Adaptive Transmit Beamforming with Space-Time ... - Semantic Scholar
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Adaptive Transmit Beamforming with Space-Time Block Coding for Correlated MIMO Fading Channels Min Lin1,2
Min Li3
Luxi Yang1
and
Xiaohu You1
1. College of Information Science and Engineering, Southeast University, Jiangsu, 210096, China 2. Nanjing Institute of Telecommunication Technology, Jiangsu, 210007, China 3. Nanjing Institute of Communication Engineering, Jiangsu, 210007, China E-mail:[email protected], [email protected] Abstract-In this paper, we present an open-loop transmit scheme for MIMO communications. We first use the uplink channel correlation matrix (UCCM) to calculate the array weight vectors for downlink beamforming (DLBF). Then based on the symbol error rate (SER) upper bound as design criterion, we derive an algorithm for adaptive power allocation among multiple beams, and thus develop the transmit scheme joint adaptive beamforming (ABF) with space-time block coding (STBC). The main benefit of the scheme is that it can almost achieve the optimal performance with low complexity. Next, using the moment generation function (MGF) approach and the Gauss-Chebyshev integration, we derive a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations. Finally, computer simulation results demonstrate the superiority of the open-loop transmit scheme. Key Words-Adaptive beamforming, Space-time block coding, Transmit scheme, MIMO communication.
I.
INTRODUCTION
Recently, beamforming (BF) and space-time coding (STC) are widely recognized as two key approaches to improve the capacity and reliability of wireless communication systems employing multiple transmit antennas. Although considering the array inter-element spacing and the required knowledge about the channel state information (CSI) at the transmitter, they are different and even contradictive, it was shown in [1] that the benefits of these two techniques can be achieved simultaneously using side information. Besides this work, there are a number of other articles that discuss the combination of BF with STBC. For example, in [2] and [3], the authors use either channel mean or channel correlation feedback, and generalized optimal transmit eigen-beamformer for widely used linear modulations, and coupled orthogonal STBC to increase the data rate without compromising the performance. However, these strategies require a considerable amount of feedback bandwidth, which may be a heavy burden in practice. In [4], the authors combined one-dimensional (1-D) beamforming with Alamouti’s STBC to achieve more performance gain for the case of more than two transmit antennas. In [5], the authors proposed a downlink transmission scheme combining two-dimensional (2-D) eigen-beamforming with Alamouti’s STBC to achieve both diversity gain as well as array gain. The analogous ideal was This work is supported by NSFC major project under Grant 60496310, NSFC under Grant 60672093, and NSF of Jiangsu province under Grant BK2005061
also presented in [6] and [7], where the pure diversity and the pure BF schemes were also analyzed and compared. But the major drawback of these methods is that they can not adaptively allocate the proper power among multiple beams according to the channel parameters, thus their performances are not optimal in a lot of environments. Besides sufficient error performance analysis was not given there. In this paper, we present an open-loop scheme for correlated MIMO fading channels, which is realistic in cellular communications due to close inter-element spacing of antenna array. We first use the UCCM to calculate the array weight vectors for DLBF. Then based on the model of equivalent scaled additive white Gaussian noise (AWGN) channel induced by the STBC, we obtain the signal-to-noise ratio (SNR) at the receiver and derive the SER upper bound. Using it as the design criterion, we derive an algorithm for adaptive power allocation, and thus develop the transmit scheme combining ABF with STBC. Although the result is a little analogous to the work of [3], that paper did not take the effect of STBC into account for beamformer design. Besides, the derivation and expression for the SER upper bound are also different. Next, utilizing the MGF approach and the Gauss-Chebyshev integration, we derive a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations. Finally, computer simulation results demonstrate that the proposed transmit scheme outperforms the conventional STBC and the other existing related typical strategies. Notation: ⋅ F denotes the Frobenius norm of a matrix, E[ ⋅
]
denotes expectation, diag (⋅) denotes a diagonal matrix,
(⋅) * , (⋅) T , and (⋅) H
denote conjugate, transpose, and Hermitian transpose, respectively. δ (⋅) represents the Kronecker delta function, Q (⋅) represents the classical definition of the Gaussian Q-function, tan(⋅) represents the tangent function, ln(⋅) represents the natural logarithm function, exp(⋅) represents the exponential function, Re[ ⋅ ] and Im[ ⋅ ] represent the real part and the imaginary part of a complex number respectively. The rest of the paper is organized as follows. In section II, the problem description is provided. In Section III, a novel method to calculate the array weight vectors for DLBF from UCCM is firstly presented, then based on the criterion of SER upper bound, an adaptive power allocation algorithm is
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
proposed, thus the space-time transmit scheme is developed. In Section IV, using the MGF approach and the GaussChebyshev integration, a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations is derived. Finally, computer simulations are given in Section V, and conclusion is drawn in Section VI. II.
PROBLEM DESCRIPTION
In this paper, for the purpose of simplicity and without loss of generality, we consider a wireless communication with N transmit antennas and a single receive antenna. However, our results can be extended to multiple receive antennas as in [3]. Figure 1 shows the structure of transmit scheme combining BF with STBC. The serial signal to be transmitted, s, is first coded with a space-time block encoder, yielding P ( P ≤ N ) branch outputs as [8] s11 s12 " s1T s 21 s 22 " s 2T S= (1) # # # # s P1 s P 2 " s PT where T denotes the number of time slots used to encode K symbols, and s ij (i ≤ P, j ≤ T ) denotes a linear combination
of the signal constellation components and their conjugates. Then the P branch coded signals pass through P transmit beamformers, and transmitted by the N elements simultaneously. Therefore, at a particular time nT, the transmitted signal matrix X nT ( N × T ) can be expressed as X nT = WFS (2) where W = [ w 1 , w 2 , " , w P ] and F = diag ( f1 , f 2 , " , f P ) , w i ( N × 1) and f i denote the array weight vector and the corresponding coefficient of the ith beamformer. Here, we suppose w i
2 F
= 1 and
P
∑f
i
2
= 1 to guarantee that the total
i =1
transmit power keeps constant for fair comparison.
s
SpaceTime Block Coding
Fig.1
( ρ1 , θ1 )
BF1 f1w1
# BFP fPwP
SpaceTime Block Decoding
+ #
(ρ L , θ L )
Transmit scheme combining BF with STBC
Suppose the channel is a cluster-based flat fading channel and there are L paths in scatter clusters and the angle-ofdeparture (AOD) of each path varies around the mean cluster AOD θ c within the cluster angle spread ∆θ , namely, the L multipath signals are uniformly distributed in the range of [θ c − ∆θ 2 , θ c + ∆θ 2] , then the instantaneous downlink
channel response vector is given by [5] L
h d (t ) =
∑ ρ (t )a l
T d
(θ l )
(3)
l =1
where θ l and ρ l (t ) denote the AOD and the fading coefficient of the lth path signal respectively. Without loss of generality, here ρ l (t ) is modeled as complex Gaussian random variable with zero mean and unit variance. If uniform linear array (ULA) is employed at the transmitter, the downlink array steering vector a d (θ l ) is a d (θ l ) = [1, exp( jβ d d cos θ l )," , exp( j( N − 1) β d d cos θ l )]T
where d is the inter-element spacing, and β d is the downlink wavenumber. Therefore, the downlink channel correlation matrix (DCCM) R d can be obtained from [9] R d = E[h dH (t ) ⋅ h d (t )] =
1 L
L
∑a
* d
(θ l ) ⋅ a Td (θ l )
(4)
l =1
The index of t will be frequently dropped from now on for notation brevity. From (2), the received signal vector can be written as YnT = h d WFS + VnT (5) where YnT and VnT denote the 1× T received signal vector and background noise vector respectively. Let g i = h d w i (i = 1,2, " , P ) (6) we find that P equivalent virtual channels can be artificially generated by selecting w i carefully. Thus, if we define G = [ f1 g 1 , f 2 g 2 , " , f P g P ] (7) and use the model of equivalent scaled AWGN channel induced by the STBC [10], the decoded signal vector considering the code rate R can be expressed as y nT =
1 G R
2 F
(8)
s nT + v nT
where y nT denotes the K × 1 complex vector after STBC decoding from the received vector YnT , and s nT denotes the input K × 1 complex vector. R is the code rate of the STBC, namely, R = K / T , and v nT is the K × 1 complex Gaussian noise with zero mean and G 2F N 0 (2 R ) variance in each real dimension. Therefore, the effective SNR at the receiver is γ=
RE S G RN 0
2 ∆ =γ s F
G
2 F
(9)
2
where γ s = E[ s ] N 0 = E S N 0 is the transmit SNR per symbol. Therefore, if we want to get the optimal performance at the receiver, the transmit beamformer must be designed to satisfy J=
max
w1 , w 2 ,", w P
E G
2 F
= max w1 ,w 2 ,",w P
s.t. w iH w j = δ (i − j )
and
P
∑ f (w i
2
i =1
P
∑f
i
2
=1
H i
Rd wi
) (10a)
(10b)
i =1
Using the method of Rayleigh quotient [11], we can get the
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solution of the constrained optimization problem as P
J =
∑δ
i fi
2
(11a)
i =1
w opt = vi i
(11b) where δ i and vi denote the ith eigenvalue and the corresponding eigenvector of the DCCM , that is R d = [ v1 , v 2 , " v N ]diag (δ 1 , δ 2 , " , δ N )[ v 1 , v 2 , " v N ] H (12) Here the N eigenvalues are arranged in a nonincreasing order. (i = 1,2, " , P)
A. Downlink Array Weight Vectors Computation
For a frequency division duplex (FDD) system, most of the previous works suppose that the DCCM R d is yielded through feedback. However, these strategies are unrealistic in practice due to the heavy feedback burden. Although many matrix transformation methods have been proposed (see [12]-[15], and references therein for example), they can not be used in our scheme due to estimation accuracy or computation load. So here, we make use of UCCM and present a new method to calculate the array weight vectors for DLBF. Analogous to (4), the UCCM can be expressed as [9] 1 L
L
∑a
H u (θ l ) ⋅ a u
(13)
(θ l )
w iH w j = δ (i − j )
Define the array weight vector space as (20) its orthogonal complement space can be obtained from E i⊥ = I N × N − E i (E iH E i ) −1 E iH (21) E i = [ w1 , w 2 , " , w i −1 ] (i = 2,3, " , N )
B. Adaptive Power Allocation
In this section, we will discuss the algorithm of adaptive power allocation. We first suppose that based on the previous mentioned calculation method for the downlink array weight vectors, N beams are formed, then our goal is to select the P beams from the N beams, and determine f i simultaneously with the SER upper bound as criterion. According to [17], based on the equivalent scaled AWGN channel induced by the STBC, the SERs of M-PSK ( M ≥ 4) , M-PAM, and M-QAM ( M ≥ 4) can be expressed as π PS , PSK ≈ 2 E Q 2 sin 2 γ M
l =1
where h u (t ) and a u (θ l ) denote the uplink channel response vector and array steering vector respectively. In order to calculate the DLBF weight vectors, we first utilize spectral theorem and decompose R u as R u = UΣU H
where U = [u1 , u 2 , " u N ] and Σ = diag (λ1 , λ 2 , " , λ N ) , which satisfy λ1 ≥ λ 2 ≥ " ≥ λ N . The N ×1 complex vector u1 can be written as u1 = A 1 exp( jφ1 ), A 2 exp( jφ 2 ), " , A N exp( jφ N ) T (15) Since a constant phase difference between the components of array weight vectors does not affect the BF performance, we can subtract φ 1 from φ i , and use the periodicity of exponential function simultaneously to obtain u1 = A 1 , A 2 exp( j∆φ 2 ), " , A N exp( j∆φ N ) T (16) where ∆φ 1 = 0 , ∆φ i = φ i − φ 1 (i = 2,3, " , N ) , which satisfies
]
[
PS , PAM =
(14)
[
]
0 ≤ ∆φ 2 ≤ " ≤ ∆φ N
(17a)
0 ≤ ∆φ i − ∆φ i −1 < 2π
(17b)
(19)
Therefore, if we project u *i (i = 2,3,", N ) to Ei⊥ , we can get the corresponding weight vectors for DLBF, that is w i = E i⊥ u *i (i = 2,3, " , N ) (22)
III. PROPOSED TRANSMIT SCHEME
R u = E[h u (t ) ⋅ h uH (t )] =
where f d and f u denote the carrier frequencies of downlink and uplink respectively. Next, we will employ the orthogonality of eigen-beamforming weight vectors as (19) to calculate w i (i = 2,3,", N ) .
Consider that the frequency difference between uplink and downlink around 10% is typical, and the antenna performance is more sensitive to the phases of array weights, we can do frequency calibration to u1 and get the 1st array weight vector as [16] f f w 1 = A1 , A 2 exp(− j∆φ 2 d ), " , A iN exp(− j∆φ N d ) (18) fu fu
2(M − 1) 6 γ E Q 2 M M − 1
3 PS ,QAM ≤ 4 E Q γ M − 1
(23a)
(23b)
(23c)
Joint considering (7) and (9), and using the well-known upper bound on the Q-function as Q(x ) ≤
x2 1 exp − 2 2
(24)
we can rewrite (23) with a unifying form as PS , bound = aE exp( −bγ s G
2 F
) = aE exp( −bγ s
N
∑f i =1
i
2
2 g i ) (25)
where 2 π sin for PSK M for PSK 3 for PAM for PAM and b = 2 M −1 for QAM 3 for QAM 2( M − 1)
1 (M − 1) a= M 2
Since R d can be considered as a perturbation of R u due to the frequency difference, we have E[ g i ⋅ g i* ] = w iH R d w i ≈ δ i ≈ λ i (i = 1,2, " , N ) (26)
(
)
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Furthermore, note that the mean of gi is zero, we can prewhite gi to g~i as g i = λi g~i
(27) ~ so that g i is independent identically distributed complex Gaussian random variable with zero mean and unit variance. Let z i = g~i
2
(28)
then zi has a chi-square distribution with 2 degrees of freedom, whose probability density function (pdf) is [17] p zi (z i ) = exp(− zi ) (29) Substitute (27), (28) and (29) into (25), straightforward computations yield PS , bound = aE exp − bγ s ∞
∫ ∫
=a " 0
∞ 0
exp − bγ s
N
=a
N
∑λ f
2 i i zi
i =1
N
∑λ f
i i
2
i =1
z i exp(− z1 )" exp(− z N )dz1 " dz N
1
∏ 1 + bγ λ f
s i i
i =1
(30)
2
N
fi 2
∏
N
(1 + bγ s λ i f i2 )
∑f
s.t.
i
2
=1
(31)
N
ln 1 + bγ s λ i f i2 + µ
∑ ( i =1
N
) ∑f
i
2
i =1
− 1
(32)
Differentiating the Lagrangian with respect to f i2 , and setting it to zero, we have f i2 = −
ˆ = 1 R u as R u K
determines
1
µ
−
1
(33)
bγ s λ i
through (33). However, for a given power
E S , this solution may not guarantee f i2 > 0 ∀i ∈ [1, N ] , we
thus have to reduce the number of operating beamformers from N to P. Toward this goal, we substitute the power constraint and obtain 1 1 − = 1 + µ P bγ s 1
P
∑ j =1
1 λ j
1 1 1 + P bγ s P
K
∑ hˆ
ˆH u (kN b )h u
(kN b ) . Here N b
is chosen
k =1
[γ th.P , γ th.P +1 ]
f P2+1 = f P2+ 2 = " = f N2 = 0 and
, set
obtain f12 , f 22 , " , f P2 through (35). Step5: With the calculated array weight vectors and the corresponding coefficients, the P adaptive beamformers are designed, and the space-time block codes are determined at the same time. IV. ERROR PERFORMANCE ANALYSIS
This section deals with the error performance analysis of the proposed transmit scheme under the three widely used modulations: M-PSK ( M ≥ 4) , M-PAM and square M-QAM ( M ≥ 4) . According to (23a), (9) and (27), the SER of M-PSK ( M ≥ 4 ) constellation signal can be approximated as π PS ,PSK ≈ 2 E Q 2 sin 2 γ s M
(34)
P
∑λ
1
j =1
f P2
j
−
1 λi
(i = 1,2, " , P)
(35)
> 0 , we can get the following Furthermore, consider lower bound on the required SNR to turn on P beams
P
∑ i =1
λi f i2 | g~i | 2 (37)
Let π γ s M
ζ = 2 sin 2
P
∑λ f
2 i i
| g~i | 2
(38)
i =1
the SER of M-PSK can be simplified to [18]
[ ( )]
PS ,PSK ≈ 2 E Q ζ =
Then the power coefficient of the ith beam is given by f i2 =
j =1
(36)
enough large to guarantee that hˆ u ( kNb ) and hˆ u ((k + 1) N b ) are uncorrelated. Step2: Decompose Rˆ u to yield the N eigenvalues in nonincreasing order and the N corresponding eigenvectors. Calculate the array weight vectors for DLBF with the method discussed in Section ⅢA. Step3: Determine the modulation and calculate γ th, P ( P = 1,2, " , N ) with (36). Step4: If γ s = E S N 0 of the given power E S is in the
Imposing the constraint, we can calculate µ , which in turn f i2
1 ∆ = γ th, P j
P
∑λ
i =1
i =1
For convenience, we change the cost function ε = ln(ε ) and define the Lagrangian as F=
1 P − b λ P
Finally, the proposed open-loop transmit scheme combining ABF with STBC can be summarized as follows. Step1: Sample the channel estimates hˆ u (i ) every N b blocks, and collect K channel response vectors to estimate UCCM
interval
Therefore, if we want to minimize the upper bound on SER to yield the power allocation among the multiple beams, we should solve the following constrained optimization problem ε = max
γs ≥
1 jπ
∫
c + j∞ c − j∞
(
Φ ζ ( s) 2s 1 − 2 s
)
−1
(39)
ds
Here we assume that c is in the region of convergence of Φ ζ ( s )(1 − 2 s )−1 / 2 , and a good choice is c=1/4. Φ ζ (s ) is the MGF of ζ , which is given by ∆
Φ ζ (s ) = E [exp(− sζ )] =
P
∏ i =1
2 π 1 + 2 sin M
sλi f i 2 γ s
−1
(40)
By setting Φ ∆ (s ) = Φ ζ (s )⋅ (1 − 2 s )−1 / 2 and s = c + jω , (39)
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can be rewritten as PS ,PSK =
1
π
∫
=
1 2π
+
1 2π
+∞
c − j∞
Φ ∆ (s) 1 ds = π 2s
1− x2 Re Φ ∆ c + jc x −1
∫
∫
+∞ −∞
Φ ∆ (c + jω ) dω c + jω
c Re[Φ ∆ (c + jω )] + ω Im[Φ ∆ (c + jω )] dω c2 + ω 2
−∞
∫
∫
c + j∞
dx 2 1 − x
+1
1 − x2 1 − x2 Im Φ ∆ c + jc x x
+1 −1
dx 2 1 − x
(41)
where ω = c 1− x 2 x . Use a Gauss-Chebyshev numerical integration rule with ν nodes, we have PS ,PSK ≈
1 2ν
ν
∑ {Re[Φ
∆
(c(1 + jτ k ))] + τ k Im[Φ ∆ (c(1 + jτ k ))]}+ Eν (42)
k =1
where τ k = tan((k − 1 2)π ν ) and Eν → 0 as ν → ∞ . In the numerical calculation, ν = 64 is good enough. Similarly, from (23b) the SER of M-PAM constellation signal can be expressed as PS ,PAM =
=
(M − 1) 2Mν
2(M − 1) 6 E Q γs 2 M M −1 ν
∑{Re[Φ
∆
P
∑ i =1
λi f i2 | g~i | 2
(c(1 + jτ k ))] + τ k Im[Φ ∆ (c(1 + jτ k ))]} + Eν
(43)
k =1
where Φ ∆ ( s) = (1 − 2 s )−1 / 2
P
∏ i =1
6 sλi f i 2 γ s 1 + 2 M −1
−1
.
For an M-ary, M = 2 n (n is even), square QAM signal, since it can be considered as two M –PAM signals on quadrature carries, its SER is given by [17] PS ,QAM = 1 − (1 − PS ,PAM ) 2 (44) where PS ,PAM is the SER of be obtained from
M –PAM signals, which can
1 3 γ PS , PAM = 21 − E Q M −1 M =
( M −1) 2 Mν
ν
∑{Re[Φ
∆
(c(1 + jτ k ))] + τ k Im[Φ ∆ (c(1 + jτ k ))]}+ Eν (45)
k =1
where Φ ∆ ( s) = (1 − 2 s )−1 / 2
P
∏ i =1
V.
two typical modulations: QPSK and 16-QAM. In addition, our tests focus on two typical cases: Case1 has the angular spread of 100, and Case2 has the angular spread of 600. First of all, with the assumption that the mean AOD is 600, the SER performances of QPSK as a function of SNR are plotted in Fig.2. As we can see, the SER difference between the proposed open-loop scheme and the eigen-beamforming scheme with perfect knowledge of DCCM, which is denoted with EBF+STBC, can be ignored. The scenario implies that our method can almost provide the optimal performance. In addition, it can also be found that the calculated results match very well with the simulated results in the two angular spread cases, thus the validity of the SER analysis based on MGF approach and the Gauss-Chebyshev integration is verified. However, compared with Monte-Carlo simulation method, it can dramatically reduce the computation time. Furthermore, the SER upper bounds are also included in the figure. The fixed SNR differences between the upper bound and the exact SER demonstrate the effectiveness of minimizing the SER bound.
3 2 1 + M − 1 sλi f i γ s
SER
1 ≈ jπ
−1
COMPUTER SIMULATIONS
In this section, we consider a 6-element ULA with half uplink wavelength inter-element spacing at the transmitter and a single antenna at the receiver, and do simulations to examine the performance of the proposed open-loop scheme. In all the simulations, we suppose f d = 1.1 f u , and consider
E s N 0 (dB)
Fig.2
SER versus SNR for QPSK
Next, we suppose the desired user moves from one end of a 1200 sector to another to cover all the possible AODs of the sector, and plot the average SER of both Case1 and Case2 versus SNR for QPSK and 16-QAM in Fig.3 and Fig.4 respectively. It can be observed that the proposed scheme outperforms the three other related strategies in the SNR region of interest. The other strategies are the conventional Alamouti’s STBC [19], the 1-D BF combined with STBC as in [4], and 2-D BF combined with STBC as in [5]. With small angular spread, the SER curve of the proposed scheme is almost identical to the 1-D BF strategy at low SNR, whereas for increasing SNR, it approaches the curve of 2-D BF strategy. On the other hand, in large angular spread environment, due to the adaptive power allocation, our scheme can achieve more diversity gain to enhance the SER performance, compared with the other relative typical strategies. Furthermore, we can see that the higher the SNR, the more performance improvement can be obtained by the proposed scheme.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Average SER
Finally, computer simulation results demonstrate that the proposed transmit scheme outperforms the conventional space-time block coding and the other existing related typical strategies. REFERENCES
E s N 0 (dB)
Average SER versus SNR for QPSK
Average SER
Fig.3.
E s N 0 (dB)
Fig.4.
Average SER versus SNR for 16-QAM
Finally, we changed the parameters, and repeated these types of simulations. The superiority of the proposed scheme can be clearly observed every time. More simulation results can not be given here due to the limitation of space. VI. CONCLUSION
Recently, the study of combing BF with STBC for downlink transmission has been an active research topic in wireless communication systems. In this paper, we present an open-loop scheme for correlated MIMO fading channels. We first use the UCCM to calculate the array weight vectors for DLBF. Then based on the model of equivalent scaled AWGN channel induced by the STBC, we obtain the SNR at the receiver and derive the SER upper bound. Using it as the design criterion, we derive an algorithm for adaptive power allocation, and thus develop the transmit scheme combining ABF with STBC. The main benefit of the scheme is that it can significantly reduce the system complexity while achieve the approximately optimal performance. Next, utilizing the MGF approach and the Gauss-Chebyshev integration, we derive a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations.
[1] G. Jongren, M. Skoglund, and B. Ottersten, “Combining beamforming and space-time block coding,” IEEE Trans. Information Theory, vol. 48,no. 3, pp. 611–627, Mar. 2002. [2] S. Zhou and G. B. Giannakis, “Optimal transmitter eigen-beamforming and space-time block coding based on channel mean feedback,” IEEE Trans. Signal Processing, vol. 50, no. 10, pp. 2599–2613, Oct. 2002. [3] ——, “Optimal transmitter eigen-beamforming and space-time block coding based on channel correlations,” IEEE Trans. Information Theory, vol. 49, no. 7, pp. 1673–1690, July 2003. [4] J. Liu, and E. Gunawan, “Combining ideal beamforming and Alamouti space-time block codes,” Electro. Lett., vol. 39, no.17, pp1258–1259, Aug. 2003. [5] Z. Lei, F. Chin, and Y.-C. Liang, “Combined beamforming with space-time block coding for wireless downlink transmission,” in Proc. IEEE Vehicular Technology Conf., Sep. 2002, pp. 2145–2148. [6] G. Dietl, J. Wang, P. Ding, M. D. Zoltowski, D. J. Love and W. Utschick “Hybrid Transmit Waveform Design Based on Beamforming and Orthogonal Space-Time Block Coding,” in Proc. IEEE ICASSP '05, Mar. 2005, pp.893-896. [7] J. Wang, P. Ding, M. D. Zoltowski,and D. J. Love, “Space-time coding and beamforming with partial channel state information,” in Proc. IEEE GLOBECOM '05, Dec. 2005, pp.3149 - 3153 [8] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, July 1999. [9] Siemens, Channel Model for Tx Diversity Simulations using Correlated Antennas, 3GPP Document TSG-RAN WG1 #15, R1-00-1067, Berlin, Germany, Aug. 2000. [10] H. Zhang and T. A. Gulliver, “Capacity and error probability analysis for orthogonal space-time block codes over fading channels,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 808–819, Mar. 2005. [11] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [12] C. Farsakh and J. A. Nossek, “Spatial covariance based downlink beamforming in an SDMA mobile radio system,” IEEE Trans. On Vehicular Technology, vol. 48, pp. 333-341, 1999. [13] Y.-C. Liang and F. Chin, “Downlink channel covariance matrix (DCCM) estimation and its applications in wireless DS-CDMA systems,” IEEE J. Sel. Areas Commun., vol. 19, no. 2, pp. 222–232, Feb. 2001. [14] B. K. Chalise, L Haering, and A Czylwik, “Robust uplink to downlink spatial covariance matrix transformation for downlink beamforming,” in Proc. IEEE Int. Conf. Communications, 2004, pp. 3010–3014. [15] L. Dong, “Robust beamforming for FDD mobile systems over Rayleigh fading channels,” in Proc. IEEE Int. Conf. on Electro. Inform. Technology, May 2005, pp.1-7. [16] W. L. Stutzman, and G. A. Thiele, Antenna theory and design, 2th ed. New York: John Wiley&Sons, 1998. [17] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw-Hill, 2001. [18] G. Taricco and E. Biglieri, “Exact pairwise error probability of space time codes,” IEEE Trans. Inf. Theory, vol. 48, no. 2, pp. 510–514, Feb.2002. [19] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp.1452–1458, Oct. 1998.
Adaptive Transmit Beamforming with Space-Time Block Coding for Correlated MIMO Fading Channels Min Lin1,2
Min Li3
Luxi Yang1
and
Xiaohu You1
1. College of Information Science and Engineering, Southeast University, Jiangsu, 210096, China 2. Nanjing Institute of Telecommunication Technology, Jiangsu, 210007, China 3. Nanjing Institute of Communication Engineering, Jiangsu, 210007, China E-mail:[email protected], [email protected] Abstract-In this paper, we present an open-loop transmit scheme for MIMO communications. We first use the uplink channel correlation matrix (UCCM) to calculate the array weight vectors for downlink beamforming (DLBF). Then based on the symbol error rate (SER) upper bound as design criterion, we derive an algorithm for adaptive power allocation among multiple beams, and thus develop the transmit scheme joint adaptive beamforming (ABF) with space-time block coding (STBC). The main benefit of the scheme is that it can almost achieve the optimal performance with low complexity. Next, using the moment generation function (MGF) approach and the Gauss-Chebyshev integration, we derive a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations. Finally, computer simulation results demonstrate the superiority of the open-loop transmit scheme. Key Words-Adaptive beamforming, Space-time block coding, Transmit scheme, MIMO communication.
I.
INTRODUCTION
Recently, beamforming (BF) and space-time coding (STC) are widely recognized as two key approaches to improve the capacity and reliability of wireless communication systems employing multiple transmit antennas. Although considering the array inter-element spacing and the required knowledge about the channel state information (CSI) at the transmitter, they are different and even contradictive, it was shown in [1] that the benefits of these two techniques can be achieved simultaneously using side information. Besides this work, there are a number of other articles that discuss the combination of BF with STBC. For example, in [2] and [3], the authors use either channel mean or channel correlation feedback, and generalized optimal transmit eigen-beamformer for widely used linear modulations, and coupled orthogonal STBC to increase the data rate without compromising the performance. However, these strategies require a considerable amount of feedback bandwidth, which may be a heavy burden in practice. In [4], the authors combined one-dimensional (1-D) beamforming with Alamouti’s STBC to achieve more performance gain for the case of more than two transmit antennas. In [5], the authors proposed a downlink transmission scheme combining two-dimensional (2-D) eigen-beamforming with Alamouti’s STBC to achieve both diversity gain as well as array gain. The analogous ideal was This work is supported by NSFC major project under Grant 60496310, NSFC under Grant 60672093, and NSF of Jiangsu province under Grant BK2005061
also presented in [6] and [7], where the pure diversity and the pure BF schemes were also analyzed and compared. But the major drawback of these methods is that they can not adaptively allocate the proper power among multiple beams according to the channel parameters, thus their performances are not optimal in a lot of environments. Besides sufficient error performance analysis was not given there. In this paper, we present an open-loop scheme for correlated MIMO fading channels, which is realistic in cellular communications due to close inter-element spacing of antenna array. We first use the UCCM to calculate the array weight vectors for DLBF. Then based on the model of equivalent scaled additive white Gaussian noise (AWGN) channel induced by the STBC, we obtain the signal-to-noise ratio (SNR) at the receiver and derive the SER upper bound. Using it as the design criterion, we derive an algorithm for adaptive power allocation, and thus develop the transmit scheme combining ABF with STBC. Although the result is a little analogous to the work of [3], that paper did not take the effect of STBC into account for beamformer design. Besides, the derivation and expression for the SER upper bound are also different. Next, utilizing the MGF approach and the Gauss-Chebyshev integration, we derive a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations. Finally, computer simulation results demonstrate that the proposed transmit scheme outperforms the conventional STBC and the other existing related typical strategies. Notation: ⋅ F denotes the Frobenius norm of a matrix, E[ ⋅
]
denotes expectation, diag (⋅) denotes a diagonal matrix,
(⋅) * , (⋅) T , and (⋅) H
denote conjugate, transpose, and Hermitian transpose, respectively. δ (⋅) represents the Kronecker delta function, Q (⋅) represents the classical definition of the Gaussian Q-function, tan(⋅) represents the tangent function, ln(⋅) represents the natural logarithm function, exp(⋅) represents the exponential function, Re[ ⋅ ] and Im[ ⋅ ] represent the real part and the imaginary part of a complex number respectively. The rest of the paper is organized as follows. In section II, the problem description is provided. In Section III, a novel method to calculate the array weight vectors for DLBF from UCCM is firstly presented, then based on the criterion of SER upper bound, an adaptive power allocation algorithm is
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proposed, thus the space-time transmit scheme is developed. In Section IV, using the MGF approach and the GaussChebyshev integration, a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations is derived. Finally, computer simulations are given in Section V, and conclusion is drawn in Section VI. II.
PROBLEM DESCRIPTION
In this paper, for the purpose of simplicity and without loss of generality, we consider a wireless communication with N transmit antennas and a single receive antenna. However, our results can be extended to multiple receive antennas as in [3]. Figure 1 shows the structure of transmit scheme combining BF with STBC. The serial signal to be transmitted, s, is first coded with a space-time block encoder, yielding P ( P ≤ N ) branch outputs as [8] s11 s12 " s1T s 21 s 22 " s 2T S= (1) # # # # s P1 s P 2 " s PT where T denotes the number of time slots used to encode K symbols, and s ij (i ≤ P, j ≤ T ) denotes a linear combination
of the signal constellation components and their conjugates. Then the P branch coded signals pass through P transmit beamformers, and transmitted by the N elements simultaneously. Therefore, at a particular time nT, the transmitted signal matrix X nT ( N × T ) can be expressed as X nT = WFS (2) where W = [ w 1 , w 2 , " , w P ] and F = diag ( f1 , f 2 , " , f P ) , w i ( N × 1) and f i denote the array weight vector and the corresponding coefficient of the ith beamformer. Here, we suppose w i
2 F
= 1 and
P
∑f
i
2
= 1 to guarantee that the total
i =1
transmit power keeps constant for fair comparison.
s
SpaceTime Block Coding
Fig.1
( ρ1 , θ1 )
BF1 f1w1
# BFP fPwP
SpaceTime Block Decoding
+ #
(ρ L , θ L )
Transmit scheme combining BF with STBC
Suppose the channel is a cluster-based flat fading channel and there are L paths in scatter clusters and the angle-ofdeparture (AOD) of each path varies around the mean cluster AOD θ c within the cluster angle spread ∆θ , namely, the L multipath signals are uniformly distributed in the range of [θ c − ∆θ 2 , θ c + ∆θ 2] , then the instantaneous downlink
channel response vector is given by [5] L
h d (t ) =
∑ ρ (t )a l
T d
(θ l )
(3)
l =1
where θ l and ρ l (t ) denote the AOD and the fading coefficient of the lth path signal respectively. Without loss of generality, here ρ l (t ) is modeled as complex Gaussian random variable with zero mean and unit variance. If uniform linear array (ULA) is employed at the transmitter, the downlink array steering vector a d (θ l ) is a d (θ l ) = [1, exp( jβ d d cos θ l )," , exp( j( N − 1) β d d cos θ l )]T
where d is the inter-element spacing, and β d is the downlink wavenumber. Therefore, the downlink channel correlation matrix (DCCM) R d can be obtained from [9] R d = E[h dH (t ) ⋅ h d (t )] =
1 L
L
∑a
* d
(θ l ) ⋅ a Td (θ l )
(4)
l =1
The index of t will be frequently dropped from now on for notation brevity. From (2), the received signal vector can be written as YnT = h d WFS + VnT (5) where YnT and VnT denote the 1× T received signal vector and background noise vector respectively. Let g i = h d w i (i = 1,2, " , P ) (6) we find that P equivalent virtual channels can be artificially generated by selecting w i carefully. Thus, if we define G = [ f1 g 1 , f 2 g 2 , " , f P g P ] (7) and use the model of equivalent scaled AWGN channel induced by the STBC [10], the decoded signal vector considering the code rate R can be expressed as y nT =
1 G R
2 F
(8)
s nT + v nT
where y nT denotes the K × 1 complex vector after STBC decoding from the received vector YnT , and s nT denotes the input K × 1 complex vector. R is the code rate of the STBC, namely, R = K / T , and v nT is the K × 1 complex Gaussian noise with zero mean and G 2F N 0 (2 R ) variance in each real dimension. Therefore, the effective SNR at the receiver is γ=
RE S G RN 0
2 ∆ =γ s F
G
2 F
(9)
2
where γ s = E[ s ] N 0 = E S N 0 is the transmit SNR per symbol. Therefore, if we want to get the optimal performance at the receiver, the transmit beamformer must be designed to satisfy J=
max
w1 , w 2 ,", w P
E G
2 F
= max w1 ,w 2 ,",w P
s.t. w iH w j = δ (i − j )
and
P
∑ f (w i
2
i =1
P
∑f
i
2
=1
H i
Rd wi
) (10a)
(10b)
i =1
Using the method of Rayleigh quotient [11], we can get the
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
solution of the constrained optimization problem as P
J =
∑δ
i fi
2
(11a)
i =1
w opt = vi i
(11b) where δ i and vi denote the ith eigenvalue and the corresponding eigenvector of the DCCM , that is R d = [ v1 , v 2 , " v N ]diag (δ 1 , δ 2 , " , δ N )[ v 1 , v 2 , " v N ] H (12) Here the N eigenvalues are arranged in a nonincreasing order. (i = 1,2, " , P)
A. Downlink Array Weight Vectors Computation
For a frequency division duplex (FDD) system, most of the previous works suppose that the DCCM R d is yielded through feedback. However, these strategies are unrealistic in practice due to the heavy feedback burden. Although many matrix transformation methods have been proposed (see [12]-[15], and references therein for example), they can not be used in our scheme due to estimation accuracy or computation load. So here, we make use of UCCM and present a new method to calculate the array weight vectors for DLBF. Analogous to (4), the UCCM can be expressed as [9] 1 L
L
∑a
H u (θ l ) ⋅ a u
(13)
(θ l )
w iH w j = δ (i − j )
Define the array weight vector space as (20) its orthogonal complement space can be obtained from E i⊥ = I N × N − E i (E iH E i ) −1 E iH (21) E i = [ w1 , w 2 , " , w i −1 ] (i = 2,3, " , N )
B. Adaptive Power Allocation
In this section, we will discuss the algorithm of adaptive power allocation. We first suppose that based on the previous mentioned calculation method for the downlink array weight vectors, N beams are formed, then our goal is to select the P beams from the N beams, and determine f i simultaneously with the SER upper bound as criterion. According to [17], based on the equivalent scaled AWGN channel induced by the STBC, the SERs of M-PSK ( M ≥ 4) , M-PAM, and M-QAM ( M ≥ 4) can be expressed as π PS , PSK ≈ 2 E Q 2 sin 2 γ M
l =1
where h u (t ) and a u (θ l ) denote the uplink channel response vector and array steering vector respectively. In order to calculate the DLBF weight vectors, we first utilize spectral theorem and decompose R u as R u = UΣU H
where U = [u1 , u 2 , " u N ] and Σ = diag (λ1 , λ 2 , " , λ N ) , which satisfy λ1 ≥ λ 2 ≥ " ≥ λ N . The N ×1 complex vector u1 can be written as u1 = A 1 exp( jφ1 ), A 2 exp( jφ 2 ), " , A N exp( jφ N ) T (15) Since a constant phase difference between the components of array weight vectors does not affect the BF performance, we can subtract φ 1 from φ i , and use the periodicity of exponential function simultaneously to obtain u1 = A 1 , A 2 exp( j∆φ 2 ), " , A N exp( j∆φ N ) T (16) where ∆φ 1 = 0 , ∆φ i = φ i − φ 1 (i = 2,3, " , N ) , which satisfies
]
[
PS , PAM =
(14)
[
]
0 ≤ ∆φ 2 ≤ " ≤ ∆φ N
(17a)
0 ≤ ∆φ i − ∆φ i −1 < 2π
(17b)
(19)
Therefore, if we project u *i (i = 2,3,", N ) to Ei⊥ , we can get the corresponding weight vectors for DLBF, that is w i = E i⊥ u *i (i = 2,3, " , N ) (22)
III. PROPOSED TRANSMIT SCHEME
R u = E[h u (t ) ⋅ h uH (t )] =
where f d and f u denote the carrier frequencies of downlink and uplink respectively. Next, we will employ the orthogonality of eigen-beamforming weight vectors as (19) to calculate w i (i = 2,3,", N ) .
Consider that the frequency difference between uplink and downlink around 10% is typical, and the antenna performance is more sensitive to the phases of array weights, we can do frequency calibration to u1 and get the 1st array weight vector as [16] f f w 1 = A1 , A 2 exp(− j∆φ 2 d ), " , A iN exp(− j∆φ N d ) (18) fu fu
2(M − 1) 6 γ E Q 2 M M − 1
3 PS ,QAM ≤ 4 E Q γ M − 1
(23a)
(23b)
(23c)
Joint considering (7) and (9), and using the well-known upper bound on the Q-function as Q(x ) ≤
x2 1 exp − 2 2
(24)
we can rewrite (23) with a unifying form as PS , bound = aE exp( −bγ s G
2 F
) = aE exp( −bγ s
N
∑f i =1
i
2
2 g i ) (25)
where 2 π sin for PSK M for PSK 3 for PAM for PAM and b = 2 M −1 for QAM 3 for QAM 2( M − 1)
1 (M − 1) a= M 2
Since R d can be considered as a perturbation of R u due to the frequency difference, we have E[ g i ⋅ g i* ] = w iH R d w i ≈ δ i ≈ λ i (i = 1,2, " , N ) (26)
(
)
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Furthermore, note that the mean of gi is zero, we can prewhite gi to g~i as g i = λi g~i
(27) ~ so that g i is independent identically distributed complex Gaussian random variable with zero mean and unit variance. Let z i = g~i
2
(28)
then zi has a chi-square distribution with 2 degrees of freedom, whose probability density function (pdf) is [17] p zi (z i ) = exp(− zi ) (29) Substitute (27), (28) and (29) into (25), straightforward computations yield PS , bound = aE exp − bγ s ∞
∫ ∫
=a " 0
∞ 0
exp − bγ s
N
=a
N
∑λ f
2 i i zi
i =1
N
∑λ f
i i
2
i =1
z i exp(− z1 )" exp(− z N )dz1 " dz N
1
∏ 1 + bγ λ f
s i i
i =1
(30)
2
N
fi 2
∏
N
(1 + bγ s λ i f i2 )
∑f
s.t.
i
2
=1
(31)
N
ln 1 + bγ s λ i f i2 + µ
∑ ( i =1
N
) ∑f
i
2
i =1
− 1
(32)
Differentiating the Lagrangian with respect to f i2 , and setting it to zero, we have f i2 = −
ˆ = 1 R u as R u K
determines
1
µ
−
1
(33)
bγ s λ i
through (33). However, for a given power
E S , this solution may not guarantee f i2 > 0 ∀i ∈ [1, N ] , we
thus have to reduce the number of operating beamformers from N to P. Toward this goal, we substitute the power constraint and obtain 1 1 − = 1 + µ P bγ s 1
P
∑ j =1
1 λ j
1 1 1 + P bγ s P
K
∑ hˆ
ˆH u (kN b )h u
(kN b ) . Here N b
is chosen
k =1
[γ th.P , γ th.P +1 ]
f P2+1 = f P2+ 2 = " = f N2 = 0 and
, set
obtain f12 , f 22 , " , f P2 through (35). Step5: With the calculated array weight vectors and the corresponding coefficients, the P adaptive beamformers are designed, and the space-time block codes are determined at the same time. IV. ERROR PERFORMANCE ANALYSIS
This section deals with the error performance analysis of the proposed transmit scheme under the three widely used modulations: M-PSK ( M ≥ 4) , M-PAM and square M-QAM ( M ≥ 4) . According to (23a), (9) and (27), the SER of M-PSK ( M ≥ 4 ) constellation signal can be approximated as π PS ,PSK ≈ 2 E Q 2 sin 2 γ s M
(34)
P
∑λ
1
j =1
f P2
j
−
1 λi
(i = 1,2, " , P)
(35)
> 0 , we can get the following Furthermore, consider lower bound on the required SNR to turn on P beams
P
∑ i =1
λi f i2 | g~i | 2 (37)
Let π γ s M
ζ = 2 sin 2
P
∑λ f
2 i i
| g~i | 2
(38)
i =1
the SER of M-PSK can be simplified to [18]
[ ( )]
PS ,PSK ≈ 2 E Q ζ =
Then the power coefficient of the ith beam is given by f i2 =
j =1
(36)
enough large to guarantee that hˆ u ( kNb ) and hˆ u ((k + 1) N b ) are uncorrelated. Step2: Decompose Rˆ u to yield the N eigenvalues in nonincreasing order and the N corresponding eigenvectors. Calculate the array weight vectors for DLBF with the method discussed in Section ⅢA. Step3: Determine the modulation and calculate γ th, P ( P = 1,2, " , N ) with (36). Step4: If γ s = E S N 0 of the given power E S is in the
Imposing the constraint, we can calculate µ , which in turn f i2
1 ∆ = γ th, P j
P
∑λ
i =1
i =1
For convenience, we change the cost function ε = ln(ε ) and define the Lagrangian as F=
1 P − b λ P
Finally, the proposed open-loop transmit scheme combining ABF with STBC can be summarized as follows. Step1: Sample the channel estimates hˆ u (i ) every N b blocks, and collect K channel response vectors to estimate UCCM
interval
Therefore, if we want to minimize the upper bound on SER to yield the power allocation among the multiple beams, we should solve the following constrained optimization problem ε = max
γs ≥
1 jπ
∫
c + j∞ c − j∞
(
Φ ζ ( s) 2s 1 − 2 s
)
−1
(39)
ds
Here we assume that c is in the region of convergence of Φ ζ ( s )(1 − 2 s )−1 / 2 , and a good choice is c=1/4. Φ ζ (s ) is the MGF of ζ , which is given by ∆
Φ ζ (s ) = E [exp(− sζ )] =
P
∏ i =1
2 π 1 + 2 sin M
sλi f i 2 γ s
−1
(40)
By setting Φ ∆ (s ) = Φ ζ (s )⋅ (1 − 2 s )−1 / 2 and s = c + jω , (39)
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
can be rewritten as PS ,PSK =
1
π
∫
=
1 2π
+
1 2π
+∞
c − j∞
Φ ∆ (s) 1 ds = π 2s
1− x2 Re Φ ∆ c + jc x −1
∫
∫
+∞ −∞
Φ ∆ (c + jω ) dω c + jω
c Re[Φ ∆ (c + jω )] + ω Im[Φ ∆ (c + jω )] dω c2 + ω 2
−∞
∫
∫
c + j∞
dx 2 1 − x
+1
1 − x2 1 − x2 Im Φ ∆ c + jc x x
+1 −1
dx 2 1 − x
(41)
where ω = c 1− x 2 x . Use a Gauss-Chebyshev numerical integration rule with ν nodes, we have PS ,PSK ≈
1 2ν
ν
∑ {Re[Φ
∆
(c(1 + jτ k ))] + τ k Im[Φ ∆ (c(1 + jτ k ))]}+ Eν (42)
k =1
where τ k = tan((k − 1 2)π ν ) and Eν → 0 as ν → ∞ . In the numerical calculation, ν = 64 is good enough. Similarly, from (23b) the SER of M-PAM constellation signal can be expressed as PS ,PAM =
=
(M − 1) 2Mν
2(M − 1) 6 E Q γs 2 M M −1 ν
∑{Re[Φ
∆
P
∑ i =1
λi f i2 | g~i | 2
(c(1 + jτ k ))] + τ k Im[Φ ∆ (c(1 + jτ k ))]} + Eν
(43)
k =1
where Φ ∆ ( s) = (1 − 2 s )−1 / 2
P
∏ i =1
6 sλi f i 2 γ s 1 + 2 M −1
−1
.
For an M-ary, M = 2 n (n is even), square QAM signal, since it can be considered as two M –PAM signals on quadrature carries, its SER is given by [17] PS ,QAM = 1 − (1 − PS ,PAM ) 2 (44) where PS ,PAM is the SER of be obtained from
M –PAM signals, which can
1 3 γ PS , PAM = 21 − E Q M −1 M =
( M −1) 2 Mν
ν
∑{Re[Φ
∆
(c(1 + jτ k ))] + τ k Im[Φ ∆ (c(1 + jτ k ))]}+ Eν (45)
k =1
where Φ ∆ ( s) = (1 − 2 s )−1 / 2
P
∏ i =1
V.
two typical modulations: QPSK and 16-QAM. In addition, our tests focus on two typical cases: Case1 has the angular spread of 100, and Case2 has the angular spread of 600. First of all, with the assumption that the mean AOD is 600, the SER performances of QPSK as a function of SNR are plotted in Fig.2. As we can see, the SER difference between the proposed open-loop scheme and the eigen-beamforming scheme with perfect knowledge of DCCM, which is denoted with EBF+STBC, can be ignored. The scenario implies that our method can almost provide the optimal performance. In addition, it can also be found that the calculated results match very well with the simulated results in the two angular spread cases, thus the validity of the SER analysis based on MGF approach and the Gauss-Chebyshev integration is verified. However, compared with Monte-Carlo simulation method, it can dramatically reduce the computation time. Furthermore, the SER upper bounds are also included in the figure. The fixed SNR differences between the upper bound and the exact SER demonstrate the effectiveness of minimizing the SER bound.
3 2 1 + M − 1 sλi f i γ s
SER
1 ≈ jπ
−1
COMPUTER SIMULATIONS
In this section, we consider a 6-element ULA with half uplink wavelength inter-element spacing at the transmitter and a single antenna at the receiver, and do simulations to examine the performance of the proposed open-loop scheme. In all the simulations, we suppose f d = 1.1 f u , and consider
E s N 0 (dB)
Fig.2
SER versus SNR for QPSK
Next, we suppose the desired user moves from one end of a 1200 sector to another to cover all the possible AODs of the sector, and plot the average SER of both Case1 and Case2 versus SNR for QPSK and 16-QAM in Fig.3 and Fig.4 respectively. It can be observed that the proposed scheme outperforms the three other related strategies in the SNR region of interest. The other strategies are the conventional Alamouti’s STBC [19], the 1-D BF combined with STBC as in [4], and 2-D BF combined with STBC as in [5]. With small angular spread, the SER curve of the proposed scheme is almost identical to the 1-D BF strategy at low SNR, whereas for increasing SNR, it approaches the curve of 2-D BF strategy. On the other hand, in large angular spread environment, due to the adaptive power allocation, our scheme can achieve more diversity gain to enhance the SER performance, compared with the other relative typical strategies. Furthermore, we can see that the higher the SNR, the more performance improvement can be obtained by the proposed scheme.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Average SER
Finally, computer simulation results demonstrate that the proposed transmit scheme outperforms the conventional space-time block coding and the other existing related typical strategies. REFERENCES
E s N 0 (dB)
Average SER versus SNR for QPSK
Average SER
Fig.3.
E s N 0 (dB)
Fig.4.
Average SER versus SNR for 16-QAM
Finally, we changed the parameters, and repeated these types of simulations. The superiority of the proposed scheme can be clearly observed every time. More simulation results can not be given here due to the limitation of space. VI. CONCLUSION
Recently, the study of combing BF with STBC for downlink transmission has been an active research topic in wireless communication systems. In this paper, we present an open-loop scheme for correlated MIMO fading channels. We first use the UCCM to calculate the array weight vectors for DLBF. Then based on the model of equivalent scaled AWGN channel induced by the STBC, we obtain the SNR at the receiver and derive the SER upper bound. Using it as the design criterion, we derive an algorithm for adaptive power allocation, and thus develop the transmit scheme combining ABF with STBC. The main benefit of the scheme is that it can significantly reduce the system complexity while achieve the approximately optimal performance. Next, utilizing the MGF approach and the Gauss-Chebyshev integration, we derive a simple and accurate numerical analysis method for the proposed scheme under three widely used modulations.
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