# On Design of Improper Signaling for SER Minimization in K

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On Design of Improper Signaling for SER Minimization in K

On Design of Improper Signaling for SER Minimization in K -User Interference Channel Hieu Duy Nguyen† , Rui Zhang†‡ , and Sumei Sun† † Institute for Infocomm Research, Singapore Department, National University of Singapore, Singapore Emails: {nguyendh, sunsm}@i2r.a-star.edu.sg, [email protected] ‡ ECE Abstract—The rate maximization for the K-user interference channels (ICs) has been investigated extensively in the literature. However, the dual problem of minimizing the error probability with given signal constellations and/or data rates of the users is less exploited. In this paper, by utilizing the additional degrees of freedom attained from the improper signaling (versus the conventional proper signaling), we optimize the precoding matrices for the K-user single-input single-output (SISO) ICs to achieve minimal transmission symbol error rate (SER). Compared to conventional proper signaling as well as other stateof-the-art improper signaling designs, our proposed improper signaling scheme is shown to achieve notable SER improvement in SISO-ICs by simulations. Our study provides another viewpoint for optimizing transmissions in ICs and further justifies the practical benefit of improper signaling in interference-limited communication systems. I. I NTRODUCTION Interference channel (IC) is a fundamental model in multiuser wireless communication and has been extensively studied to date. However, complete characterization of the capacity region of the IC corrupted by additive Gaussian noise, is in general still open, even for the simplest two-user case [1]. The recent advance in a technique so-called interference alignment (IA) has motivated numerous studies on characterizing the rate performance of ICs under the high signal-to-noise ratio (SNR) regime. With the aid of IA, the maximum achievable rates in terms of degree of freedom (DoF) have been obtained for various IC models to provide useful insights on designing optimal transmission schemes for interference-limited communication systems (see, e.g., [2] and the references therein). Another notable advancement is the use of improper Gaussian signaling (IGS) for ICs. Note that for conventional systems employing proper Gaussian signals, the real and imaginary parts of the transmitted signal have equal power and are independent zero-mean Gaussian random variables [3]. In contrast, a complex Gaussian symbol is called improper if its real and imaginary parts have unequal power and/or are correlated. Improper signals have been studied in applications such as detection and estimation [3], [4]. Studies of IGS in communication systems, however, only appeared recently. This may be due to the fact that proper Gaussian signaling (PGS) has been known to be capacity optimal for the Gaussian pointto-point, multiple-access, and broadcast channels; and as a result, it was presumably deemed to be optimal for ICs as well. In [5], it has been shown that IGS can further improve the achievable rates of the 3-user IC in high-SNR regime. Inspired by this work, subsequent studies have investigated ICs with IGS in finite-SNR regime. For instance, the achievable rate region and minimal signal-to-interference-plus-noise ratio (SINR) maximization for the K-user IC have been characterized in [6]-[9]. These works have reported a significant rate improvement of IGS over conventional PGS in terms of achievable rate under finite SNR. It comes to our attention that most of the existing works on ICs have focused on investigating the rate performance. However, a dual problem for ICs, which minimizes the transmission error probability with given users signal constellations and/or rates is less exploited. Notice that this problem may be more practically sensible for the scenarios when the users have their desired quality-of-service (QoS) in terms of data rate and error performance to be met. It is worth noting that in [10], the authors considered the problem of minimizing the mean squared error (MSE) in ICs as an indirect way to minimize the error rate. Although MSE is a meaningful criterion in practice, minimizing the MSEs in ICs does not necessarily lead to the error probability minimization. Moreover, [11] has studied the error performance for ICs based on IA. However, it is restricted only to the case of three-user ICs with at least two antennas at each node, in which each of the three user links can achieve at least one DoF. In contrast to the above prior works, in this paper, we study the problem of minimizing the users’ symbol error rates (SERs) explicitly in the K-user single-input single-output (SISO) IC by applying improper signaling over finite signal constellations. We are motivated by the results that IGS can provide rate gains for ICs over the conventional PGS [5]-[9]. It is thus expected that the additional degrees of freedom provided by improper signaling can also be exploited to improve the SER performance in ICs, even with practical (non-Gaussian) modulation schemes. Specifically, we investigate the K-user SISO IC with given signal constellations and transmission rates of the users assuming perfect channel state information (CSI). With improper signaling, we first derive a closed-form expression for the pairwise error probability (PEP) in decoding each user’s signal under a mild assumption on the Gaussian distribution of the interference that is treated as additional noise at each receiver. Based on this result, we then propose an efficient algorithm to jointly optimize the precoding matrices at all transmitters to minimize the maximum PEP/SER among all users. Through simulations, we show that a significant SER improvement can be achieved by the proposed algorithm in SISO-IC as compared to the conventional proper signaling as well as other state-of-the-art improper signaling designs. h21 n2 hK 1 II. S YSTEM M ODEL We consider a K-user SISO IC as shown in Fig. 1, where the received complex baseband signals are expressed as yk = hk1 x1 + hk2 x2 + · · · + hkK xK + nk , h11 (1) hKK nK Fig. 1: The K-user SISO IC. jθkl where hkl = |hkl |e , k, l ∈ {1, . . . , K}, is the complex coefficient for the channel from transmitter l to receiver k; xk is the transmitted symbol for user k; and nk is the additive white Gaussian noise (AWGN) at receiver k, which is assumed to be a circularly symmetric complex Gaussian (CSCG) random variable (RV) with zero mean and variance of σk2 , denoted by nk ∼ CN (0, σk2 ), k = 1, . . . , K. The transmit power of user k is assumed to be limited by Pk , i.e., E[||xk ||2 ] ≤ Pk . For convenience, we define the SNR of user k as SNRk = Pk /σk2 . At the receiver side, each user aims to decode the desired signal with minimum error probability. In this study, we assume that each receiver employs the practical single-user detection and thus the interference from all other users is treated as additional noise. However, different from the conventional setup where proper signaling is assumed, we consider the use of more general improper signals. We first define the propriety and impropriety for a complex RV as follows. Definition 2.1: Given a zero-mean complex RV α = αR + jαI and its real covariance matrix C α = E [αR αI ]T [αR αI ] , the RV α is called proper if its covariance matrix C α is a scaled identity matrix in the form of C α = pI with p > 0, which means that the real and imaginary parts αR and αI are uncorrelated and have the equal variance of p. Otherwise, we call α improper. It is worth noting that in practical communication systems, modulation schemes such as PSK (e.g., QPSK, 8PSK) and square QAM (e.g., 16QAM, 64QAM) all have proper signal constellations. For the convenience of our analysis in the sequel, we use the following equivalent real-valued representation of the complex-valued system in (1), which is essentially a K-user 2 × 2 MIMO IC with all real entries given by cos θkk − sin θkk xkR ykR = |hkk | ykI xkI sin θkk cos θkk {z } | {z } | {z } | ,y k ,J (θkk ) ,xk K X cos θkl − sin θkl xlR n (2) |hkl | + + kR sin θkl cos θkl xlI nkI l=1,l6=k {z } | {z } | {z } | nk ,J (θkl ) ,xl for k = 1, . . . , K, where “R” and “I” denote the real and imaginary parts, respectively; and nkR and nkI are independent and identically distributed (i.i.d.) real Gaussian RVs with zero mean and variance of σk2 /2, denoted as N (0, σk2 /2). Consider a normalized proper constellation for each user k, represented by a pair of real symbols inidk = [dkR dkI ]T with h the covariance matrix C dk = E dk dTk = I. For example, the symbol set for a normalized constellation from QPSK is {dk } = [1 1]T , [1 − 1]T , [−1 − 1]T , [−1 1]T . The improper symbol xk transmitted by user k can then be obtained via the following transformation xkR ak,12 dkR a = k,11 , (3) xkI ak,21 ak,22 dkI {z } | ,A k or equivalently xk = Ak dk . Here Ak is referred to as the precoding matrix, which is also called the widely linear processing [3], [4]. Specifically, Ak can also be regarded as a rotation and scaling matrix applied over any proper signal constellation of user k to obtain improper signals. Furthermore, the o power constraint for user k is re-expressed n transmit T as T r Ak Ak = a2k,11 + a2k,12 + a2k,21 + a2k,22 ≤ Pk . From (2), the real system model of the K-user IC with improper signaling can be expressed in the following form, y k = |hkk |J (θkk )Ak dk + K X |hkl |J (θkl )Al dl + nk . l=1,l6=k (4) Without loss of generality, we assume that each user k applies the decoding matrix at the receiver in the form of J (θkk )Rk . The signal after applying the decodng matrix is given by r k = RTk J T (θkk )y k = |hkk |RTk Ak dk + K X |hkl |RTk J (φkl )Al dl + RTk ñk , (5) l=1,l6=k where φkl = θkl − θkk , k, l ∈ {1, . . . , K} and l 6= k; and ñk = J T (θkk )nk , [ñkR ñkI ]T with ñkR and ñkI being i.i.d. Gaussian RVs each distributed as N (0, σk2 /2), k = 1, . . . , K. PK Denote wk = l=1,l6=k |hkl | J (φkl ) Al dl + nk as the effective noise at the receiver of user k, which includes both additive noise and interference. Then the post-processed signal in (5) can be re-expressed as r k = |hkk |RTk Ak dk + RTk wk . (6) In this paper, we consider the use of the whitening filter for −1/2 the effective noise as the decoding matrix, i.e., Rk = W k where W k is defined as W k = E wk wTk = K X σk2 |hkl |2 J (φkl )Al ATl J T (φkl ). I+ 2 (7) l=1,l6=k Considering the general Mk -ary (Mk > 1) constellations for dk of each user k, the (exact) SER expression is difficult to be obtained even assuming the conventional proper signalling, i.e., Ak is a scaled identity matrix for all k’s. Therefore, in this paper, we approximate the SER with improper signalling by an upper bound, which is obtained by applying the union bound as follows: X X 1 SERk ≤ P r{dk → d˜k |dk }P r{dk } Mk − 1 dk d˜k 6=dk X X 1 P r{dk → d˜k |dk }, (8) = Mk (Mk − 1) dk d˜k 6=dk where P r{dk → d˜k |dk } is the so-called PEP when dk is erroneously decoded as d˜k with d˜k 6= dk conditional on that dk is transmitted for user k; and in (8) we have assumed that all constellation symbols are selected with equal probability. In the rest of this paper, we consider the above SER upper bound as our performance metric. To derive the PEP, we need to make one further assumption that the interference-plus-noise term wk in (6) is Gaussian distributed. This means that for deriving the PEP of user k, we need to assume that dl ∼ N (0, I), ∀ l ∈ {1, . . . , K}, l 6= k, i.e., dl is a Gaussian random vector with zero mean and identity covariance matrix, despite of the practical Ml -ary modulation used. Under the above Gaussian assumption for the interference and hence the interference-plus-noise, after the application of the whitening filter, the interference-plus-noise RTk wk in (6) becomes a Gaussian random vector with zero mean and identity covariance matrix. Under this assumption, the optimal maximum likelihood (ML) detector for user k can be shown to be equivalent to an Euclidean-distance based detector: for each symbol transmitted by user k, its receiver finds the constellation symbol dˆk which gives the smallest distance of ||r k − |hkk |RTk Ak dˆk ||2 and declares it as the transmitted symbol. Thereby, we are able to obtain a closedform expression for the PEP as shown in the following lemma, while the accuracy of the above approximation approach for the PEP will be validated later in Section V by simulations. Lemma 2.1: Assuming the interference-plus-noise vector wk is Gaussian distributed and the Euclidean-distance based detector is used, the PEP at the receiver of user k is given by P r{dk → d˜k |dk } q ˜ |hkk | (dk − d˜k )T ATk W −1 k Ak (dk − dk ) , = Q 2 (9) where Q(x) , √1 2π R∞ x exp(−u2 /2)du. III. M AXIMUM SER M INIMIZATION In this section, we aim to minimize the maximum SER given in (8) among all K users with improper signalling. This is achieved by jointly optimizing the precoding matrices Ak ’s of all users to minimize the maximum PEP given in (9) among all different signal constellation pairs for each user k, and also over all k’s. Given a set of transmit power constraints for the users, Pk ’s, the optimization problem is thus formulated as s, min. K s Ak k=1 s.t. P r{dk → d˜k |dk } ≤ s, T r(Ak ATk ) ≤ Pk , ∀ d˜k 6= dk , k = 1, . . . , K. (10) Using (9), problem (10) is equivalent to the following problem t, max. K t Ak k=1 ˜ s.t. |hkk |2 (dk − d˜k )T ATk W −1 k Ak (dk − dk ) ≥ t, T r(Ak ATk ) ≤ Pk , ∀ d˜k 6= dk , k = 1, . . . , K. (11) Note that in the above problem, the first constraint for each user k in fact corresponds to Dk = Mk (M2 k −1) number of constraints, which are for all the different signal constellation pairs, d˜k 6= dk . We denote F k as a Dk × 2 matrix which consists of all the vectors dk − d˜k , ∀ d˜k 6= dk . For example, the corresponding F k for the normalized QPSK signal constellation is T 0 2 2 2 2 0 F k,QP SK = . (12) 2 2 0 0 −2 −2 Then the first set of constraints in (11) for each user k are equivalent to h i T |hkk |2 F k AT W −1 ≥ t, i = 1, . . . , Dk , (13) k AF k ii where [Q]ii denotes the (i, i)-th element of a matrix Q. Hence, we can express problem (11) equivalently as t, max. K t Ak k=1 h i T s.t. |hkk |2 F k ATk W −1 A F ≥ t, k k k ii T r(Ak ATk ) ≤ Pk , i = 1, . . . , Dk , k = 1, . . . , K. (14) Problem (14) can be shown to be non-convex, and thus it is in general difficult to find its optimal solution. Therefore, in the following, we propose an efficient algorithm that is guaranteed to find at least a locally optimal solution for problem (14). K First, we introduce a set of auxiliary variables B k k=1 TABLE I: M INMAX -PEP A LGORITHM and reformulate problem (14) as the following optimization problem t, Ak Kmin. k=1 , Bk K 1. t 2. k=1 s.t. [Gk ]ii ≤ t, T r(Ak ATk ) ≤ Pk , i = 1, . . . , Dk , k = 1, . . . , K. (15) In (15), we have K X Gk = |hkl |2 B Tk J (φkl )Al ATl J T (φkl )B l − |hkk |B Tk Ak F Tk − |hkk |F k ATk B k + p Pk /2I, k = 1, . . . , K. 2 PK σk 2 l=1,l6=k |hkl | 2 I + −1 ×J (φkl )Al ATl J T (φkl ) , k = 1, . . . , K. K Update Ak k=1 by solving problem (18). K Repeat steps 3 and 4 until both Ak k=1 and K B k k=1 converge within the prescribed accuracy. Set B Tk = |hkk | F k ATk IV. B ENCHMARK S CHEMES l=1,l6=k σk2 3. 4. Initialize Ak = B Tk B k , (16) In this section, we present alternative approaches in the existing literature and furthermore, the Kprecoding and decoding ma Kto optimize and R trices, i.e., A k k=1 k k=1 , for the K-user SISO-IC K X σk2 T with proper/improper signalling. It is worth pointing out that, 2 2 2 [Gk ]ii = |hkl | ||bk,i J (φkl )al,1 || ||bk,i || + 2 unlike our proposed Minmax-PEP algorithm, these benchmark l=1,l6=k schemes might not be originally introduced for minimizing the K X T T T 2 2 SER (or PEP) in ICs explicitly. + |hkl | ||bk,i J (φkl )al,2 || − 2|hkk |T r f k,i bk,i Ak , 2 l=1,l6=k (17) where bk,i is the i-th column of the matrix B k ; ak,1 and ak,2 are the first and second column of the matrix Ak ; and f k,i is the i-th row of the matrix F k , k = 1, . . . , K. Since problem (15) can be shown be convex over each to K K of the two sets Ak k=1 and B k k=1 when one of them is given as fixed, we can apply the technique of alternating optimization Specifically, Kto solve the problem iteratively. K given Ak k=1 , the solution of B k k=1 can be obtained by setting the gradient of [Gk ]ii given in (17) with respect to bk,i as zero, i.e., ∂ [Gk ]ii /∂bk,i = 0. On the other hand, K K optimizing Ak k=1 with given B k k=1 can be obtained by solving the following convex problem by using primal-dual interior point method, via, e.g., CVX [12]. t, min. K t Ak k=1 s.t. A. Proper Signaling with Power Control For the K-user SISO IC with conventional proper signaling, the p pkprecoding and decoding matrices are reduced to Ak = 2 I and Rk = I, where pk ≤ Pk . Considering the use of ML or Euclidean-distance based detection at each receiver, user k detects that the symbol dˆk is transmitted if it gives the smallest distance of ||r k − pk |h2 kk | dˆk ||2 among all signal symbols. In this case, we search over all user power allocations (p1 , . . . , pK ) subject to pk ≤ Pk and find the best (p1 , . . . , pK ) which achieves the minimum of max SERk , k = 1, . . . , K. We refer to this scheme as Proper Signalling with Power Control (PS-PC). B. MSE-based Schemes K X σk2 |hkl |2 ||bTk,i J (φkl )al,1 ||2 ||bk,i ||2 + 2 l=1,l6=k + K X |hkl |2 ||bTk,i J (φkl )al,2 ||2 l=1,l6=k − 2|hkk |T r f Tk,i bTk,i Ak ≤ t, T r(Ak ATk ) ≤ Pk , k = 1, . . . , K, i = 1, . . . , Dk , (18) To summarize, the proposed algorithm to solve problem (15) is given in Table I, which is referred to as p MinmaxPEP. Note that in Table I, we have initialized Ak = Pk /2I, k = 1, . . . , K, i.e., assuming all users to employ conventional proper signalling initially. Although there has been no existing study on directly minimizing the SERs for ICs with improper signalling, minimizing the MSEs has been considered as an alternative approach (see, e.g., [10]). Following [10], we first define the MSE matrix for r k as E k = E (r k − dk )(r k − dk )T K 2 X T σk |hkl |2 J (φkl )Al ATl J T (φkl ) I+ = Rk 2 l=1,l6=k 2 + |hkk | Ak ATk Rk − |hkk |RTk Ak − |hkk |ATk Rk + I, where we have used the following assumptions: E[dl dTk ] = 0; E[dl nTk ] = 0; and E[dk dTk ] = I. The minimization of the sum of MSEs and the maximum per-stream MSE can be expressed as the following optimization problems in (19) and (20), respectively [10]. T r {E k } k=1 T r(Ak ATk ) −1 10 ≤ Pk , k = 1, . . . , K. (19) −0.2 10 2 s.t. K X max(SER ,SER ) Kmin. K Ak k=1 , Rk k=1 0 10 s.t. max max k=1,...,K i=1,2 T r(Ak ATk ) [E k ]i,i −2 10 1 Kmin. K Ak k=1 , Rk k=1 ≤ Pk , k = 1, . . . , K. (20) The algorithms for solving the above two problems have been given in [10] and are thus omitted here for brevity. We thus denote the algorithms to solve (19) and (20) as Minsum-MSE and Minmax-MSE, respectively. Note that the convergence of these algorithms to (at least) a local optimum of (19) or (20) is guaranteed. With the obtained precoding and decoding matrices applied, the receiver of each user k finds the nearest constellation symbol dˆk to r k given in (6), i.e., which gives the smallest distance of ||r k − dˆk ||2 , and then declares it as the transmitted symbol. −0.6 10 −3 10 −5 −4 10 −5 10 −5 0 PS−PC Minsum−MSE Minmax−MSE MinIL−IA MaxSINR−IA Minmax−PEP 0 RTk J (φkl )Al = 0, rank(RTk Ak ) = 1. (21) in which k, l ∈ {1, . . . , K}, l 6= k. In (21), the first and second set of equations are for nullifying the interference and enforcing the desired signal to span exactly one dimension at each receiver, respectively. In this paper, we consider two well-established IA-based schemes, i.e., interference leakage minimization and SINR maximization [13], denoted as MinILIA and MaxSINR-IA, respectively. Note that the above IA-based algorithms will yield the precoding and decoding matrices simplified as 2 × 1 transmit and receive beamforming vectors, denoted by {v k }K k=1 and , respectively; and thus each user needs to transmit {uk }K k=1 with only one-dimensional (1D) signal constellations. Therefore, for a fair comparison with other considered improper signalling schemes in this paper which use two-dimensional (2D) signal constellations, the transmitted symbols of IAbased schemes are assumed to be drawn from the 1D PAM modulation with the same number of symbols and average transmit power as other 2D modulation schemes. For example, the corresponding constellation for the normalized QPSK is 5 10 15 20 SNR (dB) Fig. 2: Comparison of the minimized maximum SER over SNR in a two-user IC. C. IA-based Schemes It is known that the maximum sum-rate DoF of the equivalent K-user 2×2 real IC given in (2) with improper signalling is 2 when K = 2, which is achieved when each of the two users sends one data stream, based on the principle of zero-forcing (ZF) [2]. In this subsection, we consider another possible strategy to minimize the SER by applying IA-based precoder and decoder designs for the K-user SISO-IC with improper signalling. Specifically, we aim to find two sets of K K {Ak }k=1 and {Rk }k=1 such that [13] 5 the normalized 4PAM, i.e., n √ √ √ √ o −3 √25 , − √25 , √25 , 3 √25 . −1 −1 1 1 −−→ , , , 1 −1 −1 1 V. N UMERICAL RESULTS In this section, we present simulation results to compare the SER performance of the proposed scheme with other benchmark schemes in the K-user SISO-IC. First, Fig. 2 shows the results for the minimized maximum SER of the users over SNR in a two-user IC with the channel coefficients given by 1.9310e−j2.0228 0.7732ej0.5865 . 0.9249ej3.0213 2.3742ej0.2089 Here, we assume that both users apply 8PSK modulation. Therefore, the equivalent constellation for IA-based schemes is 8PAM. The SNRs of the two users are assumed to be equal and are varied from 0 to 20 dB. We observe that the proper signaling with power control (PS-PC) and MSEbased schemes result in saturated SERs for both users over SNR. This is because under such schemes, both users’ signals span two dimensions each, and thus the SER performance is interference-limited with increasing SNR. In contrast, the maximum SERs by the proposed Minmax-PEP and IA-based schemes decrease over SNR. Similar to the IA-based schemes, as the SNR increases, the precoding matrices A1 and A2 obtained from Minmax-PEP gradually converge to rank-1 matrices, i.e., each user’s precoded signal spans over only one dimension although the original signal dk is drawn from a 2D constellation (8-PSK). As a result, orthogonal transmissions of the two users are achieved at each receiver and the system SER is not limited by the interference as SNR increases. It is VI. C ONCLUSION 0 10 In this paper, we study the problem of minimizing SERs for the K-user SISO IC with improper signalling applied over practical signal modulations. We propose an efficient algorithm to jointly optimize the precoding matrices of users to directly minimize the maximum SER, by exploiting the given signal constellations and the additional degrees of freedom provided by improper signalling. Several benchmark schemes based on conventional proper signalling as well as other state-of-theart improper signaling designs are compared; and it is shown by simulations that our proposed improper signalling scheme achieves competitive or improved SER performances in SISOICs. Our study provides a different view on transmission optimization for ICs in contrast to the large body of existing works on the rate optimization of ICs. −1 10 −0.1 max(SER1,SER2,SER3) 10 −2 10 −3 10 −0.7 10 −5 0 5 10 −4 10 −5 10 −6 10 −5 PS−PC Minsum−MSE Minmax−MSE MinIL−IA MaxSINR−IA Minmax−PEP 0 5 R EFERENCES 10 15 20 25 30 SNR (dB) Fig. 3: Comparison of the minimized maximum SER over SNR in a three-user IC. also observed that Minmax-PEP performs better than IA-based schemes when SNR is less than 0 dB; however, as SNR further increases, the SER performance of Minmax-PEP becomes worse than that of IA-based schemes, which is mainly due to the use of suboptimal noise whitening filters for MinmaxPEP in order to derive the closed-form PEP expression given in (9). Thus, it is interesting to jointly optimize the decoding and precoding matrices to further improve the SER performance in our proposed improper signaling scheme, which will be left to our future work. Next, Fig. 3 compares the minimized maximum SER in a three-user IC given by 1.9310e−j2.0228 0.7732ej0.5865 0.9766ej1.1907 0.9249ej3.0213 2.3742ej0.2089 0.3009e−j1.5307 . −j0.4282 1.7628e 0.3127e−j1.4959 2.1935ej1.7364 Here users (1, 2, 3) apply (QPSK, 8PSK, 8PSK) modulations, respectively. The equivalent constellations for IAbased schemes are hence (4PAM, 8PAM, 8PAM). The users’ SNRs are set to be equal and are varied from 0 to 30 dB. It is observed that Minmax-PEP achieves substantial SER improvement over all other schemes. Note that in this case, the IA-based schemes cannot reduce SER over SNR, since there is no feasible IA solution to (21) with given 1D signal constellations for all the three users. [1] R. H. Etkin, D. N. C. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5534-5562, Dec. 2008. [2] S. A. 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