Superimposed Training Designs for Spatially Correlated MIMO-OFDM ...
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Superimposed Training Designs for Spatially Correlated MIMO-OFDM Systems Nguyen N. Tran, Member, IEEE, Hoang D. Tuan, Member, IEEE, and Ha H. Nguyen, Senior Member, IEEE
Abstract—Only one asymptotic training design for a special case of channel correlation was proposed in the literature for spatially correlated multiple-input multiple-output with orthogonal frequency-division multiplexing (MIMO-OFDM) systems. To fill this gap, this letter applies tractable semi-definite programming (SDP) to obtain the optimal superimposed training signals for the general case of channel correlation. For a more efficient computation, two approximate designs are also proposed. Simulation results demonstrate the efficiency of our approach and its advantage over the asymptotic design. Index Terms—MIMO, OFDM, spatial correlation, channel estimation, training signal design, precoder design, superimposed training.
A
I. I NTRODUCTION
CCURATE channel estimation is a major challenge in the implementation of a multiple-input multiple-output with orthogonal frequency-division multiplexing (MIMOOFDM) system. Typically, training signals are used for estimating the unknown channel [1]–[5] and various designs of training signals have been proposed for OFDM and MIMOOFDM systems in the last decade. The channels between different pairs of transmit and receive antennas are often assumed to be uncorrelated, with a diagonal covariance matrix. However, this assumption is impractical for certain wireless environments [4], [6]–[9]. Apparently, only reference [4] studies the optimal training design and channel estimation for spatially correlated frequency-selective fading MIMO-OFDM. Under some special cases of correlation, [4] provides time-multiplexing (TM) (also known as block-type) training designs in the forms of asymptotic solutions at low and high signal-to-noise ratio (SNR) regimes. This letter proposes superimposed training signal designs for spatially correlated MIMO-OFDM in the general case of channel correlation and arbitrary SNR. In particular, we extend our previous results [5], [10], [11] to the more general and challenging case of spatially correlated MIMO-OFDM
Manuscript received May 28, 2008; revised April 19, 2009 and August 3, 2009; accepted October 1, 2009. The associate editor coordinating the review of this letter and approving it for publication was A. Hjørungnes. This work is supported by the Australian Research Council under grant ARC Discovery Project 0772548. N. N. Tran was with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, NSW 2052, Australia. He is now with the Department of Computer and Electronics Engineering, University of Nebraska- Lincoln, Omaha, NE 68182 USA (email: [email protected]). H. D. Tuan is with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]). H. H. Nguyen is with the Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Dr., Saskatoon, SK, Canada S7N 5A9 (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2010.03.080661
systems. As discussed in [5], [10], [11], superimposed training allows a more flexible preprocessing of the data symbols since they can be transmitted simultaneously with the training signals. This increased flexibility leads to an improvement of the overall system performance. Furthermore, since TM training can be considered as a particular case of SP training, our optimal training design is also novel within the class of TM training. The structure of the letter is as follows. After presenting the system model in Section II, we develop semi-definite programming (SDP) to design the optimal training signals in Section III. Then the approximated closed-form solutions are derived to minimize upper bounds of the channel minimum mean-square error (MMSE) in Section IV. A power allocation for symbol detection enhancement is presented in Section V. Simulation results to show the advantages of our approach are given in Section VI. Notation: Boldface upper and lower cases denote matrices and column vectors, respectively. Superscripts 𝑇 , 𝐻 and ∗ mean transposition, Hermitian adjoint and complex conjugate operators, respectively. ⊗ stands for Kronecker matrix product. I𝑛 and 0𝑛×𝑚 are the 𝑛× 𝑛 identity matrix and the 𝑛× 𝑚 zero matrix, respectively. For a vector x, diag{x} is the diagonal matrix with the components x𝑖 of x on its diagonal. For 𝑛 matrices X𝑖 (or X(𝑖)), 𝑖 = 1, . . . , 𝑛, diag[X𝑖 ]𝑖=1,...,𝑛 (or diag[X(𝑖)]𝑖=1,...,𝑛 ) forms a block diagonal matrix by putting X𝑖 (X(𝑖)) on the 𝑖th diagonal block. Likewise, [X𝑖𝑗 ] 𝑖=1,...,𝑛 𝑗=1,...,𝑚 forms a new matrix by putting matrix X𝑖𝑗 on the (𝑖, 𝑗)th block. The symbol 𝒰𝑡 (𝒟𝑡 , resp.) denotes the set of all 𝑡 × 𝑡 unitary matrices (diagonal matrices, resp.). The vectorization operator on a matrix to form a column vector by vertically stacking the matrix columns is denoted as vec(⋅). The expectation operation with respect to the random variable x is E{x}, and also Rx is the covariance matrix of x. For a scalar 𝑥, define 𝑥+ = max(𝑥, 0). A ≥ 0 (A > 0, resp.) means that A is a positive semi-definite (definite, resp.) Hermitian matrix, while tr{A} is its trace. II. S PATIALLY C ORRELATED MIMO-OFDM S YSTEM Consider a MIMO system with 𝑡 transmit and 𝑟 receive antennas. The channel is frequency-selective and the number of significant delay paths between any antenna pair is 𝐿. The channel is assumed to be unchanged during one block of OFDM symbols of duration 𝑀 , but may change independently from block to block (i.e., quasi-static or block fading). For Rayleigh fading, the channel gains are modeled as jointly Gaussian [4]. The MIMO channel is spatially correlated at both the transmitter and receiver but the path gains for different delays are uncorrelated. Let h𝑗𝑖 =
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[ℎ𝑗𝑖 (0), ℎ𝑗𝑖 (1), . . . , ℎ𝑗𝑖 (𝐿 − 1)]𝑇 denote the channel impulse response from the 𝑖th (𝑖 = 1, . . . , 𝑡) transmit antenna to the 𝑗th (𝑗 = 1, . . . , 𝑟) receive antenna. Accordingly, the channel gain matrix H(ℓ) of the ℓth path, ℓ = 0, . . . , 𝐿 − 1, can be represented as 1/2
𝑟×𝑡 . H(ℓ) := [ℎ𝑗𝑖 (ℓ)] 𝑗=1,...,𝑟 = Σ1/2 𝑟 (ℓ)H𝑤 (ℓ)Σ𝑡 (ℓ) ∈ ℂ 𝑖=1,...,𝑡 (1) Here Σ𝑡 (ℓ) and Σ𝑟 (ℓ) are 𝑡×𝑡 and 𝑟 ×𝑟 known and invertible covariance matrices (see, e.g. [12], for their identification), in which
Σ𝑡 (ℓ) = [𝑡𝑢𝑣 (ℓ)]𝑢,𝑣=1,...,𝑡 and Σ𝑟 (ℓ) = [𝑟𝑛𝑚 (ℓ)]𝑛,𝑚=1,...,𝑟 , (2) where 𝑡𝑢𝑢 (ℓ) = 𝑟𝑛𝑛 (ℓ) = 1 and 𝑡𝑢𝑣 (ℓ) = 𝑡∗𝑣𝑢 (ℓ) (𝑟𝑛𝑚 (ℓ) = ∗ 𝑟𝑚𝑛 (ℓ), resp.) reflects the correlated fading between the 𝑢th and the 𝑣th (𝑛th and 𝑚th, resp.) elements of the transmit (receive, resp.) antenna array at the ℓth delay path. The entries of the 𝑟 × 𝑡 unknown (to-be-estimated) matrix H𝑤 (ℓ) are independent and identically-distributed (i.i.d.) circularly symmetric complex Gaussian random variables of variance 𝜎ℎ2 , i.e., 𝒞𝒩 (0, 𝜎ℎ2 ). For convenience, normalize 𝜎ℎ2 = 1/𝐿. Let 𝑁 be the number of sub-carriers. As a standard operation in OFDM, a cyclic prefix (CP) of length 𝐿 is used to eliminate ISI between consecutive OFDM blocks of length 𝑁 + 𝐿. The [𝑁 × 𝑁 discrete Fourier transform (DFT) matrix ] −1 is F𝑁 = f0 f1 . . . f𝑁 −1 = √1𝑁 [e−𝑗2𝜋𝑝𝑘/𝑁 ]𝑁 𝑝,𝑘=0 . [ ] F𝐿 Partition F𝑁 as F𝑁 = , F𝐿 ∈ ℂ𝐿×𝑁 , F𝑁 −𝐿 ∈ F𝑁 −𝐿 ℂ(𝐿−𝑁 )×𝑁 , where ] [ (3) F𝐿 = f0𝐿 f1𝐿 . . . f(𝑁 −1)𝐿 ∈ ℂ𝐿×𝑁 [ 𝑇 ]𝑇 𝑇 𝑇 = ˜f0 ˜f1 . . . ˜f𝐿−1 (4) contains the first 𝐿 rows of F𝑁 and F𝐿 F𝐻 𝐿 = I𝐿 . Note that√, f𝑘𝐿 ∈ ℂ𝐿 , 𝑘 = 0, 1, . . . , 𝑁 − 1 and , ˜fℓ = (1/ 𝑁 )(1, e−𝚥2𝜋ℓ/𝑁 , . . . , e−𝚥2𝜋ℓ(𝑁 −1)/𝑁 ). Define 𝑀 = 𝐾 + 𝑞 with 𝑞 ≥ 𝑡. The MIMO equation corresponding to the 𝑘th subcarrier (subchannel) with 𝑀 OFDM blocks (each of them has length 𝑁 + 𝐿) is Y𝑘 = H𝑓 (𝑘)U(𝑘)+W(𝑘), U(𝑘) ∈ ℂ𝑡×𝑀 , W(𝑘) ∈ ℂ𝑟×𝑀 , (5) where 𝑘 = 0, 1, . . . , 𝑁 − 1, H𝑓 (𝑘) =
𝐿−1 ∑
H(ℓ)e−𝚥2𝜋ℓ𝑘/𝑁
(6)
ℓ=0
=
[ H(f𝑘𝐿 ⊗ I𝑡 ), H = H0
. . . H𝐿−1
]
√ ˜ C(𝑘) ∈ ℂ𝑡×𝑞 and Q𝐸 ∈ ℂ𝑀×𝑞 . Specifically, P = 𝑀 𝐾 O(1 : 𝐾) ∈ ℂ𝐾×𝑀 , Q𝐸 = O𝐻 (𝐾 + 1 : 𝑀 ) ∈ ℂ𝑀×𝑞 with an arbitrary 𝑀 × 𝑀 orthogonal matrix O. Therefore, P is full rank (i.e., PP𝐻 is nonsingular), tr{PP𝐻 } = 𝑀 and PQ𝐸 = 0. Furthermore, by normalizing the total average transmit power to unity, the average training power is 𝜎𝑐2 = tr{C(𝑘)C𝐻 (𝑘)}/(𝑀 𝑡) = 1 − 𝜎𝑠2 , where 𝜎𝑠2 is the average data power. III. C HANNEL E STIMATION AND S UPERIMPOSED T RAINING D ESIGN WITH SDP To estimate H𝑓 (𝑘), 𝑘 = 0, 1, . . . , 𝑁 − 1 in (5), postmultiplying both sides of (5) by Q𝐸 gives ˜ ˜ Y𝑘 Q𝐸 = H𝑓 (𝑘)C(𝑘) + W(𝑘)Q𝐸 , C(𝑘) ∈ ℂ𝑡×𝑞 .
(9)
Now define
] [ := [Y0 Q𝐸 Y1 Q𝐸 . . . Y𝑁 −1 Q𝐸 ]∈ ℂ𝑟×(𝑁 𝑞) , := [H𝑓 (0) H𝑓 (1) . . . H𝑓 (𝑁 − ]1) ∈ ℂ𝑟×(𝑁 𝑡) , := W(0)Q𝐸 . . . W(𝑁 − 1)Q𝐸 ∈ ℂ𝑟×(𝑁 𝑞) . (10) It follows from(7) and (9) that ] [ H𝑓 = H f0𝐿 ⊗ I𝑡 f1𝐿 ⊗ I𝑡 . . . f(𝑁 −1)𝐿 ⊗ I𝑡 Y H𝑓 W
[
=
H(F𝐿 ⊗ I𝑡 ),
˜ Y = H𝑓 (0)C(0)
(11) ] ˜ . . . H𝑓 (𝑁 − 1)C(𝑁 − 1) +W. (12)
For 𝑘 = 0, 1, . . . , 𝑁 − 1, using (7) and (11) to define h𝑓 (𝑘) := := h𝑓 =
𝑇 vec(H𝑓 (𝑘)) = (f𝑘𝐿 ⊗ I𝑟𝑡 )h, h := vec(H), [ 𝑇 ]𝑇 vec(H𝑓 ) = h𝑓 (0) h𝑇𝑓 (1) . . . h𝑇𝑓 (𝑁 − 1) (F𝑇𝐿 ⊗ I𝑟𝑡 )h.
Furthermore, as Rh = diag[Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ)]ℓ=0,1,...,𝐿−1 , the covariance matrices R𝑓 (𝑘, 𝑖) := E{h𝑓 (𝑘)h𝐻 𝑓 (𝑖)}, 𝑘, 𝑖 = 0, 1, . . . , 𝑁 − 1 and R𝑓 := Rh𝑓 can be expressed by
=
𝑇 𝑇𝐻 R𝑓 (𝑘, 𝑖) = (f𝑘𝐿 ⊗ I𝑟𝑡 )diag[Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ)]ℓ (f𝑖𝐿 ⊗ I𝑟𝑡 ) 𝐿−1 ∑ (f𝑘𝐿 (ℓ)¯f𝑖𝐿 (ℓ))Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ), ℓ = 0, . . . , 𝐿 − 1. (13) ℓ=0
= =
R𝑓 = [R(𝑘, 𝑖)]𝑘,𝑖=0,1,...,𝑁 −1 (14) 𝑇 𝑇𝐻 (F𝐿 ⊗ I𝑟𝑡 )diag[Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ)]ℓ=0,1,...,𝐿−1 (F𝐿 ⊗ I𝑟𝑡 ) 𝐿−1 ∑ (˜fℓ𝑇 ⊗ I𝑟𝑡 )(Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ))(˜fℓ𝑇 𝐻 ⊗ I𝑟𝑡 ). (15) ℓ=0
(7)
is the frequency response of the 𝑘th sub-channel, and W(𝑘) is the additive white Gaussian noise (AWGN) with zero mean 2 . With super-imposed training, the matrix and variance 𝜎𝑤 U(𝑘) representing the transmitted signals has the following form: U(𝑘) = S(𝑘)P + C(𝑘), 𝑘 = 0, 1, . . . , 𝑁 − 1.
877
(8)
In (8), S(𝑘) ∈ ℂ𝑡×𝐾 contains the source data symbols whose elements are modeled as zero mean i.i.d. random variables with variance 𝜎𝑠2 , P ∈ ℂ𝐾×𝑀 is the precoder matrix, and 𝐻 𝑡×𝑀 ˜ C(𝑘) = C(𝑘)Q is the training matrix, where 𝐸 ∈ ℂ
] [ 𝑇 𝑇 ˜𝑇 ˜ ˜𝑇 Here [ 𝑇 we have𝑇used F𝐿 ⊗ I𝑟𝑡 =𝑇 f0 f1] . . . f𝐿−1 ⊗ I𝑟𝑡 = ˜f ⊗ I𝑟𝑡 ˜f ⊗ I𝑟𝑡 . . . ˜f 0 1 𝐿−1 ⊗ I𝑟𝑡 . Using (9), for 𝑘 = 0, 1, . . . , 𝑁 − 1, one has vec(Y𝑘 Q𝐸 ) =
˜ 𝑇 (𝑘) ⊗ I𝑟 )h𝑓 (𝑘) + w(𝑘), (C
(16)
where w(𝑘) = vec(W(𝑘)Q𝐸 ). For using MMSE estimator [13] to estimate the frequency response, we rewrite (12) into the vector form. With y = vec(Y), vectorizing two sides of (12) gives ¯ 𝑓 = diag[C ˜ 𝑇 (𝑘) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 , (17) ¯ 𝑓 h𝑓 + w, C y=C w = (w𝑇 (0), . . . , w𝑇 (𝑁 − 1))𝑇
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¯ 𝐻C ¯ 𝑓 = diag[(C(𝑘) ˜ C ˜ 𝐻 (𝑘)) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 . with C 𝑓 ˆ 𝑓 = (R−1 + The MMSE estimator of h𝑓 is h h𝑓 1 ¯𝐻 ¯ −1 1 ¯ with the corresponding MMSE of 2 C𝑓 C𝑓 ) 2 C𝑓 y 𝜎𝑤 𝜎𝑤 1 ¯𝐻 ¯ −1 tr{(R−1 } [13]. With the variable change 2 C𝑓 C𝑓 ) h𝑓 + 𝜎𝑤 ˜ C ˜ 𝐻 (𝑘) ∈ ℂ𝑡×𝑡 , the optimal training can be X(𝑘) = C(𝑘) solved by SDP as follows: min tr{Z} subject to
[
(18)
Z,X(𝑘)
Z
R h𝑓
R h𝑓 + R h𝑓 and
𝑁 −1 ∑
(
R h𝑓 1 2 diag[X(𝑘) 𝜎𝑤
) ⊗ I𝑟 ]𝑘 Rh𝑓
tr{X(𝑘)} = 𝑃𝑇 , 𝑃𝑇 =
] ≥ 0, (19)
𝑁 𝑀 𝑡𝜎𝑐2 .
(20)
𝑘=0
min tr{Z} subject to (20), and
Z,X(𝑘)
Z ⎢ ⎣ Rh
Let Y𝑘 (𝑖) denote the 𝑖th row of Y𝑘 Q𝐸 , H𝑓 𝑖 (𝑘) and W𝑖 (𝑘) the 𝑖th rows, respectively, of H𝑓 (𝑘) and W(𝑘)Q𝐸 in Equation (9). Then rewrite (9) as Y𝑘 (𝑖) = H𝑓 𝑖 S(𝑘) + W𝑖 (𝑘) ⇔ Y𝑘𝑇 (𝑖) = S𝑇 (𝑘)H𝑇𝑓𝑖 (𝑘) + W𝑖𝑇 (𝑘).
(22) (23)
= [Let H(𝑖) be the 𝑖th row] of H, i.e., H(𝑖) H0 (𝑖) H1 (𝑖) . . . H𝐿−1 (𝑖) with Hℓ (𝑖) being the 𝑖th row of Hℓ . It is not difficult to see that Rvec(H𝑇ℓ ) = Σ𝑟 (ℓ) ⊗ Σ𝑡 (ℓ) and the 𝑖th diagonal sub(𝑖)
matrix of Rvec(H𝑇ℓ ) is RH𝑇ℓ (𝑖) = Σ𝑟 (ℓ)Σ𝑡 (ℓ). Let RH𝑇 (𝑖) = (𝑖)
diag{RH𝑇ℓ (𝑖) }ℓ=0,1,...,𝐿−1 = diag{Σ𝑟 (ℓ)Σ𝑡 (ℓ)}ℓ=0,1,...,𝐿−1 , (𝑖)
˜ Once the solution X(𝑘) is found from (18)-(20), C(𝑘) is ˜ obtained by C(𝑘) = [X1/2 (𝑘) 0𝑡×(𝑞−𝑡) ]. Observe that as far as 𝑞 ≥ 𝑡, the MMSE does not depend on 𝑡. However, increasing 𝑡 leads to higher effective SNR, hence a better detection performance. Remark: The problem can be also solved in time-domain using SDP. Rewrite (12) as ] [ ˜ ˜ Y = H f0𝐿 ⊗ C(0) . . . f(𝑁 −1)𝐿 ⊗ C(𝑁 − 1) + W. ¯ ¯ Then y = Ch + w, where by (16), C = 𝑇 𝐻 ˜ ¯ diag[C(𝑘) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 (F𝐿 ⊗ I𝑟𝑡 ), and C C = (F𝐿 ⊗ ˜ C ˜ 𝐻 (𝑘) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 (F𝐻 × I𝑟𝑡 ). Note that I𝑟𝑡 )diag[C(𝑘) 𝐿 𝐻 Rh := E{hh } = 𝜎ℎ2 diag[Σ𝑡 (ℓ)⊗Σ𝑟 (ℓ)]ℓ=0,1,...,𝐿−1 and the ˆ = (R−1 + 12 C ¯ 𝐻 C) ¯ −1 12 C ¯ 𝐻y MMSE estimator of h is h h 𝜎𝑤 𝜎𝑤 1 ¯ 𝐻 ¯ −1 C) }. with the corresponding MMSE tr{(R−1 2 C h + 𝜎𝑤 Thus, by the variable change X(𝑘) = C(𝑘)C𝐻 (𝑘), 𝑘 = 0, 1, . . . , 𝑁 − 1, the optimal training in time-domain can also be solved by another SDP as follows:
⎡
B. General Case
where Σ𝑡 (ℓ) is the 𝑖th diagonal entry of Σ𝑡 (ℓ). 𝑇 ⊗ I𝑡 )H𝑇 (𝑖) it is true that Therefore, with H𝑇𝑓𝑖 (𝑘) = (f𝑘𝐿 𝑇 𝑇𝐻 RH𝑇𝑓𝑖 (𝑘) = (f𝑘𝐿 ⊗ I𝑡 )RH𝑇 (𝑖) (f𝑘𝐿 ⊗ I𝑡 ). ˆ 𝑇 (𝑘) of H𝑇𝑓𝑖 (𝑘) For the MMSE estimator H 𝑓𝑖 using only (23), the resulting MMSE is {( )−1 } −1 𝑇𝐻 𝑇 ˜ ˜ R 𝑇 tr + C (𝑘)C (𝑘) . Hence the total H𝑓 𝑖 (𝑘)
MMSE is
𝑟 𝑁 −1 ∑ ∑ 𝑖=1 𝑘=0
{( R−1 tr H𝑇
𝑓𝑖
)−1 } + X(𝑘) , which can (𝑘)
˜ C ˜ 𝐻 (𝑘) based on the be minimized over X(𝑘) = C(𝑘) water-filling type algorithm of [14]. V. S YMBOL DETECTION ENHANCEMENT For symbol detection, post-multiplying both sides of (5) by Q𝐷 = P𝐻 (PP𝐻 )−1 for 𝑘 = 0, 1, . . . , 𝑁 − 1 gives Y𝑘 Q𝐷 = H𝑓 (𝑘)S(𝑘) + W(𝑘)Q𝐷 ˆ 𝑓 (𝑘)S(𝑘) + H ˜ 𝑓 (𝑘)S(𝑘) + W(𝑘)Q𝐷 . =H
(24)
˜ 𝑓 (𝑘) = H𝑓 (𝑘) − H ˆ 𝑓 (𝑘) is the channel estimation Here H 𝑡×𝐾 , W(𝑘) ∈ ℂ𝑟×𝑀 . Similar to [5], error and S(𝑘) ∈ ℂ ( 𝑁 −1 Rh ) ⎥ the effective SNR of the input/output model in (24) can be ∑ ⎦ ≥ 0. 𝐻 Rh + Rh 𝜎12 (f𝑘𝐿 f𝑘𝐿 ) ⊗ X(𝑘) ⊗ I𝑟 Rh calculated as 𝑤 𝑘=0 ˆ 𝑓 (𝑘)S(𝑘)∣∣2 ) E(∣∣H SNReff = . (25) ˜ 𝑓 (𝑘)S(𝑘) + W(𝑘)Q𝐷 ∣∣2 ) IV. A PPROXIMATE T RAINING D ESIGN E(∣∣H ⎤
A. Special Case For the special case where the receive correlations at different delay paths are the same, except a scaling factor 𝜎𝑙2 , i.e., Σ𝑟 (ℓ) = 𝜎ℓ Σ𝑟 , it follows that R𝑓 (𝑘, 𝑖) = Σ𝑡 (𝑘, 𝑖) ⊗ Σ𝑟 𝐿−1 ∑ for Σ𝑡 (𝑘, 𝑖) = 𝜎ℓ (f𝑘𝐿 (ℓ)¯f𝑖𝐿 (ℓ))Σ𝑡 (ℓ). Thus, for MMSE ℓ=0
ˆ 𝑓 (𝑘) of h𝑓 (𝑘) using only equation (16), the total estimator h ˜ C ˜ 𝐻 (𝑘): MMSE is the following function in X(𝑘) = C(𝑘) { ( )−1 } 𝑁 −1 ∑ 1 ˜ ˜𝐻 tr R−1 = h𝑓 (𝑘) + 𝜎 2 (C(𝑘)C (𝑘)) ⊗ I𝑟 𝑤 𝑘=0 {( )−1 } 𝑁 −1 ∑ 1 −1 −1 tr Σ𝑡 (𝑘, 𝑘) ⊗ Σ𝑟 + 2 (X(𝑘) ⊗ I𝑟 ,(21) 𝜎𝑤 𝑘=0
which can be minimized over X(𝑘) based on the iterative bisection procedure (IBP) of [5].
By maximizing the average effective SNR of the whole MIMO-OFDM system having 𝑁 sub-carriers, the following sub-optimal power allocation is obtained with 𝛾 = 2 𝐾𝑁 𝑟𝜎𝑤 /𝑀 : √ 2 2 2 (𝐿𝜎 2 + 𝑁 𝑀 )(𝑁 𝑟𝑡 + 𝛾) + 𝐿𝛾𝜎𝑤 − 𝐿𝛾𝜎𝑤 𝑁 𝐿𝑟𝑡𝜎𝑤 𝑤 2 . 𝜎𝑐 = 2 𝑁 (𝐿𝑟𝑡𝜎𝑤 − 𝑀 𝛾) Moreover, instead of S(𝑘), one can apply a “non-redundancy precoder” F ∈ ℂ𝑡×𝑡 and replace it by FS(𝑘) to even further improve the detection performance. Let Q𝑘 = 𝑀 𝐻 H𝑓 (𝑘)}. Then the optimization problem to 2 E{H𝑓 (𝑘) 𝐾𝜎𝑤 design F is {( )−1 } 1 𝐻 min tr 𝐼 + F Q𝑘 F : tr{FF𝐻 } ≤ 𝑡. (26) F 𝜎𝑠2 The above problem can be solved as in [14, Theorem 1].
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0 Approximate, general case Approximate, special case Optimal, with SDP LS, equal−powered training
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VI. S IMULATION R ESULTS A. Estimation Performance The one-ring model in [15, E.q. (6)] is used to generate the elements of the covariance (matrices Σ𝑟 (𝑙) and ) Σ𝑡 (𝑙). Specif2𝜋 Δ ically, Σ (𝑙)(𝑛, 𝑚) ≈ 𝐽 𝑑 ∣𝑚 − 𝑛∣ , Σ𝑟 (𝑙)(𝑖, 𝑗) ≈ 𝑡 0 𝑡𝑙 𝑡 𝜆 ) ( 𝐽0 Δ𝑟𝑙 2𝜋 𝑑 ∣𝑖 − 𝑗∣ , where Δ and Δ are the angle spreads 𝑟 𝑡𝑙 𝑟𝑙 𝜆 (in radian) of the 𝑙th path at the transmitter and the receiver, respectively; 𝑑𝑡 and 𝑑𝑟 are the spacings of the transmit and receive antenna arrays, respectively; 𝜆 is the wavelength and 𝐽0 (⋅) is the zeroth-order Bessel function of the first kind. Since the average transmitted power, including the training and data powers, is normalized to unity, the received SNR in dB is 2 . The system parameters are defined as SNR = −10log10 𝜎𝑤 chosen as [1, Table I] with 𝑁 = 8 sub-carriers and 𝐿 = 3 taps. Figs. 1 and 2 show the MSE of the channel impulse ˆ 2 }, for 2×2 MIMO-OFDM system having response, E{∥h− h∥ 𝐾 = 8, 𝑞 = 𝑡, and 𝑀 = 𝐾 + 𝑞. In order to apply the approximate design for the special case of receive correlation
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matrices, the parameters are chosen such that they result in the same covariance matrix for all the delay paths at the receive antennas. In Fig. 1, they are Δ𝑡𝑙 = (𝑙/2 + 2)o , Δ𝑟𝑙 = 1o , 𝑑𝑡 = 0.5𝜆 and 𝑑𝑟 = 0.3𝜆, which yield very high correlations at both sides of the transmission. In Fig. 2, we choose average correlations with Δ𝑡𝑙 = 3(𝑙 + 2)o , Δ𝑟𝑙 = 3o , 𝑑𝑡 = 0.5𝜆 and 𝑑𝑟 = 0.3𝜆. It can be seen from Figs. 1 and 2 that, at low SNR three proposed designs (by SDP, IBP and water-filling type) yield the same estimation performance. At average SNR, the optimal training design based on SDP produces the lowest MSE. The approximate design using IBP for the special case performs as well as the SDP based design and outperforms the approximate design (for the general case). At high SNR, again all the three proposed designs perform almost the same, and their performances are better than that of the design using least square estimator (which actually ignores the MIMO correlation and gives an equal-powered training). As expected, performance difference among the three designs at medium SNR becomes larger the antenna correlation is larger (comparing between Figs. 1 and 2). Also plotted in Figs. 1 and 2 are the MSE obtained with the least-square (LS) solution. The LS solution ignores the channel and noise covariances and it is simple to show that the optimal training in this case is exactly the equalpowered training, and hence suboptimal as can be clearly seen from the two figures. B. Performance Comparison with the Design in [4] Comparisons of both channel estimation and BER performance between our optimal SP training design and the TM training design in [4] (which is for the cases of high or low SNR) are presented in Figs. 3 and 4, where the system is chosen as in [4], i.e., a 2 × 2 MIMO-OFDM system, 𝑁 = 32 sub-channels and 𝐿 = 2 taps. The exponential correlation model in [4] is also chosen where the angle spread for all taps at both transmitter and receiver is 𝜎 = 3.6∘ , the relative antenna space is △ = 1, and the average angle of departure is 𝜃¯𝑡0 = 𝜃¯𝑡1 = 7.2∘ and the average angles of arrival are 𝜃¯𝑟0 = 290.3∘, 𝜃¯𝑟1 = 332.3∘ , which give the identical transmit correlations as required by [4]. The full-rate Alamouti code together with the maximum likelihood (ML) decoding is incorporated. QPSK modulation with Gray mapping is considered. The whole matrix of the source symbols has size 𝑁 𝑡 × 𝐾 where√𝐾 = 192.√In our proposed design, the source symbols {± 𝜎𝑠2 /2 ± 𝑗 𝜎𝑠2 /2} are precoded by the proposed precoders, the precoded data matrix has size 𝑞 × 𝑀 with 𝑞 = 𝑡, 𝑀 = 𝐾 + 𝑞. For the TM system considered in [4], 𝑁 𝑡 training vectors1 are sent first in 𝑁 𝑡 time slots to estimate the fading channel and the average source symbol power is normalized to unity. The total training power 𝑃𝑇 = 𝑀 𝑁 𝑡𝜎𝑐2 as well as the LMMSE estimation are used in both designs. Fig. 3 shows that when having the same total training power, the MSE of the TM training designed in [4] is the same as that 1 The 𝑁 𝑡 training vectors are formed by horizontally stacking the training vector √designed in [4] 𝑁 𝑡 times. Therefore, [4, Equation (5)] is rewritten as 𝜌 ¯ ¯ = [A𝑇 , . . . , A𝑇 ]𝑇 , and ¯ ¯ , where ¯ r= Ah + n r = [r𝑇 , . . . , r𝑇 ]𝑇 , A 𝑀 𝑡
¯ = [n𝑇 , . . . , n𝑇 ]𝑇 are constructed by vertically stacking r, A and n for n 𝑁 𝑡 times, respectively.
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VII. C ONCLUSIONS
TM training design in [4] Proposed SP training design
Using SDP, the optimal superimposed training signal for spatially correlated MIMO-OFDM has been designed. Moreover, an approximate efficient design was proposed, which is given in a closed-form expression, for the general case of channel correlations. The designs of precoding matrices and power allocation to enhance the detection performance were also considered. Simulation results show that the proposed designs outperform the existing design. R EFERENCES
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of the optimal SP training at low and high SNRs, and worse than those of the SP training at average SNRs. However, it is interesting to observe from Fig. 4 that the BER obtained with the SP scheme is better than that with the TM scheme in [4] for all SNR levels. In particular, a significant gain of about 2.0 dB is achieved by the proposed design at the BER level of 10−5 .
[1] H. Minn and N. Al-Dhahir, “Optimal training signals for MIMO OFDM channel estimation,” IEEE Trans. Wireless Commun., vol. 5, pp. 1158– 1168, May 2006. [2] S. Ohno and G. B. Giannakis, “Optimal training and redundant precoding for block transmissions with application to wireless OFDM,” IEEE Trans. Commun., vol. 50, pp. 2113–2123, Dec. 2002. [3] J. H. Kotecha and A. M. Sayeed, “Transmit signal design for optimal estimation of correlated MIMO channels,” IEEE Trans. Signal Process., vol. 52, pp. 546–557, Feb. 2004. [4] H. Zhang, Y. G. Li, A. Reid, and J. Terry, “Optimum training symbol design for MIMO OFDM in correlated fading channels,” IEEE Trans. Wireless Commun., vol. 5, pp. 2343–2347, Sep. 2006. [5] V. Nguyen, H. D. Tuan, H. H. Nguyen, and N. N. Tran, “Optimal superimposed training design for spatially correlated fading MIMO channels,” IEEE Trans. Wireless Commun., vol. 7, pp. 3206–3217, Aug. 2008. [6] J. P. Kermoal, L. Schumacher, P. E. Mogensen, and K. I. Pedersen, “Experimental investigation of correlation properties of MIMO radio channels for indoor picocell scenario,” in Proc. IEEE VTC, pp. 14–21, Sep. 2000. [7] H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDMbased spatial multiplexing systems,” IEEE Trans. Commun., vol. 50, pp. 225–234, Feb. 2002. [8] H. Bolcskei, M. Borgmann, and A. J. Paulraj, “Impact of the propagation environment on the performance of space-frequency coded MIMOOFDM,” IEEE J. Sel Areas Commun., vol. 21, pp. 427–439, Apr. 2003. [9] M. Enescu, T. Roman, and V. Koivunen, “State-space approach to spatially correlated MIMO OFDM channel estimation,” Signal Processing, ScienceDirect, vol. 87, Mar. 2007. [10] N. N. Tran, D. H. Pham, and H. D. Tuan, “Affine precoding and decoding MIMO frequency-selective fading channel for estimation and source detection,” in Proc. IEEE ISSPIT’06, pp. 749–754,, Aug. 2006. [11] N. N. Tran, D. H. Pham, H. D. Tuan, and H. H. Nguyen, “Orthogonal affine precoding and decoding for channel estimation and source detection in MIMO frequency-selective fading channels,” IEEE Trans. Signal Process., vol. 57, pp. 1151–1162, Mar. 2009. [12] K. Werner, M. Jansson, and P. Stoica, “On estimation of covariance matrices with Kronecker product structure,” IEEE Trans. Signal Process., vol. 56, pp. 478–491, Feb. 2008. [13] S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. I— Estimation Theory. Prentice Hall PTR, 1993. [14] D. H. Pham, H. D. Tuan, B. N. Vo, and T. Q. Nguyen, “Jointly optimal precoding/postcoding for colored MIMO systems,” in Proc. ICASSP, pp. IV.745–IV.748, May 2006. [15] D. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Trans. Commun., vol. 48, pp. 502–513, Mar. 2002.
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Superimposed Training Designs for Spatially Correlated MIMO-OFDM Systems Nguyen N. Tran, Member, IEEE, Hoang D. Tuan, Member, IEEE, and Ha H. Nguyen, Senior Member, IEEE
Abstract—Only one asymptotic training design for a special case of channel correlation was proposed in the literature for spatially correlated multiple-input multiple-output with orthogonal frequency-division multiplexing (MIMO-OFDM) systems. To fill this gap, this letter applies tractable semi-definite programming (SDP) to obtain the optimal superimposed training signals for the general case of channel correlation. For a more efficient computation, two approximate designs are also proposed. Simulation results demonstrate the efficiency of our approach and its advantage over the asymptotic design. Index Terms—MIMO, OFDM, spatial correlation, channel estimation, training signal design, precoder design, superimposed training.
A
I. I NTRODUCTION
CCURATE channel estimation is a major challenge in the implementation of a multiple-input multiple-output with orthogonal frequency-division multiplexing (MIMOOFDM) system. Typically, training signals are used for estimating the unknown channel [1]–[5] and various designs of training signals have been proposed for OFDM and MIMOOFDM systems in the last decade. The channels between different pairs of transmit and receive antennas are often assumed to be uncorrelated, with a diagonal covariance matrix. However, this assumption is impractical for certain wireless environments [4], [6]–[9]. Apparently, only reference [4] studies the optimal training design and channel estimation for spatially correlated frequency-selective fading MIMO-OFDM. Under some special cases of correlation, [4] provides time-multiplexing (TM) (also known as block-type) training designs in the forms of asymptotic solutions at low and high signal-to-noise ratio (SNR) regimes. This letter proposes superimposed training signal designs for spatially correlated MIMO-OFDM in the general case of channel correlation and arbitrary SNR. In particular, we extend our previous results [5], [10], [11] to the more general and challenging case of spatially correlated MIMO-OFDM
Manuscript received May 28, 2008; revised April 19, 2009 and August 3, 2009; accepted October 1, 2009. The associate editor coordinating the review of this letter and approving it for publication was A. Hjørungnes. This work is supported by the Australian Research Council under grant ARC Discovery Project 0772548. N. N. Tran was with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, NSW 2052, Australia. He is now with the Department of Computer and Electronics Engineering, University of Nebraska- Lincoln, Omaha, NE 68182 USA (email: [email protected]). H. D. Tuan is with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]). H. H. Nguyen is with the Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Dr., Saskatoon, SK, Canada S7N 5A9 (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2010.03.080661
systems. As discussed in [5], [10], [11], superimposed training allows a more flexible preprocessing of the data symbols since they can be transmitted simultaneously with the training signals. This increased flexibility leads to an improvement of the overall system performance. Furthermore, since TM training can be considered as a particular case of SP training, our optimal training design is also novel within the class of TM training. The structure of the letter is as follows. After presenting the system model in Section II, we develop semi-definite programming (SDP) to design the optimal training signals in Section III. Then the approximated closed-form solutions are derived to minimize upper bounds of the channel minimum mean-square error (MMSE) in Section IV. A power allocation for symbol detection enhancement is presented in Section V. Simulation results to show the advantages of our approach are given in Section VI. Notation: Boldface upper and lower cases denote matrices and column vectors, respectively. Superscripts 𝑇 , 𝐻 and ∗ mean transposition, Hermitian adjoint and complex conjugate operators, respectively. ⊗ stands for Kronecker matrix product. I𝑛 and 0𝑛×𝑚 are the 𝑛× 𝑛 identity matrix and the 𝑛× 𝑚 zero matrix, respectively. For a vector x, diag{x} is the diagonal matrix with the components x𝑖 of x on its diagonal. For 𝑛 matrices X𝑖 (or X(𝑖)), 𝑖 = 1, . . . , 𝑛, diag[X𝑖 ]𝑖=1,...,𝑛 (or diag[X(𝑖)]𝑖=1,...,𝑛 ) forms a block diagonal matrix by putting X𝑖 (X(𝑖)) on the 𝑖th diagonal block. Likewise, [X𝑖𝑗 ] 𝑖=1,...,𝑛 𝑗=1,...,𝑚 forms a new matrix by putting matrix X𝑖𝑗 on the (𝑖, 𝑗)th block. The symbol 𝒰𝑡 (𝒟𝑡 , resp.) denotes the set of all 𝑡 × 𝑡 unitary matrices (diagonal matrices, resp.). The vectorization operator on a matrix to form a column vector by vertically stacking the matrix columns is denoted as vec(⋅). The expectation operation with respect to the random variable x is E{x}, and also Rx is the covariance matrix of x. For a scalar 𝑥, define 𝑥+ = max(𝑥, 0). A ≥ 0 (A > 0, resp.) means that A is a positive semi-definite (definite, resp.) Hermitian matrix, while tr{A} is its trace. II. S PATIALLY C ORRELATED MIMO-OFDM S YSTEM Consider a MIMO system with 𝑡 transmit and 𝑟 receive antennas. The channel is frequency-selective and the number of significant delay paths between any antenna pair is 𝐿. The channel is assumed to be unchanged during one block of OFDM symbols of duration 𝑀 , but may change independently from block to block (i.e., quasi-static or block fading). For Rayleigh fading, the channel gains are modeled as jointly Gaussian [4]. The MIMO channel is spatially correlated at both the transmitter and receiver but the path gains for different delays are uncorrelated. Let h𝑗𝑖 =
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[ℎ𝑗𝑖 (0), ℎ𝑗𝑖 (1), . . . , ℎ𝑗𝑖 (𝐿 − 1)]𝑇 denote the channel impulse response from the 𝑖th (𝑖 = 1, . . . , 𝑡) transmit antenna to the 𝑗th (𝑗 = 1, . . . , 𝑟) receive antenna. Accordingly, the channel gain matrix H(ℓ) of the ℓth path, ℓ = 0, . . . , 𝐿 − 1, can be represented as 1/2
𝑟×𝑡 . H(ℓ) := [ℎ𝑗𝑖 (ℓ)] 𝑗=1,...,𝑟 = Σ1/2 𝑟 (ℓ)H𝑤 (ℓ)Σ𝑡 (ℓ) ∈ ℂ 𝑖=1,...,𝑡 (1) Here Σ𝑡 (ℓ) and Σ𝑟 (ℓ) are 𝑡×𝑡 and 𝑟 ×𝑟 known and invertible covariance matrices (see, e.g. [12], for their identification), in which
Σ𝑡 (ℓ) = [𝑡𝑢𝑣 (ℓ)]𝑢,𝑣=1,...,𝑡 and Σ𝑟 (ℓ) = [𝑟𝑛𝑚 (ℓ)]𝑛,𝑚=1,...,𝑟 , (2) where 𝑡𝑢𝑢 (ℓ) = 𝑟𝑛𝑛 (ℓ) = 1 and 𝑡𝑢𝑣 (ℓ) = 𝑡∗𝑣𝑢 (ℓ) (𝑟𝑛𝑚 (ℓ) = ∗ 𝑟𝑚𝑛 (ℓ), resp.) reflects the correlated fading between the 𝑢th and the 𝑣th (𝑛th and 𝑚th, resp.) elements of the transmit (receive, resp.) antenna array at the ℓth delay path. The entries of the 𝑟 × 𝑡 unknown (to-be-estimated) matrix H𝑤 (ℓ) are independent and identically-distributed (i.i.d.) circularly symmetric complex Gaussian random variables of variance 𝜎ℎ2 , i.e., 𝒞𝒩 (0, 𝜎ℎ2 ). For convenience, normalize 𝜎ℎ2 = 1/𝐿. Let 𝑁 be the number of sub-carriers. As a standard operation in OFDM, a cyclic prefix (CP) of length 𝐿 is used to eliminate ISI between consecutive OFDM blocks of length 𝑁 + 𝐿. The [𝑁 × 𝑁 discrete Fourier transform (DFT) matrix ] −1 is F𝑁 = f0 f1 . . . f𝑁 −1 = √1𝑁 [e−𝑗2𝜋𝑝𝑘/𝑁 ]𝑁 𝑝,𝑘=0 . [ ] F𝐿 Partition F𝑁 as F𝑁 = , F𝐿 ∈ ℂ𝐿×𝑁 , F𝑁 −𝐿 ∈ F𝑁 −𝐿 ℂ(𝐿−𝑁 )×𝑁 , where ] [ (3) F𝐿 = f0𝐿 f1𝐿 . . . f(𝑁 −1)𝐿 ∈ ℂ𝐿×𝑁 [ 𝑇 ]𝑇 𝑇 𝑇 = ˜f0 ˜f1 . . . ˜f𝐿−1 (4) contains the first 𝐿 rows of F𝑁 and F𝐿 F𝐻 𝐿 = I𝐿 . Note that√, f𝑘𝐿 ∈ ℂ𝐿 , 𝑘 = 0, 1, . . . , 𝑁 − 1 and , ˜fℓ = (1/ 𝑁 )(1, e−𝚥2𝜋ℓ/𝑁 , . . . , e−𝚥2𝜋ℓ(𝑁 −1)/𝑁 ). Define 𝑀 = 𝐾 + 𝑞 with 𝑞 ≥ 𝑡. The MIMO equation corresponding to the 𝑘th subcarrier (subchannel) with 𝑀 OFDM blocks (each of them has length 𝑁 + 𝐿) is Y𝑘 = H𝑓 (𝑘)U(𝑘)+W(𝑘), U(𝑘) ∈ ℂ𝑡×𝑀 , W(𝑘) ∈ ℂ𝑟×𝑀 , (5) where 𝑘 = 0, 1, . . . , 𝑁 − 1, H𝑓 (𝑘) =
𝐿−1 ∑
H(ℓ)e−𝚥2𝜋ℓ𝑘/𝑁
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=
[ H(f𝑘𝐿 ⊗ I𝑡 ), H = H0
. . . H𝐿−1
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√ ˜ C(𝑘) ∈ ℂ𝑡×𝑞 and Q𝐸 ∈ ℂ𝑀×𝑞 . Specifically, P = 𝑀 𝐾 O(1 : 𝐾) ∈ ℂ𝐾×𝑀 , Q𝐸 = O𝐻 (𝐾 + 1 : 𝑀 ) ∈ ℂ𝑀×𝑞 with an arbitrary 𝑀 × 𝑀 orthogonal matrix O. Therefore, P is full rank (i.e., PP𝐻 is nonsingular), tr{PP𝐻 } = 𝑀 and PQ𝐸 = 0. Furthermore, by normalizing the total average transmit power to unity, the average training power is 𝜎𝑐2 = tr{C(𝑘)C𝐻 (𝑘)}/(𝑀 𝑡) = 1 − 𝜎𝑠2 , where 𝜎𝑠2 is the average data power. III. C HANNEL E STIMATION AND S UPERIMPOSED T RAINING D ESIGN WITH SDP To estimate H𝑓 (𝑘), 𝑘 = 0, 1, . . . , 𝑁 − 1 in (5), postmultiplying both sides of (5) by Q𝐸 gives ˜ ˜ Y𝑘 Q𝐸 = H𝑓 (𝑘)C(𝑘) + W(𝑘)Q𝐸 , C(𝑘) ∈ ℂ𝑡×𝑞 .
(9)
Now define
] [ := [Y0 Q𝐸 Y1 Q𝐸 . . . Y𝑁 −1 Q𝐸 ]∈ ℂ𝑟×(𝑁 𝑞) , := [H𝑓 (0) H𝑓 (1) . . . H𝑓 (𝑁 − ]1) ∈ ℂ𝑟×(𝑁 𝑡) , := W(0)Q𝐸 . . . W(𝑁 − 1)Q𝐸 ∈ ℂ𝑟×(𝑁 𝑞) . (10) It follows from(7) and (9) that ] [ H𝑓 = H f0𝐿 ⊗ I𝑡 f1𝐿 ⊗ I𝑡 . . . f(𝑁 −1)𝐿 ⊗ I𝑡 Y H𝑓 W
[
=
H(F𝐿 ⊗ I𝑡 ),
˜ Y = H𝑓 (0)C(0)
(11) ] ˜ . . . H𝑓 (𝑁 − 1)C(𝑁 − 1) +W. (12)
For 𝑘 = 0, 1, . . . , 𝑁 − 1, using (7) and (11) to define h𝑓 (𝑘) := := h𝑓 =
𝑇 vec(H𝑓 (𝑘)) = (f𝑘𝐿 ⊗ I𝑟𝑡 )h, h := vec(H), [ 𝑇 ]𝑇 vec(H𝑓 ) = h𝑓 (0) h𝑇𝑓 (1) . . . h𝑇𝑓 (𝑁 − 1) (F𝑇𝐿 ⊗ I𝑟𝑡 )h.
Furthermore, as Rh = diag[Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ)]ℓ=0,1,...,𝐿−1 , the covariance matrices R𝑓 (𝑘, 𝑖) := E{h𝑓 (𝑘)h𝐻 𝑓 (𝑖)}, 𝑘, 𝑖 = 0, 1, . . . , 𝑁 − 1 and R𝑓 := Rh𝑓 can be expressed by
=
𝑇 𝑇𝐻 R𝑓 (𝑘, 𝑖) = (f𝑘𝐿 ⊗ I𝑟𝑡 )diag[Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ)]ℓ (f𝑖𝐿 ⊗ I𝑟𝑡 ) 𝐿−1 ∑ (f𝑘𝐿 (ℓ)¯f𝑖𝐿 (ℓ))Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ), ℓ = 0, . . . , 𝐿 − 1. (13) ℓ=0
= =
R𝑓 = [R(𝑘, 𝑖)]𝑘,𝑖=0,1,...,𝑁 −1 (14) 𝑇 𝑇𝐻 (F𝐿 ⊗ I𝑟𝑡 )diag[Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ)]ℓ=0,1,...,𝐿−1 (F𝐿 ⊗ I𝑟𝑡 ) 𝐿−1 ∑ (˜fℓ𝑇 ⊗ I𝑟𝑡 )(Σ𝑡 (ℓ) ⊗ Σ𝑟 (ℓ))(˜fℓ𝑇 𝐻 ⊗ I𝑟𝑡 ). (15) ℓ=0
(7)
is the frequency response of the 𝑘th sub-channel, and W(𝑘) is the additive white Gaussian noise (AWGN) with zero mean 2 . With super-imposed training, the matrix and variance 𝜎𝑤 U(𝑘) representing the transmitted signals has the following form: U(𝑘) = S(𝑘)P + C(𝑘), 𝑘 = 0, 1, . . . , 𝑁 − 1.
877
(8)
In (8), S(𝑘) ∈ ℂ𝑡×𝐾 contains the source data symbols whose elements are modeled as zero mean i.i.d. random variables with variance 𝜎𝑠2 , P ∈ ℂ𝐾×𝑀 is the precoder matrix, and 𝐻 𝑡×𝑀 ˜ C(𝑘) = C(𝑘)Q is the training matrix, where 𝐸 ∈ ℂ
] [ 𝑇 𝑇 ˜𝑇 ˜ ˜𝑇 Here [ 𝑇 we have𝑇used F𝐿 ⊗ I𝑟𝑡 =𝑇 f0 f1] . . . f𝐿−1 ⊗ I𝑟𝑡 = ˜f ⊗ I𝑟𝑡 ˜f ⊗ I𝑟𝑡 . . . ˜f 0 1 𝐿−1 ⊗ I𝑟𝑡 . Using (9), for 𝑘 = 0, 1, . . . , 𝑁 − 1, one has vec(Y𝑘 Q𝐸 ) =
˜ 𝑇 (𝑘) ⊗ I𝑟 )h𝑓 (𝑘) + w(𝑘), (C
(16)
where w(𝑘) = vec(W(𝑘)Q𝐸 ). For using MMSE estimator [13] to estimate the frequency response, we rewrite (12) into the vector form. With y = vec(Y), vectorizing two sides of (12) gives ¯ 𝑓 = diag[C ˜ 𝑇 (𝑘) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 , (17) ¯ 𝑓 h𝑓 + w, C y=C w = (w𝑇 (0), . . . , w𝑇 (𝑁 − 1))𝑇
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¯ 𝐻C ¯ 𝑓 = diag[(C(𝑘) ˜ C ˜ 𝐻 (𝑘)) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 . with C 𝑓 ˆ 𝑓 = (R−1 + The MMSE estimator of h𝑓 is h h𝑓 1 ¯𝐻 ¯ −1 1 ¯ with the corresponding MMSE of 2 C𝑓 C𝑓 ) 2 C𝑓 y 𝜎𝑤 𝜎𝑤 1 ¯𝐻 ¯ −1 tr{(R−1 } [13]. With the variable change 2 C𝑓 C𝑓 ) h𝑓 + 𝜎𝑤 ˜ C ˜ 𝐻 (𝑘) ∈ ℂ𝑡×𝑡 , the optimal training can be X(𝑘) = C(𝑘) solved by SDP as follows: min tr{Z} subject to
[
(18)
Z,X(𝑘)
Z
R h𝑓
R h𝑓 + R h𝑓 and
𝑁 −1 ∑
(
R h𝑓 1 2 diag[X(𝑘) 𝜎𝑤
) ⊗ I𝑟 ]𝑘 Rh𝑓
tr{X(𝑘)} = 𝑃𝑇 , 𝑃𝑇 =
] ≥ 0, (19)
𝑁 𝑀 𝑡𝜎𝑐2 .
(20)
𝑘=0
min tr{Z} subject to (20), and
Z,X(𝑘)
Z ⎢ ⎣ Rh
Let Y𝑘 (𝑖) denote the 𝑖th row of Y𝑘 Q𝐸 , H𝑓 𝑖 (𝑘) and W𝑖 (𝑘) the 𝑖th rows, respectively, of H𝑓 (𝑘) and W(𝑘)Q𝐸 in Equation (9). Then rewrite (9) as Y𝑘 (𝑖) = H𝑓 𝑖 S(𝑘) + W𝑖 (𝑘) ⇔ Y𝑘𝑇 (𝑖) = S𝑇 (𝑘)H𝑇𝑓𝑖 (𝑘) + W𝑖𝑇 (𝑘).
(22) (23)
= [Let H(𝑖) be the 𝑖th row] of H, i.e., H(𝑖) H0 (𝑖) H1 (𝑖) . . . H𝐿−1 (𝑖) with Hℓ (𝑖) being the 𝑖th row of Hℓ . It is not difficult to see that Rvec(H𝑇ℓ ) = Σ𝑟 (ℓ) ⊗ Σ𝑡 (ℓ) and the 𝑖th diagonal sub(𝑖)
matrix of Rvec(H𝑇ℓ ) is RH𝑇ℓ (𝑖) = Σ𝑟 (ℓ)Σ𝑡 (ℓ). Let RH𝑇 (𝑖) = (𝑖)
diag{RH𝑇ℓ (𝑖) }ℓ=0,1,...,𝐿−1 = diag{Σ𝑟 (ℓ)Σ𝑡 (ℓ)}ℓ=0,1,...,𝐿−1 , (𝑖)
˜ Once the solution X(𝑘) is found from (18)-(20), C(𝑘) is ˜ obtained by C(𝑘) = [X1/2 (𝑘) 0𝑡×(𝑞−𝑡) ]. Observe that as far as 𝑞 ≥ 𝑡, the MMSE does not depend on 𝑡. However, increasing 𝑡 leads to higher effective SNR, hence a better detection performance. Remark: The problem can be also solved in time-domain using SDP. Rewrite (12) as ] [ ˜ ˜ Y = H f0𝐿 ⊗ C(0) . . . f(𝑁 −1)𝐿 ⊗ C(𝑁 − 1) + W. ¯ ¯ Then y = Ch + w, where by (16), C = 𝑇 𝐻 ˜ ¯ diag[C(𝑘) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 (F𝐿 ⊗ I𝑟𝑡 ), and C C = (F𝐿 ⊗ ˜ C ˜ 𝐻 (𝑘) ⊗ I𝑟 ]𝑘=0,1,...,𝑁 −1 (F𝐻 × I𝑟𝑡 ). Note that I𝑟𝑡 )diag[C(𝑘) 𝐿 𝐻 Rh := E{hh } = 𝜎ℎ2 diag[Σ𝑡 (ℓ)⊗Σ𝑟 (ℓ)]ℓ=0,1,...,𝐿−1 and the ˆ = (R−1 + 12 C ¯ 𝐻 C) ¯ −1 12 C ¯ 𝐻y MMSE estimator of h is h h 𝜎𝑤 𝜎𝑤 1 ¯ 𝐻 ¯ −1 C) }. with the corresponding MMSE tr{(R−1 2 C h + 𝜎𝑤 Thus, by the variable change X(𝑘) = C(𝑘)C𝐻 (𝑘), 𝑘 = 0, 1, . . . , 𝑁 − 1, the optimal training in time-domain can also be solved by another SDP as follows:
⎡
B. General Case
where Σ𝑡 (ℓ) is the 𝑖th diagonal entry of Σ𝑡 (ℓ). 𝑇 ⊗ I𝑡 )H𝑇 (𝑖) it is true that Therefore, with H𝑇𝑓𝑖 (𝑘) = (f𝑘𝐿 𝑇 𝑇𝐻 RH𝑇𝑓𝑖 (𝑘) = (f𝑘𝐿 ⊗ I𝑡 )RH𝑇 (𝑖) (f𝑘𝐿 ⊗ I𝑡 ). ˆ 𝑇 (𝑘) of H𝑇𝑓𝑖 (𝑘) For the MMSE estimator H 𝑓𝑖 using only (23), the resulting MMSE is {( )−1 } −1 𝑇𝐻 𝑇 ˜ ˜ R 𝑇 tr + C (𝑘)C (𝑘) . Hence the total H𝑓 𝑖 (𝑘)
MMSE is
𝑟 𝑁 −1 ∑ ∑ 𝑖=1 𝑘=0
{( R−1 tr H𝑇
𝑓𝑖
)−1 } + X(𝑘) , which can (𝑘)
˜ C ˜ 𝐻 (𝑘) based on the be minimized over X(𝑘) = C(𝑘) water-filling type algorithm of [14]. V. S YMBOL DETECTION ENHANCEMENT For symbol detection, post-multiplying both sides of (5) by Q𝐷 = P𝐻 (PP𝐻 )−1 for 𝑘 = 0, 1, . . . , 𝑁 − 1 gives Y𝑘 Q𝐷 = H𝑓 (𝑘)S(𝑘) + W(𝑘)Q𝐷 ˆ 𝑓 (𝑘)S(𝑘) + H ˜ 𝑓 (𝑘)S(𝑘) + W(𝑘)Q𝐷 . =H
(24)
˜ 𝑓 (𝑘) = H𝑓 (𝑘) − H ˆ 𝑓 (𝑘) is the channel estimation Here H 𝑡×𝐾 , W(𝑘) ∈ ℂ𝑟×𝑀 . Similar to [5], error and S(𝑘) ∈ ℂ ( 𝑁 −1 Rh ) ⎥ the effective SNR of the input/output model in (24) can be ∑ ⎦ ≥ 0. 𝐻 Rh + Rh 𝜎12 (f𝑘𝐿 f𝑘𝐿 ) ⊗ X(𝑘) ⊗ I𝑟 Rh calculated as 𝑤 𝑘=0 ˆ 𝑓 (𝑘)S(𝑘)∣∣2 ) E(∣∣H SNReff = . (25) ˜ 𝑓 (𝑘)S(𝑘) + W(𝑘)Q𝐷 ∣∣2 ) IV. A PPROXIMATE T RAINING D ESIGN E(∣∣H ⎤
A. Special Case For the special case where the receive correlations at different delay paths are the same, except a scaling factor 𝜎𝑙2 , i.e., Σ𝑟 (ℓ) = 𝜎ℓ Σ𝑟 , it follows that R𝑓 (𝑘, 𝑖) = Σ𝑡 (𝑘, 𝑖) ⊗ Σ𝑟 𝐿−1 ∑ for Σ𝑡 (𝑘, 𝑖) = 𝜎ℓ (f𝑘𝐿 (ℓ)¯f𝑖𝐿 (ℓ))Σ𝑡 (ℓ). Thus, for MMSE ℓ=0
ˆ 𝑓 (𝑘) of h𝑓 (𝑘) using only equation (16), the total estimator h ˜ C ˜ 𝐻 (𝑘): MMSE is the following function in X(𝑘) = C(𝑘) { ( )−1 } 𝑁 −1 ∑ 1 ˜ ˜𝐻 tr R−1 = h𝑓 (𝑘) + 𝜎 2 (C(𝑘)C (𝑘)) ⊗ I𝑟 𝑤 𝑘=0 {( )−1 } 𝑁 −1 ∑ 1 −1 −1 tr Σ𝑡 (𝑘, 𝑘) ⊗ Σ𝑟 + 2 (X(𝑘) ⊗ I𝑟 ,(21) 𝜎𝑤 𝑘=0
which can be minimized over X(𝑘) based on the iterative bisection procedure (IBP) of [5].
By maximizing the average effective SNR of the whole MIMO-OFDM system having 𝑁 sub-carriers, the following sub-optimal power allocation is obtained with 𝛾 = 2 𝐾𝑁 𝑟𝜎𝑤 /𝑀 : √ 2 2 2 (𝐿𝜎 2 + 𝑁 𝑀 )(𝑁 𝑟𝑡 + 𝛾) + 𝐿𝛾𝜎𝑤 − 𝐿𝛾𝜎𝑤 𝑁 𝐿𝑟𝑡𝜎𝑤 𝑤 2 . 𝜎𝑐 = 2 𝑁 (𝐿𝑟𝑡𝜎𝑤 − 𝑀 𝛾) Moreover, instead of S(𝑘), one can apply a “non-redundancy precoder” F ∈ ℂ𝑡×𝑡 and replace it by FS(𝑘) to even further improve the detection performance. Let Q𝑘 = 𝑀 𝐻 H𝑓 (𝑘)}. Then the optimization problem to 2 E{H𝑓 (𝑘) 𝐾𝜎𝑤 design F is {( )−1 } 1 𝐻 min tr 𝐼 + F Q𝑘 F : tr{FF𝐻 } ≤ 𝑡. (26) F 𝜎𝑠2 The above problem can be solved as in [14, Theorem 1].
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0 Approximate, general case Approximate, special case Optimal, with SDP LS, equal−powered training
MSE (dB)
−5
−10
−15
−20
−25 −15
−10
−5
0
5
10
15
SNR (dB) Fig. 1. Comparison of channel MSE of 2 × 2 MIMO-OFDM with high MIMO correlations. 5 Approximate, general case Approximate, special case Optimal, with SDP LS, equal−powered training
0
MSE (dB)
−5
−10
−15
−20
−25 −15
−10
−5
0
5
10
15
SNR (dB) Fig. 2. Comparison of channel MSE of 2 × 2 MIMO-OFDM with average MIMO correlations.
VI. S IMULATION R ESULTS A. Estimation Performance The one-ring model in [15, E.q. (6)] is used to generate the elements of the covariance (matrices Σ𝑟 (𝑙) and ) Σ𝑡 (𝑙). Specif2𝜋 Δ ically, Σ (𝑙)(𝑛, 𝑚) ≈ 𝐽 𝑑 ∣𝑚 − 𝑛∣ , Σ𝑟 (𝑙)(𝑖, 𝑗) ≈ 𝑡 0 𝑡𝑙 𝑡 𝜆 ) ( 𝐽0 Δ𝑟𝑙 2𝜋 𝑑 ∣𝑖 − 𝑗∣ , where Δ and Δ are the angle spreads 𝑟 𝑡𝑙 𝑟𝑙 𝜆 (in radian) of the 𝑙th path at the transmitter and the receiver, respectively; 𝑑𝑡 and 𝑑𝑟 are the spacings of the transmit and receive antenna arrays, respectively; 𝜆 is the wavelength and 𝐽0 (⋅) is the zeroth-order Bessel function of the first kind. Since the average transmitted power, including the training and data powers, is normalized to unity, the received SNR in dB is 2 . The system parameters are defined as SNR = −10log10 𝜎𝑤 chosen as [1, Table I] with 𝑁 = 8 sub-carriers and 𝐿 = 3 taps. Figs. 1 and 2 show the MSE of the channel impulse ˆ 2 }, for 2×2 MIMO-OFDM system having response, E{∥h− h∥ 𝐾 = 8, 𝑞 = 𝑡, and 𝑀 = 𝐾 + 𝑞. In order to apply the approximate design for the special case of receive correlation
879
matrices, the parameters are chosen such that they result in the same covariance matrix for all the delay paths at the receive antennas. In Fig. 1, they are Δ𝑡𝑙 = (𝑙/2 + 2)o , Δ𝑟𝑙 = 1o , 𝑑𝑡 = 0.5𝜆 and 𝑑𝑟 = 0.3𝜆, which yield very high correlations at both sides of the transmission. In Fig. 2, we choose average correlations with Δ𝑡𝑙 = 3(𝑙 + 2)o , Δ𝑟𝑙 = 3o , 𝑑𝑡 = 0.5𝜆 and 𝑑𝑟 = 0.3𝜆. It can be seen from Figs. 1 and 2 that, at low SNR three proposed designs (by SDP, IBP and water-filling type) yield the same estimation performance. At average SNR, the optimal training design based on SDP produces the lowest MSE. The approximate design using IBP for the special case performs as well as the SDP based design and outperforms the approximate design (for the general case). At high SNR, again all the three proposed designs perform almost the same, and their performances are better than that of the design using least square estimator (which actually ignores the MIMO correlation and gives an equal-powered training). As expected, performance difference among the three designs at medium SNR becomes larger the antenna correlation is larger (comparing between Figs. 1 and 2). Also plotted in Figs. 1 and 2 are the MSE obtained with the least-square (LS) solution. The LS solution ignores the channel and noise covariances and it is simple to show that the optimal training in this case is exactly the equalpowered training, and hence suboptimal as can be clearly seen from the two figures. B. Performance Comparison with the Design in [4] Comparisons of both channel estimation and BER performance between our optimal SP training design and the TM training design in [4] (which is for the cases of high or low SNR) are presented in Figs. 3 and 4, where the system is chosen as in [4], i.e., a 2 × 2 MIMO-OFDM system, 𝑁 = 32 sub-channels and 𝐿 = 2 taps. The exponential correlation model in [4] is also chosen where the angle spread for all taps at both transmitter and receiver is 𝜎 = 3.6∘ , the relative antenna space is △ = 1, and the average angle of departure is 𝜃¯𝑡0 = 𝜃¯𝑡1 = 7.2∘ and the average angles of arrival are 𝜃¯𝑟0 = 290.3∘, 𝜃¯𝑟1 = 332.3∘ , which give the identical transmit correlations as required by [4]. The full-rate Alamouti code together with the maximum likelihood (ML) decoding is incorporated. QPSK modulation with Gray mapping is considered. The whole matrix of the source symbols has size 𝑁 𝑡 × 𝐾 where√𝐾 = 192.√In our proposed design, the source symbols {± 𝜎𝑠2 /2 ± 𝑗 𝜎𝑠2 /2} are precoded by the proposed precoders, the precoded data matrix has size 𝑞 × 𝑀 with 𝑞 = 𝑡, 𝑀 = 𝐾 + 𝑞. For the TM system considered in [4], 𝑁 𝑡 training vectors1 are sent first in 𝑁 𝑡 time slots to estimate the fading channel and the average source symbol power is normalized to unity. The total training power 𝑃𝑇 = 𝑀 𝑁 𝑡𝜎𝑐2 as well as the LMMSE estimation are used in both designs. Fig. 3 shows that when having the same total training power, the MSE of the TM training designed in [4] is the same as that 1 The 𝑁 𝑡 training vectors are formed by horizontally stacking the training vector √designed in [4] 𝑁 𝑡 times. Therefore, [4, Equation (5)] is rewritten as 𝜌 ¯ ¯ = [A𝑇 , . . . , A𝑇 ]𝑇 , and ¯ ¯ , where ¯ r= Ah + n r = [r𝑇 , . . . , r𝑇 ]𝑇 , A 𝑀 𝑡
¯ = [n𝑇 , . . . , n𝑇 ]𝑇 are constructed by vertically stacking r, A and n for n 𝑁 𝑡 times, respectively.
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−15
VII. C ONCLUSIONS
TM training design in [4] Proposed SP training design
Using SDP, the optimal superimposed training signal for spatially correlated MIMO-OFDM has been designed. Moreover, an approximate efficient design was proposed, which is given in a closed-form expression, for the general case of channel correlations. The designs of precoding matrices and power allocation to enhance the detection performance were also considered. Simulation results show that the proposed designs outperform the existing design. R EFERENCES
MSE (dB)
−20
−25
−30
−35
−40
0
5
10
15
20
25
SNR (dB)
Fig. 3. Comparison of channel MSE between the proposed design and the design in [4]. −1
10
TM training design [4] Proposed SP training design −2
10
−3
BER
10
−4
10
−5
10
−6
10
−7
10
0
5
10
15
20
25
SNR (dB)
Fig. 4. Comparison of BER performance between the proposed design and the design in [4].
of the optimal SP training at low and high SNRs, and worse than those of the SP training at average SNRs. However, it is interesting to observe from Fig. 4 that the BER obtained with the SP scheme is better than that with the TM scheme in [4] for all SNR levels. In particular, a significant gain of about 2.0 dB is achieved by the proposed design at the BER level of 10−5 .
[1] H. Minn and N. Al-Dhahir, “Optimal training signals for MIMO OFDM channel estimation,” IEEE Trans. Wireless Commun., vol. 5, pp. 1158– 1168, May 2006. [2] S. Ohno and G. B. Giannakis, “Optimal training and redundant precoding for block transmissions with application to wireless OFDM,” IEEE Trans. Commun., vol. 50, pp. 2113–2123, Dec. 2002. [3] J. H. Kotecha and A. M. Sayeed, “Transmit signal design for optimal estimation of correlated MIMO channels,” IEEE Trans. Signal Process., vol. 52, pp. 546–557, Feb. 2004. [4] H. Zhang, Y. G. Li, A. Reid, and J. Terry, “Optimum training symbol design for MIMO OFDM in correlated fading channels,” IEEE Trans. Wireless Commun., vol. 5, pp. 2343–2347, Sep. 2006. [5] V. Nguyen, H. D. Tuan, H. H. Nguyen, and N. N. Tran, “Optimal superimposed training design for spatially correlated fading MIMO channels,” IEEE Trans. Wireless Commun., vol. 7, pp. 3206–3217, Aug. 2008. [6] J. P. Kermoal, L. Schumacher, P. E. Mogensen, and K. I. Pedersen, “Experimental investigation of correlation properties of MIMO radio channels for indoor picocell scenario,” in Proc. IEEE VTC, pp. 14–21, Sep. 2000. [7] H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDMbased spatial multiplexing systems,” IEEE Trans. Commun., vol. 50, pp. 225–234, Feb. 2002. [8] H. Bolcskei, M. Borgmann, and A. J. Paulraj, “Impact of the propagation environment on the performance of space-frequency coded MIMOOFDM,” IEEE J. Sel Areas Commun., vol. 21, pp. 427–439, Apr. 2003. [9] M. Enescu, T. Roman, and V. Koivunen, “State-space approach to spatially correlated MIMO OFDM channel estimation,” Signal Processing, ScienceDirect, vol. 87, Mar. 2007. [10] N. N. Tran, D. H. Pham, and H. D. Tuan, “Affine precoding and decoding MIMO frequency-selective fading channel for estimation and source detection,” in Proc. IEEE ISSPIT’06, pp. 749–754,, Aug. 2006. [11] N. N. Tran, D. H. Pham, H. D. Tuan, and H. H. Nguyen, “Orthogonal affine precoding and decoding for channel estimation and source detection in MIMO frequency-selective fading channels,” IEEE Trans. Signal Process., vol. 57, pp. 1151–1162, Mar. 2009. [12] K. Werner, M. Jansson, and P. Stoica, “On estimation of covariance matrices with Kronecker product structure,” IEEE Trans. Signal Process., vol. 56, pp. 478–491, Feb. 2008. [13] S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. I— Estimation Theory. Prentice Hall PTR, 1993. [14] D. H. Pham, H. D. Tuan, B. N. Vo, and T. Q. Nguyen, “Jointly optimal precoding/postcoding for colored MIMO systems,” in Proc. ICASSP, pp. IV.745–IV.748, May 2006. [15] D. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Trans. Commun., vol. 48, pp. 502–513, Mar. 2002.
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