1
Linear Precoding of STBC over Correlated Ricean MIMO Channels Manav R. Bhatnagar, Member, IEEE and Are Hjørungnes, Senior Member, IEEE
Abstract— A linear precoder is designed to minimize an upper bound of the pairwise error probability (PEP) when using arbitrary space-time block codes (STBC) over correlated Ricean fading channels. The proposed precoder design differs from the previously proposed precoders as follows: 1) Existing precoders are applicable when there exists only transmit correlation and no receive correlations, however, the proposed precoder is applicable to correlated Ricean MIMO channels with an invertible correlation matrix. 2) We minimize the upper bound of the PEP of nonorthogonal STBC for obtaining a precoder over correlated Ricean channels, whereas, one existing precoder minimizes an upper bound of PEP for orthogonal STBC (OSTBC). The proposed precoder outperforms two existing precoders for non-orthogonal STBC especially for highly correlated channels.
I. I NTRODUCTION It has been shown in [1]–[4] that the feedback of the channel state information (CSI) can be used to overcome multiple-input multiple-output (MIMO) channel correlations by using a linear precoder. Most of the previously proposed precoder designs consider: 1) Kronecker correlation model [1] and 2) orthogonal space-time block codes (OSTBCs) [5] based MIMO system. However, the Kronecker model does not always render the multi-path structure correctly, but it introduces artifact paths that are not present in the underlying measurement data. The accuracy of the Kronecker model has been questioned in the literature based on measurement campaigns [6]. A more useful generalized model studied in [6] considers that the receive (or transmit) correlation depends on at which transmit (or receive) antenna the measurements are performed. In addition, nonorthogonal STBCs [7]–[9] provide better coding gain than OSTBC and achieve the diversity multiplexing trade-offs [10]. Moreover, the quasi-orthogonal STBCs (QOSTBCs) [11]–[15] provide better coding gain and capacity than OSTBC with slight increase in decoding complexity. The minimum of an upper bound of the pairwise error probabilty (PEP) achieving linear precoder for space-time block codes (STBCs) are designed over Rayleigh MIMO channels in [2], [3] and over Ricean channels in [4] assuming only transmit correlations and no receive correlation. In this paper, our main contributions are as follows: 1) Assuming a coherent decoder which knows the channel coefficients and a transmitter which knows the MIMO channel Corresponding author: Manav R. Bhatnagar is with Department of Electrical Engineering, Indian Institute of Technology Delhi, Hauz Khas, IN110016 New Delhi, India, Email:
[email protected]. Are Hjørungnes is with UNIK – University Graduate Center, University of Oslo, Gunnar Randers vei 19, P. O. Box 70, NO-2027 Kjeller, Norway, email:
[email protected]. This work was supported by the Research Council of Norway projects 176773/S10 called OptiMO and 183311/S10 called M2M, which belong to the VERDIKT program.
correlations, the mean values of channel coefficients, and noise variance perfectly, we obtain an upper bound of the PEP when using linear precoding and arbitrary STBC over invertible correlated Ricean fading MIMO channels. 2) A linear precoder is designed to minimize the upper bound of the PEP. 3) We have also derived some useful properties of the proposed precoder. ℋ 𝒯 Notation: (⋅) and (⋅) stand for the Hermitian and trans∗ pose, respectively, of a vector or matrix, (⋅) provides complex conjugate, 𝔼 (⋅) represents the expectation operator, Tr {⋅} is the trace operator, 𝑰 𝑘 stands for the identity matrix of size 𝑘 × 𝑘, vec {⋅} is the vectorization operator, which stacks the columns of matrix into a column vector, and ∣⋅∣ provides the determinant of the input matrix. II. S YSTEM M ODEL Consider a MIMO system with 𝑛𝑡 transmit and 𝑛𝑟 receive antennas. Let 𝑯 be an 𝑛𝑟 ×𝑛𝑡 channel gain matrix of arbitrary mean value and circularly complex-valued Gaussian distributed random values and 𝑺 𝑘 be the 𝑏×𝑛 orthogonal or non-orthogonal STBC matrix transmitted at block k. The received data at block 𝑘 of size 𝑛𝑟 ×𝑛 is 𝒀 𝑘 = 𝑯𝑭 𝑺 𝑘 + 𝑸𝑘 , (1) where 𝑭 is an 𝑛𝑡 × 𝑏 memoryless linear precoder matrix, 𝑸𝑘 is an 𝑛𝑟 ×n matrix containing additive white complex-valued Gaussian noise (AWGN), whose elements are i.i.d. Gaussian random variables with 𝜎 2 . Let 1 × 𝑛𝑠 ] [ zero mean and variance (𝑘) (𝑘) (𝑘) (𝑘) data vector 𝒔𝑘 = 𝑠1 , 𝑠2 , . . . , 𝑠𝑛𝑠 , where 𝑠𝑖 belongs to an optimized or standard constellation, be encoded into the orthogonal or the non-orthogonal STBC, then the average power transmitted per STBC block will be 𝑎𝑛𝑠 𝜎𝑠2 , where 𝜎𝑠2 is the average power of each symbol in 𝒔𝑘 and 𝑎 is a STBC dependent constant, for example 𝑎 = 2 for [7, Ex. (6)], [8, Eq. (5)], and [9, Eq. (10)]. A. Model of Correlated Ricean MIMO Channels We assume a flat block-fading correlated Ricean channel model [1]: √ √ ( ) 𝐾 1 ¯ vec 𝑯 + vec (𝑯 F ) , (2) vec (𝑯) = 𝐾 +1 𝐾 +1 where vec (𝑯 F ) = 𝑹1/2 vec (𝑯 𝑤 ) represents the fading portion of the channel 𝑯, where 𝑹1/2 is the unique positive semi-definite matrix square root [16] of the 𝑛𝑡 𝑛𝑟 × 𝑛𝑡[𝑛𝑟 positive semi-definite autocorrelation matrix 𝑹 = ] 𝔼 vec (𝑯 F ) vecℋ (𝑯 F ) , which is assumed to be invertible and 𝑯 𝑤 is an 𝑛𝑟 × 𝑛𝑡 matrix consisting of complex circular
2
Gaussian distributed elements with zero mean and unit variance. The mean value of the channel matrix 𝑯 is given as √ 𝐾 ¯ (1+𝐾) 𝑯 where 𝐾 ≥ 0 is the Ricean factor [1]. Special
cases of this model are the Kronecker model [1] 𝑹 = 𝑹𝒯𝑡 ⊗ 𝑹𝑟 , where 𝑹𝑟 ∈ ℂ𝑛𝑟 ×𝑛𝑟 and 𝑹𝑡 ∈ ℂ𝑛𝑡 ×𝑛𝑡 are the receive and the transmit correlation matrices, respectively, and the Weichselberger model [17, Eq. (5)]. III. PEP P ERFORMANCE A NALYSIS OF P RECODED A RBITRARY STBC In this section, we present theoretical analysis of the precoded arbitrary STBC, which considers the effect of arbitrary invertible channel correlation in a Ricean MIMO channel on the PEP performance. We may write (1) into vector form as ) ( 𝒯 vec (𝒀 𝑘 ) = (𝑭 𝑺 𝑘 ) ⊗ 𝑰 𝑛𝑟 vec (𝑯) + vec (𝑸𝑘 ) . (3)
Since vec (𝑸𝑘 ) is distributed as ( 𝑸𝑘 is AWGN, ) 𝒞𝒩 0𝑛𝑟 𝑛×1 , 𝜎 2 𝑰 𝑛𝑟 𝑛 (( . If 𝑺 𝑘 , 𝑭 , and 𝑯 )are known, vec (𝒀 𝑘)) 𝒯 is distributed as 𝒞𝒩 (𝑭 𝑺 𝑘 ) ⊗ 𝑰 𝑛𝑟 vec (𝑯) , 𝜎 2 𝑰 𝑛𝑟 𝑛 . Following the procedure given in [18, Section 4.2], we may obtain the Chernoff bound of the PEP given that the channel 𝑯 and the precoder 𝑭 are known as } { Pr 𝑺 0𝑘 → 𝑺 𝑘 ∣𝑯, 𝑭 ( ) ( )ℋ ( 0 ) vecℋ (𝑯) 퓑0𝑘 −퓑𝑘 퓑𝑘 −퓑𝑘 vec(𝑯) ≤ exp − , (4) 4𝜎 2 )𝒯 ( where 퓑0𝑘 = 𝑭 𝑺 0𝑘 ⊗ 𝑰 𝑛𝑟 . By using the probability density function (p.d.f.) [19, Eq. (2.16)] and moment generating function (M.G.F.) [19, Eq. (2.16)] of a chi-square distributed random variable, (4) can be averaged over 𝑯 with the help of the procedure given in [18, Section 4.4] and the upper bound of PEP (UBPEP) can be expressed as UBPEP = where 𝜱 =
−1 −1 −1 ℋ ¯ 1 ¯ 𝑒𝐾(1+𝐾)vec (𝑯 )𝑹 (𝜱 − 1+𝐾 𝑹)𝑹 vec(𝑯 )
, (5) ∣𝑹∣ ∣𝜱∣ (( ) ) ¯ ∗𝑺 ¯ 𝒯 𝒯 ⊗𝑰 𝑛 + (1 + 𝐾)𝑹−1 and 𝑭 ∗𝑺 𝑘 𝑘𝑭 𝑟 (1 + 𝐾)
1 4𝜎 2
−𝑛𝑡 𝑛𝑟
−1 −1 −1 ℋ ¯ ¯ 𝑒𝐾(1+𝐾)vec (𝑯 )𝑹 𝜣 𝑹 vec(𝑯 ) , ˜ 𝑘=𝑺 0 −𝑺 𝑘 ∣𝜣∣ 𝑺 𝑘
¯ 𝑘= arg 𝑺
max
(6)
𝑺 0𝑘 ∕=𝑺 𝑘
(( ∗ 𝒯 ) ) ˜ 𝑺 ˜ ⊗𝑰 𝑛 + (1 + 𝐾)𝑹−1 and 𝑃 where 𝜣 = 4𝑎𝑛𝑠𝑃𝜎2 𝜎2 𝑺 𝑘 𝑘 𝑟 𝑠 is the average power used by the transmitted block 𝑭 𝑺 𝑘 . Any properly designed precoder should not perform worse than the trivial precoder, therefore, a trivial precoder is used to obtain ¯ 𝑘 in (6) from (5). The trivial precoder is expressed as follows: 𝑺 √ 𝑭 = 𝑃/ (𝑎𝑛𝑠 𝜎𝑠2 )[𝑰 max{𝑛𝑡 ,𝑏} ]𝑛𝑡 ×𝑏 , where [𝑰 max{𝑛𝑡 ,𝑏} ]𝑛𝑡 ×𝑏 is a matrix of size 𝑛𝑡 × 𝑏 taken from the upper left part of the identity matrix 𝑰 max{𝑛𝑡 ,𝑏} , where max {𝑛𝑡 , 𝑏} returns the maximum value of 𝑛𝑡 and 𝑏. If SNR → ∞, 𝑛𝑡 = 𝑏, 𝑹 is non-singular, 𝑺 𝑘 is an ¯ are finite valued, then by arbitrary STBC, and 𝐾 and 𝑯 applying the matrix inversion lemma [20, Subsection 0.7.4] in the numerator of (5) and then neglecting the identity matrix in
the resulting expression in the numerator of (5), the UBPEP can be expressed as (7) given at the top of the next page, where ¯𝑘 𝑺 = arg
min
˜ 𝑘 =𝑺 0 −𝑺 𝑘 𝑺 𝑘 𝑺 0𝑘 ∕=𝑺 𝑘
𝑰 𝑏𝑛 + 𝑟
) (( ∗ 𝒯 ) 𝑃 ˜ ˜ 𝑺 𝑘 𝑺 𝑘 ⊗𝑰 𝑛𝑟 𝑹 . 4𝑎𝑛𝑠 𝜎𝑠2 𝜎 2 (1 + 𝐾)
(8)
For a full diversity STBC by using the property of determinant ∣𝑰 + 𝑨∣ ≥ ∣𝑨∣, where 𝑨 is a positive definite matrix, in the denominator of (7) we get the following UBPEP at high SNR: −1 ℋ ¯ ¯ 𝑒(−𝐾vec (𝑯 )𝑹 vec(𝑯 )) , (9) UBPEP = ¯ ∗ ¯ 𝒯 𝒯 𝑛𝑟 −𝑛 𝑛 (4𝜎 2 (1 + 𝐾)) 𝑡 𝑟 ∣𝑹∣ 𝑭 ∗ 𝑺 𝑭 𝑺 𝑘 𝑘
where
¯ 𝑘 = arg 𝑺
min
˜ 𝑘 =𝑺 0 −𝑺 𝑘 𝑺 𝑘 𝑺 0𝑘 ∕=𝑺 𝑘
˜ ˜ ℋ 𝑺 𝑘 𝑺 𝑘 .
(10)
It can be seen from (9) and (10) that the definition of (10) which is used in [2, Section IV] is applicable only for full diversity STBCs at high SNRs. For the QOSTBCs [11]–[15], the rank of 𝑺 𝑘 𝑺 ℋ 𝑘 is two and these codes provide a diversity of only two. For these ˜ ˜ ℋ ¯ codes 𝑺 𝑘 𝑺 𝑘 = 0, therefore, the definition of 𝑺 𝑘 given in (8) cannot be used for them. From the preceding discussion, it can be seen that (8) (used in [2, Section IV] for precoder design) is valid for full-diversity STBCs at high SNRs only, however, (6) is applicable to arbitrary STBC for all SNRs. For designing precoders for the low rank non-orthogonal STBC ¯ 𝑘 are used in [4] like QOSTBCs, two alternate definitions of 𝑺 based on: 1) The minimum distance [4, Eq. (7)], where it is assumed that the minimum distance between the two different codeword matrices can be found by assuming that they differ in only one symbol out of 𝑛𝑠 original symbols (one example can be seen in [4, Subsection VIII-B.1]), and 2) the average ¯ 𝑘 is obtained by averaging all distance [4, Eq. (10)], where 𝑺 ¯ 𝑘 obtained from codeword difference matrices. In general, 𝑺 these criteria is a scaled identity matrix, which means that the precoder for a 𝑏 × 𝑛 non-orthogonal STBC is designed by approximating it as a 𝑏 × 𝑛 OSTBC in [4]. However, it is shown in Section V that a precoder designed on the basis of (5) and (6) works better than the precoder proposed in [4] for non-orthogonal STBCs. IV. P ROPERTIES AND D ESIGN OF P RECODER FOR A RBITRARY STBC Under the assumption that the transmitter knows the mean values of the channel coefficients, the channel correlations, and the noise variance perfectly, we will design a precoder for any STBC. The average power constraint{of the transmitted block } 𝑭 𝑺 𝑘 can be expressed as 𝑎𝑛𝑠 𝜎𝑠2 Tr 𝑭 𝑭 ℋ = 𝑃 . We can express the optimization problem as min UBPEP. (11) {𝑭 ∈ℂ𝑛𝑡 ×𝑏 ∣Tr{𝑭 𝑭 ℋ }=𝑃/(𝑎𝑛𝑠 𝜎𝑠2 )}
3
−1 ℋ ¯ ¯ 𝑒(−𝐾vec (𝑯 )𝑹 vec(𝑯 )) ) (( UBPEP = )−1 ) ) (( ∗ ¯𝒯 𝒯 ∗ 𝑛𝑟 𝒯 ∗ 1 ¯ 𝑭 𝑭 ⊗ 𝑰 𝑛𝑟 𝑭 𝑭 4𝜎2 (1+𝐾) 𝑺 𝑘 𝑺 𝑘 ⊗ 𝑰 𝑛𝑟 𝑹 +
As a closed-form precoder is difficult to obtain for the general case of Ricean channels with invertible correlation matrices, we focus on fast-converging numerical method to design the precoder.
(7)
−1
10
ρ=0.99999
−2
10
ρ=0.9
Property 1: If SNR → ∞, 𝑏 = 𝑛𝑡 , 𝑺 𝑘 is an arbitrary ¯ are finite valued, STBC, 𝑹 is non-singular, and 𝐾 and 𝑯 √ 𝑃 then the trivial identity scaled precoder 𝑭 = 𝑎𝑛𝑠 𝜎𝑠2 𝑰 𝑛𝑡 is the minimum UBPEP precoder. Property 2: If 𝑛 = 𝑏( = 𝑛𝑡 , the diversity of the system is ( { })) ¯𝑘 min rank{𝑹} , 𝑛𝑟×min rank{𝑭 } , rank 𝑺 . Property 1 can be proved by using Hadamard inequality [16, Eq. (5.2.4)] in (7) at SNR → ∞ and it reveals that there is no need of precoding at high SNR. Property 2 can be proved by observing the fact that, from (5), the diversity of the precoded STBC channels)depends upon (( over )correlated ) Ricean (( ℋ )𝒯 𝒯 ¯ ¯ the rank of 𝑭 𝑺 𝑘 ⊗𝑰 𝑛 𝑹 𝑭 𝑺 𝑘 ⊗𝑰 𝑛 and then 𝑟
𝑟
applying [21, Eq. (1.7.4)] and the Kronecker product properties [21] over this term. Using [21, Eqs. (1.7.4) and (1.7.5)] and Property 2 it can be shown that a low-ranked precoder may adversely affect the diversity of the MIMO system utilizing an arbitrary STBC. Therefore, Property 2 emphasizes that a precoder must always be a full-ranked matrix.
B. Precoder Design The constrained minimization problem of (11) can be converted into an unconstrained minimization problem by introducing a Lagrange multiplier 𝜇′ } { (12) ℒ (𝑭 ) = ln (UBPEP) + 𝜇′ Tr 𝑭 𝑭 ℋ .
The necessary condition for the optimality of (11) is found by setting the derivative [22] of the Lagrangian in (12) with respect to vec (𝑭 ∗ ) equal to zero. The precoder that is optimal for the optimization problem in (11) satisfies: [ {( ( ) ¯ vec𝒯 (𝑭 ) = 𝜇 vec𝒯 𝐾 (1 + 𝐾) 𝜱−1 𝑹−1 vec 𝑯 }] { } ( ) ( ) ) 𝒯 ¯ 𝑹−1 𝜱−1 𝒯 + vec𝒯 𝜫퓓, (13) 𝜱−1 ×vecℋ 𝑯 ) ( ¯ 𝑘𝑺 ¯ℋ = ⊗ 𝑰 𝑛𝑡 , 𝜫 where 퓓 = 𝑭𝑺 𝑘 (𝑰 𝑏 ⊗ 𝑲 𝑛𝑟 ,𝑛𝑡 ⊗ 𝑰 𝑛𝑟 ) [𝑰 𝑛𝑡 𝑏 ⊗ vec (𝑰 𝑛𝑟 )] , 𝑲 𝑛𝑟 ,𝑛𝑡 is the commutation matrix of size 𝑛𝑟 𝑛𝑡 × 𝑛𝑟 𝑛𝑡 [21, pp. 54–56], and 𝜇 is a real positive scalar chosen such that the power constraint is satisfied. Equation (13) can be used in a fixed point iteration for finding the precoder that solves (11). The method given in [23, Table III] can be followed for finding precoder 𝑭 . It can be seen from (13) that it is very difficult to visualize how an optimized precoder 𝑭 depends upon the general
SER
A. Properties of the Optimal Precoder
−3
10
ρ=0.5 −4
10
2
4
6
8 10 SNR [dB]
12
14
16
Fig. 1. Performance of the Golden code with the proposed precoder ∘ and with the trivial precoder ∗ over Ricean channels with invertible correlation matrices, and 𝑛𝑡 = 𝑛𝑟 = 2.
correlation matrix 𝑹 and arbitrary STBC 𝑺 𝑘 . For designing an optimized precoder, the proposed precoding solution takes into account that the employed STBC can be a low-rank matrix. Since the proposed precoder is the minimum UBPEP precoder, it also makes sure that the diversity of the MIMO system is unaffected by the optimized precoder as emphasized by Property 2. By taking into consideration the eigenmodes of the channel correlation matrix, the proposed method distributes the power over all precoder eigenmodes in a way that the optimized precoder does not waste much power over the unused eigenmodes of the low diversity STBC. In this way, the proposed precoding solution is able to provide a full-rank precoding matrix which results into much more suitable power distribution than the one designed for equivalent OSTBC [4, Eq. (35)]. In general, a precoding matrix tries to match the STBC matrix with the correlated channel [4]. It can be deduced that a precoder designed for matching OSTBC transmissions with a correlated channel cannot perfectly match a non-orthogonal STBC with the channel and vice-versa. This fact is verified by simulations in Fig.2 where the proposed precoder for a 3 × 3 algebraic STBC of [7] performs significantly better than the precoder designed [24] for a 3 × 4 OSTBC [18, Eq. (7.4.8)]. V. S IMULATION R ESULTS All simulations are obtained with 105 random channel realizations at each SNR. The Golden code [9] which utilizes an optimized 4-QAM constellation obtained from BPSK is used for transmission over Ricean (𝐾 = 10) fading MIMO channel with 𝑛𝑡 = 𝑛𝑟 = 2 and an invertible full correlation matrix [𝑹]𝑖,𝑗 = 𝜌∣𝑖−𝑗∣ , 1 ≤ {𝑖, 𝑗} ≤ 𝑛𝑡 𝑛𝑟 , 𝜌 ∈ {0.5, 0.9, 0.99999}
4
Trivial precoder Existing precoder [4] Proposed precoder
Trivial precoder Precoder designed for OSTBC Proposed precoder −1
10
−2
SER
SER
10
−2
10 −3
10
6
8
10
12 SNR [dB]
14
16
18
−5
20
Fig. 2. Performance of the STBC of [7] with the proposed precoder, the trivial precoder, and a precoder designed for 3 × 4 OSTBC over generally correlated Rayleigh channels, and 𝑛𝑡 = 𝑛𝑟 = 3.
which does not follow the Kronecker model. It can be seen from Fig. 1 that the Golden code with the proposed precoder outperforms the one with trivial precoder. For example, a SNR gain of approximately 2.0 dB is achieved at SER = 2 × 10−3 over correlated Ricean channel with 𝜌 = 0.5. It is worth noticing that the existing precoders [2], [4] cannot be used in this case, since they are applicable only for the Kronecker correlation model with transmit correlations and no receive correlation. In Fig. 2, we have shown the performance of a 3 × 3 algebraic STBC of [7] over a 3 × 3 generally correlated Rayleigh fading MIMO system with 𝜌 = 0.999. We numerically obtained a precoder for a 3 × 4 OSTBC [18, Eq. (7.4.8)] and used it for suoptimal precoding of nonorthogonal STBC [7]. It can be seen from Fig. 2 that the proposed precoder designed for the non-orthogonal STBC significantly outperforms the suboptimal precoder. Since, all three precoders perform similarly at SNR > 17.5 dB, Fig. 2 corroborates Property 1 at high SNR there is no need of precoding. Moreover, it can be seen from Fig. 2 that an optimized full rank precoder does not affect the diversity of the correlated system. This result illustrates the Property 2 given in Subsection IV-A. In Fig. 3, we have performed simulations for the Weichselberger model [17] with the following specifications. The transmit and the receive correlation matrices are given in (14) and (15), respectively, at the top of the next page, and the coupling matrix is given as ⎤ ⎡ 15.0445 0.4673 0.1099 0.0445 ⎢ 0.0327 0.0342 0.0351 0.0341 ⎥ ⎥ (16) 𝜴=⎢ ⎣ 0.0102 0.0100 0.0100 0.0099 ⎦ . 0.0058 0.0059 0.0060 0.0059
The QOSTBC of [11, Eq. (5)] is used over a 4 × 4 correlated Ricean fading MIMO channel with 𝐾 = 1 and BPSK modulation. For designing the precoder of [4] for the QOSTBC, ¯ 𝑘 is calculated by the codeword distance product matrix 𝑺 using the minimum distance design [4, Subection VIII-B.1] and it results into a scaled identity matrix. Therefore, the
−3
−1
1
3 SNR [dB]
5
7
9
10
Fig. 3. Performance of the QOSTBC [11] with 𝑛𝑡 = 𝑛𝑟 = 4, and over correlated Ricean channels with the proposed precoder, the precoder of Vu and Paulraj [4], and with the trivial precoder.
Trivial precoder Proposed precoder based on transmit correlation matrix Proposed precoder based on full correlation matrix
−1
10
−2
10
PEP
4
−3
10
−4
10
5
10
15 SNR [dB]
20
25
Fig. 4. PEP performance of the improved ABBA codes [15] with 𝑛𝑡 = 𝑛𝑟 = 4, and over correlated Ricean channels with the trivial precoder, the proposed precoder designed for transmit correlations only, and the proposed precoder designed with full correlation matrix.
performance of the QOSTBC is approximated by an OSTBC in [4]. It can be seen from Fig. 3 that the proposed precoder performs better than the existing precoder [4] over −5 dB ≤ SNR ≤ 10 dB. Since the existing precoder [4] also does not take care of the coupling matrix 𝜴, it does not provide an optimum performance over the non-Kronecker correlation model like Weichselberger model. In Fig. 4, we have plotted the PEP versus SNR performance of a 4 × 4 MIMO system employing improved ABBA code [15] over the Ricean channels with 𝐾 = 1 correlated as per the Weichselberger model (14), (15), and (16). The UBPEP is calculated from (5) at each SNR. For the scenario when the transmitter only knows the transmit correlation matrix, we design a precoder by using the proposed method. We have also designed a precoder for the transmitter that knows the transmit correlation, the receive
5
⎡
3.7767 + 𝚥0.0052 3.9327 3.7842 − 𝚥0.0020 3.6396 − 𝚥0.0027
3.6314 + 𝚥0.0046 3.7842 + 𝚥0.0020 3.9486 3.8001 − 𝚥0.0003
⎤ 3.4902 + 𝚥0.0044 3.6396 + 𝚥0.0027 ⎥ ⎥ 3.8001 + 𝚥0.0003 ⎦ 3.9630
⎡
3.9056 + 𝚥0.0016 3.9487 3.9055 − 𝚥0.0005 3.8685 − 𝚥0.0007
3.8630 + 𝚥0.0023 3.9055 + 𝚥0.0005 3.9422 3.9050 − 𝚥0.0002
⎤ 3.8263 + 𝚥0.0027 3.8685 + 𝚥0.0007 ⎥ ⎥ 3.9050 + 𝚥0.0002 ⎦ 3.9468
3.9360 ⎢ 3.7767 − 𝚥0.0052 𝑹𝑡 = ⎢ ⎣ 3.6314 − 𝚥0.0046 3.4902 − 𝚥0.0044 3.9426 ⎢ 3.9056 − 𝚥0.0016 ⎢ 𝑹𝑟 = ⎣ 3.8630 − 𝚥0.0023 3.8263 − 𝚥0.0027
correlations, and the coupling matrix perfectly. It can be seen from Fig. 4 that the precoder designed with full correlation matrix performs better than the precoder designed for transmit correlation matrix only. This simulation result justifies that we should design a precoder based on a full correlation matrix rather than a partial correlation matrix for optimizing the performance over correlated Ricean channels. VI. C ONCLUSIONS We have proposed a precoder design for orthogonal or nonorthogonal STBC over correlated Ricean MIMO channels with invertible correlation matrices where the existing precoder designs are not applicable. Some important properties of the optimized precoder are pointed out. The precoded nonorthogonal STBC is shown to achieve significant performance gains as compared to the case when a trivial precoder is used. In addition, the proposed precoder also works better than one existing precoder for the non-orthogonal codes which do no have full diversity. R EFERENCES [1] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge, UK: Cambridge University Press, 2003. [2] 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, June 2002. [3] ——, “Corrections to: ’Linear precoding for space-time coded systems with known fading correlations’,” IEEE Commun. Lett., vol. 6, no. 6, June 2002. [4] M. Vu and A. Paulraj, “Optimal linear precoders for MIMO wireless correlated channels with non-zero mean in space-time coded systems,” IEEE Trans. Signal Process., vol. 54, no. 6, pp. 2318–2332, Jun. 2006. [5] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codes for high data rate wireless communications: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998. [6] E. Bonek, H. Özcelik, M. Herdin, W. Weichselberger, and J. Wallace, “Deficiencies of a popular stochastic MIMO radio channel model,” in Proc. Int. Symp. on Wireless Personal Multimedia Communications, Yokosuka, Japan, Oct. 2003. [7] B. A. Sethuraman, B. S. Rajan, and V. Shashidhar, “Full-diversity, highrate space-time block codes from division algebras,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2596–2616, Oct. 2003. [8] M. O. Damen, A. Tewfik, and J. C. Belfiore, “A construction of a spacetime code based on number theory,” IEEE Trans. Inform. Theory, vol. 48, no. 3, pp. 753–760, Mar. 2002. [9] J. C. Belfiore, G. Rekaya, and E. Viterbo, “The Golden code: A 2 × 2 full-rate space-time code with non-vanishing determinants,” IEEE Trans. Inform. Theory, vol. 51, no. 4, pp. 1432–1436, Apr. 2005. [10] L. Zheng and D. N. C. Tse, “Diversity and multiplexing a fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inform. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.
(14)
(15)
[11] H. Jafarkhani, “A quasi-orthogonal space-time block code,” IEEE Trans. Commun., vol. 49, no. 1, pp. 1–4, Jan. 2001. [12] C. Papadias and G. Foschini, “Capacity-approaching space-time codes for system employing four transmitter antennas,” IEEE Trans. Inform. Theory, vol. 49, no. 3, pp. 726–733, Mar. 2003. [13] 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. [14] O. Tirkkonen, A. Boariu, and A. Hottinen, “Minimal nonorthogonality rate 1 space-time block code for 3+ Tx antennas,” in Proc. IEEE 6th Int. Symp. Spread-Spectrum Techniques and Applications (ISSSTA), pp. 429–432, Sep. 2000, New Jersey, USA. [15] F.-C. Zheng and A. G. Burr, “Robust non-orthogonal space-time block codes over highly correlated channels: A generalisation,” Electronics Letters, vol. 39, no. 16, pp. 1190–1191, Aug. 2003. [16] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge, UK: Cambridge University Press, 1991, reprinted 1999. [17] W. Weichselberger, M. Herdin, H. Özcelik, and E. Bonek, “A stochastic MIMO channel model with joint correlation of both link ends,” IEEE Trans. Wireless Commun., vol. 5, no. 1, pp. 90–100, Jan. 2006. [18] E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications. Cambridge, UK: Cambridge University Press, 2003. [19] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. New York, NY, USA: John Wiley & Son Inc., 2000. [20] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, UK: Cambridge University Press, 1985, reprinted 1999. [21] J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Application in Statistics and Econometrics. Essex, England: John Wiley & Sons, Inc., 1988. [22] A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2740–2746, Jun. 2007. [23] M. R. Bhatnagar, A. Hjørungnes, and L. Song, “Precoded differential orthogonal space-time modulation over correlated Ricean MIMO channels,” IEEE J. Select. Topics Signal Process., vol. 2, no. 2, pp. 124–134, Apr. 2008. [24] A. Hjørungnes and D. Gesbert, “Precoded orthogonal space-time block codes over correlated Ricean MIMO channels,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 779–783, Feb. 2007.
