Transmit Optimization with Improper Gaussian Signaling for Interference Channels

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED Transmit Optimization with Improper Gaussian Signaling for Interference Channels Yong Zeng, Student Member, IEEE, Cenk M. Yetis, Member, IEEE, Err Gunawan, Member, IEEE, Yong Liang Guan, Member, IEEE, and Rui Zhang, Member, IEEE arxiv:07.506v cs.it] Mar 03 Abstract This paper studies the achievable rates of Gaussian interference channels with additive white Gaussian noise (AWGN), when improper or circularl asmmetric complex Gaussian signaling is applied. For the Gaussian multiple-input multiple-output interference channel (MIMO-IC) with the interference treated as Gaussian noise, we show that the user s achievable rate can be expressed as a summation of the rate achievable b the conventional proper or circularl smmetric complex Gaussian signaling in terms of the users transmit covariance matrices, and an additional term, which is a function of both the users transmit covariance and pseudo-covariance matrices. The additional degrees of freedom in the pseudocovariance matrix, which is conventionall set to be zero for the case of proper Gaussian signaling, provide an opportunit to further improve the achievable rates of Gaussian MIMO-ICs b emploing improper Gaussian signaling. To this end, this paper proposes widel linear precoding, which efficientl maps proper information-bearing signals to improper transmitted signals at each transmitter for an given pair of transmit covariance and pseudo-covariance matrices. In particular, for the case of twouser Gaussian single-input single-output interference channel (SISO-IC), we propose a joint covariance and pseudo-covariance optimization algorithm with improper Gaussian signaling to achieve the Pareto-optimal rates. B utilizing the separable structure of the achievable rate expression, an alternative algorithm with separate covariance and pseudo-covariance optimization is also proposed, which guarantees the rate improvement over conventional proper Gaussian signaling. Index Terms Improper Gaussian signaling, interference channel, pseudo-covariance optimization, widel linear precoding. I. INTRODUCTION The interference channel (IC) models multi-user communication sstems where each transmitter is intended to send independent information to its corresponding receiver while causing interference to all other receivers. Although informationtheoretic stud of the IC has a long histor, characterization of its capacit region still remains an open problem in general, except for some special cases such as that with the presence of strong interference,]. For the single-input singleoutput IC, termed SISO-IC, the best achievable rate region to date is obtained b the celebrated Han-Kobaashi scheme ]. Recentl, it has been shown in 3] that a particular form of this scheme achieves within one bit to the capacit region of Y. Zeng, E. Gunawan, and Y. L. Guan are with the School of Electrical and Electronic Engineering (EEE), Nanang Technological Universit (NTU), Singapore ( ze0003ng@e.ntu.edu.sg, {egunawan, elguan}@ntu.edu.sg). C. M. Yetis was with the School of EEE, NTU. He is now with Mevlana Universit, Turke ( cenkmetis@gmail.com). R. Zhang is with the Electrical and Computer Engineering Department, National Universit of Singapore ( elezhang@nus.edu.sg). the two-user Gaussian SISO-IC with additive white Gaussian noise (AWGN). Since such capacit-approaching techniques require multi-user encoding/decoding, which are difficult to implement in practical sstems, a more pragmatic approach is to emplo single-user encoding and decoding b treating the interference as Gaussian noise at all receivers. In fact, this simplified approach has been shown to be sum-capacit optimal for Gaussian ICs when the interference level is below a certain threshold 4]. Under the assumption of single-user detection (SUD) with the interference treated as Gaussian noise, the transmit optimization problem for Gaussian ICs reduces to resource allocation among the transmitters for interference mitigation, which has received significant attention in the last few decades. Earl works on resource allocation for Gaussian ICs mostl focused on power control since SISO-IC with single-antenna terminals was considered (see e.g. 5] 7] and references therein). When transmitters/receivers are equipped with multiple antennas, the network performance can be further improved via transmit/receive beamforming. One useful technique applied for optimizing transmit beamformers is to transform the design problem into an equivalent receiver beamformer optimization problem via the so-called uplink-downlink or network dualit principle 8] ]. The transmit beamforming optimization problems for power minimization and signal-to-interferenceplus-noise ratio (SINR) balancing can also be directl solved b convex optimization techniques, such as the second-order cone programming (SOCP) 3] and semidefinite programming (SDP) 4]. For Gaussian ICs with the interference treated as Gaussian noise, the sum-rate maximization problem is in general difficult to be solved globall optimall due to its non-convexit. In 5], it was shown that finding globall optimal beamformers for the weighted sum-rate maximization (WSRMax) in the Gaussian multiple-input single-output IC (MISO-IC) is an NPhard problem. Algorithms based on the principle of interference pricing have been proposed for achieving local optimums 6], while in 7], a distributed algorithm was proposed for MISO-IC b using the virtual SINR framework. Gradient descent algorithms have also been proposed for Gaussian multiple-input multiple-output IC (MIMO-IC) over transmit covariance matrices 8] or precoding matrices 9]. More recentl, for Gaussian SISO-IC, single-input multiple-output IC (SIMO-IC) and MISO-IC, globall optimal solutions to the WSRMax problem have been obtained under the monotonic optimization framework 0] 4]. However, the complexit of such globall optimal solutions increases exponentiall with

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED the number of users, and their generalization to more general MIMO-IC remains unknown. An alternative technique for solving WSRMax problems for MIMO-ICs is via iterativel minimizing the weighted mean-square-error (MSE), which utilizes the inherent relationship between the mutual information and MSE 5] 7]. The stud of ICs with game-theoretic models has also been given in 8] and references therein. Moreover, it is worth mentioning that there has been a great deal of interest in the last few ears on studing Gaussian ICs from the degrees-of-freedom (DoF) perspective 9]. A ke technique to achieve higher DoFs than previousl believed for Gaussian ICs is interference alignment (IA) 30]. Since DoF onl provides the approximated capacit at the asmptoticall high signal-to-noise ratio (SNR), a number of IA-based precoding schemes with improved sum-rate performance at practical SNRs have been proposed in 9], 3] 33]. As for characterization of the achievable rate region for Gaussian ICs with interference treated as noise, various solutions have been obtained for the SISO-IC 34], SIMO-IC ], and MISO-IC 35] 38]. The Pareto boundar of the achievable rate region for ICs consists of all the achievable ratetuples, at each of which it is impossible to improve one user s rate, without simultaneousl decreasing the rate of at least one of other users. A traditional approach for characterizing Pareto boundaries of Gaussian ICs is via solving WSRMax problems. However, as pointed out in 36], the WSRMax approach cannot guarantee the finding of all Pareto-boundar points due to the non-convexit of the achievable rate region. An alternative method based on the concept of rate-profile was thus proposed in 36], which is able to characterize the complete Pareto boundar for ICs. Besides, the rate-profile approach generall results in optimization problems that are easier to handle than conventional WSRMax problems for ICs,36]. It is necessar to point out that in all the aforementioned works on Gaussian ICs, the transmitted signals are assumed to be proper or circularl smmetric complex Gaussian (CSCG) distributed. A ke propert of proper Gaussian random vectors (RVs) is that their second-order statistics are completel specified b the conventional covariance matrix under a zeromean assumption. In contrast, for the more general improper Gaussian RVs, an extra parameter called pseudo-covariance matrix is required for the complete second-order characterization 39] 4]. Most of the existing works on Gaussian ICs have adopted the proper Gaussian assumption without an justification. This ma be due to the common practice of modeling the additive receiver noise as proper Gaussian, and the well-known maximum-entrop theorem 39], i.e., proper Gaussian RVs maximize the differential entrop for an given covariance matrix. As a result, proper Gaussian signaling has been shown to be capacit optimal for the Gaussian point-to-point channel, multiple-access channel (MAC) and broadcast channel (BC). However, for Gaussian ICs with the interference treated as noise, improper Gaussian signaling provides a new opportunit to further improve the achievable rates over the conventional proper Gaussian signaling 4]. For instance, it was shown in 43] that improper Gaussian signaling, together with smbol extensions and IA, is able to improve the DoF for the three-user SISO-IC with timeinvariant channel coefficients at the asmptoticall high SNR. In 44,45], it was shown that the achievable rate region can be enlarged with improper Gaussian signaling even for the twouser SISO-IC at finite SNR. This is particularl interesting since it is known that for the two-user SISO-IC, no DoF gain is achievable with IA 4]. More specificall, a suboptimal scheme was proposed in 44] for the two-user SISO-IC, where the transmit covariance matrices for the equivalent real-valued MIMO-ICs are restricted to be rank-. In 45], the Paretooptimal transmit covariance matrices for the two-user SISO-IC are obtained b an exhaustive search method. The prior works 43] 45] on the stud of improper Gaussian signaling for ICs are all based on the equivalent doublesized real-valued MIMO-IC matrix b separating the real and imaginar parts of the complex-valued channels. Although an complex-valued sstem can be transformed into an equivalent real-valued sstem, as pointed out in 40,46], much of the eleganc of the sstem description is lost. Therefore, in this paper, we adopt the complex-valued channel model for studing improper Gaussian signaling in Gaussian ICs to gain new insights. The main contributions of this paper are summarized as follows: Based on existing results on improper Gaussian RVs, we derive a new achievable rate expression for the general K-user MIMO-IC, when improper Gaussian signaling is applied. Our result shows that the user s achievable rate can be expressed as a summation of the rate achievable b the conventional proper Gaussian signaling in terms of the users transmit covariance matrices, and an additional term, which is a function of both the users transmit covariance and pseudo-covariance matrices. This new result implies that the use of improper Gaussian signaling for MIMO-ICs with interference treated as noise is able to improve the achievable rate over the conventional proper Gaussian signaling with an given set of covariance matrices of transmitted signals, b further optimizing their pseudo-covariance matrices. For an given pair of signal covariance and pseudocovariance matrices at each transmitter, we consider the practical problem of generating improper transmitted signals from proper information-bearing signals. Based on existing techniques for improper RVs 46], we propose an efficient method for this implementation, named as widel linear precoding. B adopting the rate-profile method, we formulate the optimization problem for the two-user SISO-IC to characterize the Pareto boundar of the achievable rate region with improper Gaussian signaling. B appling the celebrated semidefinite relaxation (SDR) technique 47], a joint covariance and pseudo-covariance optimization algorithm is proposed, which achieves near-optimal ratepairs. Furthermore, b utilizing the separable structure of the achievable rate expression with improper Gaussian signaling, a separate covariance and pseudo-covariance optimization algorithm is also proposed, which guarantees the rate improvement over conventional proper

3 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 3 Gaussian signaling with an given transmit covariance. The rest of this paper is organized as follows. Section II studies improper Gaussian signaling for the general MIMO- IC, where a new achievable rate expression is derived and widel linear precoding is proposed. Section III focuses on the two-user SISO-IC setup, where the problem formulation for characterizing the Pareto boundar of the achievable rate region is given. In Section IV, a SDR-based joint covariance and pseudo-covariance optimization algorithm for the twouser SISO-IC is proposed. In Section V, an alternative SOCPbased algorithm b separate covariance and pseudo-covariance optimizations is presented. Section VI provides numerical results. Finall, we conclude the paper in Section VII. Notations: In this paper, scalars are denoted b italic letters. Boldface lower- and upper-case letters denote vectors and matrices, respectivel. I M denotes an M M identit matrix and the subscript M is omitted if its value is clear from the context. 0 denotes an all-zero matrix. For a square matrix S, Tr(S), S, S denote the trace, determinant and inverse of S, respectivel. S 0 and S 0 mean that S is positive semidefinite and positive definite, respectivel. C M N and R M N denote the space of M N complex and real matrices, respectivel. S K and H K denote the K K realvalued smmetric and complex-valued Hermitian matrices, respectivel. For an arbitrar matrix A, A, A T, A H and rank{a} represent the complex-conjugate, transpose, conjugate transpose and rank of A, respectivel. diag{x} represents a diagonal matrix with the elements in the main diagonal given b x. N (µ, C) represents the real-valued Gaussian RV with mean µ and covariance matrix C. CN (x, Σ) represents the CSCG RV with mean x and covariance matrix Σ. For a complex number x, x denotes its magnitude. The smbol i represents the imaginar unit, i.e., i. v] k denotes the kth element of a vector v. R{ } and I{ } represent the real and imaginar parts of complex numbers/matrices, respectivel. I(x; ) represents the mutual information between two RVs x and. II. IMPROPER GAUSSIAN SIGNALING FOR MIMO-IC Consider a K-user MIMO-IC, where each transmitter is intended to send independent information to its corresponding receiver, while possibl interfering with all other K receivers. Denote the number of transmitting and receiving antennas for each user b M and N, respectivel. Assuming the narrow-band transmission, the equivalent baseband received signal at each receiver can be expressed as k (n) H kk x k (n) + j k H kj x j (n) + n k (n), k, () where n is the smbol index, H kk C N M denotes the direct channel matrix from transmitter k to receiver k, while H kj, j k, denotes the interference channel matrix from transmitter j to receiver k; we assume quasi-static fading and thus all channels are constant over n s in () for the case of our interest; n k (n) represents the independent and identicall distributed (i.i.d.) CSCG noise vector at receiver k with n k (n) CN (0, σ I); and x k (n) C M is the transmitted signal vector from transmitter k, which is independent of x j (n), j k. In this paper, for the purpose of exposition, we assume that smbol extensions over time as in 43] are not used. Hence, x k (n) is independent over n. For brevit, n is omitted in the rest of this paper. Different from the conventional setup where proper Gaussian signaling is assumed, i.e., x k CN (0, C xk ), with C xk denoting the transmit covariance matrix, in this paper we consider the more general improper Gaussian signals, for which some preliminaries are given next. A. Preliminar for Improper Random Vectors For a zero-mean RV z C n, the covariance matrix C z and pseudo-covariance matrix C z are defined as 39] C z E(zz H ), Cz E(zz T ). () B definition, it is eas to verif that the covariance matrix C z is Hermitian and positive semidefinite, and the pseudocovariance matrix C z is smmetric. Definition. 39]: A complex RV z is called proper if its pseudo-covariance matrix C z vanishes to a zero matrix; otherwise, it is called improper. Lemma. 39]: Two zero-mean complex RVs x and z are uncorrelated if and onl if C xz 0 and C xz 0, where C xz E(xz H ) and C xz E(xz T ). A more restrictive definition than properness is known as circularl smmetric, which is defined as follows. Definition. 40]: A complex RV z is circularl smmetric if its distribution is rotationall invariant, i.e., z and ẑ e iα z have the same distribution for an real value α. For a circularl smmetric RV z, we have C z Cẑ E(ẑẑ T ) e iα Cz, α, which implies C z 0. Thus, circularit implies properness, but the converse is not true in general. However, if z is a zero-mean Gaussian RV, then properness and circularit are equivalent, as given b the following lemma. Lemma. 40]: A complex zero-mean Gaussian RV z is circularl smmetric if and onl if it is proper. For example, the commonl adopted assumption that the noise vector n k in () is zero-mean CSCG is equivalent to that n k is a proper Gaussian RV, whose pseudo-covariance matrix satisfies C nk 0. For an arbitrar complex RV z, define C z as the covariance matrix of the augmented vector z T (z ) T ] T, i.e., ( ] ] H ) ] z z Cz Cz C z E z z C z C. (3) z The augmented covariance matrix C z obviousl has some built-in redundanc for the second-order characterization of z; however, it is useful as shown in the following two theorems. Theorem. 40]: C z and C z are a valid pair of covariance and pseudo-covariance matrices, i.e., there exists a RV z with covariance and pseudo-covariance matrices given b C z and C z, respectivel, if and onl if the augmented covariance matrix C z is positive semidefinite, i.e., C z 0.

4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 4 Note that the conditions of the covariance matrix C z being Hermitian and positive semidefinite, and the pseudocovariance matrix C z being smmetric are alread implied b C z 0. Furthermore, for the improper complex Gaussian RVs, the differential entrop is in general a function of both the covariance and pseudo-covariance matrices, which can be expressed in terms of C z as shown b the following theorem. Theorem. 40]: The entrop of a complex Gaussian RV z C n with augmented covariance matrix C z is h(z) log( (πe) n C z ). (4) Theorem generalizes the entrop result for proper Gaussian RVs. If C z 0, (4) reduces to the well-known expression for proper Gaussian RVs h(z) log ( (πe) n C z ) 39]. B. Achievable Rate with Improper Gaussian Signaling In this subsection, we derive the achievable rate b improper Gaussian signaling for the K-user MIMO-IC defined in (). Denote the covariance and pseudo-covariance matrices of the zero-mean transmitted Gaussian RV x k b C xk and C xk, respectivel, i.e., C xk E(x k x H k ), Cxk E(x k x T k ), k,, K. Since x k and x j are independent for j k, then b using Lemma and the fact that independence and uncorrelatedness are equivalent for Gaussian RVs, the covariance and pseudocovariance matrices of the received signal vector k, k,, K, can be obtained as K C k E( k k H ) H kj C xj H H kj + σ I, (5) C k E( k T k ) j K H kj Cxj H T kj, (6) j where in (6), we have used the fact that the pseudo-covariance of the CSCG noise vector n k is a zero matrix. It is obvious from (5) that C k is nonsingular. Then with the augmented covariance matrix C k defined as in (3) and using the Schur complement 48], we obtain C k C Ck k C k C C k C k C k C Ck k k Ck I C Ck k C T CH k k, (7) where we have used the fact that for an invertible matrix A, A T (A ) T (A T ) ; and in the last equalit, we have used the identities A A T, AB A B, and the facts that C k is Hermitian and C k is smmetric. With the transmitted signals being Gaussian, the received signal k is also Gaussian. Then based on Theorem, the differential entrop of k C N is given b h( k ) log ( (πe) N ) C k + log I C Ck k C T CH k k. Denote s k as the interference-plus-noise term at receiver k, i.e., s k j k H kjx j +n k. Then the covariance and pseudocovariance matrices of s k are given b C sk H kj C xj H H kj + σ I, (8) j k C sk j k H kj Cxj H T kj. (9) Similarl as for k, the differential entrop of s k is h(s k ) log ( (πe) N ) Csk + log I C s Csk k C T s CH k sk. Under the assumption that interference is treated as Gaussian noise, the achievable rate at receiver k with improper Gaussian signaling can be obtained as R k I(x k ; k ) h( k ) h( k x k ) h( k ) h(s k ) log C k σ I + K j Csk log H kjc xj H H kj σ I + j k H kjc xj H H kj R k,proper ({C xj }) + I C log Ck k C T CH k k. (0) I C s k Csk C T s k CH sk The above equation shows that with improper Gaussian signaling, the achievable rate can be expressed as a summation of two terms. The first term, denoted b R k,proper ({C xj }), is the rate achievable b the conventional proper Gaussian signaling, which is a function of the transmit covariance matrices onl. The second term is a function of both the transmit covariance and pseudo-covariance matrices. B setting C xk 0, k, the second term vanishes and (0) reduces to the rate expression for the conventional case of proper Gaussian signaling. The separabilit of the achievable rate b improper Gaussian signaling provides a general method to improve the achievable rate over the conventional proper Gaussian signaling, i.e., for an given covariance matrices obtained b existing proper Gaussian signaling schemes, the rate can be improved with improper Gaussian signaling b choosing the pseudocovariance matrices that make the second term in (0) strictl positive. It is worth noting that this propert does not exist if we convert the complex-valued sstem in () to an equivalent real-valued sstem b doubling the input/output dimensions. In this paper, we are interested in characterizing the achievable rate region with improper Gaussian signaling. The achievable rate region for the K-user MIMO-IC consists of all the rate-tuples for all users that can be simultaneousl achieved under a given set of transmit power constraints for each transmitter, denoted b P k, k,..., K, i.e., { } R (r,, r K ) : 0 r k R k, k, () Tr{C xk } P k, C xk 0, k where C xk is the augmented covariance matrix of x k defined in (3). The constraint C xk 0 follows from Theorem. In Sections III-V, we will consider the transmit covariance and pseudo-covariance optimizations for achieving the Pareto boundar of the above rate region for the special case of twouser SISO-IC. C. Widel Linear Precoding In this subsection, we consider the practical problem of how to efficientl generate the transmitted signal x k at each transmitter given an valid pair of covariance matrix C xk and pseudo-covariance matrix C xk, from an information-bearing signal d k that is selected from conventional CSCG (proper Gaussian) codebooks. Without loss of generalit, we assume d k CN (0, I); thus, we have C dk I, Cdk 0, k,, K. ()

5 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 5 First, consider the conventional linear precoding given b x k U k d k, (3) where U k is the precoding matrix. Then the pseudo-covariance matrix of x k is given b C xk U k Cdk U T k 0. This implies that the conventional linear precoding is not able to map the proper Gaussian signal d k to the improper transmitted Gaussian signal x k. Since the augmented covariance matrix defined in (3) contains both the covariance and pseudo-covariance matrices, a necessar condition for a RV z k to have covariance matrix C xk and pseudo-covariance matrices C xk is that its augmented covariance matrix satisfies C zk C xk. This is ensured b the transformation ] zk ], (4) z C dk xk k d k denotes the generalized Cholesk factor of the where C xk positive semidefinite matrix C xk, which is defined b C xk C xk (C xk ) H. Since C dk I as given in (), it is eas to verif that z k in (4) satisfies C zk C xk. A common method for finding C xk is via eigenvalue decomposition (EVD). Specificall, let the EVD of C xk be expressed as C xk UDU H ; then C xk UD is obtained. However, it is worth pointing out that the above obtained C xk cannot satisf (4) in general. This is because the two vectors z k and z k in (4) need to be complex conjugate of each other; therefore, the transformation matrix C xk should be designed with more care than the conventional EVD. On the other hand, if we can find one C xk such that it has ] the following structure: C B B xk B B, (5) i.e., the upper-left (w.r.t. upper-right) block is the complex conjugate of the lower-right (w.r.t. lower-left) block, then (4) is equivalent to the following two sets of equations: z k B d k + B d k, (6) z k B d k + B d k. (7) It is eas to verif that the two equations given in (6) and (7) are consistent, i.e., (7) is simpl obtained b taking the complex conjugate on both sides of (6) and vice versa. Therefore, the remaining task is to find one C xk with the structure given b (5). To achieve this end, we define the following M M unitar matrix 46]: T ] IM ii M, TT H T H T I I M ii M. M ] A A For an real-valued matrix A R A A M M, it can be verified that the matrix TAT H has the structure given in (5), i.e., ] TAT H A A A A, (8) with A (A + A ) + i(a A )] and A (A A ) + i(a + A )]. Theorem 3. 46] There exists one form of EVD for the augmented covariance matrix C xk C M M defined in (3), which is given b C xk (TV)Λ(TV) H, (9) where V R M M is a real-valued orthogonal matrix and Λ diag{λ, λ,, λ M } contains the eigenvalues of C xk. In fact, (9) can be obtained b considering T H C xk T, which is a real-valued matrix given b T H R{C C xk T xk + C xk } I{ C xk + C ] xk } I{C xk + C xk } R{C xk C. xk } Furthermore, since C xk is Hermitian and C xk is smmetric, it can be verified that the matrix T H C xk T is smmetric as well. Therefore, its real-valued EVD can be written as T H C xk T VΛV T. (0) The EVD in (9) is then obtained b appling a unitar transformation T to (0). For an given C xk 0, all the eigenvalues are nonnegative 46], i.e., λ l 0, l,, M. Thus (9) can be written as C xk TVΛV H T H (TVΛ / T H )(TVΛ / T H ) H. Then we have ] C xk T(VΛ / )T H B B, () B B where the last equalit follows from (8) and the fact that VΛ / is a real-valued matrix. From (4) and (), it follows that to obtain the transmitted signal vector x k, which is generall improper with the covariance and pseudo-covariance matrices specified b C xk, the following precoding needs to be applied to the proper information-bearing signal d k : x k B d k + B d k, () where B and B are the corresponding blocks in TVΛ / T H as shown in (), with V and Λ obtained b the particular form of EVD in (9). Following similar terminologies used in existing literatures on improper signal processing such as 49], we refer to the precoding given in () as widel linear precoding. Note that if B 0, which is the case when C xk is block-diagonal (i.e., Cxk 0), () reduces to the conventional linear precoding for proper Gaussian signaling given b (3). Last, in terms of the realvalued representation, ] () can be re-expressed as ] R{xk } I{x k } R{B + B } I{B B } I{B + B } R{B B } ] R{dk }. I{d k } III. PARETO BOUNDARY CHARACTERIZATION FOR THE TWO-USER SISO-IC In the remaining part of this paper, we will focus on the special two-user SISO-IC case, with the aim of characterizing its Pareto rate boundar with improper Gaussian signaling b optimizing both the covariances and pseudo-covariances of transmitted signals. The input-output relationship for the twouser SISO-IC can be simplified from () as h x + h x + n, (3) h x + h x + n, where h kj h kj e iφ kj, k, j {, }, is the complex scalar channel from transmitter j to receiver k. Denote the covariance and pseudo-covariance of the transmitted signal x k b C xk E(x k x k), Cxk E(x k x k ), k,. (4) Since a phase rotation can be applied at each of the receivers with coherent demodulation, without loss of generalit, the direct channel gains h and h can be assumed to be real values. However, this assumption will not change the remaining results in this paper.

6 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 6 Note that C xk s are nonnegative real numbers equal to the transmit power values of the corresponding users, while C xk s are complex numbers in general. Due to Schur complement, the necessar and sufficient conditions stated in Theorem for the special case of two-user SISO-IC reduce to C xk 0, C xk Cx k, k,. (5) The covariance and pseudo-covariance of k can be written as C k E( k k) h k C x + h k C x + σ, (6) C k E( k k ) h C k x + h C k x. (7) For the interference-plus-noise term s k h k kx k + n k, k k, we have C sk h k k C x k + σ, Csk h C k k x k. (8) Then the achievable rate expression in (0) for the special case of two-user SISO-IC reduces to Rk SISO log C k C k Cs k C (9) sk ( log + h kk ) C xk σ + h k k + C log k C k C }{{ x k Cs } k C sk. (30) R SISO k,proper (Cx,Cx ) To characterize the Pareto boundar of the achievable rate region defined in (), we adopt the rate-profile technique proposed in 36]. Specificall, an Pareto-optimal rate-pair can be obtained b solving the following optimization problem with a given rate-profile specified b α (α, α ): (P): max. {C xk },{ C xk },R s.t. R Rk SISO α k R, k (3) 0 C xk P k, k (3) C xk C x k, k, (33) where α k denotes the target ratio between user k s achievable rate and the users sum-rate, R. Without loss of generalit, we assume that α, α > 0 and α + α. Denote the optimal solution to (P) as R, then the rate-pair (α R, α R ) must be on the Pareto boundar corresponding to the rate-profile given b (α, α ). Thereb, b solving (P) with different rateprofile parameters (α, α ), the complete Pareto boundar for the achievable rate region can be found. IV. JOINT COVARIANCE AND PSEUDO-COVARIANCE OPTIMIZATION In this section, b appling the SDR technique, we propose an approximate solution to the non-convex problem (P) where the covariance and pseudo-covariance of the transmitted signals are jointl optimized. The approach of using SDR for solving non-convex quadraticall constrained quadratic program (QCQP) has been successfull applied to find highqualit approximate solutions for various problems in communication and signal processing (see e.g. 47] and references therein). B treating R as a slack variable and substituting R SISO k in (9), (P) can be equivalentl written as (P.): max. min. log C k C k {C xk },{ C xk } k, α k Cs k C sk s.t. 0 C xk P k, k (34) C xk C x k, k. (35) (P.) is a minimum-weighted-rate maximization (MinWR- Max) problem, where the weights are related to the rate-profile α. The following result will be used for solving (P.). Lemma 3. For an set of {C xk } and { C xk } that is feasible to (P.), the following inequalities hold: C k C k σ 4 > 0, k, (36) C s k C sk σ 4 > 0, k. (37) Proof: Please refer to Appendix A. Define the following -dimensional real-valued vectors: c C x C x ] T, a h h ] T, b 0 h ] T, a h h ] T, b h 0 ] T. Then from (6) and (8), we have C k (σ + a T k c), Cs k (σ + b T k c), k,. (38) Define the following -dimensional ] complex-valued vectors: T q Cx C x, f h h ] H, g 0 h ] H, f h h ] H, g h 0 ] H. Then from (7) and (8), we have C k f H k q q H F k q, (39) C sk gk H q q H G k q, (40) where F k f k fk H and G k g k gk H, k,. B substituting (38)-(40) into (P.), we obtain the following equivalent problem (P.): max. c R,q C min. log (σ + a T k c) q H F k q k α k (σ + b T k c) q H G k q s.t. c T E k c P k, k (4) e T k c 0, k (4) q H E k q c T E k c, k (43) where e k is the kth column in the identit matrix I, and E k e k e T k. (4) and (4) correspond to the constraints (34) in (P.), and (43) is equivalent to (35). The objective function of (P.) is given b the minimum of weighted log-fraction of quadratic functions over c and q. Due to the noise power σ, the quadratics are non-homogeneous 47]. B introducing a new variable t, we obtain the homogenized quadratics 47], which ield (P.3): max. min. log (σ t + a T k c) q H F k q c,q,t k α k (σ t + b T k c) q H G k q s.t. c T E k c P k, k (44) e T k ct 0, k (45) q H E k q c T E k c, k (46) t. (47) (P.3) is equivalent to (P.) in the sense that if it has an optimal solution (c, q, t ), then (c /t, q /t ) is an optimal solution to (P.) with the same optimal value. Therefore, (P.) can be solved b solving (P.3). Next, we show that the celebrated SDR technique can be applied to find an approximate solution to (P.3). ] ] Define T t t C, Q qq H, (48) c c

7 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 7 ] ] σ σ T ] ] σ σ T A k, B a k a k (49) k b k b k ] ] 0 K k et k 0 0 e, Ê k 0 k. (50) 0 E k With the identit x H Ax Tr ( Axx H), (P.3) is recast as (P.4): max. min. log Tr(A kc) Tr(F k Q) C S 3,Q H k α k Tr(B k C) Tr(G k Q) s.t. Tr(Ê k C) P k, k (5) Tr(K k C) 0, k (5) Tr(E k Q) Tr(Ê k C), k (53) C (54) C 0, Q 0 (55) rank(c), rank(q), (56) where C denotes the (, )-th entr of C; the positive semidefinite constraints (55) and the rank- constraints (56) are due to (48). With such a reformulation, the objective function of (P.4) is now a log-fraction of linear functions of C and Q, and all the constraints (5)-(54) are also linear. With Lemma 3 and the equivalence between (P.) and (P.4), for an pair of C and Q that is feasible to (P.4), we have Tr(A k C) Tr(F k Q) σ 4 > 0, k, (57) Tr(B k C) Tr(G k Q) σ 4 > 0, k. (58) The SDR problem of (P.4) is obtained b discarding the non-convex rank- constraints in (56), and including the extra constraints (57) and (58), i.e., log Tr(A kc) Tr(F k Q) α k (P.4-SDR): max. C S 3,Q H min. k Tr(B k C) Tr(G k Q) s.t. (5) (55), (57), (58). Note that although the constraints (57) and (58) are redundant in the rank-constrained problem (P.4), in the rank-relaxed problem (P.4-SDR), the advantages of including them are twofold. First, it makes (P.4-SDR) a problem with less relaxation to (P.4). Besides, the strict positivit of (58) makes (P.4-SDR) a quasi-convex problem and hence can be solved with the standard bisection method 50]. Since an feasible solution of (P.4) is feasible for (P.4-SDR), the optimal objective value of (P.4-SDR) provides an upper bound on that of (P.4). To solve the quasi-convex problem (P.4-SDR), consider the following feasibilit problem for a fixed R: (P.5): find C S 3, Q H s.t. (5) (55), (57), (58), Tr(A k C) Tr(F k Q) e α kr (Tr(B k C) Tr(G k Q)), k. (59) (P.5) is a SDP problem, which can be efficientl solved 50]. If (P.5) is feasible, then the optimal objective value R sdr of (P.4-SDR) satisfies R sdr R; otherwise R sdr < R. Therefore, (P.4-SDR) can be solved b solving the SDP problem (P.5) together with a bisection search over R. Denote the solution to (P.4-SDR) b (C, Q ). If rank(c ) and rank(q ), then (C, Q ) is also the optimal solution to the rank-constrained problem (P.4). In this case, SDR is tight and the optimal solution to (P.3) is given b the principal components of C and Q, from which the solution to (P.) can be obtained; otherwise, we appl the following Gaussian randomization procedure customized to our problem to obtain an approximate solution to (P.) 47]. Algorithm Gaussian Randomization Procedure for (P.) Input: The solution (C, Q ) to (P.4-SDR) and the number of randomization trials L. : for l,, L] do tl : Generate N (0, C ), β l CN (0, Q ). ξ l 3: Let c l ξ l /t l, and q l β l /t l. 4: Construct a feasible point (č l, ˇq l ) for (P.) as follows: č l ] k max (0, min (c l ] k, P k )), (60) ˇq l ] k η k q l ] k, (6) { } č l ] k where η k min,, k,. q l ] k 5: end for 6: Let (ĉ, ˆq) be the solution to max. min. log (σ + a T k čl) ˇq H l F kˇq l č l,ˇq l,l,,l k α k (σ + b T k čl) ˇq H l G kˇq l Output: (ĉ, ˆq) as an approximate solution for (P.). V. SEPARATE COVARIANCE AND PSEUDO-COVARIANCE OPTIMIZATION For the algorithm proposed in the preceding section, although joint optimizations are performed over the covariances and pseudo-covariances, it is not clear whether a rate gain over conventional proper Gaussian signaling is attainable since the obtained solutions are not necessaril globall optimal. In this section, b utilizing the result that the user s achievable rate is separable as shown in (30), we propose a separate covariance and pseudo-covariance optimization algorithm for (P). Specificall, the covariances of the transmitted signals are first optimized b setting the pseudo-covariances to be zero, i.e., proper Gaussian signaling is applied. Then, the pseudocovariances are optimized with the covariances fixed as the previousl optimized values. With such a separation approach, the obtained improper signaling scheme is guaranteed to improve the rate over proper Gaussian signaling scheme. A. Covariance Optimization When restricted to proper Gaussian signaling with C x 0 and C x 0, b substituting (30) into (3), (P) reduces to (P.6): max. r r,c x,c x ( s.t. log + h C ) x σ + h α r, C x ( log + h C ) x σ + h α r, C x 0 C x P, 0 C x P. For an fixed value r, (P.6) can be transformed to the following feasibilit problem: (P.7): Find C x R, C x R s.t. h C x (σ + h C x )(e αr ), h C x (σ + h C x )(e αr ), 0 C x P, 0 C x P.

8 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 8 (P.7) is a linear programming (LP) problem, which can be efficientl solved 50]. If r is feasible to (P.7), then the optimal value of (P.6) satisfies r r; otherwise, r < r. Thus, (P.6) can be efficientl solved b solving (P.7) together with the bisection method for updating r. B. Pseudo-Covariance Optimization Denote the optimal solution to the covariance optimization problem (P.6) as {r, Cx, Cx }. B fixing the covariances as Cx and Cx, (P) is then optimized over the pseudocovariances C x and C x. B substituting the first term in the rate expression (30) with α k r, the problem for pseudocovariance optimization is formulated as (P.8): max. C x, C x,r R s.t. α r + log C C C s C s α R, α r + C log C Cs C s α R, C x Cx, C x Cx, where C, C s, C and C s are the corresponding covariance terms with the transmit covariances Cx and Cx. Again, if a given R is achievable for certain pair of C x and C x, then the optimal value of (P.8) satisfies R R; otherwise, R < R. Therefore, (P.8) can be solved via solving a set of feasibilit problems together with the bisection method. It can be easil observed that { C x 0, C x 0, R r } is feasible to (P.8). Therefore, R r is satisfied, i.e., with our proposed separate covariance and pseudo-covariance optimizations, the users sum-rate corresponding to the rate-profile given b (α, α ) with improper Gaussian signaling is guaranteed to be no smaller than that obtained with the optimal proper Gaussian signaling obtained b solving (P.6). Therefore, the remaining problem to be solved is the feasibilit problem resulting from (P.8) for a given R. B substituting C k in (7) and C sk in (8) into (P.8) and after some manipulations, the feasibilit problem for a given R can be formulated as (P.9): Find Cx C, Cx C s.t. a h C x + h C x + b C x, (6) a h C x + h C x + b C x, (63) C x Cx, (64) C x Cx, (65) C where a k s k β k C h, b k k k 4 k ( /β k)c s k h, β k k 4 k e α k(r r ), k,, k k. Since the optimal value of (P.8) satisfies R r, we can assume that R r in (P.9) without loss of optimalit. Then it follows that β k, b k 0, k. In the following, we show that (P.9) can be efficientl solved via solving a finite number of SOCP problems. First, it can be verified that if { C x, C x } is feasible for (P.9), then so is { C x e iω, C x e iω }. Therefore, without loss of generalit, we ma choose the common phase rotation ω so that C x is real and nonnegative. Denote the magnitude and phase of C x b t and θ, respectivel, i.e., Cx te iθ. Then for an fixed Fig. : Empirical ratio R / ˆR for the two-user SISO-IC over 500 random channel realizations, and with SNR0 db. value of θ, (P.9) can be transformed into a SOCP feasibilit problem given b (P.0): Find Cx R, t R s.t. a (h C x + h te iθ ) t, b a (h C x + h te iθ ) C x, b C x C x, t C x. Theorem 4. The feasibilit problem (P.9) can be optimall solved b solving a finite number of SOCP problems (P.0), each for a fixed value θ, where θ can be restricted to the following discrete set: θ {π + (φ φ ), π + (φ φ )} Θ A Θ B, where Θ A and Θ B are the solution sets for θ to the following two sets of equations with variables (θ, t) and (θ, C x ), respectivel: { a h Θ A : Cx + h te iθ + b t a h Cx + h te iθ + b Cx (66) Θ B : { a h C x + h C x e iθ + b C x a h C x + h C x e iθ + b C x (67) Proof: Please refer to Appendix B. Theorem 4 can be intuitivel explained as follows. For the feasibilit problem (P.9), if the constraint (6) is more restrictive than (63), then θ should have a value such that the left hand side (LHS) of (6) is minimized. This corresponds to θ π+(φ φ ) so that h C x and h C x are antiphase. Similar argument for θ π + (φ φ ) can be made. On the other hand, if both (6) and (63) are equall restrictive, a feasible solution tends to make both constraints satisfied with equalit, as given b (66) and (67). Θ A and Θ B correspond to the cases where either the constraint (64) or (65) is active, which can be assumed without loss of generalit as shown b Proposition in Appendix B. The elements in Θ A and Θ B can be efficientl obtained b following the steps in Appendix C. VI. NUMERICAL RESULTS In this section, we evaluate the performance of the proposed algorithms for the two-user SISO-IC with numerical exam-

9 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 9 that the mean of R / ˆR is.0, which demonstrates the high qualit of the approximate solution obtained b the SDRbased joint covariance and pseudo-covariance optimization algorithm. Fig. : Achievable rate region for the two-user SISO-IC with channel realization H (), and SNR 0 db. Fig. 3: Achievable rate region for the two-user SISO-IC with channel realization H (), and SNR 0 db. ples. Both transmitters are assumed to have the same power constraint P, i.e., P P P. SNR is defined as P/σ. For the SDR-based joint covariance and pseudo-covariance optimization algorithm, L 000 is used for the Gaussian randomization procedure in Algorithm. A. Approximation Ratio for SDR In this subsection, we evaluate the qualit of the approximate solution obtained b the SDR-based joint covariance and pseudo-covariance optimization algorithm proposed in Section IV. Denote R and R sdr as the optimal objective values of (P.4) and its relaxation (P.4-SDR), respectivel. Further denote ˆR as the objective value of (P.) corresponding to the approximate solution obtained b Algorithm. Then ˆR R R sdr, or R / ˆR R sdr / ˆR, (68) where R / ˆR is the approximation ratio. Since in general the optimal value R is difficult to be found, the upper bound R sdr / ˆR of the approximation ratio is usuall used to evaluate the qualit of the obtained approximate solutions 47]. Fortunatel, for the two-user SISO-IC considered herein, the optimal value R of (P.) and hence that of its equivalent problem (P.4)) can be obtained b the exhaustive search method proposed in 45]. With the rate-profile α setting to (/, /), the empirical ratios of R / ˆR over 500 random channel realizations at SNR 0 db are plotted in Fig., where the channel coefficients are generated from i.i.d. CSCG random variables with zero-mean and unit-variance. It is found B. Rate Region Comparison In Fig. and Fig. 3, the achievable rate regions for an example two-user SISO-IC are plotted for SNR0 db and 0 db, respectivel. The channel matrix for both plots is given b ] ] H () h h.030e i e i.645 h h 0.780e i e i The proposed improper Gaussian signaling schemes with joint and separate covariance and pseudo-covariance optimizations are compared with other existing schemes, including the optimal proper Gaussian signaling scheme b solving (P.6), the optimal improper Gaussian signaling obtained b the exhaustive search method 45], and the rank- improper Gaussian signaling scheme 44]. Both figures reveal that for the given channel H (), the achievable rate regions are significantl enlarged with improper Gaussian signaling over the conventional proper Gaussian signaling. The plots also demonstrate that the SDR-based joint covariance and pseudo-covariance optimization algorithm ields almost the optimal rates given b the exhaustive search, which is consistent with the observation in Fig.. Moreover, it is observed that the separate covariance and pseudo-covariance optimization algorithm performs close to the optimal solution, and also alwas outperforms the optimal proper signaling. It is worth remarking that, even with time-sharing (TS), improper Gaussian signaling still outperforms proper Gaussian signaling, as shown b the dashed lines in the two figures. For this particular channel realization, the Pareto boundar points of the achievable rate region with TS using improper Gaussian signaling can be obtained b the TS between the two single-user maximum rate points, and the largest rate corner point b the existing rank- scheme 44]. However, this is not alwas the case, as illustrated b the next example. Next, consider a two-user] SISO-IC channel given b H () 4.0e i e i e i e i.057. It is observed from Fig. 4 that for this particular channel realization at SNR0 db, there is no notable performance gain b using improper Gaussian signaling over proper signaling, which is in contrast to that observed in Fig. with channel H (). This is mainl due to the relativel weaker interfering link in this channel setup. For example, the interference-to-signal power gain ratio at user s receiver is given b h / h 0.05, which is much smaller than 0.53 in channel H (). Note that intuitivel, it is the non-negligible mutual interference among the users that is exploited b improper Gaussian signaling to outperform the conventional proper Gaussian signaling. 3 Therefore, for channel realization H () with almost negligible interfering link for user, no observable performance gain can be achieved b The achievable rate region with TS is obtained b taking the convex-hull operation over all the achievable rate-pairs given in (). 3 Consider the extreme case where all the interfering link gains vanish to zero and the SISO-IC reduces to K decoupled Gaussian point-to-point channels. In this case, proper Gaussian signaling is known to be optimal.

10 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED 0 Fig. 4: Achievable rate region for the two-user SISO-IC with channel realization H (), and SNR 0 db. improper Gaussian signaling. Another observation from Fig. 4 is that the rank- improper signaling scheme 44], which is based on the equivalent real-valued MIMO-IC, gives strictl smaller rate region than that b proper Gaussian signaling. In contrast, our proposed improper signaling schemes with either joint or separate covariance and pseudo-covariance optimizations, are observed to perform no worse than the optimal proper Gaussian signaling, in accordance with our previous discussion. C. Max-Min Rate Comparison The rate-profile technique used in characterizing the Pareto boundar of the achievable rate region can be directl applied for maximizing the minimum (max-min) rate of the two users without TS. Specificall, the max-min problem for the twouser SISO-IC is equivalent to solving (P) b using the rateprofile α (/, /). An alternative max-min solution with improper Gaussian signaling was proposed in 45], where based on the equivalent real-valued MIMO-IC, the transmit covariance matrix of the equivalent real-valued signal vector for each user is assumed to be of rank-. Note that the use of rank- transmission in both 44] and 45] can be justified b the fact that the total DoF of two-user MIMO-ICs exactl equals to 5]. For the ease of precoder design, zero-forcing (ZF) receivers were further applied in 45]. As a benchmark comparison, we also plot the max-min rate achievable b the simple time division multiple access (TDMA) scheme, where for simplicit, each user is assumed to access the channel for half of the time. To evaluate the average max-min rates, 500 random channel realizations are simulated, where the channel coefficients are drawn from independent zero-mean CSCG random variables. For this example, asmmetric channels are considered, where the average power values of the direct and interfering channels are and 0., respectivel, i.e., h kk CN (0, ), h k k CN (0, 0.), k,, k k. The obtained results are shown in Fig. 5. The optimal max-min rate achievable b proper Gaussian signaling and that b improper Gaussian signaling obtained b the exhaustive search method 45] are also included in the figure. It is observed that in the low SNR regime, there is no notable gain b improper Gaussian signaling over conventional proper Gaussian signaling, which Fig. 5: Average max-min rate for the two-user SISO-IC. is due to the negligible interference levels at low SNRs. As SNR increases, the max-min rate b proper Gaussian signaling saturates since the total number of data streams transmitted, which is, exceeds the total number of DoF of the twouser SISO-IC, which is. In contrast, the linear increase of the max-min rates with respect to the logarithm of SNR can be achieved either b TDMA, or b improper Gaussian signaling. It is worth remarking that over the entire SNR range, the proposed algorithms based on covariance and pseudocovariance optimizations ield close-to-optimal performance obtained b exhaustive search method. On the other hand, the rank- transmission with ZF receivers based on the equivalent real-valued MIMO-IC gives a near-optimal performance in the high-snr regime, which is expected due to the optimalit of ZF receivers at high SNR as well as the DoF optimalit of rank- transmission as pointed out in 44,45]; however, in the low and moderate SNR regime, the rank- transmission scheme results in strictl suboptimal performance, which ma be due to the noise enhancement issue associated with ZF receivers applied in 45], as well as the over-conservative number of data streams used b assuming rank- transmit covariance matrices. D. Sum-Rate Comparison In this subsection, the sum-rate maximization with improper Gaussian signaling is considered. B using the equivalent realvalued MIMO-IC of the complex-valued SISO-IC, existing sum-rate maximization algorithms in the literature, such as the one via the iterative weighted MSE minimization (WMMSE) 7], can be applied directl for maximizing the sum-rate of the two-user SISO-IC when improper Gaussian signaling is emploed. However, although the WMMSE algorithm is guaranteed to converge to a local maximum of the sum-rate, it is not guaranteed to achieve the global sum-rate maximum. With the algorithms proposed in this paper via covariance and pseudo-covariance optimization, we illustrate with the following example that our proposed algorithms strictl improve the achievable sum-rate over that b the WMMSE algorithm. In order to appl the WMMSE algorithm 7] to the sumrate maximization problem when improper Gaussian signaling is applied, we transform the complex-valued channel to the equivalent real-valued MIMO channel, similarl as in 43] 45]. Denote Q k as the transmit covariance matrix of user k in the equivalent real-valued MIMO-IC. Without loss