Applications of Robust Optimization in Signal Processing: Beamforming and Power Control Fall 2012

Transcription

1 Applications of Robust Optimization in Signal Processing: Beamforg and Power Control Fall 2012 Instructor: Farid Alizadeh Scribe: Shunqiao Sun 12/09/ Overview In this presentation, we study the applications of robust optimization in signal processing: robust beamforg in array and power control in wireless networks. First, the background of beamforg and the traditional diagonal loading (DL) method are introduced. Then, the robust beamforg designs are proposed by considering the worst-case distortionless and probabilistic constraints of the steering vector, respectively. It is shown that the robust beamforg design can be formulated as a seconder order cone programg (SOCP) problem or a semidefinite programg (SDP) problem, and then can be solved efficiently. In the last, the robust power control problem in cognitive radio networks can be formulated as a SOCP problem by considering the worst-case uncertainties in the wireless channel gains. 2 Background of Beamforg Beamforg has been widely used in wireless communications and radar systems. In practice, the performance of beamforg may undergo degradation due to the mismatch in the steering vectors. The mismatch could be a result of imperfect array calibration, look direction and signal pointing errors. Array signal processing is very sensitive to these mismatches and robust design is required [1-5]. Consider a narrow-band beamformer: y (k) = w H x (k), where k is the time index, x (k) = [x 1 (k),..., x M (k)] T C M 1 is the complex output of the array. w = [w 1,..., w M ] T C M 1 is the complex beamforg 1

2 v Target M 1 d sin r 2d sin r dr sin xm k x1 k w M Receivers d r d r w 1 Fusion center Figure 1: Uniform Linear Array Model. weights. M is the number of sensors in the array (see Figure 1). The observation vector is x (k) = s (k) a + i (k) + n (k), (1) where s (k) is the signal waveform; i (k) and n (k) are the interference and noise, respectively. Here, a the steering vector and is defined as a = [ 1, e j 2π λ dr sin θ,..., e j 2π λ (M 1)dr sin θ] T. The signal-to-interference-plus-noise ratio (SINR) is given as SINR = σ2 s w H a 2 w H R i+n w, (2) { where R i+n = E (i (k) + n (k)) (i (k) + n (k)) H} is the interference-plus-noise covariance matrix, and σ 2 s is the signal power. The beamforg vector w is chosen to maximize the SINR. The beamforg vector design problem can be formulated as w wh R i+n w s.t. w H a = 1. (3) It is well known that the solution of problem (3) is w opt = αr 1 i+na, (4) 2

3 where α = ( a H R 1 i+n a) 1. The solution is referred as the imum variance distortionless response (MVDR) beamformer. In practice, it is difficult to know the interference-plus-noise covariance matrix R i+n. So, the sample covariance matrix ^R = 1 N x (n) x H (n) (5) N n=1 is used instead of R i+n. And the sample version beamforg design problem is written as w wh ^Rw s.t. w H a = 1. (6) The solution of (6) is referred as the sample matrix inversion (SMI)-based imum variance (MV) beamforg and is written as w SMI = α^r 1 a, (7) ( 1. where α = a H ^R a) 1 For the signal free case, the optimal SINR is SINR opt = σ 2 sa H R 1 i+na. (8) The fact that sample covariance matrix is used instead of the interferenceplus noise covariance matrix would dramatically affect the beamforg performance when there is mismatch in the steering vector. The reason is as follows. In the mismatch case, the actual steering vector ã that characterizes the spatial signature of the signal is ã = a + a, (9) where is unknown complex vector that describes the mismatch of steering vector. Therefore, the SMI beamformer would view the desired signal components in the samples as interference and try to suppress them. This suppression is called signal cancellation phenomenon which will degrade the SINR. A little mismatch in the steering vector would lead to severe degradation in the SINR. One popular method to deal with the mismatch issue is diagonal loading. The basic idea of this method is to replace the sample covariance matrix ^R with the diagonal loading covariance matrix ^R dl = ξi + ^R (10) in the SMI expression (7). Here, ξ is a diagonal loading factor and I is the identity matrix. The diagonal loading method increase the variance of the artificial white noise by the amount of ξ. This modification would force the beamformer to put more effort to imize the white noise rather than the interference. As we know, when there is mismatch in the steering vector, the desired signal components are suppressed as interference. As the beamformer puts more effort in suppressing the white noise, the signal cancellation is reduced. But if ξ is too large, the beamformer fails to suppress strong interference since most effort is used to suppress the white noise. How to choose a proper diagonal loading factor ξ is not clear. 3

4 3 Uncertainty in Steering Vector: SOCP Formulation 3.1 Problem Formulation In [1], the norm of the steering vector distortion is assumed to be bounded ε, (11) where ε > 0 is some constant. The steering vector belongs to the set A (ε) = {c c = a + e, e ε}. (12) In [3], the steering vector is assumed to be lie in an n-dimensional ellipsoid ζ = {Pµ + c µ 1}. (13) The set ζ describes an ellipsoid whose center is c and whose principal semiaxes are the unit-norm left singular vectors of P scaled by the corresponding singular values. For all vectors belonging to A (ε), the absolute value of the array response should not be smaller than one w H c 1 for all c A (ε). (14) With (14), the robust beamforg can be formulated as w wh ^Rw s.t. w H c 1 for all c A (ε). (15) For each c A (ε), the constraint w H c 1 is nonconvex. Since there is an infinite number of vectors c in A (ε), there is an infinite number of constraints. Therefore, (15) is a semi-infinite nonconvex quadratic problem. We consider the worst-case constraint in (15) and rewrite the problem as w wh ^Rw s.t. c A(ε) w H c 1. (16) We rewrite the constraint in (16) as w H a + w H e 1, (17) e D(ε) where D (ε) = {e e ε}. Applying the triangle and Cauchy-Schwarz inequalities, we have w H a + w H e w H a w H e w H a ε w. (18) The equation holds with w H a > ε w and e = w w εejφ, where φ = angle { w H a }. So, we have the worst-case constraint as w H c = w H a ε w. (19) c D(ε) 4

5 The semi-infinite programg (SIP) problem (16) can be formulated as w wh ^Rw s.t. w H a ε w 1. (20) In problem (20), the constraint is nonconvex due to the absolute value operation. An important observation is that the cost function in (20) is unchanged when there is an arbitrary phase rotation in w. So if w 0 is the optimal solution to (20), we can rotate rotate the phase of w 0 without affecting the objective function, so that w H a is real. Thus, we can choose w such that Re { w H a } 0, Im { w H a } = 0. (21) So the robust beamforg problem can be rewritten as w wh ^Rw s.t. w H a ε w + 1, Im { w H a } = 0. (22) 3.2 Relationship with SMI Beamformer It is easy to see the constraint in (16) is equivalent to w H c = 1. (23) c A(ε) So the constraint in (22) is equivalent to equality constraint w H a = ε w + 1. We can rewrite the problem The Lagrange function is w wh ^Rw s.t. w H a 1 2 = ε 2 w H w. (24) H (w, λ) = w H ^Rw + λ ( ε 2 w H w w H aa H w + w H a + a H w 1 ), (25) where λ is a Lagrange multipler. Taking the gradient of H (w, λ) and equating it to zero yields With matrix inversion lemma, we get w = λ(^r + λε 2 I λaa H) 1 a. (26) λ 1a. w = ) 1a (^R + λε I) λa (^R 2 (27) H + λε 2 I 1 This indicates that the proposed robust beamforg belongs to diagonal loading methods. 5

6 3.3 SOCP Problem Implementation First, we convert the quadratic objective function in (22) to a linear one. Assume the Cholesky factorization of ^R as ^R = U H U. Then, w H ^Rw = Uw 2. Therefore, the problem (22) can be converted into the following problem By introducing τ,w τ s.t. Uw τ, ε w wh a 1, Im { w H a } = 0. (28) w = [Re ] {w} T, Im {w} T T a = [Re ] {a} T, Im {a} T T [Im ] {a} T, Re {a} T T ā = [ Re {U} Im {U} U = Im {U} Re {U} the problem (28) can be written as τ s.t. Uw τ, ε w w T a 1, τ, w ], w T ā = 0. (29) The problem (29) is a SOCP problem which can be efficiently solved. 4 Uncertainties in Steering Vector and Covariance Matrix: SDP Formulation In this section, we consider the scenario that there are uncertainties in both the steering vector and covariance matrix and the uncertainties are known to belong to a convex set U of C M H M ++. Here, H M ++ denotes the set of all Hermitian positive definite matrices of size M M. The convexity means that, for any two pairs (a 1, Σ 1 ) and (a 2, Σ 2 ) in U, η (a 1, Σ 1 ) + (1 η) (a 2, Σ 2 ) U, η (0, 1). (30) Here, we use Σ to denote the covariance. Given weight vector w, the worst-case SINR analysis problem of a finding a steering vector and a covariance matrix that imize the SINR is SINR (w, a, Σ) s.t. (a, Σ) U, (31) where the variables are a and Σ. The optimal value of the this problem is the worst-case SINR, denoted as SINR wc (w), SINR wc (w) = inf SINR (w, a, Σ). (a,σ) U 6

7 The problem (31) is a convex optimization problem as the SINR (w, a, Σ) is a convex function of a and Σ for a given weight vector w. The robust beamforg considering the worst-case SINR maximization is formulated as max SINR wc (w) s.t. w 0 (32) with variable w. The problem (32) is nonconvex. In [2], a proposition is given as below Proposition: If (a, Σ ) solve the problem sup SINR (w, a, Σ) s.t. (a, Σ) U (33) w 0 with variables a C M and Σ H M. Then (w, a, Σ ) with w = Σ 1 a satisfies the saddle-point property SINR (w, a, Σ ) SINR (w, a, Σ ) SINR (w, a, Σ), From the saddle-point property, we can show that SINR (w, a, Σ ) = w C M \ {0}, (a, Σ) U. (34) inf SINR (a,σ) U (w, a, Σ) = sup SINR (w, a, Σ ) w 0 = inf sup SINR (w, a, Σ) = sup (a,σ) U w 0 w 0 inf (a,σ) U SINR (w, a, Σ). We conclude that w = Σ 1 a solves the worst-case SINR maximization problem (32). According to (8), the SINR maximization problem (33) is equivalent to a H Σ 1 a s.t. (a, Σ) U. (35) By expanding the real and imaginary parts of a and Σ, we have a H Σ 1 a = z T R 1 z, (36) where [ ] Re {a} z = R 2M, Im {a} [ Re {Σ} Im {Σ} R = Im {Σ} Re {Σ} ] R 2M 2M. (37) Therefore, problem (35) is equivalent to z T R 1 z s.t. (z, R) V, (38) where, V is a subset of R 2M S 2M ++ V = {([ Re {a} Im {a} ], [ Re {Σ} Im {Σ} Im {Σ} Re {Σ} ]) } (a, Σ) U. (39) 7

8 Here, S 2M ++ denote the set of all 2M 2M symmetric positive definite matrices. The objective function in (38) is a matrix fractional function and is convex. V is convex since U is. Using the Schur complement, the convex problem (38) can be re-written as the following SDP problem [ ] R z t s.t. z T = 0, (z, R) V. (40) t 5 Uncertainty in Steering Vector: Probabilistic Constraints The idea of probability constraint in the beamforg is to model the steering vector statistically rather than in a deteristic way. And the beamforg maintains the distortionless response in a high probability rather than for all mismatch conditions in the uncertainty set. The probability-constrained robust beamforg problem can be written as w w H ^Rw s.t. Pr { w H ã 1 } p, (41) where p is a certain probability value. According to (9), with triangle inequality, w H (a + ) w H a w H. (42) Therefore, problem (41) can be written as w w H ^Rw s.t. Pr { w H w H a 1 } p. (43) 5.1 Gaussian Mismatch Case Let follow a complex circularly symmetric Gaussian distribution with zero mean and covariance matrix C, i.e., N C (0 M, C ). The covariance matrix C captures the second order statistics of the uncertainties in the steering vector. It is easy to show that the random variable w H (a + ) has the follow- ing complex Gaussian distribution: w H (a + ) N C (w H ) a, w 2. As C 1/2 w H is circular zero mean complex Gaussian, its real and imaginary parts are real independent identically distributed Gaussian Re { w H } N R (0, / ) C 1/2 w 2 2 Im { w H } N R (0, / ) C 1/2 w 2 2. If x and y are two real i.i.d. zero mean Gaussian random variables with variance as δ 2, then z = x 2 + y 2 is Rayleigh distributed with cumulative density function (CDF) as F (z) = 1 e z2 /2δ 2. 8

9 Thus, the constraint in (43) can be written as Pr { w H w H a ( } w H a ) = 1 exp C 1/2 w 2 p. (44) The inequality in the right hand side of (44) can be equivalently written as C 1/2 w 1 ( w H a ) 1. (45) ln (1 p) Since the cost function in (43) is unchanged when w undergoes an arbitrary phase rotation, we can choose w such that Re { w H a } 0, Im { w H a } = 0. The robust beamforg problem can be approximated by the following SOCP problem w wh ^Rw s.t. C 1/2 w 1 ln (1 p) ( w H a 1 ). (46) 5.2 Worst Mismatch Distribution Case Here, the case that the distribution of the steering vector mismatch is unknown is considered. Assume that is zero mean with given covariance. The assumptions Re { w H a } 0, Im { w H a } = 0 hold. Thus, the constraint in (43) can be written as Pr { w H < w H a 1 } p. (47) The Chebyshev inequality states that for any zero mean random variable ξ with variance σ 2 ξ and any positive real number α, Pr { ξ α} σ2 ξ α 2. (48) The constraint (47) can be lower bounded as Pr { w H < w H a 1 } C 1/2 w 2 1 (w H a 1) 2 (49) for any distribution of. The above inequality holds if C 1/2 w 2 ( < w H a ) 2. 1 The deteristic constraint is obtain as below C 1/2 w ( 1 p w H a 1 ). (50) So the robust beamforg problem can be approximated by the following SOCP problem w H ^Rw s.t. C 1/2 w w ( 1 p w H a 1 ). (51) 9

10 PU-Tx h 1 PU-Rx h n G 10 G n0 G n1 G 1n CR-Tx G 11 G nn CR-Tx CR-Rx CR-Rx CR Link 1 CR Link n Figure 2: A cognitive radio network is composed of M cognitive links randomly allocated in a cluster area. The dashed lines denote the interference channels. 6 Robust Power Control in Cognitive Radio Networks In this section, we study the robust power control problems in cognitive radio networks considering the uncertainties in the wireless channel gains. In practice, the wireless channel gains are imperfectly known or time varying. Without considering the uncertainty of these parameters, the solutions of noal problems are not robust. Consider an ad hoc cognitive radio network, i.e., there is no central control node. Assume that there are M cognitive user links randomly allocated in a cluster area, as shown in Figure 2. We assume that all the cognitive users can share the same frequency resource with primary user under the constraint that the total interference generated by cognitive users does not exceed a threshold that PU-Rx can tolerate, i.e., M h i p i I, (52) i=1 where h i denotes the channel gain between the cognitive transmitter (CR-Tx) of link i to the PU-Rx. p i represents the transmitter power of the CR-Tx of link i. I is the maximum interference level that the PU-Rx can tolerate. The signal-to-interference-noise-ratio (SINR) at the cognitive receiver (CR- Rx) of link i is SINR i = G ii p i, (53) M G ij p j + G i0 p 0 + n i j i where G ii represents the channel gain between the CR-Tx and the CR-Rx of link i. G ij is the channel gain between the CR-Tx of link j and the CR-Rx of 10

11 link i. G i0 denotes the channel gain between the primary transmitter (PU-Tx) and the CR-Rx of link i. p 0 is the transmit power of PU-Tx and n i is the background noise at the CR-Rx of link i. To guarantee the QoS requirements of cognitive users, i.e., the packet delay constraint or imum transmit rate, the SINR at each CR-Rx should be larger than a threshold, i.e., we have SINR i γ i, i, (54) where γ i is the SINR requirement at the CR-Rx of link i. Constraint (54) can be equivalently written as (D F) p m, (55) where D is a M M diagonal matrix with diagonal elements being 1 1 γ 1,..., [ ] T Vector m = N1 N G 11,..., M G MM, where Ni = G i0 p 0 + n i. F = [F ij ] with γ M. F ij = { Gij G ii, i j, 0, i = j. (56) The optimization objective is to imize the total transmit power of all CR- Txs under both the PU-Rx interference constraint and the SINR constraint of each CR-Rx. Mathematically, the problem is formulated as follows: Noal power control problem (P1) : s.t. M i=1 p i M h i p i I i=1 SINR i γ i, i, (57) where the variables are 0 p i P max for all i, and P max is the maximum transmit power for each CR-Tx. We use ellipsoid sets to describe the uncertainty. Under the worst case, the robust power control problem can be formulated as a SOCP problem. Let F i denote the uncertainty set of the ith row of matrix F. F i describes the perturbation of interfering channel gains relevant to channel gain of link i. We use an ellipsoid to describe set F i, for all i. The same method is also used in [6]. Denote the normalized channel gain from the CR-Tx of link j to the CR-Rx of link i as F ij = F ij + F ij, where F ij is the noal value, and F ij is the perturbation part. Denote the ith row of F as F i, and the corresponding perturbation part as F i. Under ellipsoid approximation, the uncertainty set of F i for F i can be written as F i = i + F i : F F ij 2 ɛ 2 i, (58) j i 11

12 where ɛ i 0 is the maximum deviation of each entry in F i. Since the channel gain vector at each CR-Rx varies independently, the row-wise structure of F i can capture the uncertainty well. Besides, the row-wise uncertainty sets of channel gains have a decomposition structure which is important in implementing the distributed algorithm. Denote the channel gain between the CR-Tx of link i and the PU-Rx as h i = h i + h i, where h i is the noal value, and h i is the corresponding deviation part. Let H be the uncertainty set of channel gain vector h, where h = [h 1,..., h M ] T. Use an ellipsoid to describe the uncertainty set H as follows { H = h + h : } h i 2 ɛ 2 0, (59) i where ɛ 0 is the maximum deviation of each entry in h. The robust power control problem that considers the channel uncertainty can be written as Robust power control problem (P2) : i p i s.t. ) ( hi + h i pi I i ) ( Fij + F ij pj j i + N i 1, i (60) p i G ii p i γ i F ij 2 ɛ 2 i, i j i h i 2 ɛ 2 0. i Since problem P2 is subject to an infinite number of constraints with respect to the sets F i (ɛ i ), for all i and H (ɛ 0 ), problem P2 is a SIP problem. A natural method to solve the SIP problem is to transform it into an equivalent problem with finite constraints. We transform the SIP problem into an equivalent problem by considering the worst case in the constraints of problem P2. With Cauchy-Schwartz inequality, we have ( max Fij F i F i F ) ij pj i = ɛ p 2 j, (61) j i j i { max h H i ( hi h i ) pi } = ɛ 0 i p 2 i. (62) Thus, under the worst case, the problem P2 is transformed as a SOCP problem with the standard form as follows 12

13 SINR CR1 CR2 CR Iteration (a) SINR CR1 CR2 CR Iteration (b) Figure 3: Performance comparisons: (a) SINR under robust algorithm; (b) SINR under non-robust algorithms. Equivalent robust problem (P3) : 1 T p s.t. ɛ 0 p 2 h T p + I (63) ( ɛ i I ei e T ) 1/2p ( ) T 2 1 i e i γ F T i p N i, i, i G ii where e i represents the ith standard basic vector, and the elements in p should satisfy 0 p i P max, for all i. The distributed algorithm is then developed to solve the SOCP problem in a distributed way. In addition, an asynchronous iterative algorithm is further proposed along with a proof of the convergence in [6]. Figure 3 gives an simulation example. Assume the SINR requirement is 10dB. It can be found that when there are channel uncertainties, under robust algorithm the SINR requirement is guaranteed while the outage events happen frequently under non-robust algorithms. References [1] S. A. Vorobyov, A. B. Gershman and Z. Q. Luo, Robust adaptive beamforg using worst-case performance optimization: A solution to the signal mismatch problem, IEEE Trans. on Signal Processing, vol. 51, no.2, pp , [2] S. J. Kim, A. Magnani, A. Mutapcic, S. Boyd and Z. Q. Luo, Robust beamforg via worst-case SINR maximization, IEEE Trans. on Signal Processing, vol. 56, no.4, pp , [3] R. G. Lorenz and S. Boyd, Robust imum variance beamforg, IEEE Trans. on Signal Processing, vol. 53, no.5, pp ,

14 [4] A. B. Gershman, N. D. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson and B. Ottersten, Convex optimization-based beamforg, IEEE Signal Processing Magazine, vol. 27, no. 3, pp , [5] S. A. Vorobyov, H. Chen and A. B. Gershman, On the relationship between robust imum variance beamformers with probabilistic and worst-case distortionless response constraints, IEEE Trans. on Signal Processing, vol. 56, no.11, pp , [6] S. Sun, W. Ni and Y. Zhu, Robust power control in cognitive radio networks: A distributed way, in Proc. IEEE International Conference on Communications (ICC), Kyoto, Japan, June, [7] A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski, Robust Optimization. Princeton University Press,