Joint Precoding and Beamforming Design for the ... - Semantic Scholar
Joint Precoding and Beamforming Design for the Downlink in a Multiuser MIMO System Roya Doostnejad, Teng Joon Lim and Elvino Sousa Department of ECE, University of Toronto, 10 King’s college Road, Toronto, ON, M5S 3G4, Canada Email:{roya,limtj,sousa}@comm.toronto.edu Phone: 416 − 978 − 8672
Abstract— Assuming perfect channel knowledge at the transmitter, we study space-time beamforming for the downlink in a multiuser multi-input multi-output (MIMO) channel while a nonlinear interference pre-cancelation is presumed at the transmitter. The antenna arrays may be employed at both the transmitter and the receivers. The optimum transmit/receive beam vectors are obtained based on a minimum mean-squared error (MMSE) criterion and a per-user power constraint. In frequency selective fading channels, where orthogonal frequency division multiplexing (OFDM) is applied, the precoding and beamforming design is extended over space and frequency dimensions as well. In fact the proposed algorithm offers a unique method for assigning frequency bins in a MIMO-OFDMA system. The bit error rate performance of the proposed algorithm is assessed by computer simulations. Index Terms— MIMO, OFDM, Beamforming, Precoding, Downlink.
I. I NTRODUCTION Precoding and beamforming design for multi-antenna broadcast channel has received significant attention as a promising solution to provide reliable high data rate communication in wireless downlink channels. This typically refers to a situation where independent data streams are to be transmitted from a multi-antenna base station (BS) to several uncoordinated receivers. In MIMO single user channels, there are a few works for joint transmit/receive beamforming design based on MMSE criterion [1], [2]. In MIMO multiuser channels, the main challenge is that the receivers cannot cooperate with each other while each user is suffering from the interference from all other users. In [3], [4], and [5], transmit preprocessing methods are proposed for the downlink of multiuser MIMO systems. In these techniques, the multiuser MIMO downlink channel is decomposed into parallel independent single user MIMO channels so that the multiuser interference is completely canceled and then single user schemes can be applied over each independent channel. However, all these techniques are highly restricted on the number of transmit/receive antennas in the sense that the number of transmit antennas must be greater than the total number of the receive antennas at different users. In practical situations where the number of transmit antennas is limited, this constraint severely restricts the number of receive antennas which results in restriction on receive diversity. In [6] zero forcing preequalization is implemented by Tomlinson Harashima Precod0-7803-9182-9/05/$20.00 ©2005 IEEE.
ing (THP) at the transmitter. Although this scheme is effective and outperforms linear pre-equalization schemes but the same restriction, as explained before, exists on the number of transmit/receive antennas. Also similar to conventional zero forcing methods, the performance is degraded in low signal-to-noise ratios (SNRs). In this work, the multiuser MMSE beamforming is combined with non-linear Dirty Paper Coding (DPC) proposed in [7] to derive a more efficient algorithm to minimize the mean squared error between transmit and receive data streams. The original idea of DPC [7], which is for single-antenna channels, indicates that if the interference is known non-causally at the transmitter but not at the receivers, a precoding strategy can be applied based on this knowledge which results the same capacity as if the interference does not exist. Since DPC is applied at the BS, the available single user algorithms can be extended to be applicable over individual single user channels. The algorithm proposed here is along with the work in [2]. The receive beam vectors are obtained based on MMSE criterion, and the transmit beam vectors are obtained through singularvalue-decomposition (SVD) of the whitened channels. In fact part of the multiuser interference is cancelled out at the BS and the residual interference at each user is minimized by transmit/receive beamforming. In this proposed scheme, there is no limitation on the number of transmit/receive antennas. Also, when the beamforming is generalized to non-spatial dimensions such as time and frequency, the proposed algorithm offers a unique method for assigning time slots in a MIMO-TDMA system, and frequency bins in a MIMO-OFDMA system, to users. The remainder of this paper is organized as follows. The problem formulation is given in the next section. The MMSE beamforming is discussed in section III. Space-Time and Space-Frequency beamforming are introduced in sections IV and V respectively. Simulation results is given in section VI, and we conclude with section VII. The notations used in this paper are as follows. We use boldface lower case letters to denote vectors, boldface upper case letters to denote matrices, and E for expectation. The superscripts ·∗ , ·T , and ·H denote conjugate, transpose and conjugate transpose respectively. I denotes the identity matrix, det(·), T r(·) and Diag(·) are abbreviates for the determinant, trace and block diagonal matrix respectively. M(i, j) represents the element in the ith row and the jth column of the matrix M.
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II. P ROBLEM F ORMULATION A. Signal Model The downlink in a multiuser MIMO channel is considered with K users, where the base station has t transmit antennas and each user receiver has ri antennas for i = 1, ..., K. Then the baseband signal received by user i is:
propagation may be avoided. This structure is also suggested and analyzed in [8] to determine the sum capacity of the broadcast channel.
DEC
yi = Hi
K
Bi xi + ni ,
i = 1, ..., K,
(1)
i=1
where Hi ∈ Cri ×t is the matrix of flat-fading channel gains between the transmitter and the ith receiver’s antennas, and ni is a ri × 1 vector of i.i.d. complex Gaussian random variables with zero mean, and the variance of σi2 , representing receiver noise for user i. It is assumed that in the ith MIMO channel, 2 each flat fading coefficient has the same variance of σh,i , and that the t.ri channels are independent. xi is the transmitted symbol vector to the user i and Bi is the transmit beamforming matrix with tr(BH i B i ) ≤ pi , when pi is the transmit power constraint for user i. The transmitter is supposed to send zi symbols to the ith user. Assuming that the signal dimensionality at the receiver has to be no smaller than the number of mutually interfering symbols to achieve an acceptable bit error probability (BEP), 1 ≤ zi ≤ min(t, ri ),
(2)
If we define the aggregate received signal vector yT = T T T , the overall channel matrix H = HT1 · · · HTK , y1 · · · yK the beamforming matrix B = [B1 · · · BK ], and the transmit T signal vector x = xT1 · · · xTK , the multi-user signal model is y = HBx + n.
(3)
Fig. 1. DFE at the Receiver
A regular subtraction at the transmitter may cause excessive power enhancement at the transmitter. The modulo-arithmetics (Tomlinson-Harashima Precoding (THP)) is used at both transmitter and receiver in order to minimize power enhancement [9], [10]. If the data symbols(x) are from an M −ary constellation, the encoder transmits x modulo−M , to constrain the symbols into the interval [−M/2, +M/2).1 The decoder has to perform a modulo−M operation to recover the original signal. To design THP, the encoding order K, ..., 1 is assumed which means that the interference caused by the users i + 1, ..., K to the user i is known prior to transmission. Hence G is a block upper triangular matrix with ones on the diagonal: ⎤ ⎡ I G1 ⎥ ⎢ 0 I G2 ⎥ ⎢ ⎥ ⎢ . .. (4) G=⎢ ⎥, ⎥ ⎢ ⎦ ⎣ 0 I GK−1 0 ... I where Gi ∈ Czi ×ζi with ζi =
K
m=i+1 zi .
B. Precoding In point-to-point communication systems or multiple access channels in which the receivers are coordinated, a minimum mean-square error decision-feedback equalizer (MMSE-DFE) can be applied to untangle the interference. The structure of DFE at the receiver is shown in Fig. 1. After each symbol is detected, it is subtracted from the received signal before the next symbol is detected. As a consequence error propagation may occur in these receivers. For multiple access channel, this is equivalent with serial interference cancelation (SIC). In the downlink channel, unlike in the other two, receivers are uncoordinated and hence it is impossible to implement joint processing techniques such as DFE. However, if the channel is known to the transmitter, the decision feedback receiver can be implemented at the transmitter as shown in Fig. 2. A well-known example of this type of “pre-equalizer” is the Tomlinson-Harashima precoder (THP), introduced in the context of equalization of ISI channels. The matrix form of THP is shown in Fig. 2, and applies to the multi-user downlink channel. This presents the same idea as DPC: if the interference is known at the transmitter, it can be pre-subtracted prior to transmission. In this way, the noise enhancement as well as error
MOD
MOD
DEC
Fig. 2. Implementation of THP
In a non-cooperative downlink channel, feedforward matrix (Fig. 2) has to be block diagonal H F = Diag(FH 1 , . . . , FK ),
(5)
where FH j is a zj × rj matrix for j = 1, ..., K. We attempt to solve the following problem: Problem statement: Using a per user power constraint, design the beamforming matrix (Bi ), and the feedforward and feedback matrices Fi and Gi respectively in order to minimize the mean squared error between the transmitted and received 1 It is well known that by implementing THP, there is still a power loss equal to M 2 /M 2 − 1 at the transmitter [9].
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data symbols for each individual user i. The design is performed to either • minimize the sum of the MSE for all the symbols transmitted to the same user or • maintain the same MSE for all the transmitted symbols to the same user. III. MMSE B EAMFORMING /P RECODING To derive the transmit/receive beamforming matrices in the downlink, because of the interference pre-cancellation at the BS with the encoding order K, ..., 1, the first user is interferencefree and can be considered as a single user. After designing the beamforming matrices for the first user, the beamforming matrices for the second user may be designed, treating the first user as known interference. This procedure can be continued for the users l = 3, ..., K sequentially, treating users i = 1, ...l − 1 as known interferences. Therefore, available single user schemes ( [1], [2]) can be extended to design the beamforming matrices for each user individually. The received signal for user i is: i Bj xj ) + ni yi = Hi (
B. Optimum Transmit Matrix With the optimal receive matrix Fi in (11), user i’s mean square error is at its minimum value for the given {Bi }, and is given by Eoi
H H H −1 = I − BH Hi Bi i Hi (Hi Bi Bi Hi + R(n+I)i ) H −1 = (I + Bi RHi Bi ) , (12) where the last line is obtained by the matrix inversion lemma. The transmit beamforming matrix Bi is designed based on the MSE criterion of the user i. • Sum-MSE Minimization Design Corollary 1: Minimizing the sum of the diagonal elements of MSE matrix Ei with respect to Bi
subject to
= Ui Σi Bopt i
(15)
(6)
where pi is the power constraint for user i, Ui denotes the zi eigenvectors associated with the zi largest eigenvalues, −1 λi , of RHi = HH i R(n+I)i Hi and Σi is a diagonal zi × zi matrix with the diagonal elements of i,l + −1/2 −1/2 i,l = µi λi,l − λ−1 , i,l
H Rri = Hi Bi BH i Hi + R(n+I)i ,
(8)
i−1 H R(n+I)i = σ 2 I + Hi ( Bj BH j )Hi
where [θ]+ = θ, θ ≥ 0; 0, θ < 0, λi,l are the zi largest eigenvalues of RHi , and µi is chosen to satisfy the transmit power constraint for user i:
(9)
zi 1/2 µi
j=1
is the covariance matrix of the noise and residual interference seen by user i. A. Optimum Receive Matrix The MSE of the lth symbol transmitted to user i is the lth diagonal element of Ei . Given a transmit beamforming matrix {Bi }, the optimal receive matrix Fi is obtained such that the diagonal elements of Ei are minimized. This is equivalent to solve: (10) min cH Ei c ∀c FH i
Differentiating cH Ei c = T r(Ei ccH ) with respect to FH i and setting the result to zero yields Fopt i
(14)
(7)
= E[(ˆ xi − xi )(ˆ xi − xi )H ], H H H = Fi Rri Fi + I − FH i Hi Bi − Bi Hi Fi ,
with
tr(BH i Bi ) ≤ pi
is given by
ˆ i = FH If the estimated data at the user i is defined as x i yi , the mean squared error matrix will be then defined as the covariance matrix of the error vector:
where
(13)
Bi
j=1
Ei
min tr[Ei ]
= R−1 r i H i Bi H −1 = (Hi Bi BH Hi Bi i Hi + R(n+I)i )
(11)
which is the linear MMSE receiver (Wiener filter). This receiver is optimum for both criteria in the problem stated in sectionII. A similar derivation is given for the single user case in [2]. Note that because of the successive nature of the design algorithm, R(n+I)i ) is known before designing the beamforming matrices Bi and Fi for user i.
=
j=1
pi +
(16)
−1/2
λj
zi
j=1
λ−1 j
(17)
Proof: This optimization problem is proved for single user case in [2]. The key difference is that the noise covariance term is replaced here by the noise-plus-interference covariance matrix. Since the derivations in [2] are valid for any arbitrary noise covariance matrix, the results are directly applicable. It can be easily shown that FH i Hi Bi becomes a diagonal matrix which means the MIMO channel for each user is diagonalized. By substituting Bi from (15) in (12) the MMSE matrix is −1 , (18) Eoi = (I + ΣH i Di Σi ) in which Di is a diagonal matrix with the elements of the largest eigenvalues of RHi in an increasing order. Therefore the MMSE matrix for each user is diagonal with diagonal elements in decreasing order. • Equal MSE Design Corollary 2: Minimizing the sum of the MSE with the same constraint as (14) while the same MSE is maintained for all the symbols transmitted to the same user is given by: = Ui Σi Q (19) Bopt i
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where Ui , Σi are defined in corollary 1 and Q could be any rotation matrix that satisfies |Q(i, k)| = |Q(i, l)|, ∀i, k, l such as the DFT matrix [2]. It can be seen that in this design, neither MIMO channel, nor the MSE matrix is diagonal. However, the MSE matrix has identical diagonal elements. In fact for any given Bi , a rotation −1 Q has matrix Q can be found so that QH (I + BH i RHi Bi ) identical diagonal elements. The sum of the diagonal elements of MSE matrix is the same regardless of Q. A similar design for single user case is also proposed in [1]. The beamforming matrix Bi is designed to minimize a weighted sum of symbols MSE. The weights can be adjusted such that all the symbols have the same MSE. In the following, a corollary to this theorem applicable to the current work is presented. Corollary 3:The MMSE beamforming matrix that ensures equal MSE for all symbols transmitted to the same user is given by: −1/2 = µ1/2 Ui Di , (20) Bopt i
p i and Ui , Di are defined the same as where µ = T r(D−1 )
to find the optimal ordering, the algorithm has to be examined over K! rearrangements which is not practical. Here, we have applied a suboptimal low complexity method. The users are ordered based on the square Frobenius norm of their channel matrix, Hi 2F for i = 1, ..., K: Hi 2F =
C. Precoding Matrix Design The feedback matrix G is obtained by MMSE solution as well. It can be easily proved that to minimize the MSE for each user symbol Gi = FH i Hi [Bi+1 , ...., BK ] , i = 1, ..., K
(22)
Table 1: The algorithm for precoding and MMSE beamforming Initialization: C = 0, Rn = σ 2 I. f or i = 1 : K R(n+I)i = Rn + Hi ∗ C ∗ HH i , −1 RHi = HH i ∗ R(n+I)i ∗ Hi , Bi = Ui Σi , H −1 Fi = (Hi Bi BH Hi Bi , i Hi + R(n+I)i ) H C = C + Bi ∗ Bi . end
f or j = K − 1 : −1 : 1 Gj = FH j Hj [Bj+1 , ...., BK ] . end
IV. S PACE -T IME S PREADING In the proposed scheme, there is no limitation on the number of users in the system. However, as the number of active users is increased, the MMSE is increased for each user too. So if there is a quality of service constraint for each user, the number of users we can accommodate is in fact limited. On the other hand, if we are allowed to increase the bandwidth by spreading each symbol over time, we can increase the number of active users without any performance loss. The beamforming is now optimized over two dimensions: space and time. Instead of assigning a beam vector to each user symbol, one t × G beamforming matrix, Φil , i = 1, ..., K, l = 1, ..., zi is assigned to each user symbol where G is the processing gain, and T = GTc is the symbol period where Tc is the chip period. Then the transmitted signal is X=
(21)
Performance of the proposed algorithm is also dependent on the precoding ordering. Defining the optimal ordering which is related to channel fading states, is an open problem. Indeed,
r,t
H2i (r, t),
when Hi (r, t) is the (r, t) element of the matrix Hi . The user which is supposed not to see any interference, is the one with the smallest channel matrix norm, and the user who sees the full interference (in this work, the Kth user) is the one with the largest channel matrix norm. This method specially makes sense, if all users have the same power constraint.
i
Corollary 1. The detailed proof for single user case is in [1]. The results are directly applicable to the current multiuser case with the same reasoning as Corollary 1. Note that the designs in both Corollary 2 and Corollary 3 provide equal MSE for all symbols transmitted to the same user, however, in the later case (20) the channel and the MSE matrix is diagonalized as is opposed to the previous one (19). In [11], a similar approach to the design proposed in (13) is applied in the downlink of a multiuser system to transmit multiple symbols to multiple users based on a per-user power constraint. However, interference pre-cancellation is not presumed at the BS and therefore R(n+I)i is dependent on other users beam vectors, and the MSE for each user is dependent on other users MSE which makes the algorithm very complicated. The authors have proposed an iterative algorithm in which each transmit matrix is obtained by fixing the others but, there is no guarantee for the convergence of the algorithm, and it may converge to a false or local optimum. Here, because of the interference pre-cancellation, R(n+I)i for i = 1, ..., K is only dependent on the other users beamforming matrices, Bl for l ≤ i − 1 (see (9)). Therefore, it is possible to design the transmit beamforming matrices, Bl sequentially for l = 1, ..., K. The algorithm for the transmit beamforming matrix design (15) is summarized in table 1. For the other two designs, it is enough to substitute the 5th line of the table by (19) or (20).
zi K
xil Φil .
(23)
i=1 l=1
The channel is assumed to be constant over at least one symbol period. The signal received by user i, over G channel uses and
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ri receiver antennas, is Yi =
zj K
xjl Hi Φjl + Ni ∈ Cri ×G
(24)
j=1 l=1
where Ni is a matrix of i.i.d. complex Gaussian random variables with zero mean and the variance of σi2 , representing receiver noise. By stacking the columns of Yi , we obtain ˜i ˜i = H y
K
˜ i, Ci x i + n
(25)
i=1
˜ i = Diag(Hi , . . . , Hi ), ˜ i = vec(Yi ), H where y ˜ i = vec(Ni ) and Ci = xi = [xi1 , ..., xizi ], n [vec(Φ1 ), . . . , vec(Φzi )]. Essentially, each column of Ci corresponds to a space-time modulation waveform for one data stream. This presents exactly the same system model as (1). Therefore we can apply the same algorithm explained in section III to design the matrices Ci , Fi , and Gi for each user. As an alternative to the above methodology, one may think of choosing a subset of users randomly in each chip time-slot (TDM fashion) and perform the original algorithm to design the proper beam vectors over each subset independently. However, by applying the algorithm once over all users as presented in signal model (25), the subsets of the users that result in the least amount of interference are automatically selected to be transmitted at the same time. In fact, each user symbol is transmitted through the subchannel with the highest gain or equivalently with the minimum amount of interference. The result of the algorithm is to assign each user symbol to only one time slot (which has the least amount of interference), rather than spreading each user symbol over all time slots. As will be shown by numerical results in section VI, even though the channels are constant during one symbol time, the algorithm benefits from selecting the simultaneous users intelligently. The other interesting point is that when we transmit multiple symbols to each user (at most min(t, ri )) and the beam vectors are designed based on either of the schemes in (15), (19), or (20), if G ≥ min(t, ri ), no two symbols for the same user will be transmitted at the same time slot. This is because the beam vectors are obtained by the largest eigenvectors of RH ˜ i . Since ˜ Hi is a block diagonal matrix with min(t, ri ) different eigenvalues and G min(t, ri ) eigenvectors, all G largest eigenvectors are orthogonal in time. The importance of this simple fact is that although we do not have interference pre-cancellation for the symbols transmitted to the same user, but there is no interference between the transmitted symbols to the same user because their beam vectors are orthogonal in time. V. S PACE -F REQUENCY S PREADING In this section, we consider the frequency selective fading channel. The frequency selective fading can severely degrade system performance by causing inter-symbol interference (ISI) and result in an irreducible bit error rate (BER) which imposes an upper limit on the data rate.
In order to transmit over frequency-selective MIMO channels, the same design principles may be applied as long as the channel can be transformed into one resembling flat fading. OFDM is a practical solution which transforms a frequency selective fading channel into parallel, possibly correlated flat fading channels. Frequency components which are separated by the channel coherence bandwidth or more may be assumed to be totally uncorrelated. Therefore the channel frequency diversity can also be exploited through the proper design of an OFDM system. An inverse fast Fourier transform is performed over a sequence of symbols which are transmitted over each antenna. If the number of FFT bins is Nf , we form a super-symbol of the original symbols and pass that through an IFFT operation: ˜= S
zi K
xil Φil ∈ CNf ×t ,
(26)
i=1 l=1
while we allow Φi to have non-zero values in any row. This implies that one symbol can occupy any frequency bin. In each OFDM block which is Nf time slots, the number of K symbols are transmitted, and the channel is supposed to be constant. At each user receive antenna, the cyclic prefix is first removed from each OFDM block and then it is fed to an OFDM demodulator. After taking Nf -point FFT we end up with Nf MIMO flat fading channels in Nf subcarriers. If we stack the signals of the ˜i, received antennas in Nf time slots, in a Nf ri × 1 vector y with the same notation as section IV, we have ⎤ ⎡ 0 Hi (0) ⎥ K ⎢ .. ˜ i(27) ˜i = ⎣ y ⎦ i=1 Ci xi + n . 0
Hi (Nf − 1)
K ˜ ˜ i, = Hi i=1 Ci xi + n
(28)
where Hi (n) for n = 0, . . . , Nf − 1 is the ri × t matrix whose each component is the nth frequency component of the channel of the user i between each single transmit-receive antenna pair. This gives the same system model as (1) and (25) and therefore the algorithm proposed in section III is applicable to design Ci , Fi , and Gi . As a result of the beamforming design, each user signal is transmitted over the channel frequency with the highest gain and the least amount of interference. The advantage of performing the optimization over all frequency bins is that, instead of spreading the signal of each user over all frequency bins, each user is only allocated to the best channel frequency (for a fixed precoding order). This is the same as what explained for space-time beamforming design in section IV. The only difference is that in the previous one, the only parameter which is changing over time-slots is the interference, while in the current case, the channel is also different over the time-slots/frequencies, and therefore, the frequency diversity may be achieved as well. VI. S IMULATION R ESULTS In this section we provide some numerical results to illustrate the performance of the proposed algorithm. In all of the simulations a Rayleigh fading channel is assumed with a zero
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Fig. 3. The performance of space-time spreading for different number of receive antennas, t = 4, G = 4, K = 8, z = 2.
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Fig. 4. The performance of space-frequency spreading compared with MMSE beamforming over flat fading channel t = 2, r = 2, Nf = 8, z × K = 16. 0
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Pe for user 2 Average Pe Pe for user 1
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mean uncorrelated complex Gaussian noise across the receive antennas (Rn = σ 2 I). The elements of the channel matrix H are generated as independent and identically distributed (i.i.d.) samples of a complex Gaussian process with zero mean and unit variance. The channel matrix is known at the transmitter. Each user is assumed to have full knowledge about its own channel and also the transmit matrix B. The results are presented based on the average probability of error (Pe ) versus signal-to-noise ratio (SNR(dB)) for each user symbol. Without loss of generality, the same power constraints, the same number of receive antennas and the same number of transmit symbols are assumed for all users (pi = p, ri = r, zi = z, for i = 1, ..., K). Here, the proposed algorithm in section III is recalled as MMSE algorithm and the design of transmit beamforming matrix for each user is based on Sum-MSE minimization (15). In Fig. 3, space-time precoding/beamforming is performed for K = 8 users to transmit two symbols for each user, when we have t = 4 transmit antennas and processing gain, G = 4 which means the signal is spread over 4 time slots. The average probability of error is compared for different number of receive antennas. It can be seen that by increasing the number of receive antennas from r = 2 to r = 4, the performance is improved by 7.5 (dB) because of receive diversity. Also for r = 4, the average Pe is compared with the Pe of the eighth user. Note that, since the precoding order is 1, ..., 8, the eighth user is the one who sees full interference from other users and may be expected the worst performance compared with other users. As is shown for low SNR, the performance for this user is about 1 − 2 (dB) less than the average. In Fig. 4, the performance of MMSE algorithm over a flat fading channel is compared with space-frequency MMSE precoding/beamforming over frequency selective channel when we have t = 2 transmit, and r = 2 receive antennas. In frequency selective channel, two-tap channels are simulated for each user with a maximum delay spread of υ = 1 chip and 8−point FFT is applied to implement OFDM. As explained in section V, once the proposed MMSE algorithm is applied to all frequency bins,
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Fig. 5. Average Pe compared with Pe for each individual user, t = 2, r = 2, K = 2, z = 1.
each user symbol is assigned to one frequency bin. We examined both cases of transmitting one symbol per user for K = 16 and two symbols per user for K = 8. It can be seen that the performance improvement over flat fading channel is about 5(dB) because of the frequency diversity which can be explored in space-frequency spreading. In Fig. 5 and Fig. 6, the average probability of error is compared with the probability of error of each individual user. The first user is the one who does not see any interference, and the second user in Fig. 5 and the eighth user in Fig. 6 are the ones who see the full interference. VII. C ONCLUSIONS A joint transmitter/receiver precoding and beamforming design is proposed for the downlink in a MIMO multiuser channel by using DPC. There is no constraint on the number of
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8
Fig. 6. Average Pe compared with Pe for each individual user in spacefrequency spreading, t = 2, r = 2, Nf = 8, K = 8, z = 2.
transmit/receive antennas and because of the interference precancellation at the BS, the single user MMSE beamforming schemes can be extended to the current multiuser channel. To increase the number of active users over the number of transmit antennas without loss of performance, the signal is spread over both time and space. The space-time multiplexing matrices are designed to minimize the MSE between the transmitted and received data vector. In frequency selective channels, where OFDM is applied, the proposed design is able to explore full frequency diversity as well; each user is allocated to the strongest frequency channel with the least amount of interference. R EFERENCES [1] H. Sampath, P. Stoica, and A. Paulraj. Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion. IEEE Trans. On Wireless Communications, 49(12):2198–2206, Dec. 2001. [2] D.P. Palomar, J.M. Cioffi, and M.A. Lagunas. Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization. IEEE Trans. on Signal Processing, 51(9):2381 – 2401, Sept. 2003. [3] A. Bourdoux and N. Khaled. Joint tx-rx optimization for mimo-sdma based on a null-space constraint. In Proc. IEEE Veh. Tech. Conf. (VTC), 1:171 – 174, Sept 2002. [4] L.U. Choi and R.D. Murch. A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach. IEEE Trans. On Wireless Communications, 3(1):20 – 24, Jan 2004. [5] Q.H. Spencer, A.L. Swindlehurst, and M. Haardt. Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels. IEEE Trans. on Signal Processing, 52(2):461 – 471, Feb. 2004. [6] C. Windpassinger, R. F. H. Fischer, T. Vencel, and J.B. Huber. Precoding in multi-antenna and multi-user communications. IEEE Trans. on wireless communications, 3(4):1305 – 1316, July 2004. [7] M. Costa. Writting on dirty paper. IEEE Trans. on Information Theory, 29(3):439–441, May 1983. [8] Wei Yu and John Cioffi. Sum capacity of gaussian vector broadcast channels. In Proc. IEEE International Symposium on Information Theory (ISIT), 2002. [9] Robert F. H. Fischer. Precoding and Signal Shaping for Digital Transmission. Wiley, New York, 2002. [10] S. Shi and M. Schubert. Precoding and power loading for multi-antenna broadcast channels. In Proc. CISS, March 2004. [11] A. J. Tenenbaum and R. S. Adve. Joint multiuser transmit-receive optimization using linear processing. In Proc. ICC’ 04, June 2004.
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Abstract— Assuming perfect channel knowledge at the transmitter, we study space-time beamforming for the downlink in a multiuser multi-input multi-output (MIMO) channel while a nonlinear interference pre-cancelation is presumed at the transmitter. The antenna arrays may be employed at both the transmitter and the receivers. The optimum transmit/receive beam vectors are obtained based on a minimum mean-squared error (MMSE) criterion and a per-user power constraint. In frequency selective fading channels, where orthogonal frequency division multiplexing (OFDM) is applied, the precoding and beamforming design is extended over space and frequency dimensions as well. In fact the proposed algorithm offers a unique method for assigning frequency bins in a MIMO-OFDMA system. The bit error rate performance of the proposed algorithm is assessed by computer simulations. Index Terms— MIMO, OFDM, Beamforming, Precoding, Downlink.
I. I NTRODUCTION Precoding and beamforming design for multi-antenna broadcast channel has received significant attention as a promising solution to provide reliable high data rate communication in wireless downlink channels. This typically refers to a situation where independent data streams are to be transmitted from a multi-antenna base station (BS) to several uncoordinated receivers. In MIMO single user channels, there are a few works for joint transmit/receive beamforming design based on MMSE criterion [1], [2]. In MIMO multiuser channels, the main challenge is that the receivers cannot cooperate with each other while each user is suffering from the interference from all other users. In [3], [4], and [5], transmit preprocessing methods are proposed for the downlink of multiuser MIMO systems. In these techniques, the multiuser MIMO downlink channel is decomposed into parallel independent single user MIMO channels so that the multiuser interference is completely canceled and then single user schemes can be applied over each independent channel. However, all these techniques are highly restricted on the number of transmit/receive antennas in the sense that the number of transmit antennas must be greater than the total number of the receive antennas at different users. In practical situations where the number of transmit antennas is limited, this constraint severely restricts the number of receive antennas which results in restriction on receive diversity. In [6] zero forcing preequalization is implemented by Tomlinson Harashima Precod0-7803-9182-9/05/$20.00 ©2005 IEEE.
ing (THP) at the transmitter. Although this scheme is effective and outperforms linear pre-equalization schemes but the same restriction, as explained before, exists on the number of transmit/receive antennas. Also similar to conventional zero forcing methods, the performance is degraded in low signal-to-noise ratios (SNRs). In this work, the multiuser MMSE beamforming is combined with non-linear Dirty Paper Coding (DPC) proposed in [7] to derive a more efficient algorithm to minimize the mean squared error between transmit and receive data streams. The original idea of DPC [7], which is for single-antenna channels, indicates that if the interference is known non-causally at the transmitter but not at the receivers, a precoding strategy can be applied based on this knowledge which results the same capacity as if the interference does not exist. Since DPC is applied at the BS, the available single user algorithms can be extended to be applicable over individual single user channels. The algorithm proposed here is along with the work in [2]. The receive beam vectors are obtained based on MMSE criterion, and the transmit beam vectors are obtained through singularvalue-decomposition (SVD) of the whitened channels. In fact part of the multiuser interference is cancelled out at the BS and the residual interference at each user is minimized by transmit/receive beamforming. In this proposed scheme, there is no limitation on the number of transmit/receive antennas. Also, when the beamforming is generalized to non-spatial dimensions such as time and frequency, the proposed algorithm offers a unique method for assigning time slots in a MIMO-TDMA system, and frequency bins in a MIMO-OFDMA system, to users. The remainder of this paper is organized as follows. The problem formulation is given in the next section. The MMSE beamforming is discussed in section III. Space-Time and Space-Frequency beamforming are introduced in sections IV and V respectively. Simulation results is given in section VI, and we conclude with section VII. The notations used in this paper are as follows. We use boldface lower case letters to denote vectors, boldface upper case letters to denote matrices, and E for expectation. The superscripts ·∗ , ·T , and ·H denote conjugate, transpose and conjugate transpose respectively. I denotes the identity matrix, det(·), T r(·) and Diag(·) are abbreviates for the determinant, trace and block diagonal matrix respectively. M(i, j) represents the element in the ith row and the jth column of the matrix M.
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II. P ROBLEM F ORMULATION A. Signal Model The downlink in a multiuser MIMO channel is considered with K users, where the base station has t transmit antennas and each user receiver has ri antennas for i = 1, ..., K. Then the baseband signal received by user i is:
propagation may be avoided. This structure is also suggested and analyzed in [8] to determine the sum capacity of the broadcast channel.
DEC
yi = Hi
K
Bi xi + ni ,
i = 1, ..., K,
(1)
i=1
where Hi ∈ Cri ×t is the matrix of flat-fading channel gains between the transmitter and the ith receiver’s antennas, and ni is a ri × 1 vector of i.i.d. complex Gaussian random variables with zero mean, and the variance of σi2 , representing receiver noise for user i. It is assumed that in the ith MIMO channel, 2 each flat fading coefficient has the same variance of σh,i , and that the t.ri channels are independent. xi is the transmitted symbol vector to the user i and Bi is the transmit beamforming matrix with tr(BH i B i ) ≤ pi , when pi is the transmit power constraint for user i. The transmitter is supposed to send zi symbols to the ith user. Assuming that the signal dimensionality at the receiver has to be no smaller than the number of mutually interfering symbols to achieve an acceptable bit error probability (BEP), 1 ≤ zi ≤ min(t, ri ),
(2)
If we define the aggregate received signal vector yT = T T T , the overall channel matrix H = HT1 · · · HTK , y1 · · · yK the beamforming matrix B = [B1 · · · BK ], and the transmit T signal vector x = xT1 · · · xTK , the multi-user signal model is y = HBx + n.
(3)
Fig. 1. DFE at the Receiver
A regular subtraction at the transmitter may cause excessive power enhancement at the transmitter. The modulo-arithmetics (Tomlinson-Harashima Precoding (THP)) is used at both transmitter and receiver in order to minimize power enhancement [9], [10]. If the data symbols(x) are from an M −ary constellation, the encoder transmits x modulo−M , to constrain the symbols into the interval [−M/2, +M/2).1 The decoder has to perform a modulo−M operation to recover the original signal. To design THP, the encoding order K, ..., 1 is assumed which means that the interference caused by the users i + 1, ..., K to the user i is known prior to transmission. Hence G is a block upper triangular matrix with ones on the diagonal: ⎤ ⎡ I G1 ⎥ ⎢ 0 I G2 ⎥ ⎢ ⎥ ⎢ . .. (4) G=⎢ ⎥, ⎥ ⎢ ⎦ ⎣ 0 I GK−1 0 ... I where Gi ∈ Czi ×ζi with ζi =
K
m=i+1 zi .
B. Precoding In point-to-point communication systems or multiple access channels in which the receivers are coordinated, a minimum mean-square error decision-feedback equalizer (MMSE-DFE) can be applied to untangle the interference. The structure of DFE at the receiver is shown in Fig. 1. After each symbol is detected, it is subtracted from the received signal before the next symbol is detected. As a consequence error propagation may occur in these receivers. For multiple access channel, this is equivalent with serial interference cancelation (SIC). In the downlink channel, unlike in the other two, receivers are uncoordinated and hence it is impossible to implement joint processing techniques such as DFE. However, if the channel is known to the transmitter, the decision feedback receiver can be implemented at the transmitter as shown in Fig. 2. A well-known example of this type of “pre-equalizer” is the Tomlinson-Harashima precoder (THP), introduced in the context of equalization of ISI channels. The matrix form of THP is shown in Fig. 2, and applies to the multi-user downlink channel. This presents the same idea as DPC: if the interference is known at the transmitter, it can be pre-subtracted prior to transmission. In this way, the noise enhancement as well as error
MOD
MOD
DEC
Fig. 2. Implementation of THP
In a non-cooperative downlink channel, feedforward matrix (Fig. 2) has to be block diagonal H F = Diag(FH 1 , . . . , FK ),
(5)
where FH j is a zj × rj matrix for j = 1, ..., K. We attempt to solve the following problem: Problem statement: Using a per user power constraint, design the beamforming matrix (Bi ), and the feedforward and feedback matrices Fi and Gi respectively in order to minimize the mean squared error between the transmitted and received 1 It is well known that by implementing THP, there is still a power loss equal to M 2 /M 2 − 1 at the transmitter [9].
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data symbols for each individual user i. The design is performed to either • minimize the sum of the MSE for all the symbols transmitted to the same user or • maintain the same MSE for all the transmitted symbols to the same user. III. MMSE B EAMFORMING /P RECODING To derive the transmit/receive beamforming matrices in the downlink, because of the interference pre-cancellation at the BS with the encoding order K, ..., 1, the first user is interferencefree and can be considered as a single user. After designing the beamforming matrices for the first user, the beamforming matrices for the second user may be designed, treating the first user as known interference. This procedure can be continued for the users l = 3, ..., K sequentially, treating users i = 1, ...l − 1 as known interferences. Therefore, available single user schemes ( [1], [2]) can be extended to design the beamforming matrices for each user individually. The received signal for user i is: i Bj xj ) + ni yi = Hi (
B. Optimum Transmit Matrix With the optimal receive matrix Fi in (11), user i’s mean square error is at its minimum value for the given {Bi }, and is given by Eoi
H H H −1 = I − BH Hi Bi i Hi (Hi Bi Bi Hi + R(n+I)i ) H −1 = (I + Bi RHi Bi ) , (12) where the last line is obtained by the matrix inversion lemma. The transmit beamforming matrix Bi is designed based on the MSE criterion of the user i. • Sum-MSE Minimization Design Corollary 1: Minimizing the sum of the diagonal elements of MSE matrix Ei with respect to Bi
subject to
= Ui Σi Bopt i
(15)
(6)
where pi is the power constraint for user i, Ui denotes the zi eigenvectors associated with the zi largest eigenvalues, −1 λi , of RHi = HH i R(n+I)i Hi and Σi is a diagonal zi × zi matrix with the diagonal elements of i,l + −1/2 −1/2 i,l = µi λi,l − λ−1 , i,l
H Rri = Hi Bi BH i Hi + R(n+I)i ,
(8)
i−1 H R(n+I)i = σ 2 I + Hi ( Bj BH j )Hi
where [θ]+ = θ, θ ≥ 0; 0, θ < 0, λi,l are the zi largest eigenvalues of RHi , and µi is chosen to satisfy the transmit power constraint for user i:
(9)
zi 1/2 µi
j=1
is the covariance matrix of the noise and residual interference seen by user i. A. Optimum Receive Matrix The MSE of the lth symbol transmitted to user i is the lth diagonal element of Ei . Given a transmit beamforming matrix {Bi }, the optimal receive matrix Fi is obtained such that the diagonal elements of Ei are minimized. This is equivalent to solve: (10) min cH Ei c ∀c FH i
Differentiating cH Ei c = T r(Ei ccH ) with respect to FH i and setting the result to zero yields Fopt i
(14)
(7)
= E[(ˆ xi − xi )(ˆ xi − xi )H ], H H H = Fi Rri Fi + I − FH i Hi Bi − Bi Hi Fi ,
with
tr(BH i Bi ) ≤ pi
is given by
ˆ i = FH If the estimated data at the user i is defined as x i yi , the mean squared error matrix will be then defined as the covariance matrix of the error vector:
where
(13)
Bi
j=1
Ei
min tr[Ei ]
= R−1 r i H i Bi H −1 = (Hi Bi BH Hi Bi i Hi + R(n+I)i )
(11)
which is the linear MMSE receiver (Wiener filter). This receiver is optimum for both criteria in the problem stated in sectionII. A similar derivation is given for the single user case in [2]. Note that because of the successive nature of the design algorithm, R(n+I)i ) is known before designing the beamforming matrices Bi and Fi for user i.
=
j=1
pi +
(16)
−1/2
λj
zi
j=1
λ−1 j
(17)
Proof: This optimization problem is proved for single user case in [2]. The key difference is that the noise covariance term is replaced here by the noise-plus-interference covariance matrix. Since the derivations in [2] are valid for any arbitrary noise covariance matrix, the results are directly applicable. It can be easily shown that FH i Hi Bi becomes a diagonal matrix which means the MIMO channel for each user is diagonalized. By substituting Bi from (15) in (12) the MMSE matrix is −1 , (18) Eoi = (I + ΣH i Di Σi ) in which Di is a diagonal matrix with the elements of the largest eigenvalues of RHi in an increasing order. Therefore the MMSE matrix for each user is diagonal with diagonal elements in decreasing order. • Equal MSE Design Corollary 2: Minimizing the sum of the MSE with the same constraint as (14) while the same MSE is maintained for all the symbols transmitted to the same user is given by: = Ui Σi Q (19) Bopt i
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where Ui , Σi are defined in corollary 1 and Q could be any rotation matrix that satisfies |Q(i, k)| = |Q(i, l)|, ∀i, k, l such as the DFT matrix [2]. It can be seen that in this design, neither MIMO channel, nor the MSE matrix is diagonal. However, the MSE matrix has identical diagonal elements. In fact for any given Bi , a rotation −1 Q has matrix Q can be found so that QH (I + BH i RHi Bi ) identical diagonal elements. The sum of the diagonal elements of MSE matrix is the same regardless of Q. A similar design for single user case is also proposed in [1]. The beamforming matrix Bi is designed to minimize a weighted sum of symbols MSE. The weights can be adjusted such that all the symbols have the same MSE. In the following, a corollary to this theorem applicable to the current work is presented. Corollary 3:The MMSE beamforming matrix that ensures equal MSE for all symbols transmitted to the same user is given by: −1/2 = µ1/2 Ui Di , (20) Bopt i
p i and Ui , Di are defined the same as where µ = T r(D−1 )
to find the optimal ordering, the algorithm has to be examined over K! rearrangements which is not practical. Here, we have applied a suboptimal low complexity method. The users are ordered based on the square Frobenius norm of their channel matrix, Hi 2F for i = 1, ..., K: Hi 2F =
C. Precoding Matrix Design The feedback matrix G is obtained by MMSE solution as well. It can be easily proved that to minimize the MSE for each user symbol Gi = FH i Hi [Bi+1 , ...., BK ] , i = 1, ..., K
(22)
Table 1: The algorithm for precoding and MMSE beamforming Initialization: C = 0, Rn = σ 2 I. f or i = 1 : K R(n+I)i = Rn + Hi ∗ C ∗ HH i , −1 RHi = HH i ∗ R(n+I)i ∗ Hi , Bi = Ui Σi , H −1 Fi = (Hi Bi BH Hi Bi , i Hi + R(n+I)i ) H C = C + Bi ∗ Bi . end
f or j = K − 1 : −1 : 1 Gj = FH j Hj [Bj+1 , ...., BK ] . end
IV. S PACE -T IME S PREADING In the proposed scheme, there is no limitation on the number of users in the system. However, as the number of active users is increased, the MMSE is increased for each user too. So if there is a quality of service constraint for each user, the number of users we can accommodate is in fact limited. On the other hand, if we are allowed to increase the bandwidth by spreading each symbol over time, we can increase the number of active users without any performance loss. The beamforming is now optimized over two dimensions: space and time. Instead of assigning a beam vector to each user symbol, one t × G beamforming matrix, Φil , i = 1, ..., K, l = 1, ..., zi is assigned to each user symbol where G is the processing gain, and T = GTc is the symbol period where Tc is the chip period. Then the transmitted signal is X=
(21)
Performance of the proposed algorithm is also dependent on the precoding ordering. Defining the optimal ordering which is related to channel fading states, is an open problem. Indeed,
r,t
H2i (r, t),
when Hi (r, t) is the (r, t) element of the matrix Hi . The user which is supposed not to see any interference, is the one with the smallest channel matrix norm, and the user who sees the full interference (in this work, the Kth user) is the one with the largest channel matrix norm. This method specially makes sense, if all users have the same power constraint.
i
Corollary 1. The detailed proof for single user case is in [1]. The results are directly applicable to the current multiuser case with the same reasoning as Corollary 1. Note that the designs in both Corollary 2 and Corollary 3 provide equal MSE for all symbols transmitted to the same user, however, in the later case (20) the channel and the MSE matrix is diagonalized as is opposed to the previous one (19). In [11], a similar approach to the design proposed in (13) is applied in the downlink of a multiuser system to transmit multiple symbols to multiple users based on a per-user power constraint. However, interference pre-cancellation is not presumed at the BS and therefore R(n+I)i is dependent on other users beam vectors, and the MSE for each user is dependent on other users MSE which makes the algorithm very complicated. The authors have proposed an iterative algorithm in which each transmit matrix is obtained by fixing the others but, there is no guarantee for the convergence of the algorithm, and it may converge to a false or local optimum. Here, because of the interference pre-cancellation, R(n+I)i for i = 1, ..., K is only dependent on the other users beamforming matrices, Bl for l ≤ i − 1 (see (9)). Therefore, it is possible to design the transmit beamforming matrices, Bl sequentially for l = 1, ..., K. The algorithm for the transmit beamforming matrix design (15) is summarized in table 1. For the other two designs, it is enough to substitute the 5th line of the table by (19) or (20).
zi K
xil Φil .
(23)
i=1 l=1
The channel is assumed to be constant over at least one symbol period. The signal received by user i, over G channel uses and
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ri receiver antennas, is Yi =
zj K
xjl Hi Φjl + Ni ∈ Cri ×G
(24)
j=1 l=1
where Ni is a matrix of i.i.d. complex Gaussian random variables with zero mean and the variance of σi2 , representing receiver noise. By stacking the columns of Yi , we obtain ˜i ˜i = H y
K
˜ i, Ci x i + n
(25)
i=1
˜ i = Diag(Hi , . . . , Hi ), ˜ i = vec(Yi ), H where y ˜ i = vec(Ni ) and Ci = xi = [xi1 , ..., xizi ], n [vec(Φ1 ), . . . , vec(Φzi )]. Essentially, each column of Ci corresponds to a space-time modulation waveform for one data stream. This presents exactly the same system model as (1). Therefore we can apply the same algorithm explained in section III to design the matrices Ci , Fi , and Gi for each user. As an alternative to the above methodology, one may think of choosing a subset of users randomly in each chip time-slot (TDM fashion) and perform the original algorithm to design the proper beam vectors over each subset independently. However, by applying the algorithm once over all users as presented in signal model (25), the subsets of the users that result in the least amount of interference are automatically selected to be transmitted at the same time. In fact, each user symbol is transmitted through the subchannel with the highest gain or equivalently with the minimum amount of interference. The result of the algorithm is to assign each user symbol to only one time slot (which has the least amount of interference), rather than spreading each user symbol over all time slots. As will be shown by numerical results in section VI, even though the channels are constant during one symbol time, the algorithm benefits from selecting the simultaneous users intelligently. The other interesting point is that when we transmit multiple symbols to each user (at most min(t, ri )) and the beam vectors are designed based on either of the schemes in (15), (19), or (20), if G ≥ min(t, ri ), no two symbols for the same user will be transmitted at the same time slot. This is because the beam vectors are obtained by the largest eigenvectors of RH ˜ i . Since ˜ Hi is a block diagonal matrix with min(t, ri ) different eigenvalues and G min(t, ri ) eigenvectors, all G largest eigenvectors are orthogonal in time. The importance of this simple fact is that although we do not have interference pre-cancellation for the symbols transmitted to the same user, but there is no interference between the transmitted symbols to the same user because their beam vectors are orthogonal in time. V. S PACE -F REQUENCY S PREADING In this section, we consider the frequency selective fading channel. The frequency selective fading can severely degrade system performance by causing inter-symbol interference (ISI) and result in an irreducible bit error rate (BER) which imposes an upper limit on the data rate.
In order to transmit over frequency-selective MIMO channels, the same design principles may be applied as long as the channel can be transformed into one resembling flat fading. OFDM is a practical solution which transforms a frequency selective fading channel into parallel, possibly correlated flat fading channels. Frequency components which are separated by the channel coherence bandwidth or more may be assumed to be totally uncorrelated. Therefore the channel frequency diversity can also be exploited through the proper design of an OFDM system. An inverse fast Fourier transform is performed over a sequence of symbols which are transmitted over each antenna. If the number of FFT bins is Nf , we form a super-symbol of the original symbols and pass that through an IFFT operation: ˜= S
zi K
xil Φil ∈ CNf ×t ,
(26)
i=1 l=1
while we allow Φi to have non-zero values in any row. This implies that one symbol can occupy any frequency bin. In each OFDM block which is Nf time slots, the number of K symbols are transmitted, and the channel is supposed to be constant. At each user receive antenna, the cyclic prefix is first removed from each OFDM block and then it is fed to an OFDM demodulator. After taking Nf -point FFT we end up with Nf MIMO flat fading channels in Nf subcarriers. If we stack the signals of the ˜i, received antennas in Nf time slots, in a Nf ri × 1 vector y with the same notation as section IV, we have ⎤ ⎡ 0 Hi (0) ⎥ K ⎢ .. ˜ i(27) ˜i = ⎣ y ⎦ i=1 Ci xi + n . 0
Hi (Nf − 1)
K ˜ ˜ i, = Hi i=1 Ci xi + n
(28)
where Hi (n) for n = 0, . . . , Nf − 1 is the ri × t matrix whose each component is the nth frequency component of the channel of the user i between each single transmit-receive antenna pair. This gives the same system model as (1) and (25) and therefore the algorithm proposed in section III is applicable to design Ci , Fi , and Gi . As a result of the beamforming design, each user signal is transmitted over the channel frequency with the highest gain and the least amount of interference. The advantage of performing the optimization over all frequency bins is that, instead of spreading the signal of each user over all frequency bins, each user is only allocated to the best channel frequency (for a fixed precoding order). This is the same as what explained for space-time beamforming design in section IV. The only difference is that in the previous one, the only parameter which is changing over time-slots is the interference, while in the current case, the channel is also different over the time-slots/frequencies, and therefore, the frequency diversity may be achieved as well. VI. S IMULATION R ESULTS In this section we provide some numerical results to illustrate the performance of the proposed algorithm. In all of the simulations a Rayleigh fading channel is assumed with a zero
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Fig. 3. The performance of space-time spreading for different number of receive antennas, t = 4, G = 4, K = 8, z = 2.
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Fig. 4. The performance of space-frequency spreading compared with MMSE beamforming over flat fading channel t = 2, r = 2, Nf = 8, z × K = 16. 0
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e
10 P
mean uncorrelated complex Gaussian noise across the receive antennas (Rn = σ 2 I). The elements of the channel matrix H are generated as independent and identically distributed (i.i.d.) samples of a complex Gaussian process with zero mean and unit variance. The channel matrix is known at the transmitter. Each user is assumed to have full knowledge about its own channel and also the transmit matrix B. The results are presented based on the average probability of error (Pe ) versus signal-to-noise ratio (SNR(dB)) for each user symbol. Without loss of generality, the same power constraints, the same number of receive antennas and the same number of transmit symbols are assumed for all users (pi = p, ri = r, zi = z, for i = 1, ..., K). Here, the proposed algorithm in section III is recalled as MMSE algorithm and the design of transmit beamforming matrix for each user is based on Sum-MSE minimization (15). In Fig. 3, space-time precoding/beamforming is performed for K = 8 users to transmit two symbols for each user, when we have t = 4 transmit antennas and processing gain, G = 4 which means the signal is spread over 4 time slots. The average probability of error is compared for different number of receive antennas. It can be seen that by increasing the number of receive antennas from r = 2 to r = 4, the performance is improved by 7.5 (dB) because of receive diversity. Also for r = 4, the average Pe is compared with the Pe of the eighth user. Note that, since the precoding order is 1, ..., 8, the eighth user is the one who sees full interference from other users and may be expected the worst performance compared with other users. As is shown for low SNR, the performance for this user is about 1 − 2 (dB) less than the average. In Fig. 4, the performance of MMSE algorithm over a flat fading channel is compared with space-frequency MMSE precoding/beamforming over frequency selective channel when we have t = 2 transmit, and r = 2 receive antennas. In frequency selective channel, two-tap channels are simulated for each user with a maximum delay spread of υ = 1 chip and 8−point FFT is applied to implement OFDM. As explained in section V, once the proposed MMSE algorithm is applied to all frequency bins,
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Fig. 5. Average Pe compared with Pe for each individual user, t = 2, r = 2, K = 2, z = 1.
each user symbol is assigned to one frequency bin. We examined both cases of transmitting one symbol per user for K = 16 and two symbols per user for K = 8. It can be seen that the performance improvement over flat fading channel is about 5(dB) because of the frequency diversity which can be explored in space-frequency spreading. In Fig. 5 and Fig. 6, the average probability of error is compared with the probability of error of each individual user. The first user is the one who does not see any interference, and the second user in Fig. 5 and the eighth user in Fig. 6 are the ones who see the full interference. VII. C ONCLUSIONS A joint transmitter/receiver precoding and beamforming design is proposed for the downlink in a MIMO multiuser channel by using DPC. There is no constraint on the number of
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Fig. 6. Average Pe compared with Pe for each individual user in spacefrequency spreading, t = 2, r = 2, Nf = 8, K = 8, z = 2.
transmit/receive antennas and because of the interference precancellation at the BS, the single user MMSE beamforming schemes can be extended to the current multiuser channel. To increase the number of active users over the number of transmit antennas without loss of performance, the signal is spread over both time and space. The space-time multiplexing matrices are designed to minimize the MSE between the transmitted and received data vector. In frequency selective channels, where OFDM is applied, the proposed design is able to explore full frequency diversity as well; each user is allocated to the strongest frequency channel with the least amount of interference. R EFERENCES [1] H. Sampath, P. Stoica, and A. Paulraj. Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion. IEEE Trans. On Wireless Communications, 49(12):2198–2206, Dec. 2001. [2] D.P. Palomar, J.M. Cioffi, and M.A. Lagunas. Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization. IEEE Trans. on Signal Processing, 51(9):2381 – 2401, Sept. 2003. [3] A. Bourdoux and N. Khaled. Joint tx-rx optimization for mimo-sdma based on a null-space constraint. In Proc. IEEE Veh. Tech. Conf. (VTC), 1:171 – 174, Sept 2002. [4] L.U. Choi and R.D. Murch. A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach. IEEE Trans. On Wireless Communications, 3(1):20 – 24, Jan 2004. [5] Q.H. Spencer, A.L. Swindlehurst, and M. Haardt. Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels. IEEE Trans. on Signal Processing, 52(2):461 – 471, Feb. 2004. [6] C. Windpassinger, R. F. H. Fischer, T. Vencel, and J.B. Huber. Precoding in multi-antenna and multi-user communications. IEEE Trans. on wireless communications, 3(4):1305 – 1316, July 2004. [7] M. Costa. Writting on dirty paper. IEEE Trans. on Information Theory, 29(3):439–441, May 1983. [8] Wei Yu and John Cioffi. Sum capacity of gaussian vector broadcast channels. In Proc. IEEE International Symposium on Information Theory (ISIT), 2002. [9] Robert F. H. Fischer. Precoding and Signal Shaping for Digital Transmission. Wiley, New York, 2002. [10] S. Shi and M. Schubert. Precoding and power loading for multi-antenna broadcast channels. In Proc. CISS, March 2004. [11] A. J. Tenenbaum and R. S. Adve. Joint multiuser transmit-receive optimization using linear processing. In Proc. ICC’ 04, June 2004.
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