Linear Precoder Design for Doubly Correlated Partially ... - IEEE Xplore

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014

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Linear Precoder Design for Doubly Correlated Partially Coherent Fading MIMO Channels Animesh Yadav, Student Member, IEEE, Markku Juntti, Senior Member, IEEE, and Jorma Lilleberg, Senior Member, IEEE

Abstract—We consider the single-user multiple-input multipleoutput (MIMO) precoder design problem for the doubly spatially correlated partially coherent Rayleigh fading channels with discrete inputs. The objective is to design a linear precoder to adapt to the degradation caused by the imperfect channel estimation at the receiver and the transmit-receive antenna correlation. The system is partially coherent so that the MIMO channel coefficients are estimated at the receiver and its error covariance matrix is fed back to the transmitter. We utilized the cutoff rate (CR) expression, an alternative to the mutual information (MI), and propose to use it as a design criterion to design the linear precoders. A linear precoder is obtained by numerically maximizing the CR with respect to the precoder matrix with a given average power constraint. Furthermore, the precoder matrix is decomposed using singular value decomposition (SVD) into the input shaping matrix, power loading matrix, and beamforming matrix. The beamforming matrix is found to coincide with the eigenvectors of the transmit correlation matrix. The power loading and input shaping matrices are solved numerically using the difference of convex (d.c.) functions programming algorithm and optimization under the unitary constraint, respectively. A 2-block alternating optimization (AO) algorithm is proposed to solve the input shaping matrix and power loading matrix iteratively. Precoders are designed to be used in conjunction with two MIMO transmission schemes: the spatial multiplexing (SM) and space–time (ST) block transmission modes. The frame error rate (FER) and average MI are used as the performance metrics to validate the performance of the newly designed CR-precoders in comparison with the conventional no-precoding case and cutoff rate optimized partially coherent constellations (PCCs). Numerical examples show that the performance gains of the designed precoders are significant compared to the CR-PCCs and conventional codewords. Index Terms—Linear precoder design, multiple-input multipleoutput (MIMO) systems, partial CSI, cutoff rate, spatial correlated channels, spatial multiplexing (SM), space–time block codes (STBC), unitary space–time codes (USTC), d.c. programming, alternating optimization.

Manuscript received March 27, 2013; revised October 24, 2013 and January 20, 2014; accepted March 27, 2014. Date of publication April 15, 2014; date of current version July 8, 2014. This paper was presented in part at the 10th International Symposium on Wireless Communication Systems (ISWCS), Ilmenau, Germany, August 27–30, 2013. This work was supported in part by the Finnish Funding Agency for Technology and Innovation (Tekes), Elektrobit, Nokia, Nokia Siemens Networks, Xilinx, Renesas Mobile Europe, and Academy of Finland. The associate editor coordinating the review of this paper and approving it for publication was D. Tuninetti. A. Yadav and M. Juntti are with the Department of Communications Engineering and Centre for Wireless Communications, University of Oulu, FI-90014 Oulu, Finland. J. Lilleberg is with the Center for Multimedia Communication, Rice University, Houston, TX 77005-1892 USA, the Centre for Wireless Communication, University of Oulu, FI-90014 Oulu, Finland, and Broadcom Corporation, 90590 Oulu, Finland. Digital Object Identifier 10.1109/TWC.2014.2317490

I. I NTRODUCTION

M

ULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) systems give high capacity and performance gains under perfect CSI at the receiver (CSIR) over single-input singleoutput (SISO) systems [1]. Better performance can be further improved by providing the perfect (exact instantaneous channel information) or partial (first- or second-order statistics of the channel) CSI at the transmitter (CSIT) [2] via channel feedback. However, the spectral efficiency and the performance of MIMO systems degrades with the quality of the channel estimates [3], [4]. Considering imperfect CSIR, multiple antenna constellations adapted to the quality of channel estimated at the receiver, called partially coherent constellations (PCCs), have been designed by maximizing the minimum Kullback-Leibler divergence (KLD) [5] and maximizing the cutoff rate (CR) [6]. Recently, their performance in the spatially correlated channels and with forward error correction (FEC) codes were studied by us in [7]. The PCC construction procedure has two-fold computational complexity. First, searching the constellations points itself, and, second, the search for a close-to-optimal, pseudoGray, bit mapping scheme [7]. Furthermore, the decoding complexity of the PCCs can be high. Because of these reasons, we consider linear precoder design as an alternative. Linear pecoders have lower decoding complexity and are more easily backward compatible with the existing conventional constellations and various wireless standards (e.g., 3GPP Long Term Evaluation [LTE] and wireless local area network [WLAN]). Precoder design has been widely investigated over the last decade with different assumptions on CSIR and CSIT to improve spectral efficiencies or link reliability for MIMO systems. The assumption on channel knowledge can include perfect CSIR and CSIT, perfect CSIR and imperfect or statistical CSIT, or imperfect CSIR and statistical CSIT. Among these designs, an assumption of the imperfect CSIR and partial CSIT is considered to be practical as channel coefficients are always estimated in real systems. Moreover, the use of second-order channel statistics such as CSIT obviates the dependence on frequent channel updates. Numerous criteria can be found in the literature to design the precoders, e.g., maximizing the mutual information [8] or CR [9], minimizing the total mean-square error (MSE) [10], minimum bit error probability (BEP) [11], maximizing the received signal-to-noise ratio (SNR) [12], and minimum Euclidean distance based designs [13]. These criteria were developed with an assumption of Gaussian input distribution which is

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practically not feasible to implement, although, capacity optimal. Furthermore, Lozano et al. [14] showed that power allocation policy to the parallel Gaussian channels with the Gaussian input is sub-optimal for discrete inputs.1 As a consequence, few works [15]–[17], [31] have considered the design of linear precoders for the discrete input over the MIMO fading channels. However, perfect CSIT and CSIR has been assumed in [15]– [17], [31] assumed perfect CSIR and second-order statistics based CSIT, and doubly correlated spatial channel. Similar to [15]–[17], [31], we consider herein the precoder design for the discrete-input continuous-output channels but for more realistic assumptions of imperfect CSIR and second-order statistics, of the estimated channel, based CSIT. Moreover, we also account for the presence of spatial correlation likewise assumed in [17], a cause for performance and capacity degradation of the MIMO systems [18], [19], both at the transmitter and receiver. A closed and tractable form of the mutual information for the MIMO fading channels with imperfect CSIR is not known [20]. Moreover, the corresponding capacity achieving input distribution is also an open problem [3], [20], [21]. Only upper and lower bounds on the mutual information are known, but those can be rather loose for certain non-Gaussian inputs distributions [3], [21]. Nevertheless, we propose to use the CR, an alternative lower bound on the mutual information, as a criterion for MIMO precoder design. The CR expression is suitable for readily available finite and discrete constellations and has a closed and analytically tractable form. In this paper, we consider the single-user MIMO precoder design for a imperfectly estimated CSI at the receiver with the assumption that second-order statistics of the CSI error covariance matrix is available at the transmitter via an errorfree feedback link. The designed linear precoder is adapted to both the quality of the channel estimate at the receiver and the transmit/receive antenna correlation. Furthermore, we consider the Kronecker channel model for correlated channel and present a detailed discerption of the MMSE channel estimation method. Then a generic closed and tractable form of CR expression, derived in [7], is utilized in the optimization problem to design the linear precoders for various MIMO transmission schemes such as spatial multiplexing (SM), space–time block codes (STBC), e.g., orthogonal STBC (OSTBC) and unitary space–time codes (USTC). However, in [22], we only considered the precoder design for STBC transmission scheme. Moreover, after the singular value decomposition (SVD) of the precoding matrix into a left singular (input-shaping) matrix, a diagonal power loading matrix and a right singular (beamforming) matrix, it is found that right singular matrix coincides with the correlation matrix of the channel. The remaining two matrices are solved using a numerical alternating optimization (AO) algorithm. In the AO algorithm, the CR expression is maximized iteratively jointly over the diagonal power loading and left singular matrices by alternating the restricted maximization over the individual matrices. The CR expression can be transformed 1 The optimal policy allocate more power to the channels with high gain to support maximum rate and finite constellation, on the other hand, might not support that high rate. Thus, power allocated to that channel must be reduced otherwise it would be wasted.

into a difference of convex (d.c.) form [23] as a function of a power loading matrix. Thus, the AO algorithm first solves a d.c. form of CR expression globally by d.c. programming approaches [23] to get the optimal power loading across the transmitting antennae for a given fixed left singular matrix. Subsequently the CR expression with respect to the left singular matrix is solved via self-tuning steepest ascent (SA) algorithm [24] for a given fixed power loading matrix. Previously, Zeng et al. [17] have also employed the similar approach to find the precoder matrices because of the similarity in the structure of the problem. However, different numerical methods were used in solving the diagonal and beamforming matrices. Performances of the CR optimized precoders are compared to those of the conventional (no precoding) methods and CR optimized partially coherent constellations (CR-PCCs) [6] designed under similar channel assumptions. To validate the usefulness of the resulting precoders, optimized for various MIMO schemes such as SM and STBC (e.g., Alamouti and unitary codes), we use frame error rate (FER), bit error rate (BER) and average mutual information (MI) as our performance metrics with turbo codes. In short, contributions of this paper includes the following: • CR expression as a criterion for designing the linear precoder matrix under channel estimation errors and spatially correlations. Most importantly, CR expression is suitable for discrete or finite-alphabet inputs and has a tractable form. • A unified framework to design the precoder for MIMO transmission, i.e., SM and STBC. • Decomposition of the CR expression with respect to the power loading matrix into a d.c. form guaranteeing global power allocations. • A 2-block AO algorithm to solve the precoder matrix iteratively with guaranteed convergence of its objective function value. • Numerical performances result of the newly designed precoder is compared to the CR-PCCs designed in [7]. They further suggest the use of CR-PCCs with SM and precoders with STBC mode of transmission. The outline of this paper is as follows. The assumptions and system model are detailed in Section II. The precoder design criterion and optimization problem are presented in Section III. The details of the numerical optimization method are presented in the Section IV. Numerical coded BER, FER, and MI results are presented in Section V. Section VI concludes the paper. Nomenclature: (·)∗ , (·)T , and (·)H denote complex conjugate, transpose, and Hermitian transpose operations, respectively. IN denotes an identity matrix of size N . Bold upper-case letters denote matrices and bold lower-case letters denote vectors. and tr denote the expectation and trace operation, respectively. The determinant of a matrix A is denoted as |A| and [A]i,j denotes its ith and jth element. The Kronecker product of two matrices A and B is denoted as A ⊗ B. A zeromean, unit-variance, circularly-symmetric, complex Gaussian random variable is denoted as CN (0, 1), and vec(A) denotes vector a which is obtained by stacking the columns of A.

YADAV et al.: LINEAR PRECODER DESIGN FOR DOUBLY CORRELATED PARTIALLY COHERENT FADING MIMO CHANNELS

Fig. 1.

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Block diagram of the MIMO system with a linear precoder F at the transmitter.

II. S YSTEM M ODEL We consider a communication system with MT transmit and MR receive antennas as shown in Fig. 1. The MIMO channel H is modeled as a block Rayleigh flat fading correlated channel with coherence time of T = Tt + Td symbol intervals, where Tt ∈ Z+ and Td ∈ Z+ are the training symbol and data symbol transmission time in discrete symbol intervals, respectively. The matrix of the received data symbols can be expressed as

where St is a known CMT ×MT orthogonal training symbol matrix with total power Pt , i.e., tr(St SH t ) = Pt = SNRt and Wt is the AWGN noise matrix during training period. −1/2 Let St = ART , where A is a unitary matrix scaled by 

Pt /tr(R−1 T ) for equal power allocation to transmit antennas. Pre-multiplying both sides of (3) by A−1 and then post−1/2 multiplying the resultant equation by RR , we get −1/2

Rd = SFH + Wd

(1)

where Rd ∈ CTd ×MR , S ∈ S ⊂ CTd ×MT is a data codeword matrix used to transmit symbol vector s = [s0 , s1 , . . . , sK−1 ]T ∈ S K , where si ∈ S, S is a set of signal alphabets such as M -level quadrature amplitude modulation (QAM) or phase shift keying (PSK), satisfying IE[|si |2 ] = 1. F ∈ CMT ×MT is a linear precoder matrix, H ∈ CMT ×MR is the channel response assumed to be constant during transmission of the codeword matrix S and Wd ∈ CTd ×MR is an additive complex circularly symmetric white Gaussian noise (WGN) matrix with zero mean and element-wise unit variance. Here, we capture the effect of data signal-to-noise ratio (SNRd ) factor in F thus, SNRd = Pd , and use the power normalization   IE SF2 = Td Pd . (2) For the sake of completeness we used the similar channel model as the one used in [7]. We model the channel with 1/2 1/2 the Kronecker model [19], i.e., H = RT Hw RR , where Hw is a spatially white matrix whose entries are independent and identically distributed (i.i.d.) CN (0, 1). Matrices RT and RR represent normalized transmit and receive correlation with MT MR eigenvalues {λiT }i=1 and {λiR }i=1 , respectively. The normal T i izations in (1) are assumed to be such that M i=1 λT = MT and M R i i=1 λR = MR . We assume that RT and RR are measured accurately at the receiver and are fed back to the transmitter via an errorfree feedback link. However, in practice correlation matrices are estimated with sufficiently high accuracy at the start of the ˆ w of transmission. Now, the receiver computes an estimate H the channel matrix Hw using the well established orthogonal training method [25]. The received training signal matrix in Tt time slots can be written as Yt =

1/2 1/2 St RT Hw RR

+ Wt

(3)

Gw = A−1 Yt RR

−1/2

= Hw + A−1 Wt RR .  

(4)

W0

The LMMSE estimation [25] of Hw is performed on (4) to obtain  

ˆ w = vec(H ˆ w ) = IM M + σ 2 R−1 ⊗ IM −1 vec(Gw ) h ce R R T T (5) −1 ˆ w = Gw [IM + σ 2 R−1 ] , where and in matrix version H ce R R −1 2 σce = tr(RT )/Pt . Similarly, channel estimation error vector and the corresponding matrix version are given by ˆ w = vec(H ˜ w = hw − h ˜ w) h   −1/2 −2 = IMR + σce RR ⊗ IMT vec(Ew )   ˜ w = Ew IM + σ −2 RR −1/2 , H ce R

(6) (7)

where Ew is a random matrix with i.i.d. complex Gaussian ˆ w . By the entries having CN (0, 1), and is independent of H ˜ ˆ orthogonality principle hw and hw are uncorrelated. The CSI model based on the model above becomes ˆ +H ˜ = R1/2 H ˆ w R1/2 + R1/2 H ˜ w R1/2 H =H T R T R ⎛ ⎞1/2 ⎜ ⎟ −2 1/2 2 = R T Gw ⎜ R−1 RR ⎟ R ⎝IMR + σce ⎠ ¯ R ·R ˆR =R

⎛

⎞1/2

1 ⎜ ⎟ −1 −2 + RT2 Ew ⎜ RR ⎟ ⎝IMR + σceRR ⎠

,

(8a)

˜R R

¯ R = [IM + σ 2 R−1 ]−1 where R ce R R −1 −1 2 σce RR ] RR .

and

ˆ R = [IM + R R

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Substituting the eigenvalue decompositions for RT = H ˆ H ¯ ˜ ¯ ˆ UT Λ T U H T , RR = UR ΛR UR , RR = UR ΛR UR and RR = H ˜ UR ΛR UR , where UT and UR are the transmit and receive unitary matrices of the eigenvalue decompositions, we get 1/2 ¯ 1/2 ˆ 1/2 H H = UT Λ T UH T Gw U R Λ R Λ R U R   ˆ ˆ H 1/2 ˜ 1/2 H + UT Λ T UH T Ew U R Λ R U R   ˜ ˜ H

ˆ ˜ ˆ H + UT HU ˜ H. = UT HU R R

(9)

After post-multiplying (1) by UR , denoting SFUT by X, and denoting Rd UR by Y, we get ˆ ˜ ˆ + XH ˜ +N Y = XH

(10)

which represents the sufficient statistics of the received signal. The unitary transformed codeword matrix {X} satisfies the same average-power constraint as the original precoded codeword matrix {SF}, and N has i.i.d. circularly symmetric CN (0, 1) entries, since it has the same distribution as Wd . Applying the vec operation to (10), we get ˆ ˜ ˆ + Zh ˜ +n y = Zh

(11)

where Z = IMR ⊗ X. The conditional probability density ˆˆ function (PDF) of the received signal conditioned on given H and Xi being sent is [5] ˆ ˆ pi (y) =p(y|Zi , h)    −1 ˆ ˆˆ H H ˆ ˜ (y−Zi h) exp −(y−Zi h) ITd MR +Zi ΣZi   , =  ˜ H  π T MR  ITd MR +Zi ΣZ i (12) ˆ ˆ ˆ ˆ H ] = (Λ ˜ = IE[vec(H)vec( ˜ R ⊗ ΛT ) is a correlation where Σ H) ˆ ˆ which is same as that of H. ˜ matrix of H In the remainder of the paper, we will assume that RT , RR , 2 are known to the receiver and the transmitter which and σce further implies knowledge of the estimation covariance matrix ˜ at the transmitter. Σ

III. P RECODER D ESIGN In this section, we derive the cutoff rate (CR) expression as a function of the estimated channel covariance matrix and adopt it as a linear precoder design criterion for a doubly correlated partially coherent channels. The CR is a lower bound on the channel capacity and determines the region of rates where a communication system can operate to achieve arbitrarily small probability of errors. Its maximization minimizes an upper bound on the codeword error probability over the ensemble of the binary channel codes assuming sequential decoding, while it is independent on a specific code [26], [27]. Although the rates larger than the cutoff rate can be achieved (e.g., by turbo or low-density parity check codes with iterative decoding [28] or optimum Viterbi decoding of convolutional codes etc.), it is still useful in predicting the system performance [29]. A. Cutoff Rate Matched to a Partially Coherent Channel In the following proposition, we use the relationship between the Bhattacharyya coefficient and the CR [6] to derive the CR expression. Proposition 1: The Bhattacharyya coefficient ρcor (i, j) between the pair of codewords Si and Sj and cutoff rate R0 , in bits per channel use, for a doubly correlated partially coherent channels with the discrete inputs and continuous output are given by (13) and (14), as shown at the bottom of the page, ˆ n are the eigenvalues of R ˜ n and λ ˜ R and respectively, where λ R R L ˆ R and {Xi } is the constellation set with corresponding R i=1 probabilities {πi }L i=1 . Proof: Derivations of the Bhattacharyya coefficient and cutoff rate follow the same line of steps as presented in [7, Appendix-II]. By replacing each codeword matrix X in  [7, Eq. (16)] with SFUT result in (13) and (14). We refer to the argument of max(·) in (14) as the CR expression and denote it as . For the equiprobable signal set with transmission probabilities {πi }L i=1 = 1/L, the linear precoder design can be now formulated as the following continuous optimization problem (for fixed L, elements of S can take any values from any constellation set, e.g., M -QAM, M -PSK) maximize F

subject to



  tr Si FFH SH = T d Pd , i

where the maximization is with respect to the precoding matrix F.

 1  1  ˜ n  2 IT + Sj FRT FH SH λ ˜ n  2 λ ITd + Si FRT FH SH d i j R R   ρcor (i, j) =

 n  1 1 H H n H H H (S − S )H λ ˜ ˆ I S λ + FR F S + S FR F S + (S − S )FR F  n=1 Td i T j T i j T i j i j R R 2 4 ⎛ ⎞   R0 = max − log ⎝ πi πj ρcor (i, j)⎠ M R 

{πi }L i=1



i

j

 



(15)

(13)

(14)

YADAV et al.: LINEAR PRECODER DESIGN FOR DOUBLY CORRELATED PARTIALLY COHERENT FADING MIMO CHANNELS

Without loss of generality we write F = VF ΛF UH F , which is the SVD of matrix F. To find a close-to-optimal linear precoder matrix F, we need to find three matrices: the inputshaping matrix (VF ), power loading matrix (ΛF ), and beamforming matrix (UF ). For the case of the Gaussian input distributions, the unitary matrix VF is always assumed to be an identity matrix [30] and moreover, for any unitary matrix VF , with non-Gaussian input, the following relationship holds: I(x; y) = I(VF x; y) in general [31]. Thus, by properly designing the unitary matrix VF , for the non-Gaussian inputs, we may improve the average mutual information. Proposition 2: The beamforming matrix UF of a linear precoder matrix F that maximizes the cutoff rate, for a doubly correlated partially coherent channel, coincides with the eigenvectors of the transmit correlation matrix RT . Proof: See Appendix A.  Using the results from Proposition 1 and Proposition 2, the Bhattacharyya coefficient can be rewritten as (16) (which can be found at the bottom of the page) where Λ = Λ2F ΛT , subject to the average power constraint i.e., tr{ΛF ΛH F } ≤ Td Pd and VF VFH = VFH VF = I. We will now rewrite the CR optimization problem equivalent to (15) with two new optimization variables, i.e., ΛF and VF as (ΛF , VF )   ≤ T d Pd subject to tr Si VF Λ2F VFH SH i H H VF VF = VF VF = I,

maximize ΛF ,VF

(17)

where the maximization is now with respect to the diagonal power loading matrix ΛF and unitary input-shaping matrix VF . The CR expression in (17) is not a convex function with respect to the matrices VF and ΛF . Furthermore, its optimization does not lead to a closed form solution for the linear precoders. Many approaches to approximate non-convex functions, e.g., gradient search, SA, etc. However, the solution is not guaranteed to be optimal. Now since two matrices are to be optimized in (17), we solve them using alternating optimization (AO) algorithm, and denote it as 2-block AO. Before addressing numerical solutions in Section IV, we consider two special cases as examples.

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a complex row vector s and the power constraint becomes tr{Λ2F } ≤ Td Pd . 2) Space–Time Codes (MT ≥ 2 and Td ≥ 2): a) Orthogonal ST codes: Herein, we use the OSTBC, designed to exploit the spatial diversity of the MIMO systems [32] and [33], as a codeword matrix. The OSTBCs are of particular interest here due to their simple decoding while providing diversity gain. Precoder design combined with OSTBC was previously studied in [34] and [35] with a perfect CSIR and statistical CSIT and in [36] and [37] with imperfect CSIR and partial CSIT (correlation matrix). However, the channel assumption in this paper is different from [36] and [37] as we have considered the covariance of the estimated channel is available as CSIT. Assume that K independent complex symbols are transmitted simultaneously over Td periods of time, i.e., the rate R of the code is K/Td , and SH S = a

K−1 

|xi |2 I

(18)

i=0

where a = 1, if S ∈ {G2 , H3 , H4 } and a = 2 if S ∈ {G3 , G4 }2 [38]. Close-to-optimal ΛF and VF matrices, with power constraint aKtr{Λ2F } ≤ Td Pd , can be obtained by solving the optimization problem (17). Furthermore, the complexity of the optimization increases with the number of transmit antennas. b) Unitary ST codes: We constrain herein the ST block code to be a unitary one. The unitary space–time codes (USTC) [39], [40] are capacity optimal signals for noncoherent channels. Moreover, they have very simple ML decoding algorithms with perfect CSIR [33]. Recently, novel unitary signals or codes were designed for the partially coherent channels in [41] and [42] and their improved form in [6]. Very little attention has been paid on the precoder design combined with the USTCs, while examples are [43] and [44]. We consider the unitary Td ×MT is a maST codes S = {Φi }L i=1 , where Φi ∈ C H trix with orthonormal columns (i.e., Φi Φi = IMT and Φi ΦH i = ITd ). In this case, close-to-optimal ΛF and VF matrices, with power constraint tr{Λ2F } ≤ Td Pd /MT , can be obtained by solving the optimization problem (17).

B. Special Cases We will now find the linear precoding matrices by solving (17) for two MIMO schemes, i.e., spatial multiplexing and space–time coding. 1) Spatial Multiplexing (MT ≥ 2 and Td = 1): In this MIMO scheme, each transmit antenna draws its symbols independently from the conventional M -QAM or M -PSK constellations. With Td = 1, each S in (16) will be replaced by

ρcor (i, j) =

M R  n=1

  ITd

C. Difference of Convex Formulation Let us now fix VF and focus on optimizing the ΛF which −1/2 ˜ i = Si VF , the CR is calculated as Λ1/2 ΛT . Substituting X 2 G , G , G , H and, H are space–time block codes with rate R of 1, 2 3 4 3 4 1/2, 1/2, 3/4 and 3/4, respectively and subscripts denote the number of transmit antennas.

 1  1  H H ˜n  2 ˜n  2  ITd + Si VF ΛVFH SH i λR  ITd + Sj VF ΛVF Sj λR  

 n H H λ ˜ + 1 (Si − Sj ) VF ΛVH (Si − Sj )H λ ˆ n  + 12 Si VF ΛVFH SH i + Sj VF ΛVF Sj R F R 4

(16)

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expression can be re-written as in (19), shown at the bottom of the page. We now decompose (19) into a d.c. form. Theorem 1: For a given VF , maximizing the cutoff rate with respect to Λ in (19) can be decomposed as a difference of two convex functions, and, therefore, optimization problem (15) can be reformulated as a d.c. programming problem as R0 = maximize {g(Λ)−f (Λ)} = − minimize {f (Λ)−g(Λ)} Λ Λ   ≤ T subject to tr ΛF ΛH P , (20) d d F where f (Λ) and g(Λ), are convex functions of Λ given as  ⎛   M R    ⎜ 1  ˜n ˜ H +X ˜ j ΛX ˜H λ ˜ i ΛX IT + 1 X f (Λ) = log⎜ i j R  d 2 ⎝L2 i,j n=1   j>i ... −1    1 ˜ H ˆn  ˜ ˜ ˜ + (X − X )Λ( X − X ) λ i j i j R  4   =B  − 1   2 M R     ˜ n  ˜ k ΛX ˜ Hλ · IT d + X   k R  k=i n=1   =Ci l=j,l>k  − 1 ⎞   2   ⎟  ˜ n  ⎟ ˜ l ΛX ˜ Hλ (21) ×  IT d + X l R    ⎠   =Cl ⎛ MR  − 1 ⎜   ˜ n  2 ˜ i ΛX ˜ Hλ g(Λ) = log⎝ ITd + X i R i,j n=1 j>i

⎞

− 1   ˜ n  2⎟ ˜ j ΛX ˜ Hλ × ITd + X ⎠. j R

1) Construction of Simplex: For a given (MT − 1)-simplex S = [p1 , p2 , . . . , pMT ], contains the feasible set D ∈ IRMT of (20), from which a prism is induced T = T (S) = {(p, t) ∈ IRMT × IR : p ∈ S}. The p1 , p2 , . . . , pMT are the vertices of S. The prism T (S) has MT edges that are vertical lines (i.e., lines parallel to the t-axis) which pass through the MT vertices of S, respectively. The initial simplex S ⊂ D is a polytope with a small num T Δ ber of vertices. We construct S = {p ∈ IRMT : M k=1 pk ≤ Td Pd , p  0}. 2) Partitioning of Simplex: For a sub-division process, also called branching, of the simplices, we use an exhaustive bisection method [45] for simplicity. The sub-division process determines the convergence of the algorithm. A simplex is bisected at the mid point over its longest edge into two subsimplices. Consider the longest edge is between vertices pi and pj for a simplex S1 with MT vertices. Now using the bisection method, simplex S1 is replaced with two new sub-simplices S2 , S3 with a shared vertex pw = (1/2)(pi + pj ). 3) Lower Bound Computation: To compute the lower bound, we reformulate the objective function (20) into a concave minimization optimization problem which is relaxed to an affine set constraint and a linear objective function. The d.c. program (20) can be reformulated into a global concave minimization problem by adding an additional variable t as ϕ(t , p ) = minimize {t − g(p)} subject to

1 Td

t,p M T 

pk ≤ P d , pk ≥ 0

k=1

f (p) − t ≤ 0, (22) 

Proof: See Appendix B. IV. N UMERICAL O PTIMIZATION

Several algorithms exist to solve the d.c. programming problem in (20), e.g., branch-and-bound (BnB) or outer approximation [45] and prismatic BnB algorithm [23]. Converting the problem into a canonical d.c. program and then use the Edge Following Algorithm [23] could be an alternative choice. More algorithms may be found in [23] and [45]. In this paper, we use the prismatic BnB algorithm to obtain the optimal power loading matrix ΛΛH due to its smaller complexity compared to the above mentioned algorithms. The prismatic BnB algorithm requires to solve the linear program (LP) at each iteration. A detail description of the algorithm is given below.

⎛ MR ⎜ 1    R0 = −log ⎝ 2 L i j n=1 I

A. Prismatic Branch-and-Bound Algorithm

Td

(23)

where p = {p1 , . . . , pMT } is a vector containing diagonal elements of ΛΛH . The objective function {t − g(p)} is con T cave and feasible set D = {(p, t) ∈ IR : (1/Td ) M k=1 pk ≤ Pd , pk ≥ 0,f (p) − t ≤ 0} is convex. Clearly, if (t , p ) is an optimal solution of (23), then p is an optimal solution to (20) as well and t = f (p ). Now we will relax the convex constraint set D to a piecewise linear convex set by outer approximating the convex function f (p) as   z T z z ) ∂f (p ) + f (p ) (24) (p − p fMi (p) = min z p ∈Mi

where Mi is the set of feasible vectors p and ∂f (pz ) is a sub-gradient of f at pz . The approximated function fMi (p) is piecewise linear approximation of f (p). Let Pi = {(p, t) : p ∈ D, fMi (p) − t ≤ 0} and F = {(p, t) : p ∈ D, f (p) − t ≤ 0}. For all p, fMi (p) ≤ f (p), implies F ⊆ Pi ,

⎞  1  1  ˜ n  2 IT + X ˜ n  2 ˜ i ΛX ˜ Hλ ˜ j ΛX ˜ Hλ ITd + X d i j R R ⎟   ⎠ 1 1 ˆ n  ˜ n + (X ˜H +X ˜ j ΛX ˜H λ ˜i −X ˜ j )Λ(X ˜i −X ˜ j )H λ ˜ i ΛX +2 X i j R R 4

(19)

YADAV et al.: LINEAR PRECODER DESIGN FOR DOUBLY CORRELATED PARTIALLY COHERENT FADING MIMO CHANNELS

and Pi is a linear piecewise approximation of the feasible set F at ith iteration index. Furthermore, max{fMi (p) − g(p)} ≤ max{f (p) − g(p)} and is a lower bound of the cutoff rate found using (20). Moreover, by adding points to set Mi , the polyhedron outer approximation P of F can be successively improved because fMi−1 (p) ≤ fMi (p) if Mi−1 ⊆ Mi , with Mi = Mi−1 ∪ {pmax,i−1 } and M1 = V (F1 ), where pmax,i−1 is a feasible vector found at (i − 1)th iteration index. Now we write the approximate polyhedron P in matrix polyhedron form [46] as Pi = {(p, t) : Ai p − ai t ≤ bi }

(25)

where 11×MT ⎤ ⎥ ⎢ −IMT ⎥ ⎢ ∂f (p1 )T ⎥ Ai = ⎢ ⎥ ⎢ .. ⎦ ⎣ . T

# ∂f p|Mi | $ 0(MT +1)×1 ai = 1(|Mi |+1)×1 ⎡ P ⎡

⎢ ⎢ bi = ⎢ ⎢ ⎣

(26a)

(26b) d

⎤

0MT ×1 ⎥ ⎥ (p1 )T ∂f (p1 ) − f (p1 ) ⎥. ⎥ .. ⎦ T



.  p|Mi | ∂f p|Mi | − f p|Mi |

(26c)

The sub-gradient of f (p) is ∇f (p) = [(∂f (p)/∂p1 ), . . . , (∂f (p)/∂pMT )]T , where ∂f (p) −1 = ∂pm log(2) qij ⎡ ⎛ ⎡ MR MR ⎢⎜ ⎢ ∂ log (|B|) 1   + ×⎣ ⎝qij⎣ ∂p 2 k=i n=1 n i,j l=j,l>k ⎤⎞⎤ ×

∂ log(|Ck |) ∂ log(|Cl |)⎥⎟⎥ + ⎦⎠⎦ ∂p ∂p m,m

T ∂ log(|B|) 1  ˜ H −1 ˜ ˜n ˜ H B−1 X ˜j λ = ΛT Xi B Xi +ΛT X j R ∂p 2   T 1 ˆn ˜i −X ˜ j )H B−1 (X ˜i −X ˜ j) λ + Λ T (X R 4 (27b)  T ∂ log (|Ck |) ˜n ˜ H C−1 X ˜k λ = ΛT X (27c) k R k ∂p M R R   M  1 1 qij = |B|−1 |Ck |− 2 |Cl |− 2 , (27d) k=i l=j,l>k

n=1

where B, Ck , and Cl are defined in (21). To compute the lower bound of the objective function t − g(p) over T ∩ P, consider the points (p, t) which satisfy the total power constraint and lead to constant values t − g(p) = αi , where αi = min {fMi (p) − g(p)}

is a local upper bound at the ith iteration. Let V denote the matrix with columns p1 , . . . , pMT . The lower bound of t − g(p) can be calculated by solving the LP in (Γi , t)[46] as minimize Λi ,t

t−

MT 

γik tk

k=1

subject to ai t − Ai Vi Γi ≤ bi ,

(29)

where Γi is a vector with elements γ1 , . . . , γMT , ti = αi + g(p), and A, a, b are given in (26). If the above LP has a feasible solution, (Γi , ti ) is an optimal solution and c its optimal value, the lower bound is given as % +∞, if LP has no feasible solution if c ≤ 0 (30) βi = α i ,  αi − c , if c > 0, and the feasible point available for updating the upper bound is pmin,i = Vi Γi . 4) Upper Bound Computation: The upper bound computation is rather simple and straightforward. Any feasible vector p represents a valid upper bound. Starting with a initial upper bound α1 = min{fM1 (p) − g(p)}, at the ith iterations, the upper bound can be computed as αi = min {fMi (p) − g(p)}

(31)

and the corresponding power vector given by pUB,i = arg minp {fMi (p) − g(p)}, where p ∈ {Mi , pmin,i }. Algorithm 1, which is presented below, finds the optimal power loading vector to the transmit antennas. Algorithm 1 Prismatic BnB Algorithm

(27a)

n

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(28)

Input: Input constellation set S ∈ {Si : 0 ≤ i ≤ L − 1}, matrix VF , Pd , MT , and tolerance . Output: Value of a cutoff rate (CR) and optimal power loading vector p across the transmitting antennas. 1: Initialization: Given tolerance > 0. Set i = 1, R ∈ {T (S1 )}. 2: Stopping criterion: if α − βi > |α| go to Step 3, otherwise STOP. 3: while R = φ do 4: Vi = T (Si ) 5: Set R(i) = φ 6: Solve LP to compute lower bound βi using (30), 7: If LP solution is feasible then pmin,i = Vi Λi and update Mi ∈ {Mi , pmin,i }. 8: Compute upper bound αi = minp∈Mi {fMi (p) − g(p)}, pUB,i = arg minp∈Mi {fMi (p) − g(p)}. 9: Update α = min{α, αi } 10: if α == αi then 11: p = pUB,i 12: if α − βi > |α| then 13: Split T (Si ) along one of its longest edge into T (Si1 ) and T (Si2 ). 14: Update R ∈ {T (Si1 ), T (Si2 )}. CR = α.

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Based on [23, Theorem 1] and the bisection sub-division process of simplices used here is exhaustive,3 Algorithm 1 converges to an optimal power loading vector. The computational complexity of the algorithm primarily depends on the complexity of the LP times the total number of iterations N . The computational complexity4 of an LP algorithm, which uses an interior-point method, is around O(n2 m), where n is number of optimization variables and m is number of constraints with m ≥ n. As mentioned above, a new constraint is added to the problem in (29) to approximate f (p) better at every iteration. Initially, an LP has (MT + 1) variables to optimize and (2MT + 1) constraints. And after N iterations, the number of constraints increases up to (2MT + N ). Now, the computational complexity of Algorithm 1 can be approximated as O((MT + 1)2 (2MT + N )). Through extensive numerical simulations we found that for MT = 2 and Pd = 10 dB, the average number of iterations required by the prismatic BnB is 70. B. Optimization Under Unitary Matrix Constraint Once we obtain the optimal matrix ΛF , the maximization of the CR w.r.t. the VF , can be reformulated as maximize VF

(VF )

subject to VF VFH = VFH VF = I.

∂ −1 = ∗ ∂VF log(2) wij & 'M ) ( R   1 ∂ log(|Ci |) ∂ log (|Cj |) × + wij 2 ∂VF∗ ∂VF∗ n i,j ∂ log (|B|) − ∂VF∗

*+ (33a)

 ∂ log (|B|) 1 ˜ n H −1 −1 = λ R X i B Xi + X H j B Xj ∗ ∂VF 2

(33b)

wij =

M R 

(33c)

|B||Ci |− 2 |Cj |− 2 . 1

1

(34)

Algorithm 2 Self-Tuning Riemennian SA Algorithm [24, Table II] Input: Unitary matrix VF . Output: Optimized unitary matrix VF = Wk+1 . 1: Initialization: k = 0, Wk = VF and μ = 1. 2: Compute the gradient of the cost function on the Euclidean space: Γ = ∂/∂VF∗ . 3: Compute the gradient direction on the Riemannian space: Gk = ΓVFH − VF ΓH . 4: if Gk , Gk I = (1/2)R{tr{Gk GH k }} ≪ 0 then 5: STOP. 6: Determine the rotation matrices: Pk := exp(−μGk ), Qk = Pk Pk . 7: while (Wk ) − (Qk Wk ) ≥ μGk , Gk I do 8: Pk := Qk , Qk = Pk Pk , μ := 2μ 9: while (Wk ) − (Pk Wk )(μ/2)Gk , Gk I do 10: Pk := exp(−μGk ), μ := μ/2 11: Update: Wk+1 = Pk Wk , k = k + 1, and go to Step 2. The convergence rate of the SA algorithm depends on the step size μ, while μ itself is objective function dependent. A small step size results in slow converge, whereas a large one speeds up the convergence. In Algorithm 2, an adaptive Armijo’s step size is used. It can be doubled or reduces to half based on the condition at step 7, and step 9 in Algorithm 2, respectively. Since the objective function (32) is nonconvex with a closed feasible set and the objective function values are upper bounded, the SA algorithm together with Armijo step size rule always converges to a local maximum. Many different initializations to VF may give better solutions. However, one cannot guarantee global maximum. The approximate overall computational complexity of the self-tuning Riemannian SA Algorithm is O(MT3 ) per iteration including the complexity for calculating the objective function and its gradient [24]. C. Alternating Optimization

) 1 ˆn H −1 + λR (Xi −Xj ) B (Xi −Xj ) VF Λ 4

∂ log (|Ci |) −1 ˜n = XH i C i Xi VF Λ λ R ∂VF∗

G = ΓVFH − VF ΓH .

(32)

This is a nonconvex optimization problem with a nonlinear constraint. To solve this problem, we use the self-tuning Riemannian steepest ascent algorithm (SA) Algorithm 2 given in [24, Table-II]. We need to calculate the complex conjugate derivative of CR w.r.t. VF Γ=

The gradient direction in Riemannian space is defined as [24]

(33d)

n 3 A sub-division process of simplices is exhaustive if after infinite divisions the simplices shrinks to a point [23, Section 3.7.2]. 4 Note that only dominant terms are included in big O notation, i.e., O(·), wherever used in the paper.

Combining Algorithms 1 and 2, we present an 2-block AO as Algorithm 3 to find the linear precoder matrix F. The simple idea underlying the AO is to replace the difficult optimization of the objective function with a sequence of easier optimization problems involving subsets of block variables, i.e., ΛF , VF . This method is simple to implement and computationally efficient. Furthermore, the objective function maximization with respect to one block variable ΛF gives a unique maximum. Whereas the objective function maximization with respect to another block variable VF gives local maximum, and always improved objective function values. Thus, we can guarantee that Algorithm 3 generates a monotonically increasing sequence of the objective values upper bonded by the log2 M , where M is the cardinality of the constellation set, and hence the sequence converges.

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Algorithm 3 2-block AO Algorithm for the CR problem (17) Input: Input constellation set S ∈ {Si : 0 ≤ i ≤ L − 1}, matrices VFinit , UF Pd , and MT . Output: R0 value and close-to-optimal linear precoder matrix F. (0) 1: Initialization: Set k = 1, VF = VFinit . 2: repeat (k) 3: Solve (20) using Algorithm 1 for ΛF for a given (k−1) . VF (k−1) , 4: Solve (32) using Algorithm 2, with input as VF (k) (k) for VF using ΛF from previous step. 5: Set k := k + 1. 6: until convergence

V. N UMERICAL R ESULTS AND D ISCUSSION Coded FER and MI are used as performance metrics to compare the performance of the CR optimized precoders with the conventional methods (no precoding) and the CR-PCCs in various MIMO transmission schemes. In all the presented examples, the precoder matrices are optimized for the true SNR value unless otherwise stated. For the computer simulation, we considered the turbo coded system with frames of length equal to 512 symbols. The exponentially correlated fading model is considered at both transmit and receive sides. The transmit and |i−j| receive correlation matrix is given by: (R{T,R} )i,j = ρ{T,R} for i, j ∈ {1, . . . , M{T,R} }, where ρ{T,R} denotes normalized correlation coefficient of the channel and satisfies 0 ≤ ρ{T,R} < 1. The values used for ρT and ρR in the simulations are 0.45 and 0.90, respectively. At the receiver, the knowledge of the channel estimation ˜ is also considered to be available to the covariance matrix Σ decoding metric as in [47]. The decoder estimates the loglikelihood ratios (LLRs) of the coded bit for soft decision decoding from the modulated symbols. Let us denote the transmitted bits of each signal b = {b1 , . . . , bN }. The LLR for the lth bit in b is given by    + ¯ ˆ X+ :X=f (b),bl =1 p Y|X , H  . L(bl ) = log  (35) ¯ −, H ˆ p Y|X − X :X=f (b),bl =0 Since our design criterion is a lower bound to the mutual information, we consider the mutual information as one of our performance metrics. The Monte-Carlo computer simulations have been used to obtain the mutual information of the conventional constellations with/without precoder and CRPCCs in the SM and STBC transmission schemes. The average mutual information with a discrete-input and continuous-output channel with imperfect CSIR is given as ' & +*  ˆ , H) p(Y|X i ˆ = πi IEp(Y|Xi ,H) . I(Y; X|H) ˆ log  ˆ j πj p(Y|Xj , H) i (36)

Fig. 2. Convergence of 2-Block AO algorithm for 2 × 2 MIMO system with L = 16, MT = 2, MR = 2, Td = 1, SNRd = 15 dB, ρT = 0.45, and ρR = 0.9.

A. Convergence of 2-Block AO Algorithm We first consider the convergence of 2-Block AO algorithm. The algorithm iteratively finds a linear precoder matrix which is used in conjunction with the SM mode of transmission with MT = 2, MR = 2, L = 16, and SNRt = 15 dB. We considered an initial power loading matrix as ΛF = diag(31.6228, 31.6228), and two different initial VFinit1 and VFinit2 unitary matrices. Fig. 2 illustrate the CR values at each iteration with both initial unitary matrices. It can be observe that, from different initial unitary matrices, the proposed 2-Block AO algorithm converge after 4 iterations to some higher objective function value compared to its values at previous iteration indices. B. Spatial Multiplexing In this subsection, we compare the coded FER performances and the average MIs of the CR optimized precoder with the conventional no-precoder case, and CR-PCCs in the SM transmission mode. In the case with/without precoders, the spatial multiplexed codeword s draws its symbols from the conventional constellations, e.g., M -QAM, independently for each MT transmit antennas. On the other hand, with the CR-PCCs, the spatial multiplexed codewords are designed simultaneously across the MT transmit antennas and their bit-mapping performed using the modified binary switching algorithm (MBSA) [7]. The coded FER performances versus SNRd for the SM transmission with precoding, no precoding and CR-PCCs, for 2 × 2 MIMO systems, are plotted in Fig. 3. In this example, the codeword vector s draws its symbols from the QPSK and 16-CR PCC constellation for the precoding/no-precoding and CR-PCCs cases, respectively. We further set the value of the training power or SNRt to be fixed and equal to 8 dB and turbo code rate (Rc ) = 1/2. In the previous example, we used the fixed training power. However, usually the training power increases with the data

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Fig. 3. Coded FER of 2 × 2 MIMO system versus SNRd for fixed SNRt = 8 dB with L = 16, MT = 2, MR = 2, Td = 1, ρT = 0.45, and ρR = 0.9.

power and their ratio remains the same. In all subsequent numerical examples we allow the training power to vary with data power with data-to-training power ratio (i.e., Pd /Pt ) of 3.5. In Fig. 4 we compare the coded FER versus SNRd performances of the SM transmission with the CR precoding, conventional precoding, fixed CR precoding, no-precoding and CR-PCCs. In the conventional precoding, the correlation of the actual channel is used as CSIT. In the fixed CR precoding, a precoder is designed at some fixed value of SNRd is used after appropriate scaling, for all other SNRd values; this is to illustrate the more practical performance. In Fig. 5 we compare the coded FER versus SNRd performances of the SM transmission with the CR precoding, no-precoding and CR-PCCs. In these two examples, we kept the simulation parameters similar to those in the previous example except the number of transmit antennas, which is MT = 2 in Fig. 4 and MT = 4 in Fig. 5. In the next example, we evaluate and compare the average MI versus SNRd performance of the CR precoded, no precoded and CR-PCCs. The results are illustrated in Fig. 6. Simulation parameters in this example are kept the same as those in Fig. 4. It can be observed form Figs. 3–6 that the spatially multiplexed CR-PCCs have significantly higher FER and average MI performance gains compared to those of conventional constellations combined with the CR optimized precoder and noprecoder cases. In the SM mode of transmission, designing the codewords5 adapted to the imperfect CSIR not only gives a good minimum distance between the codewords, which is essential to combat imperfect CSIR, but also the beamforming gains. On the other hand, the precoder combined with the conventional constellations provides beamforming gains. Thus, utilizing the available CSIT in designing the codebook is beneficial as compared to use it to design precoders. Given the simplicity and backward compatibility of the precoding approach, its performance can be very competitive in realistic applications. 5 The PPC are constructed simultaneously across the M transmit antennas T and codewords lie inside an 2MT − dimensional real sphere [6].

Fig. 4. Coded FER of 2 × 2 MIMO system versus SNRd with varying SNRt and L = 16, MT = 2, MR = 2, Td = 1, ρT = 0.45, and ρR = 0.9.

Fig. 5. Coded FER of 4 × 2 MIMO system versus SNRd with varying SNRt and L = 16, MT = 4, MR = 2, Td = 1, ρT = 0.45, and ρR = 0.9.

Conventionally, precoders are designed to adapt to the channel correlation matrix knowledge as an CSIT. The conventional precoder used in this example is numerically obtained by solving (17) with as assumption that Σ of the actual channel is available as CSIT. With this assumption, designing the conventional precoder using CR criterion is fairly equivalent to the others criteria such as maximizing capacity in [17] and [48], etc. As expected, it can be observed that the CR optimized precoder, designed to adapt to the estimated channel error ˜ gives better performance as compared to covariance matrix Σ, the conventional precoders. C. Space–Time Codes Now we will evaluate and compare the performances of the ST codes for MT = 2, MR = 2, and Td ≥ 2. Training power is also varied with the data power as Pd /Pt = 3.5. We present two examples, one for Td = 2 and another for Td = 4. In the

YADAV et al.: LINEAR PRECODER DESIGN FOR DOUBLY CORRELATED PARTIALLY COHERENT FADING MIMO CHANNELS

Fig. 6. Average mutual information of 2 × 2 MIMO system versus SNRd with varying SNRt and L = 16, MT = 2, MR = 2, Td = 1, ρT = 0.45, and ρR = 0.9.

Fig. 7. Coded FER of 2 × 2 MIMO system versus SNRd for the Alamouti codes with varying SNRt and L = 16, MT = 2, MR = 2, Td = 2, ρT = 0.45, and ρR = 0.9.

first example, we use the Alamouti codes as the OSTBC and combine it with the CR optimized precoder. In the second example, we use the USTC obtained by Giese and Skoglund [41] and combine it with the CR optimized precoder. Fig. 7 compares the coded FER versus SNRd performances of the CR precoded Alamouti codes with fixed (SNRd = 11 dB) CR precoding, conventional Alamouti codes (no precoding), Alamouti codes (with conventional precoders), and the CR optimized Alamouti codes [7]. For a fair comparison, the Alamouti codes choose their elements from the conventional 4-QAM constellation, and, on the other hand, the CR optimized Alamouti code chooses its elements from the CR optimized 16-CR PCCs, such that the data rate is the same in all three cases shown. The turbo code rate Rc = 2/3 and training power varies as Pd /Pt = 3.5. In Fig. 8, we present the average MI

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Fig. 8. Average mutual information of 2 × 2 MIMO system versus SNRd for the Alamouti codes with varying SNRt and L = 16, MT = 2, MR = 2, Td = 2, ρT = 0.45, and ρR = 0.9.

versus SNRd plots of the CR optimized Alamouti code and the conventional Alamouti codes combined with the CR optimized precoder and no-precoder. Simulation parameters used in this example are kept the same as those in Fig. 7. In the STBC transmission mode, the average MI of the CR optimized precoders are higher than those of the CR optimized constellations and conventional constellations with no precoders. Next, we evaluate the performance of the linear precoder, combined with the unitary ST codes, obtained by solving (17). Another unitary matrix designed for the partially coherent channels proposed in [42] could also be used in this example. The unitary matrices [41] are obtained by combining noncoherent [49] and coherent codes. The idea of adaptively determining the number of coherent and non-coherent component codes goes according to the weighting factor (estimation covariance or SNRt ) between their intra-subspace and intersubspace distances [41]. For the non-coherent case (SNRt = 0), the inter-subspace distance between two signals Φi and Φj depends on the principal angles between the subspaces spanned by Φi and Φj . For the coherent case (high SNRt ), two signals in the same subspace have vanishing inter-subspace and non-vanishing intra-subspace distances, since they depend on the singular values of the difference matrix Φi − Φj . In other words, higher estimation variance will involve more noncoherent components to achieve a large inter-subspace distance and less coherent component codes and vice versa. In the example above, we could have compared the performances of a purely non-coherent code (16 unitary code) to that of a signal set based on combining eight non-coherent codes with two coherent codes, and a constellation combining four non-coherent codes with four coherent codes. However, for clarity of the plots of Fig. 10, we only evaluate the performance of the USTC obtained combining four non-coherent codes with four coherent codes. The coded FER and BER versus SNRt performances of the USTC combined with precoding, no-precoding and CR optimized USTC, designed at SNRd per

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Fig. 9. Coded FER of 2 × 2 MIMO system versus SNRd for the Alamouti codes with varying SNRt and L = 16, MT = 2, MR = 2, Td = 2, and i.i.d. channel.

Fig. 10. Coded FER of 2 × 2 MIMO system versus SNRt for the USTC with varying SNRt and L = 16, MT = 2, MR = 2, Td = 4, ρT = 0.3 and ρR = 0.45.

block of 3 dB, are compared in Fig. 10 with Rc = 2/3. In the next example, we evaluate and compare the average MI of the CR optimized precoder to the CR optimized unitary constellations and conventional unitary constellations in the STBC transmission mode in Fig. 11. Simulation parameters in this example are kept same as those in Fig. 10. In the case of STBCs, which provide diversity gains, we can observe from Figs. 7, 8, 10, and 11 that the CR optimized precoders show significant FER and average MI gains as compared to the CR optimized and the conventional Alamouti and UST codes. The reason for this is that the CR optimized Alamouti codes and USTCs, designed to combat imperfect CSIR, are sub-optimal. Moreover, they transmit energy in an isotropic manner similar to their conventional counterparts. Whereas the conventional Alamouti codes and USTCs combined with

Fig. 11. Average mutual informations of 2 × 2 MIMO system versus SNRt for the USTC designed for a SNRd value of 3 dB and MT = 2, MR = 2, Td = 4, ρT = 0.30, and ρR = 0.45.

CR optimized precoder get beamforming gains along with the diversity gains. Thus, utilizing the CSIT in designing the precoders is beneficial, especially at low-to-medium SNRd ratio regime, as compared to use it to design the codewords as done in [7]. However, in the high SNRd regime, the average MI of the CR optimized Alamouti codes show better gains. At high SNRd , the impact of transmit correlation is negligible on the performance [50]. Therefore, the power allocation approaches uniform directions resulting in no beamforming gain. Furthermore, the performance for the CR optimized precoders with the conventional precoders is also plotted in Fig. 7. The conventional precoders used in this example are numerically obtained by solving (17) with the assumption that covariance Σ of the actual channel is available as an CSIT. As mentioned before, with above CSIT assumption, designing the precoders with CR criterion is fairly equivalent to the other criteria such as minimizing SER in [35], PEP in [51] etc. As expected, the performance of the CR optimized precoder is higher as compared to the conventional precoders. Finally, we compare the performances of the Alamouti codes combined with CR optimized prcoders with the CR-PCC and no precoding in Fig. 9. We assumed that the transmitter and the receiver antennas are uncorrelated and channel estimation error variance is available as an CSIT. All other simulation parameters in this example are same as those in Fig. 7. It can be observed that the CR optimized precoders show higher performance gains compared to the CR-PCCs and no precoding cases. Generally, for the uncorrelated antenna case, the CR optimized precoders should transmit in an isotropic manner (i.e., equal power allocation to the transmit antennas). However, at low SNRd regime, where channel estimation error is high, the CR optimized precoder allocate full power to one of the transmit antenna to reduce the cross terms (due to channel estimation errors) appearing while decoding. In the Alamouti case, higher the channel estimation error the worse would be the performance. However, at high SNRd regime, the performances

YADAV et al.: LINEAR PRECODER DESIGN FOR DOUBLY CORRELATED PARTIALLY COHERENT FADING MIMO CHANNELS

of the CR optimized precoder and CR-PCC are same (not shown here). VI. C ONCLUSION We have studied the single-user MIMO precoder design problem for the doubly spatially correlated partially coherent Rayleigh fading channels with discrete inputs. We have assumed that the channel is estimated at the receiver using the LMMSE estimator while the transmitter has the perfect knowledge of the estimated channel covariance matrix. The precoder matrix has been designed to adapt to the degradation caused by the imperfect channel estimation at the receiver and the presence of the spatial correlation. The CR, a lower bound on the MI, has been proposed to be use as the design criterion. A closed and tractable form of the CR expression has been utilized. The suitability of the CR expression for the discrete inputs makes it possible to design close-to-optimal precoders which can be used with the conventional constellations such as M -QAM or M -PSK. A 2-block AO algorithm has also been proposed. It iteratively maximizes the prismatic BnB and self-tuning Riemannian SA algorithms, to numerically search close-to-optimal linear precoder with a given average power constraint at the transmitter. The proposed CR optimized precoders are used with the SM and ST block transmission modes. Numerical examples are presented to compare the performances of the CR optimized precoders with conventional noprecoder case and CR optimized PCCs in terms of the FER and average MI. In the SM transmission mode, it is demonstrated through numerical examples that the use of available CSIT in designing the CR-PCCs is useful over the precoder design. However, in the ST block transmission case, CSIT can be useful in designing the precoders over the CR-PCCs. Solutions are obtained via heuristic approach and suboptimal. Thus, analyzing the optimality-gap between the solution obtained and the optimal solution is an interesting direction for future research.

⎛

  M R I    Td + ⎜ 1 R0 = −log ⎝ 2 L i j n=1

1 2



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A PPENDIX A P ROOF OF P ROPOSITION 2 ˜ i, By setting XiVF = X we can write H H H ˜ ˜H Xi VΛF UH R U Λ V X as X Λ U R U Λ T F F F i F F T F F Xi . i F H H The SVD of ΛF UF RT UF ΛF = QDQ , where D and Q are diagonal and unitary matrices, respectively. Furthermore, diagonal matrix D can be written as QH ΛUH RT UF ΛQ. Based on [52, Lemma 12], there exists a matrix M = UT ΛM such that MH RT M = D and tr{MMH } ≤ H tr{ΛF ΛH F } = tr{FF }. We can write FRT FH = VF ΛF UH RT UF Λ VFH   =QDQH

= VF QMH RT MQH VFH   ˜ =F

˜ TF ˜H. = FR

(37)

˜ = VF QMH = VF QΛH UH = V ˜ F ΛH UH , where Now, F M T M T ˜ F , means that the beamforming matrix UF of a VF Q = V linear precoder matrix must coincides with eigenvectros UT of the transmit correlation matrix RT with lower transmit power, ˜F ˜ H } ≤ tr{FFH }. tr{F

A PPENDIX B P ROOF OF T HEOREM 1 To prove the Theorem 1 we use following two lemmas: Lemma 1: [53, Lemma 1] Function f (μ, D1 , D2 ) = |(I + μ(AD1 AH + BD2 BH ))|−1 defined over positive semidefinite diagonal matrices D1 , D2 and positive number μ is a jointly log-convex function of D1 and D2 for fixed μ. Lemma 2: [53, Lemma 2] Let hi (x) and gi (x) be logconvex functions ∀ i = 1, . . . , L over IRn and ci be nonnegative

−1 ⎞  ˆ n  ˜ n + 1 (X ˜H +X ˜ j ΛX ˜H λ ˜i −X ˜ j )Λ(X ˜i −X ˜ j )H λ ˜ i ΛX X i j R R 4 ⎟ ⎠  − 1  − 1  ˜ n  2 IT + X ˜ n  2 ˜ i ΛX ˜ Hλ ˜ j ΛX ˜ Hλ ITd + X d i j R R

(39)

1  L2 i,j j>i

⎛ ⎞ 1 1        M −1 − − R , R , , 1 ˜ ˜ n  2 IT +X ˜ n 2⎠ ˜ H ˜ ˜ H ˜n 1 ˜ ˜ ˜ ˜ H ˆn  ˜ i ΛX ˜ Hλ ˜ j ΛX ˜ Hλ ⎝ M ITd+X d i j R R n=1 ITd+2 Xk ΛXk +Xl ΛXl λR+4 (Xk−Xl )Λ(Xk−Xl ) λR  · ×

k=i n=1 l=j,l>k

− 1 ,M ,R  ˜ n |− 12 |IT + X ˜ n  2 ˜ i ΛX ˜ Hλ ˜ j ΛX ˜ Hλ ITd+X d i j R R

i,j n=1 j>i

(40)

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constants. Then f (x) = log( has a d.c. decomposition



i ci (hi (x)/gi (x)))

is d.c. and

⎛ ⎞    log ⎝ ci gi (x) hj (x)⎠ − log hi (x). i

j=i

(38)

i

Now, without loss of generality, we can write (19) as shown in (39), at the bottom of the previous page. Using Lemma 2, the argument of log(·) in (39) may be expressed as (see (40) at the bottom of the previous page). By Lemma 1, ˜ n +(1/4)(X ˜ k ΛX ˜ H +X ˜ l ΛX ˜ H )λ ˜ k −X ˜ l )Λ(X ˜ k− |ITd+(1/2)(X R k l H ˆ n −1 H ˜ n −1 H ˜ n −1 ˜ ˜ ˜ ˜ ˜ Xl ) λR | , |ITd + Xi ΛXi λR | and |ITd + Xj ΛXj λR | are log-convex functions of Λ. Operations like raised to a positive index, times positive constant, sum, and product of log-convex functions preserve log-convexity. Thus, (40) is the ratio of two log-convex functions. Taking log(·) therefore gives d.c. decomposition for CR expression as f (Λ) − g(Λ), where f (Λ) and g(Λ) are defined in (21) and (22), respectively. Maximizing the negative CR expression can be written as its minimization which is (20).

ACKNOWLEDGMENT The authors would like to thank anonymous reviewers whose comments have significantly improved the paper. They also acknowledge helpful discussions with Dr. P. Komulainen and Dr. T. Le Nam, both at Centre for Wireless Communications, University of Oulu, Finland. R EFERENCES [1] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [2] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple-antenna data transmission over fading channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998. [3] M. Medard, “The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 933–946, May 2000. [4] A. Lapidoth and S. Shamai, “Fading channels: How perfect need “perfect side information” be?” IEEE Trans. Inf. Theory, vol. 48, no. 5, pp. 1118– 1134, May 2002. [5] M. J. Borran, A. Sabharwal, and B. Aazhang, “Design criterion and construction methods for partially coherent multiple antenna constellations,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4122–4133, Aug. 2009. [6] A. Yadav, M. Juntti, and J. Lilleberg, “Cutoff rate optimized space–time signal design for partially coherent channel,” IEEE Trans. Commun., vol. 60, no. 6, pp. 1563–1574, Jun. 2012. [7] A. Yadav, M. Juntti, and J. Lilleberg, “Partially coherent constellation design and bit-mapping with coding for correlated fading channels,” IEEE Trans. Commun., vol. 61, no. 10, pp. 4243–4255, Oct. 2013. [8] M. Ding and S. Blostein, “Maximum mutual information design for MIMO systems with imperfect channel knowledge,” IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 4793–4801, Oct. 2010. [9] F. Rey, M. Lamarca, and G. Vazquez, “Linear precoder design through cut-off rate maximization in MIMO-OFDM coded systems with imperfect CSIT,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1741–1755, Mar. 2010. [10] M. Ding and S. Blostein, “MIMO minimum total MSE transceiver design with imperfect CSI at both ends,” IEEE Trans. Signal Process., vol. 57, no. 3, pp. 1140–1150, Mar. 2009. [11] Z. Yan, K. Wong, and Z.-Q. Luo, “Optimal diagonal precoder for multiantenna communication systems,” IEEE Trans. Signal Process., vol. 53, no. 6, pp. 2089–2100, Jun. 2005.

[12] P. Stoica and G. Ganesan, “Maximum-SNR spatial-temporal formatting designs for MIMO channels,” IEEE Trans. Signal Process., vol. 50, no. 12, pp. 3036–3042, Dec. 2002. [13] Q. Ngo, O. Berder, and P. Scalart, “Minimum Euclidean distance based precoders for MIMO systems using rectangular QAM modulations,” IEEE Trans. Signal Process., vol. 60, no. 3, pp. 1527–1533, Mar. 2012. [14] A. Lozano, A. M. Tulino, and S. Verdú, “Optimum power allocation for parallel Gaussian channels with arbitrary input distributions,” IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 3033–3051, Jul. 2006. [15] F. Pérez-Cruz, M. Rodrigues, and S. Verdú, “MIMO Gaussian channels with arbitrary inputs: Optimal precoding and power allocation,” IEEE Trans. Inf. Theory, vol. 56, no. 3, pp. 1070–1084, Mar. 2010. [16] M. Payaró and D. Palomar, “On optimal precoding in linear vector Gaussian channels with arbitrary input distribution,” in Proc. IEEE Int. Symp. Inf. Theory, Seoul, Korea, Jun. 28–Jul. 3 2009, pp. 1085–1089. [17] W. Zeng, C. Xiao, M. Wang, and J. Lu, “Linear precoding for finitealphabet inputs over MIMO fading channels with statistical CSI,” IEEE Trans. Signal Process., vol. 60, no. 6, pp. 3134–3148, Jun. 2012. [18] M. Chiani, M. Win, and A. Zanella, “On the capacity of spatially correlated MIMO Rayleigh-fading channels,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2363–2371, Oct. 2003. [19] D. Shiu, G. J. Foshini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multi-element antenna systems,” IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000. [20] N. B. Mehta, F. Digham, A. Molisch, and J. Zhang, “Rate of MIMO systems with CSI at transmitter and receiver from pilot-aided estimation,” in Proc. IEEE Veh. Technol. Conf., Los Angeles, CA, USA, Sep. 26–29, 2004, vol. 3, pp. 1575–1579. [21] T. Yoo and A. Goldsmith, “Capacity and power allocation for fading MIMO channels with channel estimation error,” IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 2203–2214, May 2006. [22] A. Yadav, M. Juntti, and J. Lilleberg, “Linear precoder design for correlated partially coherent channels with discrete inputs,” in Proc. Int. Symp. Wireless Commun. Syst., Ilmenau, Germany, Aug. 27–30, 2013, pp. 351–355. [23] R. Horst, P. Pardolos, and N. Thoai, Introdiction to Global Optimization., 2nd ed. Boston, MA, USA: Kluwer, 2000. [24] T. Abrudan, J. Eriksson, and V. Koivunen, “Steepest descent algorithm for optimization under unitary matrix constraint,” IEEE Trans. Signal Process., vol. 56, no. 3, pp. 1134–1147, Mar. 2008. [25] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ, USA: Prentice-Hall, 1993. [26] J. Wozencraft and R. Kennedy, “Modulation and demodulation for probabilistic coding,” IEEE Trans. Inf. Theory, vol. 12, no. 3, pp. 291–297, Jul. 1966. [27] J. L. Massey, “Coding and modulation in digital communications,” in Proc. Int. Zurich Semin. Digit. Commun., Zurich, Switzerland, Mar. 1974, vol. 1, pp. E2.1–E2.4. [28] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error correcting coding and decoding: Turbo codes,” in Proc. IEEE Int. Conf. Commun., Geneva, Switzerland, May 23–26, 1993, vol. 2, pp. 1064–1070. [29] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Informationtheoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998. [30] T. T. Kim, G. Jöngren, and M. Skoglund, “Weighted space–time bitinterleaved coded modulation,” in Proc. IEEE Inf. Theory Workshop, San Antonio, TX, USA, Oct. 24–29, 2004, pp. 375–380. [31] C. Xiao, Y. Zheng, and Z. Ding, “Globally optimal linear precoders for finite alphabet signals over complex vector Gaussian channels,” IEEE Trans. Signal Process., vol. 59, no. 7, pp. 3301–3314, Jul. 2011. [32] S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [33] V. Tarokh, H. Jafarkhani, and A. Calderbank, “Space–time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [34] H. Sampath and A. Paulraj, “Linear precoding for space–time coded systems with known fading correlations,” IEEE Commun. Lett., vol. 6, no. 6, pp. 239–241, Jun. 2002. [35] K. Phan, S. Vorobyov, and C. Tellambura, “Precoder design for space–time coded systems over correlated Rayleigh fading channels using convex optimization,” IEEE Trans. Signal Process., vol. 57, no. 2, pp. 814–819, Feb. 2009. [36] S. Zhou and G. B. Giannakis, “Optimal transmitter eigen-beamforming and space–time block coding based on channel correlation,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1673–1690, Jul. 2003.

YADAV et al.: LINEAR PRECODER DESIGN FOR DOUBLY CORRELATED PARTIALLY COHERENT FADING MIMO CHANNELS

[37] L. Liu and H. Jafarkhani, “Application of quasi-orthogonal space–time block codes in beamforming,” IEEE Trans. Signal Process., vol. 53, no. 1, pp. 54–63, Jan. 2005. [38] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block coding for wireless communications: Performance results,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 451–460, Mar. 1999. [39] L. M. Marzetta and B. M. Hochwald, “Capacity of a mobile multipleantenna communication link in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 139–157, Jan. 1999. [40] B. M. Hochwald and T. L. Marzetta, “Unitary space–time modulation for multiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 46, no. 2, pp. 543–563, Mar. 2000. [41] J. Giese and M. Skoglund, “Space–time constellation design for partial CSI at the receiver,” IEEE Trans. Inf. Theory, vol. 53, no. 8, pp. 2715– 2731, Aug. 2007. [42] E. Baccarelli and M. Biagi, “Performance and optimized design of space–time codes for MIMO wireless systems with imperfect channel estimates,” IEEE Trans. Signal Process., vol. 52, no. 10, pp. 2911–2923, Oct. 2004. [43] V. K. Nguyen and Y. Xiang, “MMSE precoder for unitary space–time codes in correlated time-varying channels,” IEEE Signal Process. Lett., vol. 12, no. 8, pp. 569–572, Aug. 2005. [44] Y. Guo, S. Zhu, and Z. Liang, “Linear precoding of unitary space–time code for GLRT decoder,” IEICE Trans. Fundam. Electron., Commun. Comput. Sci., vol. E91-A, no. 2, pp. 695–699, Feb. 2007. [45] R. Horst and H. Tuy, Global Optimization: Deterministic Approaches., 3rd ed. New York, NY, USA: Springer-Verlag, 1995. [46] R. Horst, T. Q. Phong, N. V. Thoai, and J. Vries, “On solving a D.C. programming problem by a sequence of linear programs,” J. Global Optim., vol. 1, no. 2, pp. 183–203, Jun. 1991. [47] M. Wang, W. Xiao, and T. Brown, “Soft decision metric generation for QAM with channel estimation error,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1058–1061, Jul. 2002. [48] E. A. Jorswieck and H. Boche, “Channel capacity and capacity-range of beamforming in MIMO wireless systems under correlated fading with covariance feedback,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1543–1553, Sep. 2004. [49] B. Hochwald, T. Marzetta, T. Richardson, W. Sweldens, and R. Urbanke, “Systematic design of unitary space–time constellations,” IEEE Trans. Inf. Theory, vol. 46, no. 6, pp. 1962–1973, Sep. 2000. [50] E. Björnson, E. Jorswieck, and B. Ottersten, “Impact of spatial correlation and precoding design in OSTBC MIMO systems,” IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3578–3589, Nov. 2010. [51] G. Jöngren, M. Skoglund, and B. Ottersten, “Combining beamforming and orthogonal space–time block coding,” IEEE Trans. Inf. Theory, vol. 48, no. 3, pp. 611–627, Mar. 2002. [52] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2381–2401, Sep. 2003. [53] S. Srinivasan and M. Varanasi, “Constellation design for the noncoherent MIMO Rayleigh fading channel at general SNR,” IEEE Trans. Inf. Theory, vol. 53, no. 4, pp. 1572–1584, Apr. 2007.

Animesh Yadav (S’08) received the B.E. degree (with honors) from Dr. B. R. Ambedkar University, Agra, India, in 2001, the M.Tech. degree from the Indian Institute of Technology, Roorkee, India, in 2003, and Ph.D. degree from the University of Oulu, Oulu, Finland, in 2013, all in electrical engineering. During 2003–2007, he worked in Patni Computer Systems Ltd., IP Cell as Software Specialist both in India and Finland. In 2008 he joined the Centre for Wireless Communications (CWC) at University of Oulu, where he is currently holding a Post-doctoral Researcher position. His current research interests lie in the fields of communications and information theory with special emphasis on wireless communications and signal processing.

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Markku Juntti (S’93–M’98–SM’04) received the M.Sc. (Tech.) and Dr.Sc. (Tech.) degrees in electrical engineering from University of Oulu, Oulu, Finland in 1993 and 1997, respectively. He was with University of Oulu in 1992–1998. In academic year 1994–1995 he was a Visiting Scholar at Rice University, Houston, Texas. In 1999–2000 he was a Senior Specialist with Nokia Networks. Since 2000 he has been a Professor of communications engineering at University of Oulu, Department of Communication Engineering and Centre for Wireless Communications (CWC). His research interests include signal processing for wireless networks as well as communication and information theory. He is an author or co-author in some 200 papers published in international journals and conference records as well as in book WCDMA for UMTS published by Wiley. He is also an Adjunct Professor at Department of Electrical and Computer Engineering, Rice University, Houston, Texas, USA. Dr. Juntti is an Editor of IEEE T RANSACTIONS ON C OMMUNICATIONS and was an Associate Editor for IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY in 2002–2008. He was Secretary of IEEE Communication Society Finland Chapter in 1996–1997 and the Chairman for years 2000–2001. He has been Secretary of the Technical Program Committee (TPC) of the 2001 IEEE International Conference on Communications (ICC’01), and the CoChair of the Technical Program Committee of 2004 Nordic Radio Symposium and 2006 IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 2006). He is the General Chair of 2011 IEEE Communication Theory Workshop (CTW 2011).

Jorma Lilleberg (S’81–M’91) was born in Rovaniemi, Finland, in 1953. He received the DI and Lic. Tech. degrees in electrical engineering at the University of Oulu, Oulu, Finland, in 1979 and 1984, respectively, and the Doctor of Technology degree at the Tampere University of Technology, Tampere, Finland, in 1992. During 1992–1993, he worked as an acting Research Professor and Chief Scientist for signal processing at the Technical Research Center of Finland, Oulu, Finland. From August 1993 to December 2010 and January 2011 to September 2013, he has been working at Nokia and Renesas Mobile Corporation Oulu, Finland, respectively, as a Principal Scientist, Technology Fellow and Distinguished Research Leader. He was also a visiting Professor at the Chinese Academy of Science, Beijing, China, Shanghai Research Center for Wireless Communications in Shanghai, China from 2006 to 2010. He is a Docent at the University of Oulu, Oulu, Finland, an Adjunct Professor at Rice University, Houston, TX, USA, and is currently working at Broadcom Corporation as Distinguished Research Leader. Dr. Lilleberg’s research interests are in digital communications theory and application of statistical signal processing methods for digital radio networks. He has coauthored more than 90 research papers and holds more than 19 patent families.