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Multi-Stage Beamforming for Coded OFDM with Multiple Transmit and Multiple Receive Antennas Shaohua (Steven) Li, Defeng (David) Huang, Member, IEEE, K. B. Letaief, Fellow, IEEE, and Zucheng Zhou
Abstract— Conventionally, subcarrier based beamforming was used in an OFDM (Orthogonal Frequency Division Multiplexing) system under MIMO (Multiple-Input and MultipleOutput) channels to achieve optimal performance, where many DFT/IDFT (Discrete Fourier Transform/Inverse DFT) blocks, each corresponding to one antenna, are required. To reduce the complexity induced by the DFT/IDFT processing, symbol based beamforming was recently proposed with only one IDFT block at the transmitter and one DFT block at the receiver. However, this approach results in a significant performance loss especially when the number of distinct paths in the channel is large. In this paper, we propose a multi-stage beamforming (MSB) scheme for MIMO-OFDM systems employing both the principles of subcarrier level beamforming and symbol level beamforming to effectively tradeoff system performance and complexity. Using the proposed MSB scheme and with a little complexity increase compared with the symbol based beamforming, system performance can be significantly improved. Under some channel conditions, the proposed MSB scheme can achieve a performance close to the optimal subcarrier based beamforming scheme but with much lower complexity. For the MSB scheme, we propose an iterative algorithm to jointly optimize the symbol level and subcarrier level weighting coefficients. To reduce the complexity of the joint weighting coefficients calculation algorithm, we propose a reduced complexity algorithm to first obtain the symbol level weighting coefficients and then the subcarrier level weighting coefficients. Simulation results show that good performance can be achieved using the proposed weighting coefficients calculation algorithms with reasonable complexity. Index Terms— OFDM, Multiple-Input and Multiple-Output (MIMO), beamforming.
I. I NTRODUCTION RTHOGONAL frequency division multiplexing (OFDM) is a promising technology for high data rate wideband wireless communications. By splitting a broadband channel into multiple narrow ones, OFDM is robust to frequency selective fading and narrow band interference. As a result, OFDM has been adopted as the basis of
O
Manuscript received April 11, 2005; revised November 4, 2005; accepted June 9, 2006. The associate editor coordinating the review of this paper and approving it for publication is M. Saquib. This paper was presented in part at the 48th IEEE Global Telecommunications Conference, St Louis, USA, November 27-December 1, 2005. S. Li is with the Department of Electronic Engineering, Tsinghua University, Beijing, China (e-mail:
[email protected]). D. Huang is with the School of Electrical, Electronic and Computer Engineering, The University of Western Australia, Crawley, WA 6009, Australia (e-mail:
[email protected]). K. B. Letaief is with the Center for Wireless Information Technology, Electronic and Computer Engineering Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail:
[email protected]). Z. Zhou is with the Department of Electronic Engineering, Tsinghua University, Beijing, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2007.05251.
several broadband wireless communications standards such as DVB-T, IEEE 802.11a, and ETSI-BRAN HIPERLAN/2. With channel coding and interleaving, frequency diversity can be achieved in OFDM to drastically improve system performance. Multiple transmit and multiple receive antennas can also be used with OFDM to achieve space diversity, thereby, further improving system performance. For an OFDM system under MIMO (Multiple-Input and Multiple-Output) channels, frequency-domain beamforming was traditionally used [1]-[5] to achieve optimal performance, where the transmit and receive signals were handled on a subcarrier-by-subcarrier basis. In the subcarrier based beamforming, the same number of DFT/IDFT (Discrete Fourier Transform/Inverse DFT) processors as the number of receive/transmit antennas is required. Since the complexity of DFT/IDFT is a major concern for OFDM implementation [6, 7], the complexity of the subcarrier based beamforming method is huge. To reduce the number of DFT blocks required for an OFDM system with subcarrier based beamforming, some techniques based on time-domain processing [8]-[16] have been recently proposed. For example, for an OFDM system under SIMO (Single-Input and Multiple-Output) channels, in [11], time domain weighting and combining is first performed followed by one single DFT processing. This work was extended by [15] to the MIMO case using symbol based beamforming, where only one IDFT block at the transmitter and one DFT block at the receiver are used. Although system complexity can be reduced by the schemes proposed in [11] and [15], system performance degrades significantly especially when the number of distinct paths in the channel is large [15]. For an OFDM system under SIMO channels, [9] and [16] proposed eigen-analysis based pre-DFT processing techniques to improve system performance by employing more DFT blocks at the receiver. In this paper, we propose a multi-stage beamforming (MSB) scheme for MIMO-OFDM systems, where both the principle of subcarrier based beamforming and that of symbol based beamforming are employed. In contrast to [15], where only one DFT and one IDFT are employed, the number of DFT/IDFT blocks required can be very flexible (i.e., any number from one to the number of transmit antennas or the number of receive antennas) when the MSB scheme is used. Simulation results show that by increasing the number of DFT/IDFT blocks compared with the symbol based beamforming scheme proposed in [15], system performance can be significantly improved. When the number of distinct paths in the channel is small, a performance that is close to the optimal (i.e., the performance with the subcarrier based beamforming
c 2007 IEEE 1536-1276/07$25.00
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only) can be achieved using the proposed MSB scheme but with much lower complexity. One important issue in the proposed scheme is the calculation of the weighting coefficients for both the subcarrier level and symbol level beamforming. In the proposed MSB scheme, three weighting matrices are required to be optimized, while in [1] and [15], only two vectors are required. To obtain the three weighting matrices, we propose an iterative algorithm based on the maximum average SNR (Signal-to-Noise Ratio) criterion. It will be shown that the maximum average SNR criterion is equivalent to the maximum pair-wise codeword distance criterion [15]. But with the maximum average SNR criterion, the analysis becomes much simpler. In the iterative algorithm, the subcarrier level and the symbol level weighting coefficients are jointly optimized. As a result, the complexity of the iterative algorithm is huge especially when the number of subcarriers is large. To further reduce the complexity of the iterative weighting coefficients calculation algorithm, we propose a reduced complexity algorithm by calculating the symbol level weighting coefficients independent of the subcarrier level weighting coefficients calculation. We will show that by using the reduced complexity algorithm, the complexity of the weighting coefficients calculation can be significantly reduced but good performance can still be achieved. This paper is organized as follows. In Section II, the proposed MSB scheme for MIMO-OFDM systems is described. The iterative algorithm to obtain the symbol-level weighting coefficients and the subcarrier-level weighting coefficients is introduced in Section III. The reduced complexity weighting coefficients calculation algorithm is presented in Section IV. Simulation results are given in Sections V. Finally, Section VI concludes our work. Throughout this paper, the following notations will be used. IK shall denote a K × K identity matrix; 0K×P denotes a K × P all-zero matrix; (·)∗ , (·)T , and (·)H denote conjugate, transpose and Hermitian transpose, respectively. Likewise, diag(x) denotes a diagonal matrix with x on its diagonal; E( · ) denotes the expectation of ( · ); tr(·) denotes the trace of matrix ( · ); λi (·) denotes the ith largest eigenvalue of matrix ( · ) , and λmax (·) denotes the dominant eigenvalue n of the corresponding matrix. Finally, ⊕ Ai denotes a block i=0
diagonal matrix with the ith element on its diagonal given by Ai (i = 1, 2, · · · , n); and ⊗ denotes the Kronecker product.
is then given by s(l,p) =
N −1 1 u(p,n) cn ej2πln/N , N n=0
− Ng ≤ l < N, p= 1, · · · , P
(1)
where u(p,n) is the weighting coefficient of the nth subcarrier at the pth branch, Ng is the number of samples in the guard interval and j 2 = −1. We assume that (Ng + 1) < N to keep high transmission efficiency. Note that since a P -branch subcarrier based beamforming is employed, P IDFT blocks are required at the transmitter. As a result, compared to the case where only the subcarrier based beamforming is used [1] with F IDFT blocks,1 the number of IDFT blocks employed in this case can be reduced. The signals at the output of the IDFT processing are weighted based upon the principle of symbol based beamforming. For the pth branch, the signal at the lth sample and the f th transmit antenna is given by (f )
s(l,p) = ψ(f,p) s(l,p) , f = 1, · · · , F p = 1, · · · , P
(2)
where ψ(f,p) is the weighting coefficient at the f th transmit antenna of the pth branch. The output of the P branches is combined together using a combiner. In the symbol based beamforming scheme [15], only one branch is used and no combiner is required. Therefore, the complexity of the scheme proposed in [15] is lower compared with the MSB scheme. However, since subcarrier based beamforming is not used in [15], all subcarriers are treated equally without the consideration of the channel variations of different subcarriers. This results in considerable performance loss compared with the proposed MSB scheme. At the output of the combiner, the transmitted signal at the f th transmit antenna is then given by (f ) sl
=
P p=1
(f )
s(l,p) , f = 1, · · · , F.
(3)
At the receiver, the received signal at the mth receive antenna is given by (m)
rl
=
F
(m,f )
hl
(f )
∗ sl
(m)
+ zl
(4)
f =1 (m,f )
II. S YSTEM M ODEL The proposed MSB scheme for a MIMO-OFDM system is as shown in Fig. 1, where coding and interleaving are employed to achieve frequency domain diversity. In the system, there are F transmit antennas and M receive antennas. At the transmitter, after coding, interleaving and mapping, a data stream is first converted into N sub-data streams. Each subdata stream is also called a subcarrier. We assume that the data symbol at the nth subcarrier is cn . For each subcarrier, we perform a P -branch subcarrier based beamforming. After the IDFT processing, the signal at the lth sample and pth branch
where * denotes the convolution product, hl denotes the channel impulse response (CIR) between the f th transmit (m) denotes the antenna and the mth receive antenna, and zl additive white Gaussian noise (AWGN) component at the mth (m,f ) receive antenna. Here, we assume that hl has a nonzero value only for the duration 0 ≤ l < K, where K is the maximum lag of the CIR. At the receiver, the symbol based receive beamforming is first employed. The M data streams from the output of the M receive antennas are weighted and then combined to form Q 1 In [1], only subcarrier based beamforming is employed. Therefore, all processing are required to be conducted in the frequency domain. To achieve this task, multiple IDFT/DFT blocks, each corresponding to one transmit/receive antenna, are required to be used.
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Symbol-based transmit beamforming (1)
S(l,1) (2)
S(l,1)
S(l,1)
Add guard interval
IDFT
(F)
S(l,1) #F
di
ψ (1,1) ψ(2,1) ... ψ (F,1)
Subcarrier based transmit beamforming
Coding and interleaving
#2
#1 (F) S(l,P)
(2)
S(l, P)
Add guard interval
IDFT
S(l,P) (1)
S(l,P) Combiner
ψ(1,P) ψ(2,P)
...
ψ(F, P)
(a)
#1 (1)
r´l #2
r(l,1)
(2)
r´l #M
Remove guard interval
DFT
v(n,1)
(M)
r´l
ω (1,1) ω(2,1)
...
ω (M,1)
Diversity weights
Subcarrier based receive beamforming
r(l,Q)
ω (1,Q) ω (2,Q)
...
Remove guard interval
DFT
Decoding and deinterleaving
^d i
v(n,Q)
ω (M,Q)
Diversity weights
Symbol-based receive beamforming
(b) Fig. 1.
Block diagram of the MSB scheme for a coded OFDM system with MIMO antennas: (a) transmitter; (b) receiver.
branches. The received signal at the qth branch is then given by M (m) ω(m,q) rl (5) r(l,q) =
qth branch, the output of the DFT processor is given by v(n,q) =
r(l,q) e−j2πln/N .
(6)
l=0
m=1
where ω(m,q) (m = 1, · · · , M, q = 1, · · · , Q) is the weighting coefficient for the mth receive antenna at the qth branch. We further assume that the CIRs decay to zero during the cyclic extension, or K ≤ (Ng + 1) < N . After the guard interval removal for each branch, the weighted and combined signals are then applied to the DFT processors. Note that there are Q branches, and hence the number of DFT blocks required at the receiver is Q. As a result, compared with the scheme proposed in [1] with M DFT blocks, the number of DFT blocks employed at the receiver can also be reduced. For the
N −1
By substituting (1)-(5) into (6), we have the following v(n,q) =
M
ω(m,q) rn(m)
(7)
m=1
where rn(m)
=
F P f =1 p=1
ψ(f,p) u(p,n) Hn(m,f ) cn + zn(m) ,
(8)
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Hn(m,f ) =
K−1
(m,f ) −j2πln/N
hl
e
,
III. I TERATIVE W EIGHTING C OEFFICIENTS C ALCULATION A LGORITHM
(9)
l=0
A. Optimization Criterion
and zn(m) =
N −1
(m) −j2πln/N
zl
e
.
(10)
l=0
To find the weighting coefficients for the subcarrier level beamforming and the symbol level beamforming, we maximize the average SNR as follows
In order to rewrite (7) and (8) into a matrix form, we define = Hn(m,1) , Hn(m,2) , · · · , Hn(m,F ) H(m) n n = 0, 1, · · · , N − 1,
[Ω,Ψ, U] =
(11) where
and
T T (2) T (M) T Hn = (H(1) n ) , (Hn ) , · · · , (Hn ) n = 0, 1, · · · , N − 1.
η=
(m) rn(m) = cn H(m) n Ψun + zn
(12)
v(n,q) =
+
ωqT zn
(14)
(15)
(16)
The SNR of the nth subcarrier can then be represented by
SN Rn =
N0
= k1 · uH n Γn un
(17)
where Es is the average energy of the coded symbol, N0 is the variance of the noise at the nth subcarrier, k1 = Es /N0 , and ∗ T Γn = ΨH HH n Ω Ω Hn Ψ n = 0, 1, · · · , N − 1.
(18)
In order to keep the transmit power to be constant, we assume that (19) ΨH Ψ = IP uH n un
= 1 n = 0, 1, · · · , N − 1.
N −1
(22)
(20)
In the next section, we will develop an algorithm to obtain the weighting coefficients for the subcarrier based beamforming and the symbol based beamforming.
H∗n Ψ∗ u∗n uTn ΨT HTn
(24)
n=0
When the maximum likelihood sequence detection (MLSD) criterion is employed, the pair-wise error probability (PEP) can be used to denote system performance, which is further determined by the pair-wise codeword distance [15], [16]. Assume that the transmitted codeword spans over one OFDM symbol, then the PEP of deciding erroneously in favor of a coded sequence e ≡ [e0 , e1 , · · · , eN −1 ]T instead of the transmitted T coded sequence c ≡ [c0 , c1 , · · · , cN −1 ] , conditioned on the equivalent channel matrix A (defined as (A.8) in Appendix A), is given by ⎛ p(c → e|A) = Q ⎝
Q H H ∗ T uH n Ψ Hn ωq ωq Hn Ψun Es q=1
SN Rn .
Q N −1
Φ=
After the DFT processing, maximum ratio combining (MRC) is employed to achieve maximum SNR for each subcarrier. Here, we assume that Ω is normalized, that is, ΩH Ω = IQ .
N −1
H H ∗ T uH n Ψ Hn ωq ωq Hn Ψun Es N0 n=0 q=1 Q N −1 H H ∗ T uH n Ψ Hn ωq ωq Hn Ψun Es = tr N0 n=0 q=1 H = k1 · tr Ω ΦΩ (23)
η=
where
where T zn = zn(1) , zn(2) , · · · , zn(M) .
(21)
By substituting (17) into (22), we have
(13)
and (7) can be represented in the following form cn ωqT Hn Ψun
(η)
n=0
Let Ω be an M × Q matrix with the (m, q)th entry given by ω(m,q) , Ψ an F × P matrix with the (f, p)th entry given by ψ(f,p) , and U a P × N matrix with the (p, n)th entry given by u(p,n) . At the same time, we use ωi , ψi and ui to represent the ith column of Ω, Ψ, and U, respectively. With the above definitions, (8) can be rewritten as
and
arg max ΩH Ω=IQ ΨH Ψ=IP uH n un =1, f or n=0,···,N −1
⎞ Es d2 (c,e) Es d2 (c, e) ⎠ ≤ e− 4N0 2N0
(25)
where d2 (c, e) is the pair-wise codeword distance. From (25), it can be seen that minimizing the pair-wise error probability is equivalent to maximizing the pair-wise codeword distance. In Appendix A, it is shown that maximizing the average SNR is equivalent to maximizing the average pairwise codeword distance. In the following and for simplicity, we only use the maximum average SNR criterion for our discussions. B. Iterative Algorithm to Obtain Weighting Coefficients From (21), it can be seen that three weighting matrices Ψ, U and Ω are required to be calculated. Compared with [1] and [15], where only two weighting vectors are required to be optimized, the weighting coefficients calculation of the MSB scheme is much more difficult. Thus, we propose an iterative algorithm to achieve this task. In the iterative algorithm, there
LI et al.: MULTI-STAGE BEAMFORMING FOR CODED OFDM WITH MULTIPLE TRANSMIT AND MULTIPLE RECEIVE ANTENNAS
are multiple steps. In each step, two of the three matrices ( Ψ, U and Ω ) are assumed to be known and the task is to find the remaining one.2 Assume that Ψ(i) , U(i) and Ω(i) are the values for Ψ, U, and Ω at the ith iteration, respectively. The iterative algorithm is then given as follows: Initialization: Set i = 0, and initialize IP I (0) (0) Ψ = , Ω = Q 0(F −P )×P F ×P 0(M−Q)×Q M×Q
3) Derivation of Ψ given U and Ω : By putting (27) into (22), we have η = k1
Step 1: Find U Step 2: Find Ω
(i+1) (i+1)
Step 3: Find Ψ
(i)
using Ψ
(i)
using U
(i)
using U
and Ω
η = k1
N −1
≥ k1 · λmax
and Ψ
In the following, we will present further details about Step 1, Step 2 and Step 3, respectively. 1) Derivation of U given Ψ and Ω: We can achieve the maximum of η by maximizing every SN Rn (n = 0, 1, · · · , N − 1). That is,
It is easy to see that when un is the dominant eigenvector of Γn , SN Rn (n = 0, 1, · · · , N − 1) takes on the maximum value, which is given by
ΩH Ω=IQ
(28)
ΩH Ω=IQ
Assume that the eigenvalues of Φ are λq (q = 1, · · · , Q) with λ1 ≥ λ2 ≥ · · · ≥ λQ . It is well known that when ωq (q = 1, 2, · · · , Q) are the eigenvectors of Φ corresponding to the eigenvalues λq (q = 1, 2, · · · , Q), the maximum of η is achieved and is given by Q
λq (Φ).
N −1
∗ T HH n Ω Ω Hn .
(29)
q=1
From (29), it can be seen that the maximum of η is proportional to the sum of the first Q largest eigenvalues of Φ. Therefore, by increasing Q, or the number of DFT blocks at the receiver, we can collect more eigenvalues of Φ, thereby, improving system performance. Since ωq (q = 1, 2, · · · , Q) are arranged to make the corresponding eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λQ , we can expect that the marginal increase of η decreases along with the increase of Q. 2 The estimations of the three matrices are correlated, and poor estimation of one of them might have an impact on the other two. However, our ultimate objective is to minimize the bit error probability of the MSB scheme. As will be shown by the simulation results in Section V, excellent system performance can be achieved using the proposed iterative algorithm.
(31)
(32)
n=0
Since Γn is a semidefinite matrix, we have λmax (Γn ) ≤ tr(Γn ).
(33)
As a result, η = k1 ≤ k1
N −1 n=0 N −1
(27)
Ω = arg max (η)
η = k1
Γn
λmax (Γn ) tr(Γn )
n=0
2) Derivation of Ω given U and Ψ: In this case, Ω is the solution of the following function = arg max tr(ΩH ΦΩ) .
Θ=
(26)
uH n un =1
SN Rn max = k1 · λmax (Γn )
= k1 · λmax Ψ ΘΨ
(i)
where
uH n un =1
N −1 n=0 H
Go back to Loop.
(30)
λmax (Γn )
n=0
(i)
un = arg max SN Rn = arg max uH n Γn un
λmax (Γn ).
According to the theory of Weyl [17], we have
(i)
and Ω
N −1 n=0
Loop: i = i + 1 (i+1)
963
= k1 · tr
N −1
Γn
n=0 H
= k1 · tr Ψ ΘΨ . From (31) and (34), we have k1 · λmax ΨH ΘΨ ≤ η ≤ k1 · tr ΨH ΘΨ .
(34)
(35)
It can be easily proved that, when ψp (p = 1, 2, · · · , P ) are the eigenvectors of Θ corresponding to the first P largest eigenvalues, both λmax ΨH ΘΨ and tr ΨH ΘΨ are maximized. As a result, we impose the columns of Ψ to be the eigenvectors of Θ corresponding to the first P largest eigenvalues. IV. R EDUCED C OMPLEXITY W EIGHTING C OEFFICIENTS C ALCULATION A. Complexity Reduction In Step 1 of the proposed iterative algorithm shown in Section III, one eigen-value decomposition is required for each subcarrier and each iteration. As a result, the computation burden is enormous especially when N and/or the number of iterations is large. In this section, we propose a reduced complexity algorithm, where the symbol level weighting coefficients are first obtained followed by the subcarrier level weighting coefficients calculation. As a result, the subcarrier level weighting coefficients are required to be calculated only once and the complexity is significantly reduced. To obtain Ω and Ψ, we assume that the subcarrier level weighting coefficients for all the subcarriers are exactly the same in the reduced complexity algorithm. For simplicity,
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we assume that un = [1, 0, · · · , 0]T for all n. As shown in Appendix B, Φ in (24) is then given by H ¯ (IM ⊗ ψ1 ) Φ = (IM ⊗ ψ1 ) G
(36)
¯ = HH H, G
(37)
where
and ⎡
(1)
H0 , (1) H1 , .. .
⎢ ⎢ H=⎢ ⎣
(1)
(2)
H0 , (2) H1 , .. . (2)
···, ···, .. .
(M)
H0 (M) H1 .. . (M)
⎤ ⎥ ⎥ ⎥. ⎦
(38)
HN −1 , HN −1 , · · · , HN −1 Similarly, as shown in Appendix B, Θ in (32) is given by Θ=
Q
H ¯ (IF ⊗ ωq ) G (IF ⊗ ωq )
(39)
q=1
where
¯ = H H H G ⎡ ⎢ ⎢ H = ⎢ ⎣
(1)
H 0 , (1) H 1 , .. . (1)
(2)
H 0 , (2) H 1 , .. . (2)
(40) (F )
H 0 (F ) H 1 .. .
···, ···, .. .
(F )
⎤ ⎥ ⎥ ⎥ ⎦
(41) B. Complexity Consideration
H N −1 , H N −1 , · · · , H N −1 and (f )
H n
≡ Hn(1,f ) , Hn(2,f ) , · · · , Hn(M,f ) ,
n = 0, 1, · · · , N − 1 f = 1, · · · , F. (i) ψj
(42)
(i) ωj
As a result, if we define and as the jth column of Ψ(i) and Ω(i) , respectively, the iterative algorithm shown in Section III.B can then be reduced to the following: Initialization:
(0)
Set i = 0, and initialize Ψ IQ (0) . Ω = 0(M−Q)×Q M×Q Loop: i = i + 1 Step 1: Φ =
(i)
IM ⊗ ψ1
H
I = P 0(F −P )×P
¯ IM ⊗ ψ (i) . G 1
F ×P
,
Ω(i+1) (i+1)
is obtained by setting its jth column, ωj , to be the eigenvector corresponding to the jth largest eigenvalue of Φ. H Q (i) ¯ IF ⊗ ωq(i) . Ψ(i+1) G IF ⊗ ωq Step 2: Θ = q=1
a result, it can be seen that un is the dominant eigenvector of Γn . ¯ and In the above process, we only need to calculate G ¯ G once to obtain Ω and Ψ. Furthermore, after the symbol level weighting coefficients calculation, the subcarrier level weighting coefficients only need to be calculated once. As a result, using the reduced complexity algorithm, the complexity is reduced significantly compared with the algorithm proposed in Section III. In particular, the following observations can be made about the proposed MSB scheme with the reduced complexity weighting coefficients calculation algorithm: 1) When there is one antenna at the transmitter and one DFT block at the receiver, it is reduced to the scheme proposed in [11]. 2) When there is one antenna at the transmitter and multiple DFT blocks at the receiver, it is reduced to the pre-DFT processing scheme proposed in [16]. 3) When there is one IDFT block at the transmitter and one DFT block at the receiver, it is reduced to the symbol based space diversity scheme proposed in [15]. As a result, the proposed MSB scheme with the reduced complexity weighting coefficients calculation algorithm is quite general and incorporates various cases as well as previously derived methods.
(i+1)
is obtained by setting its jth column, ψj , to be the eigenvector corresponding to the jth largest eigenvalue of Θ. Go back to Loop. Ψ and Ω can be obtained using the above reduced complexity iterative algorithm. When Ψ and Ω are available, Γn (n = 0, 1, · · · , N − 1) can be calculated using (18). After that, we can obtain un by maximizing the SNR given by (17). As
The number of multiplications required in the proposed MSB scheme can be used to measure the complexity, which includes FFT transforming, weighting coefficients calculation, symbol level beamforming, and subcarrier level beamforming. In general, the number of transmit antennas and receive antennas is significantly smaller than the number of subcarriers in an OFDM system. Hence, we can neglect the complexity of the weighting coefficients calculation especially when the channel varies slowly and the reduced complexity weighting coefficients calculation algorithm is employed. When we only consider the complexity of FFT along with the symbol level and subcarrier level beamforming, the ratio of the number of multiplications used in the MSB scheme compared with the subcarrier based beamforming alone is given by (P + Q)N log2 N + P F N + QM N + P N + QN (M + F )N log2 N + (M + F )N (P + Q) log2 N + P F + QM + P + Q . (43) = (M + F ) log2 N + M + F
μ1 =
The ratio of the number of multiplications used in the MSB scheme compared with the symbol based beamforming alone is given by (P + Q)N log2 N + P F N + QM N + P N + QN 2N log2 N + (M + F )N (P + Q) log2 N + P F + QM + P + Q (44) = 2 log2 N + M + F
μ2 =
For example, when P = 2, Q = 2, N = 64, F = 4 and M = 4, the number of multiplications required by the MSB scheme is reduced by 21% compared with the scheme
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using the subcarrier based beamforming alone, and increased by 120% compared with the scheme using the symbol based beamforming alone. From (43) and (44), it can also be seen that, when log2 N >> M + F , μ1 and μ2 are close to (P + Q)/(M + F ) and (P + Q)/2, respectively. V. S IMULATION R ESULTS We consider an OFDM system with convolutional coding and QPSK modulation. The code rate of the convolutional code is 1/2, the constraint length is 7, and the generator matrix is (133,171) in octal. Soft Viterbi decoding algorithm is employed at the receiver for decoding and the number of subcarriers is 64. The interleaver employed is an 8 × 16 block bit interleaver. Thus, after coding and interleaving, each bit in the codeword is scattered across the whole OFDM symbol. In the simulations, ten OFDM symbols are grouped into one frame. We further assume that the length of the guard interval is 12. For each bit-error probability calculation, we use over 20,000 channel realizations. In our simulations, we take an OFDM system with four transmit antennas and four receiver antennas as an example. The OFDM system with the subcarrier based beamforming as shown in [1] is taken as the baseline system. In the baseline system, all the parameters are the same as those in the OFDM system using the proposed MSB scheme. In the MSB scheme, 5 iterations are used for the iterative weighting coefficients calculation. We also use an equal gain quasistatic Rayleigh fading channel model and assume that perfect channel information is available.3 When the channel is an eight-ray equal gain uncorrelated Rayleigh fading channel, the bit error rate (BER) performance of the proposed MSB scheme is shown in Fig. 2. It can be observed that, when the number of IDFT blocks at the transmitter is fixed, better performance can be achieved by using more DFT blocks at the receiver. However, the margin of the performance improvement decreases along with the increase of the number of DFT blocks. This shows that tradeoff between system performance and complexity can be easily achieved by the proposed MSB scheme. A close observation of Fig. 2, also shows that the performance of the proposed MSB scheme can be improved by increasing the number of IDFT blocks at the transmitter. Specifically, when P = 1 and Q = 1, the proposed scheme is actually the symbol based space diversity scheme as described in [15]. Fig. 2 shows that the performance of the proposed MSB scheme is always better than that of the symbol based space diversity scheme. When P = 4 and Q = 4, the performance of the MSB scheme is the same as that of the subcarrier based beamforming scheme. Fig. 3 shows the BER performance of the MSB scheme over equal gain two-ray Rayleigh fading channels. It can be seen that the BER performance can be improved by increasing either the number of transmit antennas or the number of receive antennas. However, when the number of IDFT (DFT) blocks at the transmitter and the receiver are both two, the BER performance difference between the MSB scheme and the subcarrier based beamforming scheme is very small (e.g., 3 Many schemes have been proposed to achieve a reasonable good channel estimation for an OFDM system under MIMO channels [18].
(a)
(b) Fig. 2. BER performance of the MSB scheme over eight-ray equal gain Rayleigh fading channels. (a) One IDFT block and two IDFT blocks at the transmitter, which are represented by the dotted lines and solid lines, respectively. (b) Three IDFT blocks and four IDFT blocks at the transmitter, which are represented by the dotted lines and solid lines, respectively.
0.32dB when Pe = 10−3 ). By further increasing either the number of IDFT blocks at the transmitter or the number of DFT blocks at the receiver, the performance improvement is very little. As a result, for the MSB scheme, when the number of distinct paths in the channel is small, only a small number of DFT (IDFT) blocks are required to achieve a performance close to the optimal. We conclude this section by presenting Fig. 4 that shows the BER performance of the MSB scheme with the reduced complexity weighting coefficients calculation algorithm proposed in Section IV. It can be seen that the performance is always very close to the case where the weighting coefficients calculation algorithm proposed in Section III is used. This shows that it is feasible to use the reduced complexity weighting coefficients calculation algorithm in the proposed MSB scheme.
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Symbol based beamforming
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P=1 Q=3
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Fig. 3. BER performance of the MSB scheme over two-ray equal gain Rayleigh fading channels. (a) Two IDFT blocks at the transmitter; (b) Two DFT blocks at the receiver
Fig. 4. BER performance of the MSB scheme with the reduced complexity weighting coefficients calculation algorithm over equal gain Rayleigh fading channels. Performance of the MSB scheme using the reduced complexity weighting coefficients calculation algorithm and that using the algorithm proposed in Section III are represented by the dotted lines and solid lines, respectively. (a) K = 2; (b) K = 8.
VI. C ONCLUSION A PPENDIX A In this paper, we presented an MSB scheme for an OFDM system under MIMO channels, where both subcarrier level and symbol level beamforming are employed. Compared with the cases where only the subcarrier based beamforming or only the symbol based beamforming are used, the MSB scheme can be employed to effectively tradeoff system performance and complexity. In the MSB scheme, three weighting matrices are required to be calculated. To achieve this task, an iterative algorithm was proposed, which achieves near optimal performance but with considerable complexity. To reduce this complexity, we proposed a reduced complexity weighting coefficients calculation algorithm, where the subcarrier level weighting coefficients are calculated following the symbol level weighting coefficients calculation. Simulation results have showed that good performance can be achieved by using this reduced complexity algorithm.
From (13), the received OFDM symbol for the mth receive antenna can be represented in the frequency domain by an N × 1 vector as follows T (m) (m) (m) r(m) = r0 , r1 , · · · , rN −1 = diag(c)B(m) β + z(m) (A.1) where T (A.2) c ≡ [c0 , c1 , · · · , cN −1 ] , T (m) (m) (m) z(m) = z0 , z1 , · · · , zN −1 , N −1
(A.3)
B(m) = ⊕ (H(m) n ),
(A.4)
β = (IN ⊗ Ψ)vec(U)
(A.5)
n=0
and
LI et al.: MULTI-STAGE BEAMFORMING FOR CODED OFDM WITH MULTIPLE TRANSMIT AND MULTIPLE RECEIVE ANTENNAS
where vec(·) denotes the vector representation of the corresponding matrix (·) by putting all the column vectors matrix (·) in one column. Assuming that B = (1)of (2) B , B , · · · , B(M) and Z = z(1) , z(2) , · · · , z(M) , we have r = r(1) , r(2) , · · · , r(M) = diag(c) B(1) β, B(2) β, · · · , B(M) β + Z = diag(c)B(IM ⊗ β) + Z.
V = rΩ = diag(c)B(IM ⊗ β)Ω + ZΩ (A.7)
A = B(IM ⊗ β)Ω.
(A.8)
Let aq be the qth column of A, from (A.8), we have (A.9)
According to [16], after MRC for each subcarrier at the receiver, the pair-wise codeword distance is given by d2 (c, e) =
T
When un = [1, 0, · · · , 0] (n = 0, 1, · · · , N − 1), then according to (24), Φ=
Y = [H0 ψ1 , H1 ψ1 , · · · , HN −1 ψ1 ]
where k2 is a constant and relates to the specific codeword pair. As a result, (A.12)
By substituting (A.8) into (A.12), we have (A.13)
According to (A.4) and (A.5), we can expand B(IM ⊗ β) into (A.14) as shown at the top of the next page. Given (A.14) and (24) , it follows that N −1 H ∗ ∗ ∗ T T T ξ = k2 · tr Ω Hn Ψ un un Ψ Hn Ω n=0
= k2 · tr(ΩH ΦΩ).
T
(B.2)
⎤ ⎥ ⎥. ⎦
(B.4)
T (m) Note that ψ1T Hn (m = 1, · · · , M and n = 0, 1, · · · , N − 1) is a scalar. Hence, ⎤ (1) (M) H 0 ψ1 , · · · , H 0 ψ1 ⎥ ⎢ .. .. .. Y=⎣ ⎦ . . . (1) (M) HN −1 ψ1 , · · · , HN −1 ψ1 = H (IM ⊗ ψ1 ) . ⎡
(B.5)
By substituting (B.5) into (B.3),we can obtain Φ, as defined by (36). B. Derivation of Θ (32) can be rewritten into Θ=
N −1
∗ T HH n Ω Ω Hn
=
n=0
Q
N −1
q=1
n=0
∗ T HH n ω q ω q Hn
.
(B.6) For convenience, we define T Y q = HT0 ωq , HT1 ωq , · · · , HTN −1 ωq q = 1, 2, · · · Q. (B.7) (B.6) can then be written as Θ=
Q
H
Y q Y q .
(B.8)
q=1
By substituting (12) into (B.7), we have T T (1) (F ) , · · · , ωqT H 0 ωqT H 0 ⎢ .. .. .. =⎢ ⎣ . . T . T (1) (F ) ωqT H N −1 , · · · , ωqT H N −1 ⎡
(A.15)
Compared (A.15) with (23), it can be seen that the maximum average SNR criterion and the maximum average codeword distance criterion are the same.
(B.3)
By substituting (12) into (B.2), we have T T ⎡ (1) (M) , · · · , ψ1T H0 ψ1T H0 ⎢ .. .. .. Y=⎢ ⎣ . . T . T (1) (M) ψ1T HN −1 , · · · , ψ1T HN −1
(A.10)
When the interleaver is an ideal random one, we have the following E (c − e)H (c − e) = k2 IN (A.11)
ξ = k2 · tr(ΩH (IM ⊗ β)H BH B(IM ⊗ β)Ω).
(B.1)
For convenience, we define
q=1
ξ = E d2 (c, e) = k2 · tr(AH A).
H∗n ψ1∗ ψ1T HTn .
n=0
H aH q (c − e) (c − e)aq
= tr(AH (c − e)H (c − e)A).
N −1
Φ = YH Y
where
Q
A. Derivation of Φ
Then, we have
= diag(c)A + ZΩ
aq = B(IM ⊗ β)ωq q = 1, 2, · · · , Q.
A PPENDIX B D ERIVATION OF Φ AND Θ FOR R EDUCED C OMPLEXITY W EIGHTING C OEFFICIENTS C ALCULATION
(A.6)
Let V be an N × Q matrix with the (n, q)th entry given by v(n,q) . (7) in a matrix form is then as follows
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Y q
⎤ ⎥ ⎥ . (B.9) ⎦
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⎡ ⎢ ⎢ B(IM ⊗ β) = ⎢ ⎣
(1)
H0 Ψu0 , (1) H1 Ψu1 , .. .
(1)
(2)
H0 Ψu0 , (2) H1 Ψu1 , .. .
(2)
···, ···, .. .
(M)
H0 Ψu0 (M) H1 Ψu1 .. . (M)
⎤ ⎥ ⎥ ⎥ ⎦
HN −1 ΨuN −1 , HN −1 ΨuN −1 , · · · , HN −1 ΨuN −1 T T T (1) (2) (M) , uT0 ΨT H0 , ···, uT0 ΨT H0 uT0 ΨT H0 ⎢ T T T ⎢ (2) (M) T T T T ⎢ uT1 ΨT H(1) H H , u Ψ , · · · , u Ψ 1 1 1 1 1 =⎢ ⎢ .. .. .. .. ⎢ . . ⎣ . T . T T (1) (2) (M) uTN −1 ΨT HN −1 , uTN −1 ΨT HN −1 , · · · , uTN −1 ΨT HN −1 ⎡
T
= [ H0 Ψu0 , H1 Ψu1 , · · · , HN −1 ΨuN −1 ] .
T (f ) Note that ωqT H n (f = 1, · · · , F and n = 0, · · · , N −1) is a scalar. Hence, ⎤ ⎡ (1) (F ) H 0 ω q , · · · , H 0 ω q ⎥ ⎢ .. .. .. Y q = ⎣ ⎦ . . . (1) (F ) H N −1 ωq , · · · , H N −1 ωq = H (IF ⊗ ωq ) (B.10) where H is defined as (41). By substituting (B.10) into (B.8), we can obtain Θ, as defined by (39). R EFERENCES [1] K. K. Wong, R. S. K. Cheng, K. B. Letaief, and R. D. Murch, “Adaptive antennas at the mobile and base stations in an OFDM/TDMA system,” IEEE Trans. Commun., vol. 49, no. 1, pp. 195-206, Jan. 2001. [2] P. Xia, S. Zhou, and G. B. Giannakis, “Adaptive MIMO-OFDM based on partial channel state information,” IEEE Trans. Signal Processing, vol. 52, no. 1, pp. 202-213, Jan. 2004. [3] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization,” IEEE Trans. Signal Processing, vol. 51, no. 9, pp. 2381-2401, Sept. 2003. [4] A. Pascual-Iserte, A. I. Perez-Neira, and M. A. Lagunas, “On power allocation strategies for maximum signal to noise and interference ratio in an OFDM-MIMO system,” IEEE Trans. Wireless Commun., vol. 3, no. 3, pp. 808-817, May 2004. [5] Y. Pan, K. B. Letaief, and Z. Cao, “Dynamic resource allocation with adaptive beamforming for MIMO/OFDM systems under perfect and imperfect CSI,” in Proc. IEEE Wireless Commun. and Networking Conf. 2004, pp. 93-97. [6] E. Grass, K. Tittelbach-Helmrich, U. Jagdhold, A. Troya, G. Lippert, O. Kruger, J. Lehmann, K. Maharatna, K. F. Dombrowski, N. Fiebig, R. Kraemer, and P. Mahonen, “On the single-chip implementation of a Hiperlan/2 and IEEE 802.11a capable modem,” IEEE Personal Commun., vol. 8, no. 6, pp. 48-57, Dec. 2001. [7] M. Speth, S. Fechtel, G. Fock, and H. Meyr, “Optimum receiver design for OFDM-based broadband transmission–part II: a case study,” IEEE Trans. Commun., vol. 49, no. 4, pp. 571-578, Apr. 2001. [8] M. Budsabathon, Y. Hara, and S. Hara, “Optimum beamforming for preFFT OFDM adaptive antenna array,” IEEE Trans. Veh. Technol., vol. 53, no. 4, pp. 945-955, July 2004. [9] S. Hara, M. Budsabathon, and Y. Hara, “ A pre-FFT OFDM adaptive antenna array with eigenvector combining,” in Proc. IEEE Int. Conf. Commun. 2004, vol. 4, pp. 2412-2416. [10] M. I. Rahman, K. Witrisal, S. S. Das, F. H. P. Fitzek, O. Olsen, and R. Prasad, “Optimum pre-DFT combining with cyclic delay diversity for OFDM based WLAN systems,” in Proc. IEEE Veh. Technol. Conf. 2004, vol. 4, pp. 1844-1848. [11] M. Okada and S. Komaki, “Pre-DFT combining space diversity assisted COFDM,” IEEE Trans. Veh. Technol., vol. 50, no. 2, pp. 487-496, Mar. 2001. [12] G. Barriac and U. Madhow, “Space-time communication for OFDM with implicit channel feedback,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3111-3129, Dec. 2004.
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (A.14)
[13] J. Yang and Y. Li, “Low complexity OFDM MIMO system based on channel correlations,” in Proc. IEEE Global Telecommun. Conf. 2003, vol. 2, pp. 591-595. [14] S. B. Slimane, “A low complexity antennas diversity receiver for OFDM based systems,” in Proc. IEEE Int. Conf. Commun. 2000, vol. 4, pp. 11471151. [15] D. Huang and K. B. Letaief, “Symbol-based space diversity for coded OFDM systems,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 117127, Jan. 2004. [16] D. Huang and K. B. Letaief, “Pre-DFT processing using eigen-analysis for coded OFDM with multiple receive antennas,” IEEE Trans. Commun., vol. 52, no. 11, pp. 2019-2027, Nov. 2004. [17] P. Lancaster and M. Tismenetsky, The Theory of Matrices with Applications, Second Edition. New York: Academic, 1985. [18] Y. Li, “Simplified channel estimation for OFDM systems with multiple transmit antennas,” IEEE Trans. Wireless Commun., vol. 1, no. 1, pp. 67-75, Jan. 2002. Shaohua (Steven) Li received the B.S.E.E and M.S.E.E degrees in electronic engineering from Harbin Institute of Technology, Harbin, China, in 2000 and 2002, respectively, and the Ph.D. degree in the department of Electronic Engineering, Tsinghua University, Beijing, China, in 2006. His research interests include wireless communications, OFDM, MIMO, Beamforming, space-timefrequency processing and digital implementation of communication systems. Defeng (David) Huang (M’01-S’02-M’05) received the B. E. E. E. and M. E. E. E. degree in electronic engineering from Tsinghua University, Beijing, China, in 1996 and 1999, respectively, and the Ph.D. degree in electrical and electronic engineering from the Hong Kong University of Science and Technology (HKUST), Kowloon, Hong Kong, in 2004. From 1998, he was an assistant teacher and later a lecturer with Tsinghua University. Currently, he is a lecturer with School of Electrical, Electronic and Computer Engineering at the University of Western Australia. His research interests include broadband wireless communications, OFDM, OFDMA, cross-layer design, multiple access protocol, and digital implementation of communication systems. Dr. Huang serves as an Editor for the IEEE Transactions on Wireless Communications. He received the Hong Kong Telecom Institute of Information Technology Postgraduate Excellence Scholarships in 2004.
LI et al.: MULTI-STAGE BEAMFORMING FOR CODED OFDM WITH MULTIPLE TRANSMIT AND MULTIPLE RECEIVE ANTENNAS
Khaled B. Letaief (S’85-M’86-SM’97-F’03) received the BS degree with distinction in Electrical Engineering from Purdue University at West Lafayette, Indiana, USA, in December 1984. He received the MS and Ph.D. Degrees in Electrical Engineering from Purdue University, in August 1986, and May 1990, respectively. From January 1985 and as a Graduate Instructor in the School of Electrical Engineering at Purdue University, he has taught courses in communications and electronics. From 1990 to 1993, he was a faculty member at the University of Melbourne, Australia. Since 1993, he has been with the Hong Kong University of Science and Technology where he is currently Chair Professor and Head of the Electronic and Computer Engineering Department. He is also the Director of the Hong Kong Telecom Institute of Information Technology as well as the Director of the Center for Wireless Information Technology. His current research interests include wireless and mobile networks, Broadband wireless access, OFDM, CDMA, and Beyond 3G systems. In these areas, he has published over 280 journal and conference papers and given invited talks as well as courses all over the world. Dr. Letaief served as consultants for different organizations and is currently the founding Editor-inChief of the IEEE Transactions on Wireless Communications. He has served on the editorial board of other prestigious journals, including the IEEE Journal on Selected Areas in Communications — Wireless Series (as Editor-in-Chief) and the IEEE Transactions on Communications. He has been involved in organizing a number of major international conferences and events. These include serving as the Technical Program Chair of the 1998 IEEE Globecom Mini-Conference on Communications Theory, held in Sydney, Australia as well as the Co-Chair of the 2001 IEEE ICC Communications Theory Symposium, held in Helsinki, Finland. In 2004, he served as the Co-Chair of
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the IEEE Wireless Communications, Networks and Systems Symposium, held in Dallas, USA as well as the Co-Technical Program Chair of the 2004 IEEE International Conference on Communications, Circuits and Systems, held in Chengdu, China. He is the Co-Chair of the 2006 IEEE Wireless Ad Hoc and Sensor Networks Symposium, held in Istanbul, Turkey. In addition to his active research and professional activities, Professor Letaief has been a dedicated teacher committed to excellence in teaching and scholarship. He received the Mangoon Teaching Award from Purdue University in 1990; the Teaching Excellence Appreciation Award by the School of Engineering at HKUST (four times); and the Michael G. Gale Medal for Distinguished Teaching Highest university-wide teaching award and only one recipient/year is honored for his/her contributions). He is a Fellow of IEEE, an elected member of the IEEE Communications Society Board of Governors, and an IEEE Distinguished lecturer of the IEEE Communications Society. He also served as the Chair of the IEEE Communications Society Technical Committee on Personal Communications as well as a member of the IEEE ComSoc Technical Activity Council. Zucheng Zhou graduated from the department of Radio Electronics at Tsinghua University, Beijing, China, in 1964. Since 1964, he has been with Tsinghua University, where he is a professor of Electronic Engineering Department and Director of the CAD laboratory of the State Key Laboratory on Microwave and Digital Communications. His major research fields are the design methodology and design automation of integrated circuits and systems, design of integrated circuits for communication.
