J k2 þ η1ðkÞ2 ka ? k2 rb PN

ð46Þ

Therefore, SINRðkÞ ðkÞ

1 þSINR ) SINRðkÞ 4 SINRðk  1Þ

4

The upper bound of SINR SINRðkÞ o

SINRðk  1Þ 1 þ SINRðk  1Þ ð47Þ

ðkÞ

ð40Þ

ð1Þ where ηð0Þ 1 ¼ 1 4 η1 . Combining (36), (37), and (39), we find that the following iterative relationship between η1 and P S holds step by step: ð1Þ ð0Þ 1 o 1

If

Property 3. SINRðkÞ 4 SINRðk  1Þ for each iteration, and SINR has an upper bound.

ðk  1Þ ηðkÞ ) 1 o η1

1

P Sð0Þ ¼

ð43Þ

 2  2 ðkÞ ðk  1Þ Therefore, ) aðkÞ  aS  o aðk  1Þ  aS  .  ðkÞ η1 o 2η1 ðkÞ   η1 ¼ 0, a  aS ¼ 0, the actual SV is obtained.□

SINRðkÞ

LX þ1

where μ3 is a positive constant. If we rewrite

2M ka J k ¼ 2M  pffiffiffiffiffiffiffiffiffiffiffiffiffi ka J k2 þ η1ðkÞ2 ka ? k2

wðkÞH RwðkÞ ¼ aðkÞH R  1 RR  1 aðkÞ ¼ aðkÞH R  1 aðkÞ    1 1 H H a J þ ηðkÞ ¼ ða J þ ηðkÞ Q S ΣS Q H S þ Q N ΣN Q N 1 a? Þ 1 a?

ðkÞ2 H H MðaHJ Q S ΣS Q H S a J þ η1 a ? Q N ΣN Q N a ? Þ 2

Proof.    ðkÞ 2 a  aS  ¼ 2M  2 RefaðkÞH aS g 8pffiffiffiffiffi ffi 9 < M ða þ ηðkÞ a ÞH pffiffiffiffi Ma J = J ? 1   ¼ 2M  2 Re  ðkÞ :  ka J k ; a J þ η1 a ? 

Proof.

S

¼

215

MP S r b  MP S

ðkÞ 1

is achieved if η □

¼ 0: ð48Þ

3.3. The proposed iterative RCB approach 2 The proposed IRCB2 adds another updating variable SV per step on the basis of IRCB1. The first step is the same as IRCB1. From the second step, we use að1Þ as a new initial signal SV, which means the k-th initial signal SV is aðk  1Þ . The difference is that the initial value of signal SV per step is equal to að0Þ for IRCB1. Furthermore, if aðkÞ goes closer to actual signal SV, we should decrease ε per step, which is easy ðkÞ ðkÞ to be implemented as εðkÞ ¼ β εðk  1Þ ; 0 o β r1 per step. We summarize the implementation of IRCB2 as

εð0Þ ¼ ε;

að0Þ ¼ a^ S ;

for k ¼ 1; 2; …

P Sð0Þ ¼

1 að0ÞH R  1 að0Þ

216

Y. Li et al. / Signal Processing 118 (2016) 211–220

pffiffiffiffiffiffiffi

εðkÞ ¼ βðkÞ εðk  1Þ ; aðkÞ ¼ aðk  1Þ  P ðkÞ S ¼

0oβ

ðkÞ

!1

R  1Þ τðkÞ P ðk S

1 aðkÞH R  1 aðkÞ

;

M εðkÞ τðkÞ ¼ pffiffiffiffiffi pffiffiffiffiffiffiffi M  εðkÞ

r1;

þI

wðkÞ ¼

aðk  1Þ ;

For IRCB2, the convergence weight is k

pffiffiffiffiffi ðkÞ Ma  aðkÞ ¼  aðkÞ 

S

R  1 aðkÞ

a

ðkÞ 2 a? Þ

¼ μða J þ η

Therefore, the IRCB1 belongs to the class of DL and IRCB2 does not.

ð49Þ

where

8 ðiÞ ði  1Þ ðiÞ > =r a þ 1 > < τ PS ; ðiÞ ðiÞ ðkÞ η2 ¼ ∏ η^ 2 ; η^ 2 9 τðiÞ P ðiS  1Þ =P N þ1 > > i¼0 : 1; k

i40

ð50Þ

i¼0

ðkÞ r 1 4r ðiÞ a 4r L þ 1 ⪢P N ; i4 0, it is obvious that 0 o η2 o  1Þ ηðk o1; k 4 1. 2

P ðkÞ S ¼

2 ka J k2 þ ηðkÞ2 2 ka ? k  1 1 M ka J k2 þ η2ðkÞ2 ka ? k2 rb PN



 2 2M ka J k  ðkÞ  a aS  ¼ 2M  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ka J k2 þ η2ðkÞ2 ka ? k2 1 MP S ka J k2 r 2b ¼ 1 þSINRðkÞ 1 ka k2 þ 1 ηðkÞ2 ka k2 J ? rb PN 2 SINRðkÞ

ð51Þ

ð52Þ

ð53Þ

Eqs. (51), (52), and (53) have the same forms as Eqs. (40), (43), and (46), respectively. Therefore, the IRCB2 has the same properties with IRCB1.

3.4.3. The choice of parameter ε The parameter τ is defined in the implementation of IRCB1 in Section 3.2 It can be seen from (35) that, if τ 4 0 holds, the three properties of IRCB1 will always hold, which indicates that our iterative progress will always converge if ε o M.Therefore, 2 the ε does no longer subject to the constraint aðkÞ  a^ S  r ε, any 0 o ε o M is suitable for IRCB1. So does IRCB2, that is, any 0 o εðkÞ o M is suitable for IRCB2. If we define a varying εðkÞ instead of fixed ε per ðkÞ iterative which satisfies  ðkÞ  step for IRCB1, the ε a  a^ S 2 ¼ εðkÞ per step is a reasonable choice for both IRCB1 and IRCB2. It is hard to obtain the optimum parameter εðkÞ . But we could estimate an approximate  2 value of εðkÞ . Substituting (30) and (34) into aðkÞ  aS  ¼ εðkÞ , we obtain pffiffiffiffiffi pffiffiffiffiffi   2  M ða J þ ηðkÞ a Þ Ma J  ? 1     ¼ εðkÞ ð56Þ     ka J k   a J þ ηðkÞ  1 a?  Substituting εð0Þ and η1ð0Þ ¼ 1 into (56), using ka ? k=ka J k as an intermediate variable, we have "  # 2  2 εð0Þ εðkÞ 1= 1   1 ¼ 1= 1  ð57Þ 1 þ ηðkÞ2 1 2M 2M Because εðiÞ2⪡2M, make the approximation   i ¼ðiÞ0; 2k, we can ðiÞ ðiÞ ε ε εðiÞ2  1 þ εðiÞ . Then (57) 1= 1  2M  1 þ 2M ¼ 1 þ εM þ 4M 2 M can be simplified as

3.4. Further remarks on proposed beamformers

εðkÞ  η1ðkÞ2 εð0Þ

3.4.1. The IRCB2 is better than IRCB1 under certain conditions Using superscript ðcÞ to denote the converge value, subscripts 1 and 2 denote the IRCB1 and IRCB2, respec-

For IRCB2, it has a similar relationship:

tively. The difference between IRCB1 and IRCB2 is that ηðcÞ 1

ðcÞ only depends on P ðcÞ S , whereas η2 depends on the iterative ðiÞ

product of ∏ki ¼ 0 η^ 2 , thus ηðcÞ 2 could tend to zero whereas ðcÞ 1

ðcÞ does not. Therefore, we may have P ðcÞ S2 4 P S1 ,  2  2  ðcÞ   ðcÞ  ðcÞ ðcÞ a2  aS  o a1  aS  , and SINR2 4 SINR1 under certain

η

ðcÞ conditions. Once ηðcÞ 2 tends to zero, a2 will only have a J component and is equal to or very close to the actual signal SV, which will let the main beam peak of beampattern point to the actual DOA of SOI.

3.4.2. The IRCB1 belongs to the class of DL and IRCB2 does not For IRCB1, the convergence weight is  1Þ  1 wðkÞ ¼ ðR þ τP ðk IÞ a^ S S

S

ð55Þ

aðkÞH R  1 aðkÞ

Similar to IRCB1, it is easy to prove that the IRCB2 has the following formula: ðkÞ

wðkÞ ¼ R  1 ∏ ðτðiÞ P Sði  1Þ R  1 þ IÞ  1 a^ S i¼1 ( ) 1 k 1 1 k 1 ; …; Q a^ S ¼ Q diag ∏ ∏ ði  1Þ ði  1Þ r1 i ¼ 1 τðiÞ P rM i ¼ 1 τðiÞ P =r 1 þ 1 =r M þ 1

ð54Þ

εðkÞ  η2ðkÞ2 εð0Þ

ð58Þ

ð59Þ

According to the discussion above, when iteration stops, ηðkÞ 1 tends to a stable value between 0 and 1, and ηðkÞ 2 tends to zero. Therefore, for IRCB1, we could choose a fixed value less than εð0Þ per step, which can be expressed as εðkÞ ¼ β1 εð0Þ , 0 o β1 o 1. For IRCB2, we could choose a smaller value per step, which can be expressed as ðkÞ ðk  1Þ εðkÞ ¼ βðkÞ , 0 o β2 r 1. For further simplification, we 2 ε ðkÞ can choose a fixed equal value β1 ¼ β 2 ¼ β. Eq. (59) gives ðkÞ an interpretation for we use a parameter β on the implementation of IRCB2 in Section 3.3. 3.4.4. Low SNR case We have made a stopping criterion 1 in Section 3.2 as ðk  1Þ  1Þ ðP ðkÞ Þ=P ðk oδ S  PS S

ð60Þ

Unfortunately, the estimated power of SOI (14) is actually a sum of P S and P N =M, as shown in (28). If SNR is very low, P S ⪡P N =M, the estimated power of SOI will not go to

Y. Li et al. / Signal Processing 118 (2016) 211–220

stabilization, aðkÞ might even converge to interferences. Therefore, we add another stopping criterion 2 as [18]    

       H θW ^  ^   ^ H ^ θW ^  ^   ðkÞH ^ ^  a θ ; a θ  a θ  a aðθ Þ o min a^ θ^ þ 2 2 ð61Þ ^ where θ is the estimated DOA of SOI, θW is the uncertainty range of SOI's DOA . This stopping criterion only works at very low SNR when aðkÞ converge to noise peak or interference direction that is beyond the SOI's DOA uncertainty range. 3.4.5. The unified framework for IRCB1 and IRCB2 We make a unified framework for both IRCB1 and IRCB2 as

εð0Þ ¼ ε;

að0Þ ¼ a^ S ;

P Sð0Þ ¼

for k ¼ 1; 2; …

εðkÞ ¼ βεOLD ; aðkÞ ¼ aOLD  P ðkÞ S ¼

1 að0ÞH R  1 að0Þ

pffiffiffiffiffiffiffi

M εðkÞ τðkÞ ¼ pffiffiffiffiffi pffiffiffiffiffiffiffi M  εðkÞ R

!1

þI ðk  1Þ

τðkÞ P S

aOLD ;

pffiffiffiffiffi ðkÞ Ma  aðkÞ ¼  aðkÞ 

1 aðkÞH R  1 aðkÞ

If ð60Þ or ð61Þ is satisfied; stop:

w¼

R  1 aðk  1Þ

217

only several important performances with CFRCB and IRMVB. A ULA of 16 omnidirectional antennas with halfwavelength inter-element spacing d is employed. We assume that the Δg, Δϕ, Δxm , Δym , m ¼ 1; 2; …; M are statistically independent, zero-mean, Gaussian random variables. The perturbation parameters Δg, Δϕ=π , Δxm =d, and Δym =d are different for each independent simulation, without loss of generality, supposing they have a same statistical standard deviation σ Δ . There are five interferences with directions and interference to noise ratios of [401, 5 dB], [601, 10 dB], [1001, 30 dB], [1201, 20 dB], and [1401, 0 dB], respectively. The additive noise is a spatially white Gaussian process and has a variance, which is equal to 1. Actual DOA of desired signal is 801. Estimated DOA is 821 except example 5. The DOA uncertainty range is θW ¼ 81. The actual norm bound of the unknown mismatch between aS and a^ S is calculated by  2 εOPT ¼ minψ a^ S ejψ  aS  [15]. The εCFRCB , εIRMVB , εIRCB1 , and εIRCB2 are norm bounds used in simulation for four beamformers. The β for both IRCB1 and IRCB2 are fixed to 0.5. The parameter δ in (60) and the δ of IRMVB in the iteration stopping condition are all set to 0.01. Snapshots number is 100. 100 Monte-Carlo runs are performed except examples 2 and 4. 4.1. Example 1: SINR versus SV error bound

aðk  1ÞH R  1 aðk  1Þ

where εOLD ¼ εð0Þ , aOLD ¼ að0Þ for IRCB1 and εOLD ¼ εðk  1Þ , aOLD ¼ aðk  1Þ for IRCB2. This approach is suboptimum, because the ε and β are not optimized. 3.4.6. The computational complexity The dominant computational complexity of the proposed two approaches is determined by the inversion of diagonal loaded R and is equal to OðM 3 Þ per iterative step. 3.4.7. Similarity and difference among IRCB, CFRCB, and IRMVB From the point of fundamental principle, IRCB, CFRCB, and IRMVB are based on the same CF problem (9). From the point of solution, CFRCB and IRMVB are the same while IRCB is fundamentally different. From the point of SOI's SV estimation: IRCB1 and CFRCB are similar, both of them could not achieve actual SV; IRCB2 and IRMVB are similar, both of them could not achieve but could tend to actual SV. From the point of DL class, both IRCB1 and CFRCB belong to the class of DL while both IRCB2 and IRMVB do not. From the point of user's parameter: ε for CFRCB should be large enough; ε for IRMVB should be small enough; any 0 o ε o M is suitable for IRCB. 4. Simulation results The primary motivation of the work is to propose an efficient closed form useful in studying the RCB problem. The two proposed beamformers are similar to CFRCB in [17] and IRMVB in [18] under a nondegenerate ellipsoidal uncertainty set signal model case. This paper compares

The first example evaluates SINR versus initial value of SV error bound ε. σ Δ ¼ 0, and SNR varies from  30 dB to 50 dB. The actual bound is εOPT ¼ 3:8. Fig. 1(a) and (b) represents the simulation results of IRCB1 and IRCB2, respectively. The εIRCB1 and εIRCB2 vary from 0.5 to 15. The result of these two beamformers is very similar: the SINR tends to stable through all the SNR if εIRCB1 4 1, which reveals two properties: both IRCB1 and IRCB2 are able to perform well when εIRCB1 and εIRCB2 are less than εOPT , and they have a stable property through a wide bound range 1–15, as we remarked in Section 3.4.3. Therefore both εIRCB1 and εIRCB2 can be set to 4, not optimally, but suitably, in all the following examples. 4.2. Example 2: iterative convergence property The second example evaluates the convergence properties, σ Δ ¼ 0 and SNR¼20 dB. Fig. 2(a), (b), and (c) indicates that when the iterative steps increase, the estimated power of SOI grows larger and tends toward the actual power of SOI, the error between the estimated SV and the actual SV grows smaller and tends to zero, and the SINR increases to a stable value, respectively. Additionally, the proposed IRCB1 and IRCB2 have a faster convergence speed than IRMVB underthe parameters set above. The  2 SV error, minψ aðkÞ ejψ  aS  in Fig. 2(b), has two features: first, the minimum value of error is larger than 0. Because, when we use R^ with finite snapshots, aðkÞ will converge to 1 a SV that minimizes aH R^ a, but not actual SV that  1 minimizes aH R a. Second, the value of error grows slightly after the minimum value. Because the  it reaches 2  condition aH S ai =M ⪡1 in Section 3.1 indicates that aS may have certain ai components. If the convergence parameter

218

Y. Li et al. / Signal Processing 118 (2016) 211–220

Output SINR (dB)

40 30

20

ε=7 ε=9 ε=11 ε=13 ε=15

OPT ε=0.5 ε=1 ε=3 ε=5

50

Estimated power of SOI (dB)

60

20 10 0 −10

15 10 5 IRMVB−Ps(k) 0

(k)

IRCB1 −Ps

(k)

IRCB2 −Ps

−20 −5

−30

−20

−10

0

10

20

30

40

50

1

2

3

4

Input SNR (dB)

6

7

8

9

10

3

ε=7 ε=9 ε=11 ε=13 ε=15

OPT ε=0.5 ε=1 ε=3 ε=5

50 40 30

IRMVB−min ||a(k)ejψ−a ||2 ψ

2.5

s

(k) jψ

(k)

20 10 0 −10

2

IRCB1 −min ||a e −a || ψ

2 jψ minψ||a e −as||

60

Output SINR (dB)

5

Iterative steps (k)

s

(k) jψ

2

IRCB2 −minψ||a e −as||

2 1.5 1 0.5

−20 −30

−20

−10

0

10

20

30

40

0

50

1

2

3

4

Input SNR (dB)

5 6 7 Iterative steps (k)

8

9

10

Fig. 1. SINR versus SV error bound: (a) IRCB1 and (b) IRCB2. 20

greater interference component that increases P ðkÞ S , which will lead to the overestimation of SOI's power and degradation of SINR performance. Therefore, a suitable parameter δ should not be based on a value that is too small. 4.3. Example 3: SINR and power of SOI versus SNR

Output SINR (dB)

δ in (60) is too small, the estimated SV may include a 10 0 −10 (k)

IRMVB−SINR −20

In the third example, both array perturbations and pointing error are taken into consideration, σ Δ ¼ 0:05. There is no fixed actual SV norm bound value for each run, however we know that εOPT 4 3:8 for each run due to pointing error of 21 plus array perturbations. SNR varies from 30 dB to 50 dB. Fig. 3(a) evaluates SINR versus SNR. Results show that the IRMVB, IRCB1, and IRCB2 with fixed ε are able to perform stably at different SV errors through all the SNR. The performance of CFRCB decreases greatly at εCFRCB o εOPT . Another problem is that, the SINR of IRCB2 is lower than IRMVB at SNR 4 30 dB. If εIRCB2 increases to 9, the SINR of IRCB2 will exceed IRMVB at SNR 430 dB, but show a tiny decrease at 10 dBo SNR o30 dB, as shown in Fig. 1(b). Therefore, experience suggests that choice in practice, and under the same other conditions εIRCB2 could be set to a larger value as the SNR increases. Fig. 3(b) evaluates the estimated power of SOI versus SNR. Results show that both the proposed IRCB1 and IRCB2 have an accurate power estimation property if SNR is larger than  10 dB. Eq. (28) reveals that 10log10 ðP N =MÞ   12 dB is the threshold where the signal spatial power spectrum is larger than the noise spatial power spectrum. The power of SOI is overestimated

IRCB1 −SINR(k) (k)

IRCB2 −SINR −30

1

2

3

4

5

6

7

8

9

10

Iterative steps (k)

Fig. 2. Iterative convergence simulations: (a) estimated power of SOI versus iterative steps; (b) SV error of SOI versus iterative steps; (c) SINR versus iterative steps.

for the IRMVB at low SNR and is underestimated for the CFRCB at high SNR. One reason is that when SNR is low, the aðkÞ of IRMVB may have a certain interference component that increases power estimation at each step. Therefore, the power of SOI is overestimated. This overestimation could be reduced by increasing δ appropriately. On the other hand, when the SNR is high, the CFRCB without a large enough εCFRCB will cause SOI cancellation, which leads to an underestimation of SOI's power. 4.4. Example 4: array beam pattern gain Example 4 verifies two properties through array beam pattern gain, σ Δ ¼ 0 and SNR ¼20 dB. Fig. 4 shows that the

Y. Li et al. / Signal Processing 118 (2016) 211–220

0

CFRCB IRMVB IRCB1 IRCB2

60

40

OPT

IRCB1

CFRCB

IRCB2

−10

Pattern Gain (dB)

Output SINR (dB)

50

IRMVB

30 20 10

219

0 −5 −10 70

80

90

−20

−30

−40

0 −10

−50

−20 −30

−20

−10

0

10

20

30

40

20

50

40

60

80

100

120

140

160

180

4

5

Phi (degrees)

Input SNR (dB)

Fig. 4. Array beam pattern gain simulation.

40 30

OPT

IRCB1

CFRCB

IRCB2

30

IRMVB

20

20

Output SINR (dB)

Estimated power of SOI (dB)

50

10 0 −10 −20

10 OPT CFRCB IRMVB IRCB1 IRCB2

0 −10

−30 −30

−20

−10

0

10

20

30

40

−20

50

Input SNR (dB)

−5

−4

−3

−2

−1

0

1

2

3

Pointing Error (degrees)

Fig. 3. (a) SINR versus SNR. (b) Estimated power of SOI versus SNR.

Fig. 5. SINR versus pointing error.

4.5. Example 5: SINR versus pointing error

35 30 Output SINR (dB)

main beam peak of IRCB1 is closer to actual DOA than CFRCB, because the ηðkÞ 1 o1 at the final iterative step reduces the distance between estimated SV and actual SV of SOI. But IRCB1 cannot point to actual DOA because ηðkÞ 1 does not tend to zero. As expected, IRCB2 could point the main beam peak to actual signal DOA, because ηðkÞ 2 is a ðkÞ continuous multiplication of k numbers of η^ 2 o 1, which lets ηðkÞ 2 approach to zero.

25

OPT

IRMVB

CFRCB

IRCB1

IRCB2

20 15 10 5 0

The fifth example evaluates the SINR versus pointing error, σ Δ ¼ 0 and SNR¼20 dB. The initial norm bound of CFRCB, IRCB1, and IRCB2 is 4, which corresponds to about pointing error of 21. The results in Fig. 5 verify that the performance of CFRCB decreases greatly when εCFRCB o εOPT , while three other beamformers exhibit stable performance in the DOA uncertainty range θW ¼ 81, regardless of whether the norm bound is larger or smaller than εOPT .

7 dB, 5 dB, and 4 dB, respectively, while CFRCB decreases more than 20 dB. The proposed two beamformers have a better SINR performance relative to perturbations.

4.6. Example 6: SINR versus perturbations

5. Conclusion

Before beamforming proceeds, there is usually a calibration procedure implemented on the array. This calibration can decrease but not eliminate array perturbations. In the sixth example, small perturbations standard deviation σ Δ varies from 0 to 0.1 at SNR¼20 dB. The results in Fig. 6 show that, as σ Δ increases from 0 to 0.1, the SINR performance of IRMVB, IRCB1, and IRCB2 decreases about

We have derived an approximate closed-form solution of the CFRCB in this paper. Two novel beamformers, IRCB1 and IRCB2, have been proposed based on this closed-form solution. The first IRCB1, similar to CFRCB, belongs to the robust constrained approach category, and is classified to DL approach. The second IRCB2, similar to IRMVB, belongs to the SV updating approach category. Theoretical analysis

−5

0

0.02

0.04

0.06

0.08

0.1

Perturbations standard deviation Fig. 6. SINR versus perturbations standard deviation.

220

Y. Li et al. / Signal Processing 118 (2016) 211–220

and simulation results have been presented to validate the iterative convergence properties on signal power estimation, SV estimation, and SINR performance for two proposed beamformers. In addition, the proposed beamformers are not subjected to the SV error norm bound constraint, and exhibit stable performance through a wide SV error bound range.

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