Adaptive Space Modulation With Partial CSIT in ... - Semantic Scholar

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Adaptive Space Modulation With Partial CSIT in Spatially Correlated Fading Channels Mehdi Maleki, Student Member, IEEE, and Hamid Reza Bahrami, Member, IEEE

Abstract—The problem of adaptive space modulation (ASM) design using partial channel state information at the transmitter (CSIT) is investigated. ASM relies on adaptive modulation of information symbols in the spatial domain using multiple antennas based on some sort of CSIT. The CSIT is exploited to adjust the transmission such that the distances between the receive constellation vectors are increased to reduce the probability of error. We assume that the spatial correlation matrices are available as the partial CSIT and introduce two different approaches to design ASM. In the first approach, we consider that all the transmit antennas are active at each transmit interval and, using an upper bound on the average probability of error, design the transmit vectors based on the spatial correlation matrices to improve the performance. In the second approach and to avoid interantenna synchronization, we assume that only one transmit antenna is active at each transmit interval, and we modify a space shift keying (SSK) transmission by transmitting different complex values from different transmit antennas. Finally, we derive a power allocation to improve further the performance in nonstationary channels. By simulation, we show the effectiveness of the proposed approaches in improving the performance in terms of the symbol error rate (SER). Index Terms—Adaptive space modulation (ASM), multipleinput–multiple-output (MIMO), partial channel state information at the transmitter (CSIT), space shift keying (SSK), spatial modulation (SM).

I. I NTRODUCTION

U

SING the space domain to modulate the transmitted signal in a multiple-input–multiple-output (MIMO) system is a novel technique that has been recently investigated in the literature. In such systems, multiple antennas are employed to modulate spatially the information [1]–[8]. Such techniques can be collectively referred to as space modulation. The simplest form of space modulation is space shift keying (SSK), in which only one transmit antenna is active in each transmit interval, and the index of the active antenna is determined by the index of the transmitted data symbol [4]. Moreover, the combination of SSK with amplitude and phase modulation (APM) schemes, which is also known as spatial modulation (SM), is proposed in [5] and [6]. Several advantages of SSK transmission over other Manuscript received May 9, 2013; revised September 2, 2013 and November 28, 2013; accepted December 30, 2013. Date of publication January 14, 2014; date of current version September 11, 2014. The review of this paper was coordinated by Prof. S.-H. Leung. The authors are with the Department of Electrical and Computer Engineering, The University of Akron, Akron, OH 44325-3904 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2014.2300102

types of MIMO transmissions are addressed in [4], [5], [7], and [9]–[14]. In contrast to SSK, some space modulation schemes, involving more than one transmit antenna for transmission, is introduced. For example, in [15] and [16], the generalization of SSK and SM transmission schemes is proposed. Another type of space modulation is space–time shift keying, in which both time and space are exploited for data modulation, and each data symbol is modulated using an individual space–time code dispersion matrix [17]–[21]. In all the given schemes, no channel state information at the transmitter (CSIT) is assumed. However, since the performance of such schemes highly depends on the relative position of the received constellation vectors, which are functions of the underlying MIMO channel, having some sort of CSIT can greatly improve the design of space modulation schemes. As the structure of transmit signal changes with channel state variations, the techniques that rely on the CSIT to tailor the space modulation transmission can be collectively called adaptive space modulation (ASM). The effect of full and imperfect CSIT on the performance of space modulation has been investigated in [22], where two different approaches have been introduced to manipulate the receive constellation space to improve the performance of the system. The advantage of such schemes over other types of MIMO transmission with CSIT has been also investigated in [22]. While generally having full CSIT can improve the performance of ASM further, our focus in this paper is on the use of partial CSIT as it requires less feedback from the receiver and is thereby more practical. As far as the literature is concerned and to the best of our knowledge, there are only a few works in the literature on the use of partial CSIT to improve the performance of space modulation transmission. For example, in [23] and [24], an adaptive SM scheme is proposed such that the optimum order of the APM for each antenna based on the received SNR is fed back to the transmitter. In this type of adaptive SM, no power allocation or rotation is considered for the transmitted signal, and only channel information is used to adapt the modulation dimension. There are several works in the literature on the design of optimal transmission for MIMO systems having transmit and receive correlation matrices at the transmitter (see, e.g., [25] and [26], and the references therein). In this paper, we use the Kronecker model [27] to characterize a spatially correlated MIMO channel and assume that only the spatial correlation matrices are available as partial CSIT. Having this information, we derive the union bound of the average symbol error rate (SER) as the function of the correlation matrices, and we

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MALEKI AND BAHRAMI: ASM WITH PARTIAL CSIT IN SPATIALLY CORRELATED FADING CHANNELS

Fig. 1.

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Block diagram of a MIMO system with SSK transmission.

define the optimization problem to design the constellation vectors. We show that the methods proposed in [22] for signal design having full CSIT can be extended to the case of ASM design having partial CSIT in a correlated MIMO channel. Therefore, we introduce two different ASM signal design methods with partial CSIT. In the first method, called multiantenna ASM (MA-ASM), all the transmit antennas are considered active in each transmit interval and are used to transmit one specific data symbol. Such signal design has been done in [28] where, by deriving the average probability of error of an SM system, a metric to design an optimal precoding scheme as a function of spatial correlation matrices is derived. The precoder matrix is then obtained by an exhaustive search. In contrast to [28], we derive a closed-form solution for MA-ASM to reduce the complexity and to improve the performance of the system. In the second method, called single-antenna ASM (SA-ASM), which can be interpreted as an adaptive SSK transmission, to avoid interantenna synchronization (IAS) and to reduce the number of required RF chains, only one antenna is assumed active in each transmit interval and that antenna is used to transmit a constellation symbol. One can think of this scheme as a way to allocate different amplitude and phases to different transmit antennas in a proper way based on partial CSIT. A simpler scheme is proposed in [29] where, by assuming partial CSIT, optimal power allocation across antennas is derived for the case of two transmit antennas; however, the same transmission phase is considered for both antennas. In [30], a phase adaptation technique for SSK transmission in multiple-input–single-output systems is proposed. However, the proposed scheme does not consider power allocation and is not applicable to MIMO systems with multiple receive antennas. In this paper, we propose an optimal amplitude (power) and phase allocation scheme for SA-ASM with two transmit antennas, and we extend this method to the case of multiple transmit antennas while, at the same time, we impose no restriction on the number of receive antennas. The remainder of this paper is organized as follows. In Section II, SSK system model is introduced, and an upper bound on its average probability of error with partial CSIT is derived. In Section III, by using this upper bound, the proposed ASM designs based on partial CSIT are derived. Later in Section IV, assuming that the channel is nonstationary, i.e., the channel statistics vary with time, we propose a power allocation to improve further the performance in such channels. In Section V, by some simulations, the performance of the

proposed methods is illustrated. It is shown that, in a spatially correlated MIMO environment, a significant performance improvement in terms of the SER is obtained. Finally, Section VI concludes this paper. We will use the following notations throughout this paper. (·)T and (·)H represent transpose and conjugate transpose operations, respectively. Lowercase or uppercase italic letters denote scalar variables. Vectors and matrices are distinguished by lowercase and uppercase bold letters, respectively.  ·  denotes the Euclidean norm. e{·} denotes the real part of a complex number. The mean of a function of a random variable x is represented by Ex {·}. diag(x1 , x2 , . . . , xN ) represents an N × N diagonal matrix whose diagonal elements are x1 , x2 , . . . , xN , respectively. Moreover, the sth smallest value of discrete function f (x) is represented by min{f (x), s}.

II. S PACE S HIFT K EYING A. System Model The block diagram of a system that relies on SSK transmission is shown in Fig. 1. The received signal for SSK transmission over a frequency-flat MIMO channel with Nt transmit and Nr receive antennas can be written as yj =

√

γHxj + n,

j = 1, 2, . . . , Nt

(1)

where H is the Nr × Nt MIMO channel matrix, γ is the received SNR, n is the additive white zero-mean complex Gaussian noise vector with unitary covariance matrix, and xj is the jth SSK constellation symbol, i.e., jth entry ↓

xj = [0, . . . , 0, 1, 0, . . . , 0]T

(2)

which means that only the jth element of vector xj is “1” and that the rest of the entries are zero. The optimal detector to minimize the SER for such a transmission is a maximumlikelihood detector, which is given by [22]  H  √ γ ˆi = arg max e hi y− hi (3) 2 i where y is the received signal, hi is the ith column of the channel matrix H, and ˆi is the estimated constellation symbol

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(transmit antenna) index. The union bound for the probability of error can be written as [4]   Nt  Nt γ 1  Hδ k, l  Q PM |H ≤ Nt 2 k=1 l=1

≤

k=l

Nt  Nt  γ 1  exp − Hδ l, k 2 2Nt 4 k=1 l=1

(4)

k=l

where PM |H is the probability of error conditioned on H, and δ k, l = xk − xl is the difference between the kth and the lth constellation vectors. The second inequality is the result of applying the Chernoff bound as an upper bound for the Q-function.

˜l,i k is where λR, i is the ith diagonal element of ΛR , and u ˜ l, k , which is a zero-mean Gaussian the ith element of vector u vector with the following covariance matrix:  2

 2  1/2 ˜ l, k = E u ˜H ˜ l, k u δ l, k  INr . (10) Σ ˆ T l, k = σH By substituting (9) in (7), we have N   Nt Nt r      1  γ  l, k 2 ui  EH¯ exp − λR, i ˜ PM |R, T ≤ . 2Nt 4 i=1 l=1 k=1,k=l

(11)

B. Union Bound on Average SER

Based on (10), the random variables u ˜l,i k ’s are independent. Therefore, the expectation in (11) can be simplified further as N  Nt  Nt  r  γ 1  Mwl, k − λR, i PM |R, T ≤ (12) i 2K 4 l=1 i=1

We consider the well-known and broadly accepted Kronecker model [27] to characterize a spatially correlated MIMO channel. The channel matrix can be written as

ul,i k |2 is an exponentially distributed random where wil, k = |˜ variable with parameter (1/var{˜ ul,i k }), and Mwl, k (·) is the

H=R

H/2

ˆ 1/2 HT

l=1

(5)

where R = RH/2 R1/2 and T = TH/2 T1/2 are the Nr × Nr receive and Nt × Nt transmit spatial correlation matrices, reˆ is an uncorrelated Gaussian matrix with indespectively. H pendent and identically distributed zero-mean entries whose 2 variances are σH ˆ . We assume that R, T, and the distribution of ˆ H are known at the transmitter. This assumption significantly reduces the feedback overhead as R and T change much less frequently than MIMO channel matrix H. In the following, we derive the average error probability using the method based Δ ¯ = on moment-generating function (MGF) [31]. By defining H ˆ the channel matrix can be rewritten as RH/2 H, ¯ 1/2 H = HT

Nt Nt

 γ   1  (7) EH¯ exp − ul, k 2 2Nt 4 l=1 k=1,k=l

Δ ¯ 1/2 ¯ (i) where ul, k = [ul,1 k , . . . , ul,Nkr ]T , ul,i k = h δ l, k , and h (i) T ¯ ul, k is a zero-mean Gaussian random vector is the ith row of H. with the following covariance matrix:  2

 2  1/2 T = σ δ (8) Σl, k = E ul, k uH  l, k  R. ˆ l, k H

Considering the singular value decomposition (SVD) of H and defining u ˜l, k = R1/2 RH/2 as R1/2 RH/2 = VR ΛR VR H −H/2 ul,k , we can write VR R ul, k 2 =

Nr  i=1

   l, k 2 ˜i  λR, i u

i

MGF of wil, k . By replacing the MGF of an exponentially distributed random variable in (12), we obtain N   Nt  Nt r  2 −1  1   1/2  PM |R, T ≤ 1 + γβi T δ l, k  2Nt k=1 i=1 l=1

k=l

(13) 2 where βi = (σH ˆ λR, i /4). For the case of uncorrelated receive antennas with equal average gains (i.e., R = INr ), the upper bound can be expressed as

PM |R, T ≤

Nt  Nt  −Nr γ 2 1  1/2 1 + σH δ k, l 2 . ˆ T 2Nt 4 k=1 l=1

k=l

(14)

(6)

¯ is a zero-mean Gaussian matrix. The variance of the where H 2 2 2 entries on the ith row of this matrix is σH ˆ σRi , where σRi is the ith diagonal entry of R. The union bound on the average SER can be then written as PM |R, T ≤

k=l

(9)

Moreover, for the special case of uncorrelated transmit and receive antennas (i.e., T = INt and R = INr ), the upper bound can be expressed as N r  2 (Nt − 1) PM ≤ (15) 2 2 2 + γσH which can be interpreted as the union bound for the probability of error of SSK over Rayleigh fading channels [4]. III. A DAPTIVE S PACE M ODULATION In ASM, the signal constellation vectors are designed based on the CSIT in such a way that the probability of error at the receiver is minimized. Therefore, instead of assuming the transmit vectors as in (2), here, we consider more general constellation vectors pi (i = 1, 2, . . . , Nt ) that should be found to minimize the union bound on the probability of error. From (13), it can be noticed that the probability of error is a function of T1/2 δ l, k , which can be looked at as the distance between the two vectors T1/2 pl and T1/2 pk . The premise of ASM is to design the matrix P = [p1 , p2 , . . . , pNt ] to maximize the

MALEKI AND BAHRAMI: ASM WITH PARTIAL CSIT IN SPATIALLY CORRELATED FADING CHANNELS

distances between vectors {T1/2 pi }i=1, 2,..., Nt , thus reducing the probability of error. Therefore, the problem of ASM design can be generally expressed as P∗ =

arg min P:

1 Nt

tr{PPH }=1

(16)

N r 

 (1 + γβi λmax x)

Nr  

−1 βm + γλmax x

−1

.

m=1

i=1

Finding all the solutions to (21) is not easy; however, one straightforward solution is that, for every l and k, there be only unique l = l and k  = k such that 

i=1

k=1 k=l

−1

(22)

where P∗ is the optimal constellation matrix, the constraint (1/Nt )tr{PPH } = 1 is for preserving the transmit power, and N   Nt  Nt r  2 −1    1/2  . (17) 1 + γβi T δ k, l  AP = l=1

where σ is a constant, and

f (x) =

{AP }

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|αl − αk | = |αl − αk | αk  αk αl = α  .

(23)

l

We solve the problem in (16) with two different approaches. First, we assume that there is no additional constraint on the matrix P and find out P∗ in its most general form. As this general form results in transmit constellation vectors whose nonzero entries span all the transmit antennas, in the second approach and to avoid IAS, we impose a constraint on P such that each of its columns, pi (i = 1, 2, . . . , Nt ) includes only one nonzero entry.

Some 2-D constellations such as phase-shift keying (PSK) and quadrature amplitude modulation (QAM) satisfy (21). For example, if we consider PSK, one can consider the coefficients as αk = exp(j(2π/Nt )k), which satisfies (23). In this case, the union bound for the probability of error of MA-ASM with partial CSIT can be simplified to PM |R, T 

A. MA-ASM Signal Design Without imposing any constraint on the structure of matrix P, designing ASM leads to maximizing T1/2 δ k, l 2 = (pl − pk )H T(pl − pk ) for k = l, which means that pl − pk should be a factor of the eigenvector corresponding to the largest eigenvalue of T. Therefore, without loss of generality, we have

−1  Nt −1  Nr  π 1  k . 1 + 4γβi λmax sin2 2 Nt k=1 i=1 (24)

For the special case of uncorrelated receive antennas (R = INr ), we have

PM

−Nr  Nt −1  π 1  2 2  k . 1 + γσHˆ λmax sin 2 Nt

(25)

k=1

p∗l = αl vmax

l = 1, 2, . . . , Nt

(18)

where vmax is the eigenvector corresponding to the largest eigenvector of T, and αl ’s are complex numbers satisfying the power constraint. By substituting (18) in (17), we have N  Nt  Nt r     (1) 2 −1 1 + γβi λmax |αl − αk | (19) AP = l=1

i=1

k=1 k=l

where λmax is the largest eigenvalue of T. The optimization problem for ASM design in this case can be then expressed as   ∗ ∗ = arg min A1P α1 , . . . , α N t (α1 ,...,αNt )

s.t.

Nt 1  |αl |2 = 1. Nt

(20)

l=1

Equation (20) is not a convex optimization problem, and even numerical methods cannot guarantee to find the global minimum of AP . To solve this optimization problem, we apply the Lagrange multiplier method (LMM). Taking derivative of the Lagrangian function and setting it to zero lead to the following: Nt  k=1, k=l

 1−

αk αl



B. SA-ASM Signal Design Although compared with the regular SSK, applying MAASM significantly improves the performance of the system; the fact that it uses all the transmit antennas in each transmit interval increases the complexity of the transmitter. On the other hand, in SSK, only one of the transmit antennas is active in each transmit intervals, alleviating the need for IAS. Therefore, in our second approach to the design of ASM with partial CSIT, we consider the constellation vectors with only one nonzero entry. This constraint results in one active transmit antenna in each transmit interval. The resulted ASM design can be interpreted as a modified version of SSK as it bears all the properties of SSK with the added advantage of reduced probability of error. Since, in this case, P is a diagonal matrix, (17) can be written as (2) AP

=

N  Nt  Nt r   l=1

  f |αl − αk |2 = σ,

k=1 k=l

 2 −1  1/2  1 + γβi T (pl xl − pk xk )



i=1

(26) l = 1, . . . , Nt (21)

where pi (i = 1, 2, . . . , Nt ) is the nonzero entry of the ith transmit constellation vector pi . Our objective is to find pi ’s

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(2)

such that AP in (26) is minimized. We consider the following structure for the transmit correlation matrix T: ⎡ σ2 ρ σ σ ··· ρ σ σ ⎤ 1

⎢ ρ2, 1 σ2 σ1 T=⎢ .. ⎣ . ρNt , 1 σNt σ1

1, 2 1 2 σ22

.. . ρNt , 2 σNt σ2

··· .. . ···

1, Nt 1 Nt

ρ2, Nt σ2 σNt ⎥ ⎥ .. ⎦ . 2 σN t

(27)

where ρl, k = ρk, l . For the general case, finding the closed form for P minimizing (26) is not straightforward. However, for the special case of Nt = 2, a closed-form solution can be obtained, which can help us develop an iterative algorithm for the case of Nt > 2. We first consider the case of Nt = 2 and then extend the proposed design to the case of Nt > 2. 1) Nt = 2: The problem of optimal power allocation (i.e., finding real pi ’s) for the case of Nt = 2 and Nr = 1 is investigated in [29] where LMM is used to find the power allocation coefficients (r1 , r2 ), which maximize the following distance metric: √ σ ¯ 2 = σ12 r1 + σ22 r2 − 2ρσ1 σ2 r1 r2

(28)

subject to the normalized transmit power constraint r1 + r2 ≤ 2. In [29], it is shown that the pair of (r1 , r2 ) that maximizes (28) is always located on the boundary, and the optimal power allocation for different cases is as follows: ⎧ ⎨ σ1 > σ2 ⇒ (r1∗ , r2∗ ) = (2, 0) (29) σ = σ2 ⇒ (r1∗ , r2∗ ) = (1, 1) ⎩ 1 σ1 < σ2 ⇒ (r1∗ , r2∗ ) = (0, 2) where (r1∗ , r2∗ ) denotes the optimal power allocation. Equation (29) means that the transmission is done only from the antenna whose channel gain vector has a larger variance, and in the case of equal variance, an equal power allocation is the optimal solution. In [29], such power-allocation scheme is referred to as on–off SSK (OO-SSK). Although the performance of OO-SSK is better than the regular SSK, it is possible to exploit partial CSIT in a more intelligent way to improve the performance further. By considering complex pi ’s, we use partial CSIT to find out the optimum amplitudes (power allocation) and phases. Considering (27) for Nt = 2, the optimization problem can be written as 2    (p∗1 , p∗2 ) = arg max T1/2 (p1 x1 − p2 x2 ) (p1 ,p2 )

s.t. |p1 |2 + |p2 |2 = 2.

(30)

If pi ’s are positive real numbers, the problem is the same as in [29]. However, for real pi ’s, i.e., pi = ri exp(jϕi ), ϕi = kπ, we have 2    1/2 T (p1 x1 − p2 x2 ) = r12 σ12 + r22 σ22 − 2ρ1, 2 r1 r2 σ1 σ2 cos(ϕ1 − ϕ2 ).

(31)

Therefore, from (31), we can notice that for maximizing the right hand side, we should have ϕ1 − ϕ2 = π. Solving (30)

using LMM, the optimal values of pi ’s are  ⎧ σ 2 −σ 2 ⎪ ⎪ p∗1 = −r1 = − 1 + " 2 2 12 2 2 2 2 ⎨ σ −σ ( 1 2 ) + 4ρ1, 2 σ1 σ2  σ 2 −σ 2 ⎪ ⎪ ⎩ p∗2 = r2 = 1 + " 2 2 22 1 2 2 2 (σ1 −σ2 ) + 4ρ1, 2 σ2 σ1

(32)

where we have considered ϕ1 = 0 as only the relative phase difference ϕ1 − ϕ2 is important. In this case, we have  2   σ ¯ 2 = T1/2 (p1 x1 − p2 x2 )  # 2  2 2 2 = σ1 + σ2 + (σ1 − σ22 ) + 4 ρ21, 2 σ12 σ22 (33) which satisfies the following inequality:   

 σ ¯ 2  σ12 + σ22 + σ12 − σ22  = max 2σ12 , 2σ22 .

(34)

Due to (34), one can ensure that the maximum of σ ¯ 2 in our proposed approach is larger than the one in (28), i.e., the case where only power allocation is considered. 2) Nt > 2: Finding the optimal pi ’s in this case is generally difficult. We therefore present a suboptimal algorithm to design SA-ASM using the result that we obtained for the case of Nt = 2. Algorithm 1 is the algorithm used to find the suboptimal pi ’s. In each iteration, the first step is to find a pair of transmit vectors with minimum distance and to apply the optimal design for the case of Nt = 2 proposed earlier for these two vectors while keeping other transmit vectors unchanged. Since any change in the two selected transmit vectors affects the distances of corresponding constellation vectors from other vectors, it is not guaranteed that this always increases the minimum distance and thereby improves the SER performance. To ensure that the probability of error never increases, we define the following function for the lth vector in the transmit constellation space: N  Nt r     2 2 −1 2 1+γβi r σl +σk −2ρl, k rσl σk . err(r, l) = k=1,k=l

i=1

(35) Equation (35) is related to the terms in (26) associated to the lth transmit antenna. Assuming l∗ and k ∗ to be the indices of the vectors chosen in one of the iterations, p∗l and p∗k are, respectively, replaced with ql∗ and qk∗ only if err(ql∗ , l∗ ) + err(qk∗ , k ∗ ) ≤ err(pl∗ , l∗ ) + err(pk∗ , k ∗ ) (36) " where by assuming ml∗ , k ∗ = (p2l∗ + p2k∗ /2) $⎛ ⎞ % % 2 −σ 2 σ ∗ ∗ % l k ⎠ ql∗ =−g ∗ · ml∗ ,k∗ &⎝1+ # 2 2 2 2 2 2 (σl∗ −σk∗ ) + 4 ρl∗ ,k∗ σl∗ σk∗ $⎛ ⎞ % % 2 −σ 2 σ % k∗ l∗ ⎠. qk∗ = g ∗ · ml∗ , k∗ &⎝1+ # 2 2 2 (σl∗ − σk∗ ) + 4 ρ2l∗ , k∗ σl2∗ σk2∗ (37)

MALEKI AND BAHRAMI: ASM WITH PARTIAL CSIT IN SPATIALLY CORRELATED FADING CHANNELS

Algorithm 1 SA-ASM design with partial CSIT (Nt > 2) Initialization Step: Start with p1 = · · · = pNt = 1, s = 1 Main Steps: 1. Calculate σl = pl σl , l = 1, 2, . . . , Nt 2. Find

(l∗ , k ∗ ) = arg min σl2 + σk2 − 2ρl, k σl σk s l, k

3. 4. 5. 6.

+ err(pk∗ , k ∗ ), E2 = E1 + 1 Set E1 = err(pl∗ , l∗ )" ∗ ∗ Calculate ml , k = (p2l∗ + p2k∗ /2) Calculate ql∗ and qk∗ using (37) with g ∗ = 1 Find g ∗ = arg min {err(g · ql∗ , l∗ ) + err(g · qk∗ , k ∗ )} g∈{−1, 1}

7. Compute ql∗ = g ∗ · ql∗ , qk∗ = g ∗ · qk∗ 8. If err(ql∗ , l∗ ) + err(qk∗ , k ∗ ) ≤ E1 −s=1

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arbitrary small number), is met or if s > (Nt (Nt − 1))/(2), the Algorithm will stop; otherwise, it continues with the next iteration. With the means of simulation, we will further investigate the number of required iterations for Algorithm 1 in Section V. IV. A DAPTIVE S PACE M ODULATION D ESIGN FOR N ONSTATIONARY C HANNEL S TATISTICS In Section III, we design ASM schemes for a channel with fixed spatial correlation matrices. However, in practice, not only the channel changes but also the stochastic properties of the channel such as the spatial correlation matrices may vary with time. We can model such channel as a nonstationary random process. In such case, our strategy should be to design the ASM transmission over a time interval that the correlation matrices are assumed constant and then to distribute the available transmit power in such a way that the union bound on the SER is minimized. To do this, we first approximate the union bound in (17) as a function of minimum distance, i.e., AP ≈

∗

− E2 = err(ql∗ , l ) + err(qk∗ , k )

Else −s=s+1 End If 9. If (E1 − E2 /E1 ) > ε and s ≤ (Nt (Nt − 1))/(2) − Go to step 1 Else − Stop the Algorithm, Output p1 , . . . , pNt End If To improve further the algorithm and in case that, in an iteration, (36) is not satisfied for the two transmit vectors with minimum distance, we consider the pair with the second smallest distance and continue this process until we find a pair that satisfy (36). Another small improvement in the algorithm comes by multiplying ql∗ and qk∗ by g = −1. This does not affect the distance between the lth and the kth vectors; however, it affects the distance of either of them with respect to the rest of the transmit vectors. Therefore, if the correlation matrix is assumed real (as in [29]), it is sufficient to check which of g = +1 or g = −1 leads to a smaller minimum distance and thereby to a better performance (Algorithm 1, Step 6). Note that, for the case of a complex correlation matrix, to find more accurate results, it is advantageous to search over a wider range of possible candidates considering g = exp(jθ), 0 ≤ θ < 2π. However, even in this case, considering two discrete values for g will reduce the search complexity while providing an acceptable performance improvement. As far as the complexity is concerned, the main computational load in each iteration of Algorithm 1 is due to finding the minimum pair in Step 1 having the complexity order of O(Nt2 ), and the calculation of E1 and E2 in Steps 3 and 8 with the complexity order of O(Nt Nr ). In Step 9 of Algorithm 1, if the convergence condition, i.e., (E1 − E2 /E1 ) < ε (ε is an

(1 + γm ωi, m )−1

(38)

i=1

∗

− pl ∗ = ql ∗ , pk ∗ = q k ∗

Nr 

1/2

where ωi, m = βi, m minl, k {Tm δ k, l 2 }, and we have added the subscript m to show a particular channel state over which the statistical properties of the channel remain unchanged. Since the ASM design for a channel with constant statistics (i.e., constant correlation matrices) has been discussed earlier, we assume the transmit vectors for a particular channel state m are known. By proper allocation of the available transmit powerγ over different channel states, one can minimize the average AP . We consider a total of M channel states, and to simplify the problem, we minimize an upper bound on the average AP such that the optimization problem can be written as  M   ∗ ∗ ∗ −Nr p(m)(1+γm ωm ) (γ1 , γ2 , . . . , γM ) = arg min (γ1 ,γ2 ,...,γM ): M

+

m=1

p(m)γm =γ

m=1

(39) where ωm = mini {ωi, m }, γm is the allocated power to the mth channel state, and p(m) is the probability of the occurrence of the mth state. Applying LMM, the power distribution over channel states can be obtained as a water-filling such that + r γm = aN m [λ − am ]

(40)

(−1/N +1)

r , and λ is the water-filling constant where am = ωm that should be numerically calculated such that it satisfies the + p(m)γ transmit power constraint, i.e., M m = γ. t=1

V. S IMULATION R ESULTS Here, we provide numerical results to show the performance of the proposed ASM transmission schemes. We assume transmission over Rayleigh block fading channels having a coherence time larger than the symbol transmission time. σH ˆ =1

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Fig. 2.

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SER performance of SSK, SA-ASM, and MA-ASM with partial CSIT.

Fig. 3. Simulated and theoretical (union bound) SER performance of SSK, SA-ASM, and MA-ASM (Nr = 2).

and the channel noise is assumed additive white Gaussian. Unless otherwise stated, the PSK-based design is assumed for all the MA-ASM cases. The correlation matrix model considered for most of the cases is ⎡ ⎤ ρσ1 σ2 · · · ρn−1 σ1 σn σ12 n ⎢ ⎥ .. .. .. .. Cn×n = + ⎣ ⎦ . n . . . 2 2 σi ρn−1 σn σ1 ρn−2 σn σ2 . . . σ i=1

n

(41) where n = Nt for the transmit correlation matrix (C = T), and n = Nr for the receive correlation matrix (C = R). Throughout the simulations, except otherwise stated, we assume σi = 1 + 0.1(i − 1), ρ = 0.5 for T and, in the case of correlated receiver antennas, ρ = 0.4 for R. For the case of uncorrelated receiver antennas, we consider R = INr . The SER performance of SA-ASM and MA-ASM assuming partial CSIT is shown in Fig. 2. Uncorrelated and correlated receive antennas are assumed for the cases of Nt = Nr = 2 and Nt = Nr = 4, respectively. For comparison, the performance of SSK transmission with no CSIT is also shown. From the figure, it is observed that, in all cases, a significant performance improvement is obtained when the proposed schemes are applied. For example, in the case of Nt = Nr = 2 with uncorrelated receive antennas, MA-ASM provides a 7–8 dB performance gain at midrange SNRs compared with SSK, whereas SA-ASM achieves a performance gain of 4–5 dB at midrange SNRs. For the case of Nt = Nr = 4 with correlated receive antennas, using MA-ASM, a performance improvement of about 6–7 dB is obtained, whereas SA-ASM provides a gain of 2–3 dB. As expected, the performance of MA-ASM is significantly superior to that of SA-ASM due to the fact that, in MA-ASM, all the transmit antennas can be used to send a transmit vector, which gives us a degree of freedom to construct the transmit constellation space. In Fig. 3, we show the SER performance of SSK, SA-ASM, and MA-ASM obtained from simulation results along with

Fig. 4. SER performance of SSK, SA-ASM, and MA-ASM versus correlation coefficient ρ(Nr = 2).

the analytical union bound in (14). The receive antennas are assumed uncorrelated. From the figure, we can notice that the union bound and simulation results behave similarly with a performance gap of about 2 dB over a wide range of SNRs. Moreover, as shown in the figure, MA-ASM and SA-ASM provide performance gains of about 4 and 2 dB, respectively, compared with SSK. By comparing Fig. 3 with Fig. 2, we can conclude that the performance improvement of SA-ASM and MA-ASM reduces by increasing the number of transmit antennas. The SER performance of SSK, SA-ASM, and MA-ASM for different correlation coefficients (ρ) [see (41)] is shown in Fig. 4. The receive antennas are considered uncorrelated and all the diagonal entries of the transmit correlation matrix are assumed the same to only account for the effect of the

MALEKI AND BAHRAMI: ASM WITH PARTIAL CSIT IN SPATIALLY CORRELATED FADING CHANNELS

Fig. 5. SER performance comparison of SSK, SA-ASM, and OO-SSK [29] (Nt = 2).

correlation coefficient. From the figure, we can observe that, for both cases of Nt = 2 and Nt = 4, increasing ρ degrades the performance of SSK, whereas the performance of SA-ASM remains almost constant. More interestingly, by increasing ρ, the MA-ASM scheme performs better. In other words, with highly correlated transmit antennas, applying SA-ASM or MAASM provides a larger performance improvement compared with the case of small correlation coefficients. As it is shown in Fig. 4, the performance of SSK, SA-ASM, and MA-ASM is close at no or low correlation, whereas the performance gap becomes larger at a high correlation coefficient. This behavior is due to the fact that, at small ρ, having the transmit correlation matrix T as CSIT does not provide much useful information to design efficiently the transmission signal and thereby improve the performance. In Fig. 5, the performance of the OO-SSK transmission scheme proposed in [29] is compared with that of the proposed SA-ASM scheme for the case of uncorrelated receive antennas. Two different 2×2 MIMO channel cases are assumed. In Case I, it is assumed that ρ = 0.5 and σ12 = σ22 = 1, and in Case II, ρ = 0.7, σ12 = 1, and σ22 = 2. The receive correlation matrix for both cases is assumed to be identity matrix. Fig. 5 shows the advantage of SA-ASM over OO-SSK, when the variances of two transmit antennas are the same. As shown in Case I, a performance improvement of 5–6 dB is obtained at midrange SNRs, whereas OO-SSK performs the same as regular SSK with no CSIT. As shown in the figure, in Case II, although OO-SSK provides significant performance improvement over SSK, the proposed SA-ASK is still superior as it provides about 1 dB gain at midrange SNRs. In Fig. 6, the performance of MA-ASM (QAM-based) is compared with the spatial multiplexing transmission scheme with partial CSIT [26]. Here, we pick a precoding scheme designed based on the pairwise error probability (PEP) criterion because it has the best bit-error-rate performance among the proposed precoding schemes in [26]. We assume three cases

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Fig. 6. SER performance comparison of MA-ASM and spatial multiplexing with PEP-based precoding in [26] (Nt = 4).

with different transmit correlation matrices constructed by T = VDVH , where V is a singular matrix associated to the SVD of a matrix with structure similar to the right side of (41) with σi = 1 (i = 1, 2, . . . , Nt ) and ρ = 0.5. We also consider three different D such that D = (Nt /3(3κ + 2) · diag(9κ, 3, 2, 1) for κ = 2, 3, and 9 to obtain three distinct transmit correlation matrices with different singular values and the same singular vectors. Fig. 6 shows the performance of MA-ASM for M = 64 modulation size and PEP-based precoding scheme with quadratic PSK (QPSK) modulation for each data stream. From the figure, one can conclude that, when the ratio of the largest singular value of T and other singular values (i.e., condition number) is small, spatial multiplexing with PEP-based precoding outperforms MA-ASM. However, by increasing this ratio, the performance of MA-ASM improves significantly, and it outperforms the spatial multiplexing with precoding. For example, for κ = 9 at the SNR of 24 dB, the performance of MA-ASM is about 4 dB better than that of the spatial multiplexing with precoding. In Fig. 7, we compare the performance of MA-ASM with precoded space–time block coding (STBC) transmission schemes. We consider the two orthogonal STBCs (OSTBCs) with rates of 1/2 and 3/4 proposed in [32], and one full-rate quasi-orthogonal STBC (QOSTBC) in [33]. The transmitter is assumed to have four antennas in all cases. To fix the transmission rate, modulation sizes of 8, 8, 64, and 16 are assumed for MA-ASM, full-rate QOSTBC, rate-1/2 OSTBC, and rate-3/4 OSTBC, respectively. We employ the precoder design method introduced in [25] for the OSTBC schemes and the one proposed in [34] for the QOSTBC scheme. In Fig. 7, it is observed that MA-ASM significantly outperforms the precoded rate-1/2 STBC by a margin of about 5 dB, whereas it outperforms the precoded rate-3/4 OSTBC by a gain of 1dB. However, the performance of MA-ASM is almost the same as the precoded full-rate QOSTBC. Due to the fact that STBC generally requires a more complicated transmitter

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Fig. 7. SER performance comparison of MA-ASM and precoded STBCs (Nt = 4).

Fig. 8. SER performance comparison of SA-ASM and AS.

compared with MA-ASM, MA-ASM seems to be a better choice compared with the precoded STBC schemes. In the case of precoded STBCs, to design the precoder, one needs to find the power-allocation matrix using water-filling. This further results in higher transmission complexity compared with MA-ASM. Furthermore, in the case of QOSTBC due to coupled symbols at the receiver side, a joint symbol detection is required, which complicates the detection. To evaluate the performance of the SA-ASM scheme, we compare it with antenna selection (AS) in Fig. 8. This is a reasonable comparison as, in both of these schemes, only one antenna is active in each transmit interval. We consider two different cases assuming a constant transmission rate. In the first case, we compare the probability of error of the AS with 16-QAM (M1 = 16) and that of SA-ASM having 16 transmit antennas (Nt = 16) and no APM (M2 = 1). To make the

Fig. 9. SER performance comparison of SSK, SA-ASM, and OH-SM for two different correlation coefficients ρ (Nt = 8, Nr = 2). Note that for SA-ASM, the performance curves for ρ = 0.4 and ρ = 0.8 almost overlap.

comparison more interesting, we compare both schemes with a scheme obtained by a combination of AS and SA-ASM (hybrid scheme), i.e., we first pick a set of nt best antennas and then apply the SA-ASM to the selected antennas. In this case, to fix the transmission rate, we assume nt = 4 and a QPSK modulation (M3 = 4). As shown in Fig. 8, the performance of SA-ASM is almost 1 dB better than AS for the SNRs between 5 and 10 dB. The performance of the hybrid scheme is significantly better (by almost 4 dB) in this SNR range. However, due to higher diversity order of AS, as the SNR increases, the AS finally outperforms these schemes. However, at low SNR regimes, the hybrid version of AS and SA-ASM is a better candidate for signal design. For the second case, we assume Nt = 8 antennas and 64-QAM (M1 = 64) for the AS, 8-PSK modulation (M2 = 8) for SA-ASM, and 16-QAM (M3 = 16) for the hybrid scheme with nt = 4. As it is shown in the figure, the performances of SA-ASM and the hybrid scheme are about 3–4 dB better compared with that of the AS for the midrange SNRs, and as the SNR increases, the performance gap between these schemes reduces. In Fig. 9, we compare the performance of SA-ASM with optimal hybrid SM (OH-SM), which has the best performance among the proposed schemes in [24]. We assume a transmit correlation matrix with equal diagonal elements. The performance is shown for two different values of the correlation coefficient (ρ = 0.4 and ρ = 0.8). The average transmission rate is assumed 3 bits per channel use and eight available transmit antennas are assumed for all cases. Moreover, note that, for a fair comparison, a version of OH-SM is used that assumes the availability of the correlation matrix at the transmitter. Fig. 9 shows that SA-ASM outperforms OH-SM in all cases. For example, at SNR = 20 dB, the performance of SA-ASM is almost 1 (for ρ = 0.4) and 3 dB (for ρ = 0.8) better than that of the OH-SM. This means that the performance gap between the two schemes increases with the correlation

MALEKI AND BAHRAMI: ASM WITH PARTIAL CSIT IN SPATIALLY CORRELATED FADING CHANNELS

Fig. 10. Average number of iterations in Algorithm 1.

coefficient. Moreover, since the OH-SM requires an exhaustive search over all possible APM types and sizes for all the transmit antennas, its complexity is generally higher than the proposed SA-ASM. To study further the complexity of the proposed algorithm for SA-ASM, since the signal design depends on the SNR and the number of transmit and receive antennas, we compute the number of required iterations versus the number of transmit antennas (Nt ), assuming two different values for the number of receive antennas Nr and at four different SNRs. We generate the correlation matrix randomly several times and average out the number of required iterations. Fig. 10 shows the results indicating that, by increasing the number of transmit antennas, the number of iterations increases. The reason is that, by increasing Nt , the size of the constellation space and thereby the optimization problem grow; therefore, more iterations are required to converge to an optimal solution. Moreover, in Fig. 10, it is also observed that the number of iterations is almost independent of the SNR, whereas it increases with the number of receive antennas Nr . Note that, according to Fig. 10, the convergence rate of the algorithm is generally fast. For example, in the extreme case of Nt = Nr = 16, the algorithm only requires 125 iterations. Figs. 11 and 12 show the performance of the proposed ASM transmission schemes with Nt = 4 and Nr = 2 when the channel variance changes with time. In Fig. 11, we assume the case of uncorrelated receive antennas with ten possible values 2 for the channel variance as σH ˆ = 1.5, 1, 2, 1.6, 3, 0.7, 0.9, 1.2, 0.1, 2.3, each of which occurs with probabilities p = 0.1, 0.05, 0.2, 0.1, 0.02, 0.03, 0.05, 0.2, 0.1, 0.15, respectively (Case A). Fig. 12 shows the SER performance for the case of correlated 2 receive antennas with five different channel variances as σH ˆ = 5, 2, 0.1, 3, 0.5 with equal probability (Case B). In each case, water-filling as in (40) is applied to find the optimal power allocation. For Case A, applying power allocation provides a gain of 4–5 dB for both SA-ASM and MA-ASM, whereas for

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Fig. 11. SER performance comparison of SA-ASM and MA-ASM with water-filling for Case A.

Fig. 12. SER performance comparison of SA-ASM and MA-ASM with water-filling for Case B.

Case B, a gain of 2–4 dB is obtained. This difference in the gains is due to the fact that, for Case A, the water-filling is performed over a larger number of channel states compared with Case B. VI. C ONCLUSION The problem of applying partial CSIT to design ASM transmission schemes has been investigated in this paper. We considered the Kronecker model to characterize a spatially correlated MIMO channel and assumed the transmit and receive correlation matrices are available as partial CSIT. By deriving an upper bound for the probability of error of SSK having partial CSIT, two different ASM approaches were introduced.

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In the first approach, called MA-ASM, in each transmit interval, all the transmit antennas are employed to modulate spatially data symbols. While in the second approach, called SA-ASM, only one transmit antenna is active in each transmit interval, which alleviates the need for IAS. Using the upper bound on the probability of error, both approaches find the transmit constellation vectors such that a more favorable distance property is obtained at the receiver. While SA-ASM is less complex, MA-ASM provides a better performance and can be served as a good measure for evaluating the performance of any space modulation scheme. Simulation results showed that both proposed ASM approaches based on partial CSIT provide a significant performance gain in terms of the probability of error compared with SSK and other proposed techniques in the literature. We also considered a MIMO channel with timevariant statistics and derived a water-filling power allocation for this case. The simulation results shows that this power allocation can further improve the performance of the system. R EFERENCES [1] Y. A. Chau and S.-H. Yu, “Space modulation on wireless fading channels,” in Proc. IEEE VTC Fall, Atlantic City, NJ, USA, Oct. 2001, vol. 3, pp. 1668–1671. [2] M. D. Renzo, H. Haas, and P. M. Grant, “Spatial modulation for multipleantenna wireless systems: A survey,” IEEE Commun. Mag., vol. 49, no. 12, pp. 182–191, Dec. 2011. [3] M. D. Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, “Spatial modulation for generalized MIMO: Challenges, opportunities, and implementation,” in Proc. IEEE, Jan. 2014, vol. 102, no. 1, pp. 56–103. [4] J. Jeganathan, A. Ghrayeb, L. Szczecinski, and A. Ceron, “Space shift keying modulation for MIMO channels,” IEEE Trans. Wireless Commun., vol. 8, no. 7, pp. 3692–3703, Jul. 2009. [5] R. Y. Mesleh, H. Haas, S. Sinanovic, C. W. Ahn, and S. Yun, “Spatial modulation,” IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 2228–2241, Jul. 2008. [6] J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Spatial modulation: Optimal detection and performance analysis,” IEEE Commun. Lett., vol. 12, no. 8, pp. 545–547, Aug. 2008. [7] R. Mesleh, H. Haas, C. W. Ahn, and S. Yun, “Spatial modulation—A new low-complexity spectral efficiency enhancing technique,” in Proc. CHINACOM, Beijing, China, Oct. 2006, pp. 1–5. [8] M. Maleki, H. R. Bahrami, A. Alizadeh, and N. H. Tran, “On the performance of spatial modulation: Optimal constellation breakdown,” IEEE Trans. Commun., vol. 62, no. 1, pp. 144–157, Jan. 2014. [9] M. D. Renzo and H. Haas, “Bit error probability of space shift keying MIMO over multipleaccess independent fading channels,” IEEE Trans. Veh. Technol., vol. 60, no. 8, pp. 3694–3711, Oct. 2011. [10] M. D. Renzo and H. Haas, “Bit error probability of SM-MIMO over generalized fading channels,” IEEE Trans. Veh. Technol., vol. 61, no. 3, pp. 1124–1144, Mar. 2012. [11] M. D. Renzo and H. Haas, “A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami-m fading channels,” IEEE Trans. Commun., vol. 58, no. 9, pp. 2590–3603, Sep. 2010. [12] S. S. Ikki and R. Mesleh, “A general framework for performance analysis of space shift keying (SSK) modulation in the presence of Gaussian imperfect estimations,” IEEE Commun. Lett., vol. 16, no. 2, pp. 228–230, Feb. 2012. [13] E. Basar, U. Aygolu, E. Panayirci, and H. V. Poor, “Performance of spatial modulation in the presence of channel estimation errors,” IEEE Commun. Lett., vol. 16, no. 2, pp. 176–179, Feb. 2012. [14] M. D. Renzo, D. D. Leonardis, F. Graziosi, and H. Haas, “Space shift keying (SSK-) MIMO with practical channel estimates,” IEEE Trans. Commun., vol. 60, no. 4, pp. 998–1012, Apr. 2012. [15] J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Generalized space shift keying modulation for MIMO channels,” in Proc. IEEE Int. Symp. PIMRC, Cannes, France, Sep. 2008, pp. 1–5. [16] J. Fu, C. Hou, W. Xiangy, L. Yan, and Y. Hou, “Generalised spatial modulation with multiple active transmit antennas,” in Proc. IEEE GLOBECOM, Miami, FL, USA, Dec. 2010, pp. 839–844.

[17] S. Sugiura, S. Chen, and L. Hanzo, “Coherent and differential spacetime shift keying: A dispersion matrix approach,” IEEE Trans. Commun., vol. 58, no. 11, pp. 3219–3230, Nov. 2010. [18] S. Sugiura, S. Chen, and L. Hanzo, “A universal space-time architecture for multiple-antenna aided systems,” IEEE Commun. Surveys Tuts., vol. 14, no. 2, pp. 401–420, 2nd Quart., 2012. [19] S. Sugiura, S. Chen, and L. Hanzo, “Generalized space-time shift keying designed for flexible diversity-, multiplexing- and complexitytradeoffs,” IEEE Trans. Wireless Commun., vol. 10, no. 4, pp. 1144–1153, Apr. 2011. [20] S. Sugiura, “Dispersion matrix optimization for space-time shift keying,” IEEE Commun. Lett., vol. 15, no. 11, pp. 1152–1155, Nov. 2011. [21] F. Babich, A. Crismani, M. Driusso, and L. Hanzo, “Design criteria and genetic algorithm aided optimization of three-stage-concatenated spacetime shift keying systems,” IEEE Signal Process. Lett., vol. 19, no. 8, pp. 543–546, Aug. 2012. [22] M. Maleki, H. R. Bahrami, S. Beygi, M. Kafashan, and N. Tran, “Space modulation with CSI: Constellation design and performance evaluation,” IEEE Trans. Veh. Technol., vol. 62, no. 4, pp. 1623–1634, May 2013. [23] P. Yang, Y. Xiao, Y. Yu, and S. Li, “Adaptive spatial modulation for wireless MIMO transmission systems,” IEEE Commun. Lett., vol. 15, no. 6, pp. 602–604, Jun. 2011. [24] P. Yang, Y. Xiao, Q. T. L. Li, Y. Yu, and S. Li, “Link adaptation for spacial modulation with limited feedback,” IEEE Trans. Veh. Technol., vol. 61, no. 8, pp. 3808–3813, Oct. 2012. [25] 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. [26] H. R. Bahrami and T. Le-Ngoc, “Precoder design based on correlation matrices for MIMO systems,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3579–3587, Dec. 2006. [27] D. Gesbert, H. Bolcskei, D. A. Gor, and A. Paulraj, “Outdoor MIMO wireless channels: Models and performance prediction,” IEEE Trans. Commun., vol. 50, no. 12, pp. 1926–1934, Dec. 2002. [28] T. Handte, A. Muller, and J. Speidel, “BER analysis and optimization of generalized spatial modulation in correlated fading channels,” in Proc. IEEE VTC Fall, Anchorage, AK, USA, Sep. 2009, pp. 1–5. [29] M. D. Renzo and H. Haas, “Improving the performance of space shift keying (SSK) modulation via opportunistic power allocation,” IEEE Commun. Lett., vol. 14, no. 6, pp. 500–502, Jun. 2010. [30] J. Jeganathan, “Space shift keying modulation for MIMO channels,” M.S. thesis, Dept. Elect. Eng., Concordia Univ., Montreal, QC, Canada, Aug. 2008. [31] M. K. Simon and M. Alouini, Digital Communication over Fading Channels. New York, NY, USA: Wiley, 2004. [32] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: Performance results,” IEEE J. Select. Areas Commun., vol. 17, no. 3, pp. 451–460, Mar. 1999. [33] N. Sharma and C. B. Papadias, “Improved quasi-orthogonal codes through constellation rotation,” IEEE Trans. Commun., vol. 51, no. 3, pp. 332–335, Mar. 2003. [34] A. Medles and A. Alexiou, “New design for linear precoding over STBC in the presence of channel correlation,” IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1203–1207, Apr. 2007.

Mehdi Maleki (S’13) received the B.Sc. and M.Sc. degrees in electrical engineering from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2006 and 2009, respectively. He is currently working toward the Ph.D. degree with The University of Akron, Akron, OH, USA. He is a Research Assistant with the Wireless Communications Laboratory, Department of Electrical and Computer Engineering, The University of Akron. His research interests include digital communications, digital signal processing, multiuser communications, multiple-input–multiple-output wireless systems, cooperative communications, and cognitive radio networks. Mr. Maleki regularly serves as a Reviewer for IEEE T RANSACTIONS and J OURNALS, as well as major conferences.

MALEKI AND BAHRAMI: ASM WITH PARTIAL CSIT IN SPATIALLY CORRELATED FADING CHANNELS

Hamid Reza Bahrami (M’11) received the B.Sc. degree from Sharif University of Technology, Tehran, Iran; the M.Sc. degree from the University of Tehran; and the Ph.D. degree from McGill University, Montreal, QC, Canada, in 2008, all in electrical engineering. From 2007 to 2009, he was a Scientist with Wavesat Inc. Montreal. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, The University of Akron, Akron, OH, USA. His research interests include wireless communications, information theory, and applications of signal processing in communications. Dr. Bahrami has served as a member of technical program committees of numerous IEEE conferences, including the IEEE Global Communications Conference and the IEEE International Conference on Communications. He is currently a member of the IEEE Communications and Vehicular Technology Societies. He served as the Editor for the Transactions on Emerging Telecommunications Technologies.

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