MIMO Systems, Theory and Applications 270 The above equation implies that local minimum solution does not exist and the optimum solution with minimum square error is definitely determined as well as in Eq. (8). Thus, by differentiating this equation respect to w t2 , we can obtain 22 22 22 22 2 2 11 2 1 44 2 HHHH ttttt HHH tt t Ee E Es E E EEEs ⎡⎤ ⎡⎤ ⎡⎤ ⎡⎤ ⎡⎤ ∇=− + ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦ ⎡⎤ ⎡⎤ ⎡⎤ + ⎢⎥ ⎣⎦ ⎣⎦ ⎣⎦ w HHw HHww HHw HHww HHw (14) After substituting this equation for Eq.(10) and removing the expectation operation, Eq.(10) is reduced to 2 22 22 1121 2 2 1 (1) () ()() () 2 () HHH tt ttt mm mem ms m μ ∗ ⎛⎞ += + − ⎜⎟ ⎝⎠ ww Hr HwwHw r (15) where r 2 (m)=H(w t1 (m)s 1 +w t2 (m)s 2 ). The optimum weight matrix W t is obtained by updating weight vectors of these two recursive equations, i.e., Eqs. (11) and (15). The above discussion on 2×2 MIMO system is easily extended to N t × 2 or 2 × N r MIMO system, i.e., for N t ×2 MIMO system, the received signal at the virtual receiver can be given as 11 12 11 1 1111 12 222 12 2 2 12 t t tt tt H ttN o t HH tt H o ttN t tN tN ww ww sss sss ww ww ∗∗ ∗∗ ⎡⎤ ⎡⎤ ⎡⎤ ⎢⎥ ′ ⎡ ⎤ ⎡⎤ ⎡⎤ ⎡⎤ ⎢⎥ == ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎣⎦ ′ ⎢⎥ ⎢⎥ ⎢⎥ ⎣ ⎦ ⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦ ⎢⎥ ⎣⎦ w HH HHw w w " ## " (16) where w t1 = (w t11 , ⋅ ⋅ ⋅ , w tNt1 ) T and w t2 = (w t12 , ⋅ ⋅ ⋅ , w tNt2 ) T . From this equation, it is clear that optimum weight matrixes for N t ×2 MIMO system are obtained by the same way as 2×2 MIMO case, since channel autocorrelation matrix H H H is given as N t × N t matrix. For case of 2×N t MIMO system, since the autocorrelation matrix H H H is given as 2×2 matrix, the same discussion as 2×2 MIMO case can be applied. In addition, the proposed method can be applied to case where the rank of channel matrix is more than two, e.g., when the rank of channel matrix is 3, optimum weight matrix is obtained by minimizing the error function defined so that the third weight vector w t3 is orthogonal to both the first and second weight vectors of w t1 and w t2 , where the weight vectors obtained in the previous calculation, i.e., w t1 and w t2 , are used as the fixed vectors in this case. Thus, it is obvious that this discussion can be extended to case of channel matrix with the rank of more than 3. In the proposed method, the parameter convergence speed depends on initial values of weight coefficients. When continuous data transmission is assumed, the convergence time becomes faster by employing weight vectors in last data frame as initial parameters in current recursive calculation. 2.3 Simulation results We evaluate the performance of a MIMO system using the proposed algorithm by computer simulation. For comparison purpose, obtained eigenvalues, bit error rate (BER) and capacity performance of the E-SDM systems using the proposed algorithm are compared to cases Iterative Optimization Algorithms to Determine Transmit and Receive Weights for MIMO Systems 271 with SVD. Simulation parameters are summarized in Table 1. QPSK with coherent detection is employed as modulation/demodulation scheme. Propagation model is flat uncorrelated quasistatic Rayleigh fading, where we assume that there is no correlation between paths. In the iterative calculation, an initial value of weight vector is set to (1, 0, 0, ⋅ ⋅ ⋅ , 0) T for both w t1 and w t2 . The step size of μ is set to 0.01 for w t1 and 0.0001 for w t2 , respectively. A frame structure consisting of 57 pilot and 182 data symbols in Fig.3 is employed. For simplicity, we assume that channel parameters are perfectly estimated at the receiver and sent back to the transmitter side in this paper. Number of users 1 Number of data streams 1, 2 (Number of the transmit antennas × Number of the receive antennas) (2×2), (3×2), (4×2), (2×3), (2×4) Data modulation /demodulation QPSK / Coherent detection Angular spread (Tx & Rx Station) 360° Propagation model Flat uncorrelated quasistatic Ralyleight fading Table 1. Simulation parameters Figure 4 shows the first and second eigenvalues measured by the proposed method as a function of the frame number in 2×2 MIMO system, where these eigenvalues are obtained by using channel matrix and the transmit and receive weights determined by the proposed algorithm. Figure 4 also shows eigenvalues determined by the SVD method. In Fig. 4, although the first eigenvalue obtained by the proposed method occasionally takes slightly smaller value than that of SVD, the proposed method finds almost the same eigenvectors as the theoretical value obtained by SVD. Figure 5 shows BER performance of N t x2 MIMO diversity system using the maximum ratio combining (MRC) as a function of transmit signal to noise power ratio, where average gain of channel is unity. Figure 6 also shows BER performance of 2xN r MIMO MRC diversity system. In Figs. 5 and 6, the data stream is transmitted by the first eigenpath. Therefore, it can be seen that both methods (LMS, SVD) achieve almost the same BER performance. This result suggests that the eigenvector corresponding to the highest eigenvalue is correctly detected as the first weight vector, i.e., the first eigenpath. It can be also qualitatively explained that the highest eivenvalue is first found as the most dominant parameter determining the error signal. Figures 7 and 8 show BER performance of N t ×2 and 2×N r E-MIMO, respectively. The number of data streams is set to two, since the rank of channel matrix is two. Based on the BER minimization criterion [1], the achievable BER is minimized by multiplying the transmit signal by the inverse of the corresponding eigenvalue at the transmitter. In Figs. 7 and 8, we can see that both methods (LMS and SVD) achieve almost the same BER performance. Figures 9 and 10 show the MIMO channel capacity in case of two data streams. In this paper, for simplicity, MIMO channel capacity is defined as the sum of each eigenpath channel capacity which is calculated based on Shannon channel capacity in AWGN channel [3]; C = log 2 (1+SNR) [bit/s/Hz] (17) MIMO Systems, Theory and Applications 272 The transmit power allocation for each eigenpath is determined based on the water-filling theorem [3]. In Figs.9 and 10, it can be seen that the E-SDM system with the proposed method achieves the same channel capacity as that of the ideal one (SVD). Fig. 4. Measured eigenvalues Fig. 5. Bit error rate performance (1 data stream, N t ×2) Iterative Optimization Algorithms to Determine Transmit and Receive Weights for MIMO Systems 273 Fig. 6. Bit error rate performance (1 data stream, 2×N r ) Fig. 7. Bit error rate performance (2 data stream, N t ×2) MIMO Systems, Theory and Applications 274 Fig. 8. Bit error rate performance (2 data stream, 2×N r ) Fig. 9. Channel capacity performance (N t ×2) Iterative Optimization Algorithms to Determine Transmit and Receive Weights for MIMO Systems 275 Fig. 10. Channel capacity performance (2×N r ) 3. Iterative optimization of the transmitter weights under constraint of the maximum transmit power for an antenna element in MIMO systems 3.1 System model Figure 11 shows MU-MIMO system considered in this paper, where K antenna elements and single antenna element are equipped at the Base Station (BS) and Mobile Station (MS), respectively. Single antenna is assumed for each Mobile Station (MS). The number of users in SDMA is N. The receive signal at receive antenna Y=[y 1 , ⋅ ⋅ ⋅ ,y N ] T is expressed as HH H rt r =+YWHWXWn (18) where superscript T and superscript H denote transpose and Hermitian transpose, respectively. H is N×K complex channel metrics, W t is N×K complex transmit weight matrices, W r =diag(w 1 , ⋅ ⋅ ⋅, w N ) is receive weight metrics, X=[x 1 , ⋅ ⋅ ⋅,x N ] T is transmit signal, and μ=[n 1 , ⋅ ⋅ ⋅,n N ] T is noise signal. The average power of transmit signal is unity (i.e., E[x i 2 ] =1), where E[ ] denotes ensemble average operation) and there is no correlation between each user signal (i.e., E[x i1 x i2 ] =0), the condition to keep the total average transmit power to be less than or equal to P th is given as 2 11 NK i j th ij wP == ≤ ∑∑ (19) where w ij denotes the transmit weight of antenna #j for user #i. Then, the condition to constrain the average transmit power per each antenna to be less than or equal to p th is given as MIMO Systems, Theory and Applications 276 2 1 N i j th i wp = ≤ ∑ j ∀ (1 ≤ j ≤ K) (20) Base Station Maximum permissible transmit power per antenna: th p #1 #K H User #1 User #N Base Station Maximum permissible transmit power per antenna: th p Maximum permissible transmit power per antenna: th p #1 #K H User #1 User #N Fig. 11. MU-MIMO Systems ・・・ ・・・・ H H t W r W ˆ Base Station User #1 1 n 1 w 1 ˆ n N n ˆ 1 x N x 1 ˆ y N y ˆ 1 y #1 #K Virtual Channel & Receiver Root Nyquist Filter Modulated Signal Root Nyquist Filter Modulated Signal Root Nyquist Filter Root Nyquist Filter weight control Root Nyquist Filter User #N N n N w N y Root Nyquist Filter ・・・ ・・・・ H H t W r W ˆ Base Station User #1 1 n 1 w 1 ˆ n N n ˆ 1 x N x 1 ˆ y N y ˆ 1 y #1 #K Virtual Channel & Receiver Root Nyquist Filter Modulated Signal Root Nyquist Filter Modulated Signal Root Nyquist Filter Root Nyquist Filter weight control Root Nyquist Filter User #N N n N w N y Root Nyquist Filter Fig. 12. System configurations Pilot Data symbolsN p symbolsN d Pilot Data symbolsN p symbolsN d Fig. 13. Frame format 3.2 Transmitter and receiver model Figure 12 shows the system configuration of the transmitter and receiver in MU-MIMO system considered in this paper, where the number of transmit antennas and the number of receive antennas are K and 1, respectively. A virtual channel and virtual receiver are equipped with the transmitter to estimate mean square error at the receiver side, where Iterative Optimization Algorithms to Determine Transmit and Receive Weights for MIMO Systems 277 ˆ r W =diag( 1 ˆ w, ⋅ ⋅ ⋅ , ˆ w N ) and ˆ n =[ 1 ˆ n , . . . , ˆ n N ] T denote the virtual receive weight and the virtual noise, respectively. We assume that the average power of additive white Gaussian noise (AWGN) is known to the transmitter, i.e., we assume 2 i ˆ nE ⎡ ⎤ ⎣ ⎦ =E[n i 2 ]. Then, the receive signal at the virtual receiver ˆ Y is given as ˆ ˆˆ ˆ HH H rt r =+YWHWXWn (21) The transmit weights are optimized by minimizing the error signal between transmit and receive signals at the virtual receiver under constraints given as Eqs.(19) and (20). Figure 13 shows a frame format assumed in this paper, where each frame consists of N p pilot symbols and N d data symbols. Pilot symbols are known and used for optimizing the receive weights on the receiver side. 3.3 Weight optimization a. Problem Formulation The transmit weights are optimized by minimizing the mean square error between transmit and receive signals at the virtual receiver under constraint given as Eqs. (19) and (20). From Eq.(21), the error signal between transmit signal X and receive signal at the virtual receiver ˆ Y is given as ˆ ˆˆ ˆ HH H rt r e =−=− −XYXWHWXWn (22) where e=[e 1 , . . . ,e N ]. From Eqs.(19) and (20), the problem to minimize the mean square error under two constraints can be formulated as the following constrained minimizing problem; Minimize 2 ()Ee ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ W Subject to 2 11 () 0 NK ij th ij gwP == = −≤ ∑∑ W (23) 2 1 () 0 N jijth i hwp = = −≤ ∑ W j ∀ where ⋅ denotes vector norm. W is N×(N+K) complex matrix defined as W=[W t , ˆ r W ]. b. A EIPF based Approach for Weight Optimization By introducing the extended interior penalty function (EIPF) method into the problem shown in Eq.(23), this problem can be transformed into the following non-constrained minimizing problem [11]; Minimize {} 2 () () ()Ee r ⎡⎤ +Φ +Ψ ⎢⎥ ⎣⎦ W WW Subject to 2 1 () 2() if ( ) () if ( ) g g g g ε ε ε ε − ⎧ − ≤ ⎪ Φ= ⎨ − > ⎪ ⎩ W W W W W MIMO Systems, Theory and Applications 278 1 () () K j j ψ = Ψ= ∑ W W 2 1 j () 2() j if ( ) () if ( ) j j h j h h h ε ε ε ψ ε − ⎧ − ≤ ⎪ = ⎨ ⎪ − > ⎩ W W W W W Here, ε(<0) and r(>0) denote the design parameters for non-constrained problem. In Eq.(24), ()Φ W and ()Ψ W increase rapidly as approaches to the boundary. When g(W) = ε and h j (W)=ε, the continuity of () Φ W and () Ψ W is guaranteed as well as derivatives of these two functions. Thus, Eq. (24) can be minimized by using the Steepest Descent method; W is updated as {} 2 (1) () () ()()mmEer μ ⎛⎞ ⎡⎤ += −∇ +Φ +Ψ ⎜⎟ ⎢⎥ ⎣⎦ ⎝⎠ WWwW WW (28) where μ is a step size to adjust the updating speed. ∇ w denotes a gradient with respect to W, which is defined as 11 1 1 1 ˆ ˆ K NNK N www ww w ⎡ ⎤ ∂∂∂ ⎢ ⎥ ∂∂∂ ⎢ ⎥ ⎢ ⎥ ∇= ⎢ ⎥ ∂∂ ∂ ⎢ ⎥ ⎢ ⎥ ∂∂ ∂ ⎣ ⎦ W 0 0 " #%# % " (29) where j denotes an imaginary unit and {}{} Re() Im() ij i j i j j w ww ∂∂ ∂ =+ ∂ ∂∂ , (30) {}{} ˆˆ ˆ Re() Im() ii i j ww w ∂∂ ∂ =+ ∂∂ ∂ , (31) When W is updated as in Eq. (28) at every symbols, Eq. (28) can be reduced to {} 2 (1) () () ()()mm r μ ⎡ ⎤ += −∇ +Φ +Ψ ⎢ ⎥ ⎣ ⎦ W WW eW WW. (32) 3.4 Performance evaluation Performance of MU-MIMO system using the considered algorithm is evaluated by computer simulation. Simulation parameters are shown in Table 2. As a channel model, we consider a set of 8 plane waves transmitted in random direction within the angle range of 12 degrees at the BS. Each of the plane waves has constant amplitude and takes the random phase distributed from 0 to 2π. All users are randomly distributed with a uniform distribution in a range of the coverage area of a BS. Channel states and distribution of users [...]... Transmit and Receive Weights for MIMO Systems Fig 15 SINR vs γ (K=4, SNRmax=10dB) Fig 16 SINR vs γ (K=8, SNRmax=10dB) 281 282 Fig 17 Bit Error Rate Performance (K=4) Fig 18 Bit Error Rate Performance (K=8, N=3) MIMO Systems, Theory and Applications Iterative Optimization Algorithms to Determine Transmit and Receive Weights for MIMO Systems 283 4 Conclusion We proposed optimization algorithms of transmit and. .. indexes (7 and 303 Beamforming Based on Finite-Rate Feedback Original IA, k Good IA, Πgood (k ) Bad IA, Πbad (k ) Codeword [0.8 393 − j 0. 293 9, −0.1677 + j 0.4256]T 1 7 1 0.20 19] T 2 1 4 [−0.2065 + j 0.3371, 0 .91 66 + j 0.0600]T 3 6 2 [0.3478 − j 0.3351, 0.2584 + j 0.8366] T 4 8 8 [0.10 49 + j 0.6820, 0.6537 + j 0.3106] T 5 2 5 [0.0347 − j 0.2716, 0. 093 5 − j 0 .95 72]T 6 4 3 0.47 19] T 7 3 6 [−0. 798 3 + j 0.3232,... Bold upper and lower case letters are used to denote matrices and column vectors, respectively (·) T , and (·)∗ refer to transpose and conjugate transpose, respectively · and · F stand for vector 2-norm and matrix Frobenius norm, respectively I N refers to the N × N identity matrix CN (μ, σ2 ) stands for the circularly symmetric complex Gaussian distribution with mean μ and covariance σ2 Pr and denote... substituting (8) into (7) and taking expectation, the average SER of the beamforming system can be written as PE PE = 1 π ( M −1) π M 0 exp − gPSK γS max Hc sin2 θ c∈C 2 dθ, (9) 288 MIMO Systems, Theory and Applications where the expectation is respect to the channel matrix H To find an upper bound on the average SER, we first study the expectation term in the right-hand-side of (9) , as shown in the following... IA and the original IA Furthermore, we can see that the analytical result (71) is tight Figure 9 and 10 depict the SER simulation results QPSK and 16-QAM modulations are used in (2,1,8) and (4,2,64) systems respectively In the simulations, thirty-six symbols are transmitted in a block, and ideal coherent detection is adopted In Figure 9, the SER of ‘good IA’ is much lower than that of ‘bad IA’, and. .. c|2 < z ≥ 1 − Nc (1 − z) Nt −1 , C2 ≤ z ≤ 1, ( 29) c∈C where C2 is defined in (17) The inequalities (28) and ( 29) are both tight We now treat then as approximations, and substitute ( 29) into (28) After performing the integration, the right-hand-side of (15) is obtained We then apply the results in Lemma 1 and 2 to the SER analysis Setting t = gPSK γS / sin2 θ and after some manipulations, (10) becomes exp... γS Hw where γS 2 , (|s|2 )/N0 (38) ( 39) is the average symbol SNR In the system, the beamforming vector w is determined by feedback information The receiver conveys the feedback information to the transmitter via a low-rate feedback link, which 296 MIMO Systems, Theory and Applications consists of the five blocks at the bottom of Figure 6 The ‘Index permutation’ and ‘Inverse permutation’ blocks are... , (50) where p is a parameter of the BSC; Nc = 2B ; and dH (i − 1, j − 1) denotes the Hamming distance between the binary representations of i − 1 and j − 1 The BSC satisfies (43) if p > 0 Hence, a beamforming system based on finite-rate feedback can only achieve a diversity gain of Nr , if the feedback channel is a BSC  298 MIMO Systems, Theory and Applications Remark 2(Comparison with STBC) With error-free... Optimization, Theory and Practice, 3rd Edition, Wiley-Interscience, 199 6 0 12 Beamforming Based on Finite-Rate Feedback Pengcheng Zhu1 , Lan Tang2 , Yan Wang3 , Xiaohu You4 1,3,4 National Mobile Communications Research Laboratory Southeast University, Nanjing, 210 096 4 School of Electrical Science and Engineering Nanjing University, Nanjing, 210 093 P R China 1 Introduction Multiple-input multi-output (MIMO) ,... λ1 ⎢ ⎥⎢ ⎥ ∗ H H = u1 , · · · , u Nt ⎣ ⎦⎣ ⎦ u∗ t λ Nt N (21) (22) where λ1 ≥ · · · ≥ λ Nt ≥ 0 and u1 , · · · , u Nt denote the eigenvalues and the eigenvectors, respectively We have the following inequality Hc 2 = c∗ H∗ Hc = Nt ∗ ∑ λ n | u ∗ c |2 ≥ λ1 | u1 c |2 n n =1 (23) 290 MIMO Systems, Theory and Applications Substituting (23) into (21), one obtains −N gCB t Nr ≤ lim t Nt Nr ∗ exp − tλ1 · max . Transmit and Receive Weights for MIMO Systems 281 Fig. 15. SINR vs. γ (K=4, SNR max =10dB) Fig. 16. SINR vs. γ (K=8, SNR max =10dB) MIMO Systems, Theory and Applications. antennas. MIMO Systems, Theory and Applications 280 Figures 17 and 18 show BER performance as a function of SNR max , where the number of users is set to 1∼3 for K=4 in Fig.17, and set to. [bit/s/Hz] (17) MIMO Systems, Theory and Applications 272 The transmit power allocation for each eigenpath is determined based on the water-filling theorem [3]. In Figs .9 and 10, it can be