Constant Modulus Algorithms via Low-Rank Approximation

Transcription

1 CBMM Memo No. 77 April 2, 28 Constant Modulus Algorithms via Low-Rank Approximation A. Adler M. Wax Abstract We present a novel convex-optimization-based approach to the solutions of a family of problems involving constant modulus signals. The family of problems includes the constant modulus the constrained constant modulus, as well as the modified constant modulus the constrained modified constant modulus. The usefulness of the proposed solutions is demonstrated for the tasks of blind beamforming blind multiuser detection. The performance of these solutions, as we demonstrate by simulated data, is superior to existing methods. Keywords: Constant modulus, convex optimization, trace norm. This work was supported by the Center for Brains, Minds Machines (CBMM), funded by NSF STC award CCF-2326.

2 Constant Modulus Algorithms via Low-Rank Approximation Amir Adler, Member, IEEE, Mati Wax, Fellow, IEEE Abstract We present a novel convex-optimization-based approach to the solutions of a family of problems involving constant modulus signals. The family of problems include the constant modulus the constrained constant modulus, as well as the modified constant modulus the constrained modified constant modulus. The usefulness of the proposed solutions is demonstrated for the tasks of blind beamforming blind multiuser detection. The performance of these solutions, as we demonstrate by simulated data, is superior to existing methods. Index Terms Constant modulus, constrained constant modulus, modified constant modulus, constrained modified constant modulus. convex optimization, trace norm. I. INTRODUCTION Constant modulus algorithms are based on exploiting the constant modulus of the desired signal. They are used in a variety of areas in signal processing ranging from blind equalization blind beamforming to blind multiuser detection. The constant modulus (CM) algorithm was first introduced in the works of Godard [] Triechler Agee [2] on blind equalization. In these works, the linear equalizer weight vector was computed by exploiting the constant modulus of the desired signal, without any explicit learning of the channel impulse response, was therefore referred as blind. Subsequently, Triechler Larimore [3] introduced a CM algorithm for extracting a desired constant modulus signal in the presence of non-constant-modulus interfering signals. It is referred to as blind beamforming, since the only information exploited is the constancy of the modulus of the desired signal. The extension of the CM algorithm to allow additional linear constraints to be imposed on the weight vector, referred to as linear constrained constant modulus (LCCM) algorithm, was introduced by Rude Griffith [4]. Miguez Castro [] then proposed a LCCM algorithm for multiuser CDMA communications, aimed at detecting the data of one user in the presence of the data of the other users, acting as interference. It is referred to as blind multiuser detection (MUD) since the only information exploited is the constant modulus of the CDMA signals the spreading code of the desired user. Note that since all the signals are constant-modulus, hence indistinguishable by the CM algorithm, the LCCM algorithm was necessary in order to single out the desired signal. Another extension of the CM algorithm was proposed in [6], referred to as the modified constant modulus (MCM) algorithm. This modification was motivated by the insensitiveness of the A. Adler adleram@mit.edu. M. Wax matiwax@gmail.com. This work was supported by the Center for Brains, Minds Machines (CBMM), funded by NSF STC award CCF CM algorithm to the phase of the signal carrier, was aimed at enabling improved performance for high order QAM constellation for carrier phase synchronization. In all these algorithms, as well as in more recent developments [7], [8], the computation of the weight vector is based on a multidimensional non-convex cost function with multiple local minima [9], making global minimization very challenging. In this paper we present a novel framework based on convex formulations of the CM CCM cost functions, as well as of the MCCM the linearly constrained MCM (LCMCM) cost functions. The solutions assure global optimality are parameter free, i.e, they do not contain any tuneable parameter do not require any a-priori parameter setting. The performance of these solutions is better than the existing CCM based solutions, reaching the theoretical performance limit with a much lower number of samples. The rest of the paper is organized as follows. The problem formulation is presented in section II. Section III describes the convex CM solution. Section IV describes the convex CCM solution. Sections V VII present the convex solutions for the MCM for the LCMCM, respectively. Section VII discusses the blind beamforming problem, while section VIII discusses the blind multiuser detection problem. The performance analysis is presented in section IX. Finally, section X presents the conclusions. II. PROBLEM FORMULATION Let the received signal x(t) be a P complex vextor given by: x(t) = a s (t) + K a k s k (t) + n(t), () k=2 where s (t) is a desired constant modulus (CM) signal, {s k (t)} K k=2 are non-constant-modulus interfering signals, {a k } K k= are unknown vectors, n(t) is the P noise vector. We further assume that the number of signals obeys K P that the vectors {a k } K k= are linearly independent. The constant modulus problem can be formulated as follows: Given the received vectors {x(t n )} N,, find a P weight vector w such that the linear combiner output y(t) = w H x(t), where H denotes the conjugate transpose, provides a good estimate of the CM signal s (t). Assuming, without loss of generality, that the square of the modulus of the desired signal s (t) is R, the common CM cost function for estimating the linear combiner weight w is given by minimization of the sample-average of the deviation of the linear combiner power output from R:

3 2 ŵ = argmin w N ( w H x(t n ) 2 R) 2. (2) This is a fourth order minimization problem in the vector w, as such does not admit a closed form solution. Moreover, as shown in [3]-[4], it is a non-convex problem (i.e. it has multiple local minima), making global minimization very challenging. We next show how to reformulate the CMA as a convex optimization problem, which assures global optimality. III. CONVEX CONSTANT MODULUS ALGORITHM First, we rewrite the linear combiner power output, denoted by z(t), as : z(t) = w H x(t) 2 = w H x(t)x(t) H w (3) = tr(ww H x(t)x(t) H ) = tr(wx(t)x(t) H ), where tr() denotes the trace of the bracketed matrix W denotes the P P positive semidefinite (PSD) rank- matrix: We can now rewrite (2) as: Ŵ = argmin W W = ww H. (4) N z(t n ) R 2, (a) z(t n ) = tr(wx(t n )x(t n ) H ), n =,..., N, (b) W, rank W =, (c) (d) where W denotes the PSD constraint. Note however, that since the rank constraint (d) is not convex, the minimization problem is not convex. A commonly-used convex relaxation surrogate to the rank- constraint is to minimize the trace norm (nuclear norm), defined as the sum of the singular values of the matrix. Recalling that W is a PSD matrix, it follows that its trace norm is given by tr(w). This implies that we can reformulate the CM problem as the following convex optimization problem: Ŵ = argmin{( W N z(t n ) R 2 ) + tr(w)}, (6a) tr(wx(t n )x(t n ) H ) = z(t n ), n =,..., N, (6b) W. (6c) Since (6) is a convex optimization problem, we can use any of the convex optimization solvers [], [] to solve for Ŵ. We use the following properties of the trace operator tr(): (i) cyclic shift: tr(abcd) = tr(bcda) = tr(cdab) = tr(dabc); (ii) tr(a) = a for any scalar a. With Ŵ at h, a straightforward way to estimate the weight vector w is by the rank- approximation of Ŵ: Ŵ λ v v H, (7) where λ denotes the largest eigenvalue of Ŵ, v denotes the eigenvector of Ŵ corresponding to λ. Using this rank- approximation, we estimate the weight vector w as: ŵ = v. (8) IV. CONVEX LINEARLY CONSTRAINED CMA In many scenarios involving CM signals, it may be desired to impose additional constraints on the weight vector w in the form of the following linear constraint: w H C = v H, (9) where C v are P J matrix J vector, respectively, assumed to be known. This problem is referred to as LCCM. To incorporate the linear constraint (9) into our convex CMA formulation, we first rewrite it as w H c j = v j, j =,..., J, () where c j denotes the j-th column of C v j denotes the j-th element of v. Now, using the properties of the trace operator (9), we have tr(ww H c j c H j ) = tr(c H j ww H c j ) = tr(v j v H j ) = v j 2, () which implies that we can rewrite the linear constraint () as, tr(wc j c H j ) = v j 2, j =,..., J,. (2) The convex LCCM cost function can now be formulated as: Ŵ = argmin{( z(t n ) R 2 ) + tr(w)}, (3a) W N tr(wx(t n )x(t n )) H ) = z(t n ) n =,..., N, (3b) tr(wc j c H j ) = v j 2 j =,..., J, (3c) W. (3d) V. CONVEX MODIFIED LINEARLY CONSTRAINED CMA The CM cost function is insensitive to the phase of the signal carrier. Therefore, in the presence of an unknown phase rotation the resulting estimated signal will also be rotated. A phase-sensitive modification of the CM cost function, referred to as the MCM, was introduced in [6], aimed at enabling improved performance for high order QAM carrier phase synchronization. To introduce the MCM cost function, let w R, w I y R (t), y I (t) denote, respectively, the real imaginary parts of w y(t). Using this notation, the MCM cost function can be written as: ŵ = argmin w N [( y R (t n ) 2 R R ) 2 + ( y I (t n ) 2 R I ) 2 ], (4a)

4 3 where R R = E s R(t) 4 E s R (t) 2, R I = E s I(t) 4 E s I (t) 2, (4b) (4c) where s R (t) s I (t) denote the real imaginary parts of s(t). To reformulate the MCM cost function as a convex optimization problem, we first rewrite y R (t) y I (t) as where y R (t) = w T x (t), y I (t) = w T x 2 (t). w = x (t) = x 2 (t) = [ wr (a) (b) ], w I (c) [ ] xr (t), x I (t) (d) [ ] xi (t). (e) x R (t) Using the properties of the trace operator, we have z (t) = y R (t) 2 = w T x (t)x T (t) w = tr( Wx (t)x T 2 (t)), (6a) z 2 (t) = y I (t) 2 = w T x 2 (t)x T 2 (t) w = tr( Wx 2 (t)x T 2 (t)), (6b) where W denotes the 2P 2P rank- matrix W = w w T. (7) With this notation we can rewrite the MCM cost function as ˆ w = argmin [(z (t n )) R R ) 2 +(z 2 (t n ) R I ) 2 ], (8a) w N subject to z (t n ) = tr( Wx (t n )x T (t n )), n =,..., N, (8b) z 2 (t n ) = tr( Wx 2 (t n )x T 2 (t n )), n =,..., N, (8c) W, rank W =. (8d) (8e) Using the trace norm as a surrogate for the non-convex rank- constraint, we can reformulate the MCM cost function as the following convex optimization problem: ˆ W = argmin N W [(z (t n ) R R ) 2 +(z 2 (t n ) R I ) 2 ]+tr( W), (9a) z (t n ) = tr( Wx (t n )x T (t n )), n =,..., N, (9b) z 2 (t n ) = tr( Wx 2 (t n )x T 2 (t n )), n =,..., N, (9c) W, (9d) We next show that for a symmetric constellation, the solution W of this optimization problem has two different rank- solutions corresponding to two linearly independent vectors. To this end, note that from (d) (e) we have x 2 (t) = Gx (t), (2) where G is the 2(P + L ) 2(P + L ) block matrix [ ] I G =. (2) I This enables us to rewrite z 2 (t) as z 2 (t) = tr( Wx 2 (t)x T 2 (t)) = tr( WGx (t)x T (t)g T ), (22) or alternatively, using the properties of the trace operator, as z 2 (t) = tr( Wx (t)x T (t)), (23) where W is given by W = G T WG, (24) which, by using (4), can be rewritten as where w is given by W = w w T, (2) w = G T w, (26) which implies that w w are linearly independent. Now, since the optimization process yields z (t) R R, z 2 (t) R I, since in a symmetrical constellation we have it follows that (27a) (27b) R R = R I (28) z (t) z 2 (t). (29) This implies, as can be easily verified, that tr( Wx (t)x T (t) tr( Wx (t)x T (t), tr( Wx 2 (t)x T 2 (t) tr( Wx2 (t)x T 2 (t). (3a) (3b) which shows that both W W are feasible solutions, corresponding to two rank- solutions given by w w. Thus, given the solution ˆ W of (9), we can estimate vectors w w, from the rank-2 approximation of ˆ W, as: ˆ w = v ˆ w = v 2 (3a) (3b) where v v 2 denote the two largest eigenvectors of ˆ W. As in the CM formulation, we can incorporate additional

5 4 linear constraints on the vector w. This problem is referred to as the Constrained Modified Constant Modulus (CMCM). To incorporate the constraints () into our convex MCM formulation, let c Rj, c Ij v Rj, v Ij denote, respectively, the real imaginary parts of c j v j. Using this notation we can rewrite () as where This implies that w T c j = v Rj, j =,..., J, (32a) w T c 2j = v Ij, j =,..., J, (32b) c j = c 2j = [ ] crj, (32c) c Ij [ cij c Rj ]. (32d) w T c j c T j w = v 2 R j, j =,..., J, (33a) w T c 2j c T 2 j w = v 2 I j, j =,..., J, (33b) which, using the properties of the trace operator, can be rewritten tr( W T c j c T j ) = v 2 R j, j =,..., J, (34a) tr( W T c 2j c T 2 j ) = v 2 I j, j =,..., J,. (34b) Thus, combining these two equation, we can rewite () as tr( W T c j c T j ) + tr( W T c 2j c T 2 j ) = v j 2, j =,..., J, (34c) The convex formulation of CMCM is therefore given by ˆ W = argmin{ N subject to W [(z (t n ) R R ) 2 +(z 2 (t n ) R I ) 2 ]+tr( W), (3a) z (t n ) = tr( Wx (t n )x T (t n )), n =,..., N, (3b) z 2 (t n ) = tr( Wx 2 (t n )x T 2 (t n )), n =,..., N, (3c) tr( W T c j c T j ) + tr( W T c 2j c T 2 j ) = v j 2, j =,..., J, (3d) W, (3e) VI. CONVEX CMA FOR BLIND BEAMFORMING Consider an antenna array composed of P antennas with arbitrary locations arbitrary directional characteristics. Assume that a desired signal s (t) is impinging on the array from an unknown direction-of-arrival θ that K other interfering signals s k (t), k = 2,..., K, are impinging on the array from unknown directions-of-arrival θ 2,..., θ K. All the signals are assumed to be narrow-b, namely that the array aperture, denoted by d, obeys d << c/b, where c is the speed of light, B is the signals bwidth. Under these assumptions, the P array vector x(t) of the complex envelopes of the received signals can be written as: x(t) = a(θ )s (t) + K a(θ k )s q (t) + n(t), (36) k=2 where a(θ ) is the P steering vector of the array toward the desired CM signal s (t), a(θ q ) is the P steering vector of the array toward the interfering signal s q (t), n(t) is the P noise vector. Since (36) is in the form of (), the estimation of the desired CM signal s (t) from the received array vectors {x(t n )} N, can be readily done using the unconstrained convex CMA (6), summarized in Algorithm, or the convex MCM (9), summarized in Algorithm 2. Algorithm Convex CMA Blind Beamforming : Input: Received array vectors {x(t n)} N, 2: Solve: Ŵ = argmin{ z(t n) R 2 + tr(w)}, W N tr(wx(t n)x(t n) H ) = z(t n), n =,..., N, W. 3: Compute: v = Largest Eigenvector of Ŵ 4: Output: ŵ = v. Regarding the constrained algorithms (6) (3), it is worth while to point out some useful linear constraints, which are special cases of (9). An example for such a constraint is the well-known look direction constraint: w H a(θ) =, (37) constraining w to have a unity gain in the direction θ. Another example is the constraint, w H B =, (38) constraining w to be orthogonal to the columns of B. One example for such a B is B = a(θ), (39) assuring deep nulls in the direction θ. This may be desired, for example, in case a strong interference is known to be impinging from direction θ the desire is to put a deep null in this direction. Another example is B = [v K+,..., v P ], (4) where v i is the eigenvector of the array covariance matrix ˆR = N x(t n)x(t n ) H corresponding to the i-th eigenvalue. This constraints w to be orthogonal to the noise subspace, i.e., to be confined to the K-dimensional signal subspace [2]. This low-dimensional confinement reduces the number of degreesof-freedom of w, thereby improving the solution performance, especially in challenging conditions such as small number of samples low signal-to-noise ratio.

6 Algorithm 2 Convex Modified CMA Blind Beamforming : Input: Received array vectors {x(t n)} N, 2: Set: [ ] xr(t) x (t) =, x I(t) 3: Solve: ˆ W = argmin N W x 2(t) = [ ] xi(t). x R(t) [(z (t n) R R) 2 +(z 2(t n) R I) 2 ]+tr( W), z (t n) = tr( Wx (t n)x T (t n)), n =,..., N, z 2(t n) = tr( Wx 2(t n)x T 2 (t n)), n =,..., N, W, 4: Compute: v = Largest Eigenvector of ˆ W : Set: [ ] wr ˆ w = = v w. I 6: Output: ŵ = w R + jw I. VII. CONVEX LINEARLY CONSTRAINED CMA FOR BLIND MULTIUSER DETECTION Consider a symbol-synchronous un-coded CDMA system with K simultaneous users. Let s k denote the P spreading code of user k, assuming that the spreading codes are normalized, i.e, s k =. Let b k (t) denote the the transmitted QAM symbols to user k, let B denote constellation alphabet from which b k (t) take its value. Assuming multipath free propagation, the P vector x(t) of the complex envelopes of the signals received by a single user, say user, after filtering by a chip-pulse matched filter sampled at chip rate, can be written as [] x(t) = A b (t)s + K A k b k (t)s k + n(t), (4) k=2 where A k s k are the received amplitude the spreading code, respectively, of the k-th user. To insure identifiability, we further assume that the number of users obeys K L, that the signals spreading codes {s k )} K k= are linearly independent. The estimation of user symbols {b (t n )} N, from the received vectors {x(t n )} N,, is referred to as blind multiuser detection. Note that (4) is in the form of (), with the difference that here all the other interfering signals {b k (t)} K k=2 are also constant modulus. In this case the unconstrained algorithms (6) (3) will capture the desired signal only if it is stronger than the interfering signals. To assure capturing of the desired signal in this case it necessary to incorporate the following desired user unit gain constraint: w H s =, (42) Incorporating this constraint in the convex CCM (3), as summarized in Algorithm 3, or in the convex CMCM (3) readily solves the blind multiuser detection problem. Algorithm 3 Convex LCCM Blind Multiuser Detection : Input: Chip-matched filter vectors {x(t n)} N,, desired user signature s. 2: Solve: Ŵ = argmin{ W N z(t n) R 2 + tr(w)}, tr(wx(t n)x(t n) H ) = z(t n), n =,..., N, tr(ws s H ) =, W. 3: Compute: v = Largest Eigenvector of Ŵ 4: Output: ŵ = v. In case of multipath propagation the situation is slightly more complicated. Assuming a maximum delay of LT c, where T c is the chip duration, the (P + L ) vector x(t) of the complex envelopes of the signals received by a single user, say user, after filtering by a chip-pulse matched filter sampled at chip rate, can be written as [], K x(t) = A b (t)s h +u (t)+ (A k b k (t)s k h k +u k (t))+n(t), k=2 (43) where S k is the (P + L ) L matrix whose columns are shifted versions of the spreading code s k, s k S k = s k , (44) s k h k is the L vector of the channel response of user k, u k (t) is the inter-symbol interference (ISI) for user k from the adjacent symbols, n(t) is the (L + P ) noise vector. To insure identifiability in this case, we further assume that the number of users obeys 2K (L + P ). The blind multiuser detection in this case can be similarly solved using the constrained algorithms described above, by using the following linear constraints: where c is given by w H c =, (4) c = S h, (46) with h S assumed to be known. Since h is typically unknown in practice, it is estimated from the data, as discussed in [4]-[]. Another useful constraint is given by, w H C =, (47)

7 6 constraining w to be be orthogonal to the columns of C. One example for such a C, presented here for simplicity for the multipath free case is given by C = s k, (48) assuring deep nulls towards spreading code s k. This may be desired, for example, in case a strong interference is known to have a spreading code s k, it desired to mitigate it by a deep null. Another example, again presented for simplicity for the multipath free case is C = [v K+,..., v P ] (49) where v i is the eigenvector corresponding to the i-th eigenvalue of the covariance matrix ˆR = N x(t n)x(t n ) H. This constraints w to be orthogonal to the noise subspace [29], i.e., to be confined to the K-dimensional signal subspace spanned by the vectors {x(t n )} N. This confinement to a low-dimensional subspace reduces the number of degrees-offreedom of w, thereby improves the solution performance, especially in challenging conditions such as small number of samples low signal-to-noise ratio. VIII. PERFORMANCE ANALYSIS A. Blind Beamforming Performance Evaluation In this section we present blind beamforming simulation results illustrating the performance of the proposed solution, referred to as Trace Norm. The performance is compared to the Recursive Least Squares (RLS) [3] the Unscented Kalman Filter (UKF) [8] the Constrained CM-RLS (CCM-RLS) [4] solutions. The desired signal was simulated as a unit power QPSK signal. The interfering signals were simulated as complex Gaussian with zero mean unit variance. The noise was simulated as a complex Gaussian with zero mean covariance σ 2 ni. The performance measure employed is the signal-tointerference-plus-noise ratio (SINR) at the beamformer output: SINR = w H R ss w w H R nn w + w H R ii w, () where R ss = a(θ )a(θ ) H, R nn = σni, 2 R ii = q j= a(θ j)a(θ j ) H are the CM signal, noise interference covariance matrices, respectively. All presented results are averaged over experiments, employ a uniform linear array (ULA) with P = 6, unless specified differently. Before discussing the results of the simulated experiments, we would like to discuss the computation time of the Trace Norm solution its suitability to real time communication systems. To this end, we evaluated the computation time of the simulated experiments using the MATLAB CVX [] toolbox 2. The computation time for a typical scenario with N = samples, consisting of a CM signal impinging from 2 on a Uniform Linear Array (ULA) with P = 6 3 interferers impinging from 4, 4, all having 2 Using an Intel Core i793k, 32GB RAM, desktop computer. SNR of db, is second. Since the average speed-up factor between CVX-based implementation a real-time implementation, as analyzed in [6], is (single processor), this implies a ms in real-time implementation, which is highly suitable to packet-based communications, especially in low mobility. Experiment evaluates the ratio between the largest (λ ) the second largest (λ 2 ) eigenvalues of Ŵ, which is a good measure for the goodness of the rank- approximation of the trace norm solution of Ŵ. We evaluated this ratio by simulating times 3 each of the following scenarios: a CM signal in the presence of,, or 2 interferers, all signals having equal power, at SNR of db or 2dB (σn 2 =. or., respectively). For the case of no interference, the ratio λ λ 2 exceeded 6 with probability, implying a perfect rank- result. Fig. (a) presents the results for the cases of 2 λ interferers, reveals that with probability, λ 2 for SNR = db, λ λ 2 for SNR = 2dB. These results demonstrate the goodness of the rank- approximation of Ŵ. Experiment 2 evaluates the performance of the solution in the presence of two CM signals: The first from 2 with unit power, the second from, attenuated in each trial by a rom attenuation, uniformly distributed between db to db. Fig. (b) presents the averaged array pattern, over experiments, demonstrates the capture effect of the solution: the algorithm captures always the strongest CM signal, cancels the weaker Experiment 3 compares the SINR of the, UKF RLS, in the presence of interferers. Note that the UKF the RLS are sequential algorithms, i.e., operating on the samples sequentially, from the first to the last, say n, while the is a batch algorithm operating on all the n samples simultaneously. In the first scenario we simulated a CM signal impinging from 2, with 3 interfering signals impinging from 4,, 4, noise variance σ 2 n =.. The results are presented in Fig. 2(a) demonstrate that the Trace- Norm solution yields better SINR converges after samples, as compared to the UKF RLS, which require 2, 7 samples, respectively, to converge. Fig. 2(b) presents the performance with an additional interferer from 6. In this case convergence of the UKF RLS is slower (, 3, samples, respectively), whereas the Trace- Norm is essentially invariant to the additional interferer, surpasses UKF RLS with only samples. The array pattern of the Trace Norm with 2 samples (averaged over, experiments), is depicted in Fig. 2(c). The rejection of all 4 interferers is clearly visible. Experiment 4 evaluates the robustness of the proposed linearly-constrained solution for the case of a look direction error. In this scenario, the constraint is w H ã(θ) = where a(θ) = a(θ) + (θ E ), with being the steering vector error 3 Each solution treated different transmitted symbols, different noise realization, different interfering signals waveforms.

8 7.9.8 Interferers, SNR = db Interferers, SNR = 2dB 2 Interferers, SNR = db 2 Interferers, SNR = 2dB Pattern st CM Signal 2nd CM Signal.7 Prob(λ / λ 2 )> X.6..4 Gain [db] X [λ / λ 2 ] Angle [Degrees] Fig.. (a) The ratio between the first second largest eigenvalues of Ŵ. (b) Averaged array pattern of the solution, over experiments, with two CM signals: unit power from 2, attenuated by rom attenuation (db to db) from UKF RLS Samples UKF RLS Samples Fig. 2. (a) Output SINR of the, UKF RLS (p =, λ =.98, δ =.) vs. number of samples (N), with noise variance σ 2 n =., CM signal at 2 ; 3 interferers at 4, 4 ; (b) with a 4-th interferer at 6. Pattern CM Signal Interferer Interferer 2 Interferer 3 Interferer Gain [db] Angle [Degrees] Fig. 3. Array pattern of the solution, with the 4 interferers (Mismatch range [-,+] Degrees) CCM-RLS (Mismatch range [-,+] Degrees) (Mismatch range [,+] Degrees) CCM-RLS (Mismatch range [,+] Degrees) Samples Fig. 4. SINR vs. steering angle error of the LCCMA CCM- RLS [].

9 8 Gain [db] Pattern CM Signal Interferer Interferer 2 Interferer 3 Constraint Constraint Angle [Degrees] Fig.. LCCMA array pattern, with null constraints at 3, 6 3 interferers (N = 2 samples, SNR = db). 2 2 Signal-Subspace Constrained Samples Fig. 6. SINR of the vs. the Signal-Subspace Constrained Trace- Norm (P = 32 elements) in the presence of a CM signal at 2, 2 interferers at 4, 2 : SNR = db (lower curves); SNR = db (middle); SNR = db (upper). component, resulting from a steering angle error θ E. Fig. 3(a) compares output SINR performance of the LCCMA to the CCM-RLS [4] for the following scenario: the CM signal of interest is impinging from 2 two CM interferers are impinging from 4 6, all QPSK modulated with SNR of 2dB. Performance was evaluated for a steering angle mismatch error uniformly distributed in the range of [, + ] [, + ]. The robustness of the Trace Norm algorithm is clearly visible, in addition to the faster convergence rate, as compared to CCM-RLS. Experiment demonstrates the ability of the LCCMA to generate deep nulls in the array pattern in predefined directions, using the constraint (),(2). The simulated scenario includes a CM signal at 2, 3 interferers from 4, 4 (σ 2 n =.). The nulls are constrained to directions 3 6. The resulting array pattern, averaged over, experiments, is depicted in Fig. 3(b). Clearly visible is the rejection of all interferers, as well as the deep nulls in the specified directions. Experiment 6 demonstrates the performance advantage of the Trace Norm LCCMA over the CMA when the constraint (),(3) is imposed. The simulated scenario includes a CM signal impinging from 2 on a P = 32 elements ULA, with 2 interferers impinging from 4 2. The SNR per array element is varied between db to db. The constraint (),(3) forces the beamforming vector to be confined to the 3-dimensional signal subspace. Fig. 3(c) shows SINR results vs. the number of samples (N). The results demonstrate the advantage of the Trace Norm LCCMA over the CMA for all signal-to-noise ratios (excluding a minor disadvantage for SNR=dB N > 3 samples). B. Blind MUD Performance Evaluation In this section we present blind MUD simulation results illustrating the performance of the proposed Trace Norm solution, as compared to the Linearly Constrained CMA RLS (LCCMA-RLS) [4], The Minimum Output Energy (MOE- MUD) [7], the Subspace-based blind multiuser detector (SUB-MUD) [7], the Minimum Mean Squared Error with Tikhonov Reguralization (MMSE-Tikhonov) [8]. The MOE-MUD detector under the constraint w H s = is given by [7]: ŵ = (s H ˆR s ) ˆR s, () where ˆR is the estimated covariance matrix. The blind subspace-based MUD requires explicit knowledge of the number of users K, is given by [7]: ŵ = (s H U s Λ s U H s s ) U s Λ s U H s s, (2) where U s Λ s are computed from the eigenvalue decomposition of the estimated covariance ˆRU = UΛ. U s includes the K leading eigenvectors of U, spans the signal subspace. Similarly, Λ s includes the K leading eigenvalues of Λ. The Minimum Mean Squared Error with Tikhonov Regularization is given by [8]: ŵ = [s H ( ˆR + αi) s ] ( ˆR + αi) s, (3) where α = m tr( ˆR) (m =. in our experiments). All users were simulated as QPSK signals, spread by Gold sequences with P = 3 chips. The desired user was simulated with unit power, whereas the interfering users were simulated with amplitudes A k >. The noise was simulated as a complex Gaussian with zero mean covariance σni. 2 The performance measure employed is the SINR at the MUD detector output: SINR = w H R ss w w H R nn w + w H R ii w, (4) where R ss = s s H, R nn = σ 2 ni, R ii = K k=2 A2 k s ks H k, are the desired signal, noise multiple-access interference (MAI) covariance matrices, respectively. All presented results

10 SINR Limit CCM-RLS MOE-MUD MMSE-TIKHONOV SUB-MUD Number of Symbols Fig. 7. MUD output of the solution vs. CCM-RLS, MOE, MMSE-TIKHONOV Subspace-based MUD. P=3 chips, K= users, QPSK modulation, SNR=dB, Interference-to-Signal-Ratio=dB. -2 SINR Limit CCM-RLS MOE-MUD MMSE-TIKHONOV SUB-MUD Number of Symbols Fig. 8. MUD output of the solution vs. CCM-RLS, MOE, MMSE-TIKHONOV Subspace-based MUD. P=3 chips, K= users, QPSK modulation, SNR=dB, Interference-to-Signal-Ratio=dB. were averaged over experiments. Experiment evaluates MUD SINR output as a function of the number of symbols (2 to 2). Figure 7, presents the results with K = users, interference-to-signal ratio of db, SNR of db, demonstrating the superior performance of the. Figure 8, presents the results with K = users, further demonstrating the superior performance of the, in this challenging scenario. The SINR limit was computed using the ground truth covariance matrix R = K k= A2 k s ks T k + σ2 ni, by computing w using the MMSE detector with ˆR = R. Experiment 2 evaluates MUD SINR output as a function of the number of symbols (2 to 2), in the presence of spreading code mismatch due to multi-path propagation: the code of the desired user was distorted by the channel h = [.92,,.,,.2], resulting in an average correlation of.97 between the correct distorted codes. Figure 9, presents the results with K = users, interference-to-signal ratio of db, SNR of db, demonstrating the superior performance of the. Figure, presents the results with K = users, further demonstrating the superior performance of the, in this challenging scenario. Experiment 3 evaluates SINR performance of the Trace- Norm solution with the multi-path constraints (4)-(46), in the presence of the channel h = [.92,,.,,.2], as depicted in Fig., with K = users, interference-to-signal ratio of db, SNR of db. Fig. 2 demonstrates the results with K = users a more challenging channel h = [.8,.2,.,.4,.,.] (of longer delay spread). Fig. 3, presents the constellation at the detector output SINR Limit CCM-RLS -2 MOE-MUD MMSE-TIKHONOV SUB-MUD Number of Symbols Fig. 9. MUD output in the presence of spreading code mismatch due to multi-path propagation with the channel h = [.92,,.,,.2]. Spreading codes of P=3 chips, K= users, QPSK modulation, SNR=dB, Interference-to-Signal-Ratio=dB. IX. CONCLUSIONS We have presented a new convex-optimization-based approach to the constant modulus problem, to the related problems of linearly constrained constant modulus modified constant modulus. This approach is based on casting these problem as rank- matrix minimization problems, then transforming them to convex optimization problems by replacing the rank- constraint by its convex surrogate - the minimization of the trace norm. As solutions to convex optimization problems, the proposed solutions are free from the local minima problem hindering the existing solutions. We have demonstrated the effectiveness of the proposed solutions in simulated experiments of typical scenarios in blind beamforming blind multiuser detection. In all these experiments the proposed solution have shown superior per-

11 SINR Limit CCM-RLS MOE-MUD MMSE-TIKHONOV SUB-MUD Number of Symbols Fig.. MUD output in the presence of spreading code mismatch due to multi-path propagation with the channel h = [.92,,.,,.2]. Spreading codes of P=3 chips, K= users, QPSK modulation, SNR=dB, Interference-to-Signal-Ratio=dB. 4 SINR Limit Number of Symbols Fig. 2. SINR performance with multi-path constraints (4)- (46), the channel h = [.8,.2,.,.4,.,.], spreading codes of P=3 chips, K= users, QPSK modulation, SNR=dB, Interferenceto-Signal-Ratio=dB (a). (b) SINR Limit Number of Symbols Fig.. SINR performance with multi-path constraints (4)- (46), the channel h = [.92,,.,,.2], spreading codes of P=3 chips, K= users, QPSK modulation, SNR=dB, Interference-to-Signal- Ratio=dB. formance over the existing solutions, especially in challenging conditions of low number of samples/symbols. REFERENCES [] D. Godard, Self-recovering equalization carrier tracking in twodimensional data communication systems, IEEE Transactions on Communications, vol. 28, no., pp , Nov 98. [2] J. Treichler B. Agee, A new approach to multipath correction of constant modulus signals, IEEE Transactions on Acoustics, Speech, Signal Processing, vol. 3, no. 2, pp , Apr 983. [3] J. Treichler M. Larimore, New processing techniques based on the constant modulus adaptive algorithm, IEEE Transactions on Acoustics, Speech, Signal Processing, vol. 33, no. 2, pp , Apr 98. [4] Michael John Rude, A Linearly Constrained Adaptive Algorithm for Constant Modulus Signal Processing, Ph.D. thesis, Los Angeles, CA, USA, 99, AAI [] J. Miguez L. Castedo, A linearly constrained constant modulus approach to blind adaptive multiuser interference suppression, IEEE Communications Letters, vol. 2, no. 8, pp , Aug 998. Fig. 3. performance with multi-path constraints (4)-(46), the channel h = [.8,.2,.,.4,.,.], spreading codes of P=3 chips, K= users, QPSK modulation, SNR=dB, Interference-to- Signal-Ratio=dB: (a) Detector input; (b) detector output (2 symbols, SINR = 2.6dB). [6] Changjiang Xu, Guangzeng Feng, Kyung Sup Kwak, A modified constrained constant modulus approach to blind adaptive multiuser detection, IEEE Transactions on Communications, vol. 49, no. 9, pp , Sep 2. [7] L. Lau, R. C. de Lamare, M. Haardt, Robust adaptive beamforming algorithms using the constrained constant modulus criterion, IET Signal Processing, vol. 8, no., pp , July 24. [8] M. Z. A. Bhotto I. V. Baji, Constant modulus blind adaptive beamforming based on unscented kalman filtering, IEEE Signal Processing Letters, vol. 22, no. 4, pp , April 2. [9] A. Leshem A. J. van der Veen, On the finite sample behavior of the constant modulus cost, in 2 IEEE International Conference on Acoustics, Speech, Signal Processing. Proceedings (Cat. No.CH37), 2, vol., pp vol.. [] S. Boyd L. Venberghe, Convex Optimization, Cambridge University Press, 24. [] Shuiwang Ji Jieping Ye, An accelerated gradient method for trace norm minimization, in Proceedings of the 26th Annual International Conference on Machine Learning. 29, ICML 9, pp , ACM.

12 [2] H.L Van Trees, Optimum array processing, part IV of Detection, estimation, modulation theory, Wiley, 24. [3] Yuxin Chen, T. Le-Ngoc, B. Champagne, Changjiang Xu, Recursive least squares constant modulus algorithm for blind adaptive array, IEEE Transactions on Signal Processing, vol. 2, no., pp , May 24. [4] L. Wang R. C. de Lamare, Constrained constant modulus rls-based blind adaptive beamforming algorithm for smart antennas, in 27 4th International Symposium on Wireless Communication Systems, Oct 27, pp [] M. Grant S. Boyd, CVX: Matlab software for disciplined convex programming, version 2., Mar. 24. [6] J. Mattingley S. Boyd, Cvxgen: a code generator for embedded convex optimization, Optimization Engineering, vol. 3, no., pp. 27, Jan 22. [7] X. Wang H. V. Poor, Wireless Communication Systems: Advanced Techniques for Signal Reception, Prentice Hall PTR, Upper Saddle River, NJ, USA, st edition, 29. [8] L. Hu, X. Zhou, L. Zhang, Blind multiuser detection based on tikhonov regularization, IEEE Communications Letters, vol., no., pp , May 2.