IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
in the CFO estimation can still be achieved. We have also studied the optimal free null subcarrier placement problem for practical OFDM systems with guard bands. We have demonstrated that the proposed null subcarrier placement significantly improves the performance of the CFO estimation.
2115
Performance of Variable-Power Adaptive Modulation With Space–Time Coding and Imperfect CSI in MIMO Systems Xiangbin Yu, Shu-Hung Leung, Wai Ho Mow, Senior Member, IEEE, and Wai-Ki Wong
ACKNOWLEDGMENT The authors would like to thank Prof. T. J. Lim from the University of Toronto for the insightful discussion that they had with him and for his valuable suggestions. R EFERENCES [1] H. Liu and U. Tureli, “A high-efficiency carrier estimator for OFDM communications,” IEEE Commun. Lett., vol. 2, no. 4, pp. 104–106, Apr. 1998. [2] X. Ma, C. Tepedelenlioglu, G. Giannakis, and S. Barbarossa, “Non-dataaided carrier offset estimators for ofdm with null subcarriers: Identifiability, algorithms, and performance,” IEEE J. Sel. Areas Commun., vol. 9, no. 12, pp. 2504–2515, Dec. 2001. [3] M. Ghogho, A. Swami, and G. Giannakis, “Optimized null-subcarrier selection for CFO estimation in OFDM over frequency-selective fading channels,” in Proc. IEEE GLOBECOM, Nov. 2001, vol. 1, pp. 202–206. [4] U. Tureli, D. Kivanc, and H. Liu, “Experimental and analytical studies on a high-resolution OFDM carrier frequency offset estimator,” IEEE Trans. Veh. Technol., vol. 50, no. 2, pp. 629–643, Mar. 2001. [5] K. Sathananthan and C. Tellambura, “Probability of error calculation of OFDM systems with frequency offset,” IEEE Trans. Commun., vol. 49, no. 11, pp. 1884–1888, Nov. 2001. [6] C. Y. Wong, R. Cheng, K. Lataief, and R. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999. [7] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [8] Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High-Speed Physical Layer in the 5 GHz Band, IEEE Std. 802.11a-1999, Sep. 1999. Supplement to IEEE Std. 802.11-1999, pp. 104–106. [9] J. Medbo, H. Hallenberg, and J.-E. Berg, “Propagation characteristics at 5 GHz in typical radio-LAN scenarios,” in Proc. IEEE Veh. Technol. Conf.—Spring, May 1999, vol. 1, pp. 185–189.
Abstract—The performance analysis of multi-input–multi-output (MIMO) systems with M-ary quadrature amplitude modulation (MQAM) and a space–time block code (STBC) over flat Rayleigh fading channels for imperfect channel state information (CSI) is presented. In this paper, the optimum fading gain switching thresholds for attaining maximum spectrum efficiency (SE) subject to a target bit-error rate (BER) and an average power constraint are derived. It is shown that the Lagrange multiplier in the constrained SE optimization does exist and is unique for imperfect CSI and for single-input–single-output (SISO) systems under perfect CSI. On the other hand, the Lagrange multiplier will be unique if the existence condition for MIMO under perfect CSI is satisfied. Numerical evaluation shows that the variable-power (VP) adaptive modulation (AM) with STBC provides better SE than its constant-power (CP) counterpart. Index Terms—Adaptive modulation (AM), multi-input–multi-output (MIMO) system, space–time coding, spectrum efficiency (SE), variable power (VP).
I. I NTRODUCTION Adaptive modulation (AM) is a powerful technique for improving the spectrum efficiency (SE), which can take advantage of the timevarying nature of wireless channels to transmit data at higher rates under favorable channel conditions and to maintain the bit error rate (BER) by varying the transmit power and symbol rate under poor channel conditions [1]–[3]. The multiple-antenna approach is another well-known SE technique with diversity and/or coding gain. In particular, space–time coding in a multiantenna system provides effective transmit diversity for combating fading effects [4], [5]. Therefore, the effective combination of AM and multiple-antenna technique has received much attention in the literature [6]. Most of the above systems, however, employ constant-power (CP) AM schemes. This CP approach restricts the systems performance because the freedom of a variable power (VP) has been ignored. VP in AM has been considered in [7]–[9], which are mostly singleinput–single-output (SISO) systems. For the optimization of the SE for most of the aforementioned schemes, the Lagrange multiplier technique is used to integrate constraints to the optimization problem. However, the existence and uniqueness of the Lagrange multiplier have yet to be studied in the literature. Furthermore, no practical algorithm for computing the Lagrange multiplier has been developed. The notations we use throughout this paper are as follows. Bold uppercase and lowercase letters denote matrices and column vectors, respectively. The superscripts (·)H , (·)T , and (·)∗ denote the Manuscript received March 21, 2007; revised April 15, 2008, May 26, 2008, and July 1, 2008. First published July 25, 2008; current version published April 22, 2009. This work was supported in part by the China Postdoctoral Science Foundation under Grant 2005038242 and in part by CityU under Grant 7001851. The review of this paper was coordinated by Dr. C. Yuen. X. Yu was with the City University of Hong Kong, Kowloon, Hong Kong. He is now with the Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (e-mail:
[email protected]). S.-H. Leung and W.-K. Wong are with the City University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected]; eewkwong@ ciytu.edu.hk). W. H. Mow is with the Hong Kong University of Science and Technology, Kowloon, Hong Kong (e-mail:
[email protected]). Digital Object Identifier 10.1109/TVT.2008.2002543
0018-9545/$25.00 © 2008 IEEE
Authorized licensed use limited to: NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS. Downloaded on March 04,2010 at 06:44:23 EST from IEEE Xplore. Restrictions apply.
2116
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
Fig. 1. Block diagram of the VP AM MIMO system with space–time coding.
Hermitian transposition, transposition, and complex conjugation, respectively.
II. S YSTEM M ODEL
f (ρ) = exp −
We consider a wireless multiantenna communication system with N transmit antennas and K receive antennas operating over a flat and quasi-static Rayleigh fading channel represented by a K × N fading channel matrix H = {hkn }. The complex element hkn denotes the channel gain from the nth transmit antenna to the kth receive antenna, which is assumed to be constant over a frame of P symbols and varied from one frame to another. The channel gains are modeled as independent complex Gaussian random variables with zero mean and variance 0.5 per real dimension. A complex orthogonal space–time block code (STBC), which is represented by an L × N transmission matrix D, is used to encode P input symbols into an N -dimensional vector sequence of L time slots. The matrix D is a linear combination of P symbols satisfying the complex orthogonality DH D = b(|d1 |2 + · · · + |dP |2 )IN , where IN is the N × N identity matrix, {di }i=1,...,P are the P input symbols, and b is a constant that depends on the STBC transmission matrix [5]. Accordingly, the transmission rate of the STBC is r = P/L. Utilizing the complex orthogonality of the STBC, the decoded signal vector y after space–time block decoding can be written as [5] z y = bH2F d +
H2F S b2 H4F E {|di |2 } = . 2 bHF N0 W rN N0 W
(2)
For a Rayleigh fading channel, H2F is a central χ2 -distributed random variable with 2N K degrees of freedom. Using [10, eqs. (2), (1), and (110)] and the transformation of random variable, the probability density function (pdf) of ρ is given by
f (ρ) =
N rρ ρ
N K
1 N rρ exp − ρΓ(N K) ρ
NK N rρ (N rρ/ρ)a , Aa,N K (μ) ρ Γ(a)ρ
ρ≥0
(4)
a=1
ˆ kn h∗ } is where μ = |c|2 is the squared correlation, and c = E{h kn the correlation between the actual complex gain hkn and N K−1channel (N ˆ kn . Aa,N K (μ) = (1 − μ) K−a) μ(a−1) its estimate h a−1 is a Bernstein polynomial with the following properties: N K A (μ) = 1, and A (1) = δ(a − N K). When the a + 1,N K+1 a,N K a=0 ˆ kn = hkn , the squared correlation channel is perfectly known, i.e., h μ = 1, and using Aa,N K (1) = δ(a − N K), the pdf in (4) is equal to (3). Practically, training data are used to estimate the channel gains at the training block time via such as minimum mean squared error estimation (MMSEE). Then, the channel gains of nontraining blocks can be obtained through linear prediction with these channel estimates [13]. Based on the MMSEE and prediction, as shown in Appendix A, the correlation between the channel gain hkn and its estimate can be expressed as c = μ0 μp . The prediction with the channel estimates of the training block attributes to μ0 and μp = ερ/(1 + ερ) is dependent of the average SNR, where ε is related to the number of training data.
(1)
where d = [d1 , . . . , dP ]T . Each symbol in the transmission matrix has an average power LS/(bN P ), where S is the average transmitted power radiated from the N transmit antennas. H2F is the squared z is a P × 1 noise vector, whose elements are Frobenius norm. independent and identically distributed complex Gaussian random variables with zero mean and variance bH2F N0 W , where W is the system bandwidth. Thus, the instantaneous SNR per symbol after STBC decoding is expressed as ρ=
[11, eq. (46)] and [12, eq. (3)], the pdf of ρ with estimation errors can be expressed as
,
ρ≥0
(3)
where ρ = S/(N0 W ) is the average SNR per receive antenna, and Γ(.) is the Gamma function. When the channel estimation errors are modeled as complex Gaussian random variables, making use of the equations in
III. VP AM FOR MIMO W ITH S PACE –T IME C ODING U NDER I MPERFECT CSI A multi-input–multi-output (MIMO) system with VP AM and STBC (VP-AM-STBC-MIMO) is depicted in Fig. 1. The adaptive modulator in the system employs square M-ary quadrature amplitude modulation (MQAM). For discrete-rate MQAM, the constellation size Ml is defined as {M0 = 0, M1 = 2, and Ml = 22l−2 , l = 2, . . . , q − 1}, where M0 denotes no data transmission. The instantaneous SNR range is divided into q fading regions with switching thresholds {ρ0 , ρ1 , . . . , ρq−1 , ρq ; ρ0 = 0, ρq = ∞}. The MQAM of the constellation size Ml is used for modulation when ρ falls in the lth region [ρl , ρl+1 ). Based on {ρl }, a control policy is used to adjust the transmit power to meet the BER requirement under the constraint of average power S. The control information will instantaneously be returned to the transmitter via perfect feedback. Let Sρ be the power control for the instantaneous SNR ρ. The received SNR is given as ρSρ /S. According to [7] and [10], the BER of the square MQAM is approximately given by
BERρ ≈ 0.2 exp
1.5(Sρ /S)ρ − Ml − 1
.
(5)
The exact BER is tightly bounded from above by this approximated BER for Ml ≥ 4. Let the target BER be BER0 . For meeting this target,
Authorized licensed use limited to: NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS. Downloaded on March 04,2010 at 06:44:23 EST from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
(5) can be written as
0.2 exp
−
1.5(Sρ /S)ρ Ml − 1
= BER0 ,
ρ ∈ [ρl , ρl+1 ).
(6)
2117
and (12) yields the optimal switching thresholds {ρl , l = 1, . . . , q − 1}. Then, based on these optimal switching thresholds and using (4), we can calculate the probability that the instantaneous SNR ρ falls in the lth region [ρl , ρl+1 ), which is denoted by Pl , as
By using (6), the power adaptation is governed by
ρl+1
Sρ /S = −[2(Ml −1)/(3ρ)] ln(5BER0 ) = (Ml −1)/(Bρ)
(7)
η=
l=1
rml
f (ρ)dρ.
(8)
Sρ f (ρ)dρ = S.
L(ρ1 , ρ2 , . . . , ρq−1 ) =
⎡
ρl+1
rml
l=1
+ λ ⎣1 −
f (ρ)dρ ρl ρl+1
q−1
l=1
⎤
(Ml − 1)f (ρ)/(Bρ)dρ⎦ . (10)
ρl
Taking partial derivatives of the objective function with respect to {ρl , 1 ≤ l ≤ q − 1} and equating the partial derivatives to zero give the optimal switching thresholds as follows: For ρ1
N rρl+1 ρ
(13)
η=
q−1
rml
NK Aa,N K (μ)
Γ(a)
a=1
N rρl · Γ a, ρ
N rρl+1 − Γ a, ρ
.
(14)
We define the ensemble average BER for the VP-AM-STBC-MIMO system as
BER =
An auxiliary objective function is defined with a Lagrange multiplier λ and the use of (7) for the constraint (9) as
ρ
−Γ a,
a−1 −t where the incomplete Gamma function Γ(a, v) = ∞ e dt [14]. v t Using (13), the average SE as defined in (8) can be expressed as
(9)
ρl
Γ a,
Γ(a)
a=1
l=1
ρl
ρl+1
q−1
=
We now derive the optimal switching thresholds to maximize the SE in (8) subject to the target BER0 and the average power constraint q−1
NK Aa,N K (μ) N rρl
ρl+1
l=1
f (ρ)dρ ρl
where ρ ∈ [ρl , ρl+1 ), and B = −1.5/ ln(5BER0 ). We set Sρ /S = 0 for Ml = 0. For the discrete-rate adaptive scheme, the SE is defined as the ensemble average of the effective transmission rate. Therefore, the SE of the VP adaptive MQAM scheme can be given by q−1
Pl =
q−1
q−1
ρl+1
ml
l=1
BERl f (ρ)dρ
ml Pl
(15)
l=1
ρl
where BERl is a BER expression of MQAM of size Ml . The ensemble average BER will be dependent on the expression of BERl . If we use (5), which is an approximate formula used for the derivation of power control, for BERl of (15), we have BER = BER0 . That is, using (5) for BERl in (15) shows that the VP-AM system can perfectly achieve the target BER. The approximate formula in (5) is good for deriving a practical power control scheme but may not be so accurate for the evaluation of BER performance. In order to evaluate the BER performance of the VP-AM system, we use the following accurate BERl [7], [10]: BERl =
ζlj erfc
κlj (Ml − 1)/(Bml )
j
∂L/∂ρ1 = −rm1 f (ρ1 ) − λ [−(M1 − 1)f (ρ1 )/(Bρ1 )] = 0
where ζlj and κlj are constants for Ml [10]. Then, substituting (13) into (15) gives
gives ρ1 = λk1 = λ(M1 − 1)/(Brm1 ).
(11) BER =
For ρl , l = 2, . . . , q − 1
q−1
ml
l=1
∂L/∂ρl = r(ml−1 − ml )f (ρl ) − λ [(Ml−1 − Ml )f (ρl )/(Bρl )] = 0 gives ρl = λkl = λ(Ml − Ml−1 )/ [Br(ml − ml−1 )] .
(12)
Substituting (7), (11), and (12) into (9) yields a formula for finding the Lagrange multiplier λ. For imperfect channel state information (CSI) with the pdf defined in (4) and for perfect CSI with N K = 1, it is shown in Appendix B that the Lagrange multiplier for the optimal switching thresholds does exist and is unique. However, the Lagrange multiplier for N K > 1 under perfect CSI can exist if the condition of existence N r(Mq−1 − 1)/[(N K − 1)Bρ] > 1 is satisfied. When the Lagrange multiplier exists, its value is unique. In Appendix B, we propose to use Newton’s method, which has the quadratic convergence rate, to find the Lagrange multiplier. Applying the obtained λ to (11)
·
ζlj erfc
κlj (Ml − 1)/(Bml )
j
NK Aa,N K (μ)
a=1 q−1
l=1
Γ(a) ml
[Γ(a, N rρl /ρ) − Γ(a, N rρl+1 /ρ)]
NK Aa,N K (μ) a=1
Γ(a)
· [Γ(a, N rρl /ρ) − Γ(a, N rρl+1 /ρ)] .
(16)
The above equation is a closed-form expression of the ensemble average BER of the VP-AM-STBC-MIMO system for imperfect CSI. When μ = 1, (14) and (16) will yield the closed-form expressions of the SE and average BER of the system with perfect CSI. IV. N UMERICAL E VALUATION In this section, we use the derived theoretical performance formulas to evaluate the SE and average BER of the VP-AM-STBC-MIMO
Authorized licensed use limited to: NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS. Downloaded on March 04,2010 at 06:44:23 EST from IEEE Xplore. Restrictions apply.
2118
Fig. 2. SE of the VP-AM-STBC-MIMO scheme with multiple transmit antennas and one receive antenna for imperfect CSI and perfect CSI.
Fig. 3. SE of the CP-AM-STBC-MIMO scheme with multiple transmit antennas and one receive antenna for imperfect CSI and perfect CSI.
over a Rayleigh fading channel. Its CP counterpart is called CP-AMSTBC-MIMO. The set of MQAM constellations is {Ml }l=0,1,...,5 = {0, 2, 4, 16, 64, 256}. The target BER equals BER0 = 0.001. Different STBCs, such as G2 , G3 , H3 , G4 , and H4 , are adopted for evaluation and comparison. In the following figures, xTyR denotes a MIMO system with x transmit antennas and y receive antennas. The constant ε in μp of the correlation c for imperfect CSI is set equal to 1. In the following figures, the average SNR is ρ = S/(N0 W ). In Fig. 2, we plot the theoretical SE of the VP-AM-STBC-MIMO versus ρ for different transmit antennas and one receive antenna, where imperfect CSI (μ = μ20 μ2p ) and perfect CSI (μ = 1) cases are considered. Under imperfect CSI, μ20 = 0.3, 0.7 is employed for evaluation. As shown in Fig. 2, the 2T1R system of using G2 provides a larger SE than that of the 4T1R system using half-rate G4 or 3/4-rate H4 because G2 is a full rate code. For the same reason, the systems using H4 have a larger SE than that using the G4 code. From the result, it is observed that the larger the value of μ20 , the bigger the SE. For comparison, we also plot the SE of the CP-AM-STBC-MIMO in Fig. 3. The CP-AM-STBC-MIMO scheme with two transmit antennas can effectively improve the SE. Comparing the results of Fig. 2 and those of Fig. 3, it is found that the SE of the VP system is obviously
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
Fig. 4. Average BER of VP-AM-STBC-MIMO with multiple transmit antennas and two receive antennas for imperfect CSI and perfect CSI.
higher than that of the CP system. The former having better SE than the latter verifies that VP in accordance with the CSI is an effective means to increase the SE. It is observed that the SE curves in Fig. 3 are not as smooth as those in Fig. 2 because the switching thresholds for CP systems are obtained from the upper bound of BER (see [3, eqs. (28) and (30)]), whereas the switching thresholds for our VP systems are fully optimized. In Fig. 4, we plot the average BER of VP-AM-STBC-MIMO versus ρ for different transmit antennas and two receive antennas. It is observed that the VP-AM scheme using the G3 code performs better than that using the G2 code because the former has greater diversity than the latter. Most importantly, the average BER of the MIMO systems with moderate SNR is maintained less than the target BER0 , whereas the SE effectively increases with the SNR. This result verifies the effectiveness of VP-AM-STBC-MIMO on SE. However, the BER of the MIMO system at low SNR is above the target BER because the BER bound in (5) for power control is no longer an upper bound for Ml = 2, which is, however, often used at low SNR. It is noteworthy to mention that no result is given for 3T2R at large SNR because the condition of existence for the Lagrange multiplier is not satisfied. V. C ONCLUSION The control strategy of a VP adaptive MQAM MIMO system with STBC in Rayleigh fading channels has been presented. The optimal switching thresholds for the control strategy are derived. It is shown that the Lagrange multiplier for the constrained optimization of SE does exist and is unique for imperfect CSI and for SISO under perfect CSI. On the other hand, there is a condition of existence for the Lagrange multiplier for N K > 1 under perfect CSI. The evaluation of SE shows that the VP-AM scheme is an effective means of increasing the SE and performs better than the CP counterpart. A PPENDIX A ˆ kn h∗ } G ENERAL E XPRESSION OF c = E{h kn We assume that the system is using MMSEE with training data to estimate the channel gains and prediction to find the channel gain estimates of nontraining blocks. The signal model of the channel gain vector hl at the lth training block, where its [(k − 1)N + n]th element is the channel gain hkn , can be described as xl = Tl hl + zl
(17)
Authorized licensed use limited to: NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS. Downloaded on March 04,2010 at 06:44:23 EST from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
where xl is the received signal vector, Tl is the matrix containing the training data, and zl is the white Gaussian channel noise vector. The covariance matrices of hl and zl are equal to Chh = I and Czz = N0 W I, respectively, where I is an identity matrix. We assume TH l Tl = εSI, where ε is a constant, depending on the number of training data. Then, the MMSEE channel estimate can be expressed as [15, eqs. (12) and (26)]
ˆT ˆT , . . . , h ˆ kn = gH h h kn 1 J
T
TH 1 x1
.
(19)
T
, . . . , TH J xJ
T T
hT1 , . . . , hTJ
(a − 1)Bρ ·
q−1
−
ul e
T
h∗kn
v=0
N rA1,N K (μ) ul E1 F (λ) = Bρ q−1
q−1 N rAa,N K (μ)
(a − 1)Bρ
F (λ) = =
a=1
·
q−1
−
ul e
N rkl λ ρ
The inner summation of (21) can be rewritten as
λkl+1
ρa−2 exp(−N rρ/ρ)dρ λkl
∞ ρa−2 exp(−N rρ/ρ)dρ (22)
λkl
where ρq = ∞ is applied, and u1 = M1 − 1 = 1 and ul = Ml − Ml−1 > 0 for l = 2, . . . , q − 1 are defined. Substituting (22) into (21) and utilizing the indefinite integral
v=0
Γ(v + 1)ρv
− 1 (24)
a−2
N rkl λ ρ
N rkl − ρ
< 0.
(25)
N rA1,N K (μ) E1 (0+ ) Bρ NK N rAa,N K (μ) a=2
a−2 (N rkl λ)v
Γ(a)Bρ
+
l=1
N rkl λ ρ
F (0+ ) =
ul
−
ul e
N rkl λ ρ
NK N rAa,N K (μ)
ρa−2 exp(−N rρ/ρ)dρ = 1. (21)
q−1
(23)
The derivative F (λ) is negative because B, N , K, r, λ, kl , ul , ρ, and Aa,N K (μ) are all positive. Thus, F (λ) is a strictly monotonically decreasing function of λ. From (24), we can obtain
l=1 λkl+1
=
l=1
l=1
λkl
l=1
=1
∂F (λ) ∂λ
(20)
(Ml − 1)
Γ(v + 1)ρv
∞
(Ml − 1)
·
a−2 (N rkl λ)v
where F (∞) = −1. Differentiating F (λ) with respect to λ gives
Substituting (7), (11), (12), and (4) into (9) yields
q−1
N rkl λ ρ
where E1 (z) = z [exp(−t)/t]dt denotes the first-order exponential integral function [14]. In the following, we express solving the Lagrange multiplier from (23) as finding the root of F (λ) = 0 with
a=2
h∗kn
NK N rAa,N K (μ)
l=1
A PPENDIX B P ROPERTIES OF L AGRANGE M ULTIPLIER
a=1
+
N rkl λ ρ
l=1
where μp = (ερ)/[1 + (ερ)] with ρ = S/(N0 W ), and μ0 = H E{[hT1 , . . . , hTJ ]T h∗kn } depends on the prediction mechanism gkn and the correlations of the channel gains.
ρ
l=1
+
εS/(N0 W ) H gkn = E 1 + εS/(N0 W ) = μ0 μp
a q−1 NK Aa,N K (μ) N r
NK
1/(N0 W ) c= 1 + εS/(N0 W )
BΓ(a)
q−1
(18)
ˆ kn h∗ } is given as Therefore, the correlation c = E{h kn
H gkn
N rA1,N K (μ) ul E1 Bρ
a=2
Based on the channel gain estimates of J training blocks, the ˆ kn of nontraining blocks can be obtained by channel gain estimate h using the prediction gain vector gkn with the J obtained channel gain estimates as [13]
·E
obtains
−1
−1 ˆ l = C−1 + TH C−1 Tl TH h l zz l Czz xl hh 1/(N0 W ) = TH l xl . 1 + εS/(N0 W )
2119
xn xn−k n! + x exp(tx)dx = exp(tx) (−1)k k+1 t t (n − k)! n
n
k=1
(a − 1)Bρ
(Mq−1 − 1) − 1. (26)
According to the definition of E1 (x), we have E1 (0+ ) = ∞. This result is also valid for N K = 1 (SISO system) with μ = 1 (perfect CSI). As a result, F (λ) is monotonically decreased from ∞ to −1 as λ increases from 0+ to ∞. Thus, F (λ) = 0 has a unique solution for λ > 0. We propose to use Newton’s method to iteratively find the root of the monotonic F (λ) because it has the quadratic convergence rate when the second derivative of F (λ) is nonzero. Newton’s method is described as λn+1 = λn − F (λn )/F (λn )
(27)
where λn is the Lagrange multiplier value at the nth iteration, and the initial value λ0 may be set equal to 0.001. F (λn ) and F (λn ) are computed by (24) and (25), respectively. For the perfect CSI case, using Aa,N K (μ = 1) = δ(a − N K) yields (28), shown at the top of the next page.
Authorized licensed use limited to: NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS. Downloaded on March 04,2010 at 06:44:23 EST from IEEE Xplore. Restrictions apply.
2120
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
⎧ q−1 K−2 N ⎪ (N rkl λ)v Nr l ⎪ ul exp − N rk λ · − 1, for N K > 1 v ⎨ (N K−1)Bρ ρ Γ(v+1)ρ F (λ) =
l=1
q−1 ⎪ r ⎪ ul E1 (rkl λ/ρ) − 1, ⎩ Bρ
v=0
(28) for N K = 1
l=1
For N K > 1, from (28), we can get
(Mq−1 −1)N r Nr −1. ul −1 = (N K −1)Bρ (N K −1)Bρ q−1
F (0) =
(29)
l=1
As shown in (29), F (0) of the perfect CSI case may not be positive when K and/or ρ are large such that (Mq−1 − 1)N r/[(N K − 1)Bρ] − 1 < 0. Under this condition, we cannot find a Lagrange multiplier as a root of F (λ). Hence, the condition of existence of the Lagrange multiplier for N K > 1 under perfect CSI is (Mq−1 − 1)N r/ [(N K − 1)Bρ] > 1.
(30)
When the condition is satisfied, the Lagrange multiplier obtained from F (λ) = 0 is unique. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments. R EFERENCES [1] S. T. Chung and A. J. Goldsmith, “Degrees of freedom in adaptive modulation: A unified view,” IEEE Trans. Commun., vol. 49, no. 9, pp. 1561–1571, Sep. 2001. [2] A. Maaref and S. Aissa, “Capacity of space–time block codes in MIMO Rayleigh fading channels with adaptive transmission and estimation errors,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2568–2578, Sep. 2005. [3] M. S. Alouini and A. J. Goldsmith, “Adaptive modulation over Nakagami fading channels,” Wirel. Pers. Commun., vol. 13, no. 1/2, pp. 119–143, May 2000. [4] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block coding for wireless communications: Performance results,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 451–460, Mar. 1999. [5] H. Shin and J. H. Lee, “Performance analysis of space–time block codes over keyhole Nakagami-m fading channels,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 351–362, Mar. 2004. [6] D. V. Duong and G. E. Øien, “Optimal pilot spacing and power in rateadaptive MIMO diversity systems with imperfect CSI,” IEEE Trans. Wireless Commun., vol. 6, no. 3, pp. 845–851, Mar. 2007. [7] A. J. Goldsmith and S. G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218–1230, Oct. 1997. [8] A. Gjendemsjø, G. E. Øien, and H. Holm, “Optimal power control for discrete-rate link adaptation schemes with capacity-approaching coding,” in Proc. IEEE Global Telecommun. Conf., St. Louis, MO, Nov./Dec. 2005, pp. 3498–3502. [9] Z. Zhou, B. Vucetic, M. Dohler, and Y. Li, “MIMO systems with adaptive modulation,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1828–1842, Sep. 2005. [10] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001, ch. 2–5. [11] M. J. Gans, “The effect of Gaussian error in maximal ratio combiners,” IEEE Trans. Commun. Technol., vol. COM-19, no. 4, pp. 492–500, Aug. 1971. [12] B. R. Tomiuk, N. C. Beaulieu, and A. A. Abu-Dayya, “General forms for maximal ratio diversity with weighting errors,” IEEE Trans. Commun., vol. 47, no. 4, pp. 488–492, Apr. 1999.
[13] A. Stamoulis, S. N. Diggavi, and N. Al-Dhahir, “Intercarrier interference in MIMO OFDM,” IEEE Trans. Signal Process, vol. 50, no. 10, pp. 2451–2464, Oct. 2002. [14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic, 2000, ch. 8/9. [15] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice–Hall, 1993, ch. 12.
Interleaved Random Space–Time Coding for Multisource Cooperation Rong Zhang and Lajos Hanzo, Fellow, IEEE
Abstract—In this paper, we propose a novel distributed interleaved random space–time code (IR-STC) designed for multisource cooperation (MSC) employing various relaying techniques, namely, amplify–forward, decode–forward, soft decode–forward, and differential decode–forward. We introduce a two-phase communication regime for our IR-STC-aided MSC and propose a novel structured embedded (SE) random interleavergeneration method. We also characterize the achievable performance of our proposed IR-STC design in conjunction with various relaying techniques communicating over different intersource Nakagami-m fading channels. Index Terms—Cooperative communications, interleaver division multiplexing, relaying, space–time coding.
I. I NTRODUCTION Cooperative diversity [1]–[4] relying on a distributed (virtual) multiple-input–multiple-output (MIMO) system is capable of eliminating the correlated-fading-induced spatial diversity gain erosion of colocated MIMO elements. Hence, this novel technique is capable of improving the achievable performance while supporting a high throughput, as well as providing improved cell-edge coverage [5]. It has the potential to beneficially combine the traditional infrastructurebased wireless networks and the ad hoc wireless network philosophy [6]. Recently, the cooperative multiple access channel has attracted substantial research interests, where multiple sources forming a cluster of cooperating nodes communicate with the destination, which is also known as multisource cooperation (MSC) [7]–[9]. Manuscript received June 6, 2008; revised August 31, 2008 and October 18, 2008. First published October 31, 2008; current version published April 22, 2009. This work formed part of the Core 4 Research Programme of the Virtual Center of Excellence in Mobile and Personal Communications, Mobile VCE (www.mobilevce.com) and was supported by the Engineering and Physical Sciences Research Council (PSRC). Fully detailed technical reports on this work are available to Industrial Members of Mobile VCE. The review of this paper was coordinated by Dr. G. Bauch. The authors are with the School of Electronics and Computer Science, University of Southampton, Southampton SO171BJ, U.K. (e-mail: rz05r@ecs. soton.ac.uk;
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2008.2008416
0018-9545/$25.00 © 2008 IEEE
Authorized licensed use limited to: NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS. Downloaded on March 04,2010 at 06:44:23 EST from IEEE Xplore. Restrictions apply.
