MU-MIMO Decomposition Transmission with Limited Feedback
MU-MIMO Decomposition Transmission with Limited Feedback Cheng WANG and Ross D. Murch1 Department of Electronic & Computer Engineering The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong E-mail: [email protected], [email protected] Abstract — Downlink multi-user multiple-input multiple-output (MU-MIMO) transmission techniques have gained much attention recently because of their potential to significantly increase system capacity compared to single-user transmission by separating multiple users in the space domain through appropriate signal processing. Unfortunately, these techniques require accurate channel state information at the transmitter (CSIT) for their proper operation. In practice however perfect CSIT is hard to obtain due to capacity-limited feedback channel in FDD systems for example. In this paper we investigate systems employing MU-MIMO decomposition transmission in the downlink with limited feedback. In particular, the essential information that enables the scheme to deal with inter-user interference is identified and the optimal codebook design in terms of minimizing the residual inter-user interference power is derived. Simulation results show that in general limited feedback has a significant impact on the mutual information achieved by MU-MIMO decomposition transmission.
I. INTRODUCTION Downlink multi-user multiple-input multiple-output (MU-MIMO) transmission techniques have received much attention recently because of their potential to significantly increase system capacity compared to single-user transmission by separating multiple users in the space domain through appropriate signal processing. A limitation of these downlink MU-MIMO techniques lies in that accurate channel state information (CSI) at the base station (BS) is crucial for them to operate properly so as to realize their significant capacity potential. In practice however, channel state information is difficult to be known perfectly at the BS due to the limited feedback channel capacity in FDD systems for instance. Therefore it is important to investigate multi-user MIMO transmission with limited feedback. That is, how to do the feedback and what is the impact of limited feedback on the achievable performance. In previous work, research on feedback strategies has been first considered for single-user point-to-point scenario and [2] provides an excellent overview of these strategies. Nearly all these schemes show that it is preferred to design the transmit signal at the receiver rather than at the transmitter, where a codebook is designed which contains several precoding matrices. Both the transmitter and the receiver know the codebook, so for each channel realization the receiver selects one precoding matrix from the codebook according to a certain criterion and sends back the index of the selected precoding matrix to the transmitter. It has been shown that the performance achieved with a small number of feedback bits is already very close to that achieved with perfect CSIT. Recently limited feedback has also been considered for the multi-user transmission scenario, but mainly for single-antenna
1
We acknowledge support of Hong Kong RGC grant HKUST 6164/04E
users. Because of the presence of inter-user interference, the situation for multi-user transmission with limited feedback is much more intricate than the single-user transmission case. In [3] a random beamforming scheme is proposed for MU-MISO system. However the scheme needs a very large number of users to achieve good performance. When the number of users is comparable to the number of transmit antennas, the performance is very poor. In [4], [5], the scenario with the number of users comparable to the number of transmit antennas is considered for MU-MISO system. In [4], the capacity degradation due to limited feedback and the required feedback load are analyzed when random vector quantization (RVQ) is used. While in [5], the optimal codebook design in terms of minimizing the residual inter-user interference for zero-forcing transmission is derived. Both show that for multi-user transmission to perform well a high feedback requirement is needed compared to single-user transmission. In this paper we consider limited feedback for the scenario where both the BS and the users are equipped with multiple antennas and multiple data streams are allowed to be transmitted to each user. In particular, we consider a multi-user MIMO system where the MU-MIMO decomposition scheme [1] is used for downlink transmission. In the framework, each user independently sends back selected but useful information of its own channel to the BS and let the BS select suitable precoding weights based on the feedback from all the users. We first identify the essential information that the BS needs to deal with the inter-user interference and then an optimal codebook design is derived in terms of minimizing the residual inter-user interference power. The remainder of the paper is organized as follows. System model is introduced in section II. In section III, we identify the essential information to be sent back and the optimal codebook design is derived accordingly. Simulation results and discussion are provided in section IV. Finally conclusions are drawn in section V. II. SYSTEM MODEL We consider the downlink of a narrow-band FDD multiuser MIMO system with a single BS equipped with M transmit antennas and K users each of which has N receive antennas. At the BS, the transmit data symbol vector of each user, i.e. b k , k = 1, , K , is passed through certain transmit precoding matrix Wk , k = 1, , K before it is launched into the downlink channel. Then the received signal at the kth user is given by K
y k = H k ∑ Wi b i + n k , k = 1, i =1
,K
(1)
where H k is the N × M channel matrix of user k with zero-mean unit-variance independent and identically distributed (i.i.d.) complex Gaussian entries, n k ∈ CN (0, I ) is the additive white Gaussian noise vector and I is the identity matrix. The sum mutual information of the downlink system can then be expressed as I=
K
Ik ∑ k =1
(2)
caused by limited feedback is derived accordingly. III.A MU-MIMO Decomposition Transmission Scheme
The objective of this scheme [1] is to choose the precoding matrices Wk , k = 1, , K in a way such that each user receives no interference from the other users. Equivalently, the scheme decomposes the multi-user system into multiple parallel single-user systems. Thus the precoding matrix can be written in the form as
where the mutual information of user k is given by −1 K H H H H I k = log 2 det I + Wk H k I + H k ∑ Wi Wi H k H k Wk = 1 , ≠ i i k (3) and H represents the hermitian operation. Note that K is the largest number of users that we would like to serve simultaneously. It may or may not be equal to the total number of users in a practical system, which may be larger than K, and therefore other multiple access techniques can be used as well. In a FDD system the uplink and the downlink channel are not reciprocal due to the frequency separation between them, thus feedback from the mobile to the BS is essential so that the broadcast channel can be known. Ideally each Wk is a continuous function of all H 1 , , H K such that appropriate signal processing can be performed to separate the simultaneous users in space. In practice however one user cannot know the channels of other users, therefore it is impossible for one user to choose a suitable precoding matrix depending on its own channel. As a result, unlike the single-user point-to-point scenario, instead of choosing the precoding matrix at the receiver, it is desirable for each user to independently send back some useful information of its own channel and let the BS decide the precoding matrices for all the users based on the independent feedback from all of them. As a consequence the proposed framework is as follows. If log 2 Q -bit feedback is allowed for each user, the single-user channel space can be partitioned into Q regions
~ H = {H 0 , , H Q-1} along with Q partition centroids ~ F = {F 0 , , F Q−1} . Note that in the rest of the paper the
number in the superscript and subscript indicates the index of the region and the index of the user respectively. Each user then independently selects the region its channel belongs to according to a certain criterion and the index of the selected region is fed back to the BS. At the BS, the transmit precoding matrix of each user is chosen according to the feedback information from all K users. Note that since the focus of the paper is to investigate the impact of limited feedback on the performance of MU-MIMO transmission, in the rest of the paper we shall assume each user has perfect CSI of its own channel and the feedback channel is error-free and delay-free.
III. LIMITED FEEDBACK APPROACH In this section we first briefly introduce the downlink MU-MIMO decomposition scheme, which is suboptimal but efficient and with low-complexity. Then the essential information for the scheme to deal with the inter-user interference is identified. Finally the optimal codebook design in terms of minimizing the residual inter-user interference
Wk = Vk G k ,
(4)
where Vk is an orthonormal basis of the joint null-space K
∩i=1,i≠k Ν (H i )
and it can be computed by singular value
decomposition as H1 H k −1 ~ Σ k − Hk = = Uk , Uk 0 H k +1 H K
[
]
~ 0 VkH . 0 VkH
(5)
And G k , k = 1, , K are non-zero matrices which can be designed alone by some criteria. The total power constraint PT of the BS can then be expressed as K
K
k =1
k =1
∑ tr(WkH Wk ) = ∑ tr(G kH G k ) ≤ PT , where tr (•) represents the matrix trace operation. Note that a sufficient condition to ensure the existence of the null-space is M > ( K − 1) N . III.B Identification of the Essential Information Denote the singular value decomposition of H k as
[
]
Λ Hk = Uk ,Uk k 0
0 VkH , k = 1, 0 VkH
,K ,
(6)
λ kN } , λ k1 ≥ ≥ λ kN > 0 are the where Λ k = diag {λ k1 non-zero singular values of H k . The joint null-space
Ν (H −k ) is the orthogonal complement of the row-space of
H −k , which is fully determined by the row-spaces of H i , i ≠ k .
( )
it is suffice to know the Therefore, to get Ν H −k orthonormal basis of the row space of each H i , i ≠ k . As a result, the useful information each user should send back to the BS is their individual Vk . To see this mathematically, denote V1H H V Vk− = kH−1 , Vk +1 H VK
(7)
and the received signal at the kth user can be expressed as
then we can rewrite H −k as H 1 U 1 Λ1 V1H H H U Λ V k −1 k −1 k −1 k −1 H k− = = H H U k + 1 k +1 Λ k +1 Vk +1 H H K U K Λ K VK U1 Λ 1 =
yk =
U k +1 Λ k +1
k
ˆ elements of H k V k are also i.i.d. zero-mean unit-variance complex Gaussian random variables. The second term is the residual inter-user interference caused by limited feedback, which is the main cause of performance degradation. Thus our aim is to minimize the residual interference, the power of which is given by
V VkH−1 H Vk +1 H U K Λ K VK
PT K ˆ ˆ tr H k Vi ViH H kH . ∑ KL i =1,i ≠ k
Ak =
(8) Clearly we can see that H −k and Vk− share the same row-space and thus the null-space. As a result Vk , k = 1, , K is the most important information which enables the BS to deal with the inter-user interference. Therefore we decide that each user should send back the information about its individual Vk . III.C Codebook and Feedback Strategy Design After the essential information is identified, it is also important to decide how to efficiently convey the identified essential information to the BS. This includes the design of how to partition the single-user channel space and how each user determines which region its channel belongs to. Since the BS only has information about Vk , k = 1, , K , it is thus impossible to design G k to match the equivalent single-user channel H k Vk . Also no power allocation can be performed either. Thus G k , k = 1, , K should be chosen as PT I L , k = 1, , K , where KL L = rank Vk = M − ( K − 1) N under the assumption of i.i.d. channels [1]. Note that the precoding matrix of each user is independent of its own channel. If the feedback channel has no capacity constraint, i.e. perfect knowledge of Vk , k = 1, , K is available at the BS, inter-user interference can be eliminated completely. In practice however only limited feedback is allowed, the BS only ˆ , k = 1, , K . As a has an estimate of Vk , k = 1, , K , i.e. V k result there will be residual inter-user interference seen by each ~ˆ ˆ− = V ˆ ˆ ˆ ˆ H , V V V V user. Let V k k −1 k +1 K 1 k be ˆ ˆ − and V an orthonormal basis of the row-space of V k k be an ~ ˆ ˆ ˆ − , i.e. V , V orthonormal basis of the null-space of V k k k
scaled identity matrices G k =
( )
[
(9)
where the first term is the desired signal, since the elements of H k are i.i.d. zero-mean unit-variance complex Gaussian ˆ has orthonormal columns, the random variables and V H 1
U k −1 Λ k −1
K PT ˆ H k ∑ Vi b i + n k , KL i =1,i ≠ k
PT ˆ H k Vk b k + KL
]
ˆ = spans C M . The precoding matrices become W k
PT ˆ Vk KL
(10)
ˆ for any i , i ≠ k . Note that Vˆ i is a function of V k ˆ is given by Therefore the criterion for choosing V k K
∑
min
ˆ V k i =1,i ≠ k
ˆ ˆ tr H k Vi ViH H kH
.
(11)
~ˆ Since Vi contains an orthonormal basis of the row-space ~ˆ ˆ − , without loss of generality V of V i can be chosen as i
[Vˆ , x ,
, x ( K −2 ) N
]
, where x1 ,
[Vˆ , x ,
, x ( K −2 ) N
]
ˆ − . Then we spans the row-space of V i
, x ( K −2 ) N are arbitrary ˆ and such that orthonormal vectors that are orthogonal to V k k
k
1
1
have
ˆ ˆ tr H k Vi ViH H kH ~ˆ ~ˆ = tr H k I − Vi ViH H kH , H H ˆ ˆ = tr H H − H V V H H
(
−
k
k
k
H k x1x1H H kH
k
k
for any i , i ≠ k ,
k
− H k x ( K −2 ) N x (HK −2 ) N H kH
−
(12)
)
ˆ becomes thus the selection criterion for choosing V k
(
)
ˆ V ˆH H max tr H k V k k Hk . ˆ V k
(13)
Note that for the special case with N = 1 , the criterion reduces to the one derived in [5] for multi-user MISO system. To see the intuition behind the criterion, we have
(
)
ˆ V ˆH H tr H k V k k Hk ˆ H V Λ2V H V ˆ = tr V
( = tr (Λ V k
2 k
k
k
k
k
H ˆ ˆ H k Vk Vk Vk
) )≤ ∑λ γ N
i =1
2 ki
, ki
(14)
where the inequality comes from [6] and γ k1 ≥ ≥ γ kN ≥ 0 ˆ V ˆH are the eigenvalues of VkH V k k Vk . From [7] we know that
M=4, N=2, K=2 35
γ ki = cos 2 θ ki , where θ k1 ≤ ≤ θ kN , are the principle angles ˆ and V . The closer between the subspaces spanned by V k k these two subspaces, the smaller the principle angles, as a result the larger the γ ki , i = 1, N and therefore the smaller the residual inter-user interference. As a consequence the objective of the codebook design is Q −1
q
q=0
qH
Vq
)
(
H H HV q H ∈ H q Pr H ∈ H
H
s.t. V q V q =I N for q = 0,1,
q
Sum Ergocid Capacity in bits/channel−use
E tr (V ∑ Hmax ,
30
),
, Q −1 .
The design problem can then be solved by a two-step method as follows,
(
)
(
H V q = arg max E tr V q H H HV q H ∈ H q Pr H ∈ H q V
H
s.t. V q V q =I N . q
given V q ,
H q = H : tr V q H H HV q ≥ tr V j H H HV j ,
∀j ∈ [0,
),
(16)
Step 2) Determine the optimal partition H q = 0, , Q − 1 , as H
q
H
, q − 1, q + 1, , Q − 1]}
(17)
Thus the codebook can be obtained by iteratively solving the above two steps. The algorithm is guaranteed to converge since at each step the objective quantity is always increased because we choose the updated parameters to maximize the other fixed parameters. However a local optimal may be found, thus we take several random initial partitions and the one that gives the maximum objective quantity in (15) is selected. Since H E tr V q H H HV q H ∈ H q , H = tr V q E H H H H ∈ H q V q
[
(18)
]
H
the solution V q of the first step subject to V q V q =I N should be chosen as the N dominant eigenvectors of
[
E HH H H ∈H
q
] corresponding to the N largest eigenvalues
[6]. For every channel realization, each user sends back the index chosen according to
(
H
)
q *k = arg max tr V q H Hk H k V q , k = 1, q = 0,1, ,Q −1
,K ,
(19)
ˆ = V q*k and the BS decides the transmit precoding then V k
25
20
15
10
5
(15)
Step 1) Determine the optimal V q given a certain partition H q , q = 0, , Q − 1 , as
MU Perfect CIST MU 11bits MU 9bits MU 7bits SU Perfect CSIT SU 7bits
0 −4
0
4
8
12
16
20
24
28
32
PT in dB
Fig. 1. Sum ergodic mutual information comparison with different number of feedback bits per user, M=4, N=2, K=2
matrices based on (q1* , q2* , , q*K ) . Note that when it happens that more than 1 user selects the same partition, single-user ˆ and each user transmission is used, where Wk = (PT N )V k uses 1 K of the time for transmission due to time division. Since even if we let the codebook of each user to be a rotated version of each other, although this can prevent the case for exactly identical estimated channel row space, this does not change the fact that the users’ row spaces are very close to each other given a reasonable number of feedback bits. So in this case even with perfect channel state information, it is not good to use multi-user MIMO decomposition scheme [9].
IV. NUMERICAL RESULTS AND DISCUSSION In this section we provide numerical results to see the impact of limited feedback on the system performance in terms of sum ergodic mutual information. Note that as we have mentioned that when more than 1 user selects the same partition, single-user transmission is used. However for all the figures in the simulation section, the sum ergodic mutual information of MU-MIMO decomposition transmission with limited feedback that we show is averaged over the simulation trials when multi-user transmission is used in order to study the impact of limited feedback. In Fig.1, the performance of a system with a BS equipped with 4 transmit antennas and 2 users each has 2 receive antennas is investigated for both MU-MIMO decomposition transmission and TDMA single-user transmission†. From the figure we can see that single-user transmission with 7bits feedback already performs very near that achieved with perfect CSIT. While the performance achieved by MU-MIMO decomposition transmission with 11bits feedback is still far away from that achieved with perfect CSIT and is worse than that achieved by single-user transmission. Considerably more feedback bits are needed for MU-MIMO decomposition †
Each user uses half of the time with full transmit power. The codebook for single-user transmission is obtained using the scheme proposed in [8].
M=4, N=3, K=2
M=3, N=2, K=2 40
30
MU Perfect CIST MU 3bits SU Perfect CSIT 35
25 Sum Ergodic Capacity in bits/channel−use
Sum Ergodic Mutual Information in bits/channel−use
MU Perfect CSIT MU 3bits SU Perfect CSIT
20
15
10
30
25
20
15
10
5 5
0 −4
0
4
8
12
16 20 PT in dB
24
28
32
36
40
Fig. 2. Sum ergodic mutual information comparison with different number of feedback bits per user, M=3, N=2, K=2
transmission to outperform single-user transmission. So with moderate number of feedback bits, single-user transmission is preferred. In Fig. 2, we provide the performance of a system with a BS equipped with 3 transmit antennas and 2 users each with 2 receive antennas. In this special case we can see that contrary to what we observed in Fig. 1, the performance achieved by MU-MIMO decomposition transmission with 3bits feedback is quite close to that provided by knowing perfect CSIT. In fact, this phenomenon is valid for any systems with N = M − 1, K = 2 . We show the reason as follows. For systems with N = M − 1, K = 2 , the mutual information achieved by user 1 can be expressed as P I1 = log 2 1 + T 2 where Vˆ 1 and
P ˆ ˆ ˆ ˆ V1H H1H I + T H1 V2 V2H H1H H1V1 , (20) 2 ˆ V2 are both M × 1 vectors. Denote the eigenvalue decomposition of H 1 Vˆ 2 Vˆ 2H H 1H as −1
ˆH H ˆ V H HV 2 1 1 2 0 ˆ ˆ H H1 V2 V2H H 1H = X X , 0 the mutual information can then be re-expressed as −1 P ˆ P ˆ ˆ ˆ I 1 = log 2 1 + T V1H H 1H I + T H 1 V2 V2H H 1H H 1 V1 2 2
(
= log 2 1 + 1 P ˆH H ˆ T V H H V +1 PT ˆ H H 2 2 1 1 2 V1 H 1 X 1 2
X H H Vˆ 1 1 1
(21)
0 −4
0
4
8
12
16 20 PT in dB
24
28
32
36
40
Fig. 3. Sum ergodic mutual information comparison with different number of feedback bits per user, M=4, N=3, K=2
1 a PT 2 = log 2 1 + z
z H , PT 2 P 2 2 1 2 P = log 2 1 + z1 + T z2 + + T z N 2 2 a
(22) 2 ˆH H ˆ where a = V 2 H1 H1V2 + PT
ˆH H and z = V 1 H 1 X is a vector
with i.i.d. CN (0,1) entries. This means z i , i = 1, , N are i.i.d. chi-squre distributed with 2 degrees of freedom. Similar analysis applies for the mutual information achieved by user 2. From the above analysis we can see that in this special case the residual interference affects only one of the chi-square distributed random variables, this explains why the performance loss is small. In addition, as the number of transmit antennas increases, the impact of the residual interference gets smaller as it affects a smaller percentage. And Fig. 3 numerically verifies this point as we can see that with the same number of feedback bits, the performance achieved with limited feedback achieves a higher percentage of that achieved with perfect CSIT in a system with M = 4 , N = 3 than a system with M = 3 , N = 2 . Note however, although in this system setting where MU-MIMO decomposition with only a few bits feedback performs close to that achieved with perfect CSIT, single-user transmission performs better than MU-MIMO decomposition even when perfect CSIT is known. It is because in this system setting, MU-MIMO decomposition transmission scheme sacrifices too many degrees of freedom to cancel the inter-user interference. Furthermore, similar analysis and results apply for cases with K = 2, N > M − N , where the residual interference has less detrimental impact compared to other system settings. 2
V.
CONCLUSIONS
In this paper, multi-user systems employing MU-MIMO decomposition transmission in the downlink with limited feedback are investigated. We identify the essential information that the BS needs to deal with the inter-user interference and propose an optimal way to convey the identified essential information to the BS so as to minimize the residual inter-user interference power. We found that in system settings which are in favor of MU-MIMO decomposition transmission when perfect CSIT is available, i.e. where MU-MIMO decomposition transmission outperforms single-user transmission when perfect CSIT is available, considerable more feedback bits are needed for MU-MIMO decomposition to perform well, which is caused by the detrimental impact of inter-user interference. As a result, if only very limited number of feedback bits is allowed single-user transmission is preferred. When the system allows much more feedback bits and can afford the complexity (as the number of feedback bits increases, the codebook size and search space increase exponentially), then MU-MIMO decomposition should be used. Although from this conclusion we may doubt the feasibility of multi-user MIMO schemes in practical wireless systems, there are still some possibilities of improving the performance. One aspect is that in a system with a large number of users, multi-user diversity in feedback quality can be exploited to improve the system performance with a fixed number of feedback bits. Since it is the residual inter-user interference that causes the major performance degradation, we may only allow ˆ V ˆ HHH exceeds a certain those users whose tr H k V k k k threshold to do feedback. In this way, inter-user interference can be better eliminated and a byproduct of this approach is that the overall feedback overhead is reduced. Also, the channel we considered in this paper is assumed to be uncorrelated, in practice however spatial correlation may exist. From [9] we know that in very highly correlated channels, the information of the spatial correlation matrix of the channel alone is sufficient for MU-MIMO techniques to provide much better performance than single-user transmission, therefore we may exploit the information contained in the correlation matrix in conjunction with those provided by limited feedback to improve the system performance for moderately correlated channels.
(
)
REFERENCES [1]
[2]
[3]
[4] [5] [6]
Lai-U Choi and R.D.Murch, “A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach,” IEEE Trans. Wireless Commun., vol. 3, No. 1, pp. 20-24, Jan. 2004. D. J. Love and R. W. Heath Jr., “What is the value of limited feedback for MIMO channels,” IEEE Communication Magazine, vol. 42, no. 10, pp. 54-59, Oct. 2004. M. Sharif, B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. Inform. Theory, vol. 51, no. 2, pp. 506-522, Feb. 2005.. N. Jindal, “MIMO Broadcast Channels with Finite Rate Feedback,” in IEEE Proc. GLOBECOM’05, vol. 3, pp.1520-1524, Nov. 2005. C. Wang and R. D. Murch, “Investigation into MU-MISO transmission with limited feedback,” in IEEE Proc. WCNC’06, Apr. 2006. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, NY, 1987.
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[9]
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I. INTRODUCTION Downlink multi-user multiple-input multiple-output (MU-MIMO) transmission techniques have received much attention recently because of their potential to significantly increase system capacity compared to single-user transmission by separating multiple users in the space domain through appropriate signal processing. A limitation of these downlink MU-MIMO techniques lies in that accurate channel state information (CSI) at the base station (BS) is crucial for them to operate properly so as to realize their significant capacity potential. In practice however, channel state information is difficult to be known perfectly at the BS due to the limited feedback channel capacity in FDD systems for instance. Therefore it is important to investigate multi-user MIMO transmission with limited feedback. That is, how to do the feedback and what is the impact of limited feedback on the achievable performance. In previous work, research on feedback strategies has been first considered for single-user point-to-point scenario and [2] provides an excellent overview of these strategies. Nearly all these schemes show that it is preferred to design the transmit signal at the receiver rather than at the transmitter, where a codebook is designed which contains several precoding matrices. Both the transmitter and the receiver know the codebook, so for each channel realization the receiver selects one precoding matrix from the codebook according to a certain criterion and sends back the index of the selected precoding matrix to the transmitter. It has been shown that the performance achieved with a small number of feedback bits is already very close to that achieved with perfect CSIT. Recently limited feedback has also been considered for the multi-user transmission scenario, but mainly for single-antenna
1
We acknowledge support of Hong Kong RGC grant HKUST 6164/04E
users. Because of the presence of inter-user interference, the situation for multi-user transmission with limited feedback is much more intricate than the single-user transmission case. In [3] a random beamforming scheme is proposed for MU-MISO system. However the scheme needs a very large number of users to achieve good performance. When the number of users is comparable to the number of transmit antennas, the performance is very poor. In [4], [5], the scenario with the number of users comparable to the number of transmit antennas is considered for MU-MISO system. In [4], the capacity degradation due to limited feedback and the required feedback load are analyzed when random vector quantization (RVQ) is used. While in [5], the optimal codebook design in terms of minimizing the residual inter-user interference for zero-forcing transmission is derived. Both show that for multi-user transmission to perform well a high feedback requirement is needed compared to single-user transmission. In this paper we consider limited feedback for the scenario where both the BS and the users are equipped with multiple antennas and multiple data streams are allowed to be transmitted to each user. In particular, we consider a multi-user MIMO system where the MU-MIMO decomposition scheme [1] is used for downlink transmission. In the framework, each user independently sends back selected but useful information of its own channel to the BS and let the BS select suitable precoding weights based on the feedback from all the users. We first identify the essential information that the BS needs to deal with the inter-user interference and then an optimal codebook design is derived in terms of minimizing the residual inter-user interference power. The remainder of the paper is organized as follows. System model is introduced in section II. In section III, we identify the essential information to be sent back and the optimal codebook design is derived accordingly. Simulation results and discussion are provided in section IV. Finally conclusions are drawn in section V. II. SYSTEM MODEL We consider the downlink of a narrow-band FDD multiuser MIMO system with a single BS equipped with M transmit antennas and K users each of which has N receive antennas. At the BS, the transmit data symbol vector of each user, i.e. b k , k = 1, , K , is passed through certain transmit precoding matrix Wk , k = 1, , K before it is launched into the downlink channel. Then the received signal at the kth user is given by K
y k = H k ∑ Wi b i + n k , k = 1, i =1
,K
(1)
where H k is the N × M channel matrix of user k with zero-mean unit-variance independent and identically distributed (i.i.d.) complex Gaussian entries, n k ∈ CN (0, I ) is the additive white Gaussian noise vector and I is the identity matrix. The sum mutual information of the downlink system can then be expressed as I=
K
Ik ∑ k =1
(2)
caused by limited feedback is derived accordingly. III.A MU-MIMO Decomposition Transmission Scheme
The objective of this scheme [1] is to choose the precoding matrices Wk , k = 1, , K in a way such that each user receives no interference from the other users. Equivalently, the scheme decomposes the multi-user system into multiple parallel single-user systems. Thus the precoding matrix can be written in the form as
where the mutual information of user k is given by −1 K H H H H I k = log 2 det I + Wk H k I + H k ∑ Wi Wi H k H k Wk = 1 , ≠ i i k (3) and H represents the hermitian operation. Note that K is the largest number of users that we would like to serve simultaneously. It may or may not be equal to the total number of users in a practical system, which may be larger than K, and therefore other multiple access techniques can be used as well. In a FDD system the uplink and the downlink channel are not reciprocal due to the frequency separation between them, thus feedback from the mobile to the BS is essential so that the broadcast channel can be known. Ideally each Wk is a continuous function of all H 1 , , H K such that appropriate signal processing can be performed to separate the simultaneous users in space. In practice however one user cannot know the channels of other users, therefore it is impossible for one user to choose a suitable precoding matrix depending on its own channel. As a result, unlike the single-user point-to-point scenario, instead of choosing the precoding matrix at the receiver, it is desirable for each user to independently send back some useful information of its own channel and let the BS decide the precoding matrices for all the users based on the independent feedback from all of them. As a consequence the proposed framework is as follows. If log 2 Q -bit feedback is allowed for each user, the single-user channel space can be partitioned into Q regions
~ H = {H 0 , , H Q-1} along with Q partition centroids ~ F = {F 0 , , F Q−1} . Note that in the rest of the paper the
number in the superscript and subscript indicates the index of the region and the index of the user respectively. Each user then independently selects the region its channel belongs to according to a certain criterion and the index of the selected region is fed back to the BS. At the BS, the transmit precoding matrix of each user is chosen according to the feedback information from all K users. Note that since the focus of the paper is to investigate the impact of limited feedback on the performance of MU-MIMO transmission, in the rest of the paper we shall assume each user has perfect CSI of its own channel and the feedback channel is error-free and delay-free.
III. LIMITED FEEDBACK APPROACH In this section we first briefly introduce the downlink MU-MIMO decomposition scheme, which is suboptimal but efficient and with low-complexity. Then the essential information for the scheme to deal with the inter-user interference is identified. Finally the optimal codebook design in terms of minimizing the residual inter-user interference
Wk = Vk G k ,
(4)
where Vk is an orthonormal basis of the joint null-space K
∩i=1,i≠k Ν (H i )
and it can be computed by singular value
decomposition as H1 H k −1 ~ Σ k − Hk = = Uk , Uk 0 H k +1 H K
[
]
~ 0 VkH . 0 VkH
(5)
And G k , k = 1, , K are non-zero matrices which can be designed alone by some criteria. The total power constraint PT of the BS can then be expressed as K
K
k =1
k =1
∑ tr(WkH Wk ) = ∑ tr(G kH G k ) ≤ PT , where tr (•) represents the matrix trace operation. Note that a sufficient condition to ensure the existence of the null-space is M > ( K − 1) N . III.B Identification of the Essential Information Denote the singular value decomposition of H k as
[
]
Λ Hk = Uk ,Uk k 0
0 VkH , k = 1, 0 VkH
,K ,
(6)
λ kN } , λ k1 ≥ ≥ λ kN > 0 are the where Λ k = diag {λ k1 non-zero singular values of H k . The joint null-space
Ν (H −k ) is the orthogonal complement of the row-space of
H −k , which is fully determined by the row-spaces of H i , i ≠ k .
( )
it is suffice to know the Therefore, to get Ν H −k orthonormal basis of the row space of each H i , i ≠ k . As a result, the useful information each user should send back to the BS is their individual Vk . To see this mathematically, denote V1H H V Vk− = kH−1 , Vk +1 H VK
(7)
and the received signal at the kth user can be expressed as
then we can rewrite H −k as H 1 U 1 Λ1 V1H H H U Λ V k −1 k −1 k −1 k −1 H k− = = H H U k + 1 k +1 Λ k +1 Vk +1 H H K U K Λ K VK U1 Λ 1 =
yk =
U k +1 Λ k +1
k
ˆ elements of H k V k are also i.i.d. zero-mean unit-variance complex Gaussian random variables. The second term is the residual inter-user interference caused by limited feedback, which is the main cause of performance degradation. Thus our aim is to minimize the residual interference, the power of which is given by
V VkH−1 H Vk +1 H U K Λ K VK
PT K ˆ ˆ tr H k Vi ViH H kH . ∑ KL i =1,i ≠ k
Ak =
(8) Clearly we can see that H −k and Vk− share the same row-space and thus the null-space. As a result Vk , k = 1, , K is the most important information which enables the BS to deal with the inter-user interference. Therefore we decide that each user should send back the information about its individual Vk . III.C Codebook and Feedback Strategy Design After the essential information is identified, it is also important to decide how to efficiently convey the identified essential information to the BS. This includes the design of how to partition the single-user channel space and how each user determines which region its channel belongs to. Since the BS only has information about Vk , k = 1, , K , it is thus impossible to design G k to match the equivalent single-user channel H k Vk . Also no power allocation can be performed either. Thus G k , k = 1, , K should be chosen as PT I L , k = 1, , K , where KL L = rank Vk = M − ( K − 1) N under the assumption of i.i.d. channels [1]. Note that the precoding matrix of each user is independent of its own channel. If the feedback channel has no capacity constraint, i.e. perfect knowledge of Vk , k = 1, , K is available at the BS, inter-user interference can be eliminated completely. In practice however only limited feedback is allowed, the BS only ˆ , k = 1, , K . As a has an estimate of Vk , k = 1, , K , i.e. V k result there will be residual inter-user interference seen by each ~ˆ ˆ− = V ˆ ˆ ˆ ˆ H , V V V V user. Let V k k −1 k +1 K 1 k be ˆ ˆ − and V an orthonormal basis of the row-space of V k k be an ~ ˆ ˆ ˆ − , i.e. V , V orthonormal basis of the null-space of V k k k
scaled identity matrices G k =
( )
[
(9)
where the first term is the desired signal, since the elements of H k are i.i.d. zero-mean unit-variance complex Gaussian ˆ has orthonormal columns, the random variables and V H 1
U k −1 Λ k −1
K PT ˆ H k ∑ Vi b i + n k , KL i =1,i ≠ k
PT ˆ H k Vk b k + KL
]
ˆ = spans C M . The precoding matrices become W k
PT ˆ Vk KL
(10)
ˆ for any i , i ≠ k . Note that Vˆ i is a function of V k ˆ is given by Therefore the criterion for choosing V k K
∑
min
ˆ V k i =1,i ≠ k
ˆ ˆ tr H k Vi ViH H kH
.
(11)
~ˆ Since Vi contains an orthonormal basis of the row-space ~ˆ ˆ − , without loss of generality V of V i can be chosen as i
[Vˆ , x ,
, x ( K −2 ) N
]
, where x1 ,
[Vˆ , x ,
, x ( K −2 ) N
]
ˆ − . Then we spans the row-space of V i
, x ( K −2 ) N are arbitrary ˆ and such that orthonormal vectors that are orthogonal to V k k
k
1
1
have
ˆ ˆ tr H k Vi ViH H kH ~ˆ ~ˆ = tr H k I − Vi ViH H kH , H H ˆ ˆ = tr H H − H V V H H
(
−
k
k
k
H k x1x1H H kH
k
k
for any i , i ≠ k ,
k
− H k x ( K −2 ) N x (HK −2 ) N H kH
−
(12)
)
ˆ becomes thus the selection criterion for choosing V k
(
)
ˆ V ˆH H max tr H k V k k Hk . ˆ V k
(13)
Note that for the special case with N = 1 , the criterion reduces to the one derived in [5] for multi-user MISO system. To see the intuition behind the criterion, we have
(
)
ˆ V ˆH H tr H k V k k Hk ˆ H V Λ2V H V ˆ = tr V
( = tr (Λ V k
2 k
k
k
k
k
H ˆ ˆ H k Vk Vk Vk
) )≤ ∑λ γ N
i =1
2 ki
, ki
(14)
where the inequality comes from [6] and γ k1 ≥ ≥ γ kN ≥ 0 ˆ V ˆH are the eigenvalues of VkH V k k Vk . From [7] we know that
M=4, N=2, K=2 35
γ ki = cos 2 θ ki , where θ k1 ≤ ≤ θ kN , are the principle angles ˆ and V . The closer between the subspaces spanned by V k k these two subspaces, the smaller the principle angles, as a result the larger the γ ki , i = 1, N and therefore the smaller the residual inter-user interference. As a consequence the objective of the codebook design is Q −1
q
q=0
qH
Vq
)
(
H H HV q H ∈ H q Pr H ∈ H
H
s.t. V q V q =I N for q = 0,1,
q
Sum Ergocid Capacity in bits/channel−use
E tr (V ∑ Hmax ,
30
),
, Q −1 .
The design problem can then be solved by a two-step method as follows,
(
)
(
H V q = arg max E tr V q H H HV q H ∈ H q Pr H ∈ H q V
H
s.t. V q V q =I N . q
given V q ,
H q = H : tr V q H H HV q ≥ tr V j H H HV j ,
∀j ∈ [0,
),
(16)
Step 2) Determine the optimal partition H q = 0, , Q − 1 , as H
q
H
, q − 1, q + 1, , Q − 1]}
(17)
Thus the codebook can be obtained by iteratively solving the above two steps. The algorithm is guaranteed to converge since at each step the objective quantity is always increased because we choose the updated parameters to maximize the other fixed parameters. However a local optimal may be found, thus we take several random initial partitions and the one that gives the maximum objective quantity in (15) is selected. Since H E tr V q H H HV q H ∈ H q , H = tr V q E H H H H ∈ H q V q
[
(18)
]
H
the solution V q of the first step subject to V q V q =I N should be chosen as the N dominant eigenvectors of
[
E HH H H ∈H
q
] corresponding to the N largest eigenvalues
[6]. For every channel realization, each user sends back the index chosen according to
(
H
)
q *k = arg max tr V q H Hk H k V q , k = 1, q = 0,1, ,Q −1
,K ,
(19)
ˆ = V q*k and the BS decides the transmit precoding then V k
25
20
15
10
5
(15)
Step 1) Determine the optimal V q given a certain partition H q , q = 0, , Q − 1 , as
MU Perfect CIST MU 11bits MU 9bits MU 7bits SU Perfect CSIT SU 7bits
0 −4
0
4
8
12
16
20
24
28
32
PT in dB
Fig. 1. Sum ergodic mutual information comparison with different number of feedback bits per user, M=4, N=2, K=2
matrices based on (q1* , q2* , , q*K ) . Note that when it happens that more than 1 user selects the same partition, single-user ˆ and each user transmission is used, where Wk = (PT N )V k uses 1 K of the time for transmission due to time division. Since even if we let the codebook of each user to be a rotated version of each other, although this can prevent the case for exactly identical estimated channel row space, this does not change the fact that the users’ row spaces are very close to each other given a reasonable number of feedback bits. So in this case even with perfect channel state information, it is not good to use multi-user MIMO decomposition scheme [9].
IV. NUMERICAL RESULTS AND DISCUSSION In this section we provide numerical results to see the impact of limited feedback on the system performance in terms of sum ergodic mutual information. Note that as we have mentioned that when more than 1 user selects the same partition, single-user transmission is used. However for all the figures in the simulation section, the sum ergodic mutual information of MU-MIMO decomposition transmission with limited feedback that we show is averaged over the simulation trials when multi-user transmission is used in order to study the impact of limited feedback. In Fig.1, the performance of a system with a BS equipped with 4 transmit antennas and 2 users each has 2 receive antennas is investigated for both MU-MIMO decomposition transmission and TDMA single-user transmission†. From the figure we can see that single-user transmission with 7bits feedback already performs very near that achieved with perfect CSIT. While the performance achieved by MU-MIMO decomposition transmission with 11bits feedback is still far away from that achieved with perfect CSIT and is worse than that achieved by single-user transmission. Considerably more feedback bits are needed for MU-MIMO decomposition †
Each user uses half of the time with full transmit power. The codebook for single-user transmission is obtained using the scheme proposed in [8].
M=4, N=3, K=2
M=3, N=2, K=2 40
30
MU Perfect CIST MU 3bits SU Perfect CSIT 35
25 Sum Ergodic Capacity in bits/channel−use
Sum Ergodic Mutual Information in bits/channel−use
MU Perfect CSIT MU 3bits SU Perfect CSIT
20
15
10
30
25
20
15
10
5 5
0 −4
0
4
8
12
16 20 PT in dB
24
28
32
36
40
Fig. 2. Sum ergodic mutual information comparison with different number of feedback bits per user, M=3, N=2, K=2
transmission to outperform single-user transmission. So with moderate number of feedback bits, single-user transmission is preferred. In Fig. 2, we provide the performance of a system with a BS equipped with 3 transmit antennas and 2 users each with 2 receive antennas. In this special case we can see that contrary to what we observed in Fig. 1, the performance achieved by MU-MIMO decomposition transmission with 3bits feedback is quite close to that provided by knowing perfect CSIT. In fact, this phenomenon is valid for any systems with N = M − 1, K = 2 . We show the reason as follows. For systems with N = M − 1, K = 2 , the mutual information achieved by user 1 can be expressed as P I1 = log 2 1 + T 2 where Vˆ 1 and
P ˆ ˆ ˆ ˆ V1H H1H I + T H1 V2 V2H H1H H1V1 , (20) 2 ˆ V2 are both M × 1 vectors. Denote the eigenvalue decomposition of H 1 Vˆ 2 Vˆ 2H H 1H as −1
ˆH H ˆ V H HV 2 1 1 2 0 ˆ ˆ H H1 V2 V2H H 1H = X X , 0 the mutual information can then be re-expressed as −1 P ˆ P ˆ ˆ ˆ I 1 = log 2 1 + T V1H H 1H I + T H 1 V2 V2H H 1H H 1 V1 2 2
(
= log 2 1 + 1 P ˆH H ˆ T V H H V +1 PT ˆ H H 2 2 1 1 2 V1 H 1 X 1 2
X H H Vˆ 1 1 1
(21)
0 −4
0
4
8
12
16 20 PT in dB
24
28
32
36
40
Fig. 3. Sum ergodic mutual information comparison with different number of feedback bits per user, M=4, N=3, K=2
1 a PT 2 = log 2 1 + z
z H , PT 2 P 2 2 1 2 P = log 2 1 + z1 + T z2 + + T z N 2 2 a
(22) 2 ˆH H ˆ where a = V 2 H1 H1V2 + PT
ˆH H and z = V 1 H 1 X is a vector
with i.i.d. CN (0,1) entries. This means z i , i = 1, , N are i.i.d. chi-squre distributed with 2 degrees of freedom. Similar analysis applies for the mutual information achieved by user 2. From the above analysis we can see that in this special case the residual interference affects only one of the chi-square distributed random variables, this explains why the performance loss is small. In addition, as the number of transmit antennas increases, the impact of the residual interference gets smaller as it affects a smaller percentage. And Fig. 3 numerically verifies this point as we can see that with the same number of feedback bits, the performance achieved with limited feedback achieves a higher percentage of that achieved with perfect CSIT in a system with M = 4 , N = 3 than a system with M = 3 , N = 2 . Note however, although in this system setting where MU-MIMO decomposition with only a few bits feedback performs close to that achieved with perfect CSIT, single-user transmission performs better than MU-MIMO decomposition even when perfect CSIT is known. It is because in this system setting, MU-MIMO decomposition transmission scheme sacrifices too many degrees of freedom to cancel the inter-user interference. Furthermore, similar analysis and results apply for cases with K = 2, N > M − N , where the residual interference has less detrimental impact compared to other system settings. 2
V.
CONCLUSIONS
In this paper, multi-user systems employing MU-MIMO decomposition transmission in the downlink with limited feedback are investigated. We identify the essential information that the BS needs to deal with the inter-user interference and propose an optimal way to convey the identified essential information to the BS so as to minimize the residual inter-user interference power. We found that in system settings which are in favor of MU-MIMO decomposition transmission when perfect CSIT is available, i.e. where MU-MIMO decomposition transmission outperforms single-user transmission when perfect CSIT is available, considerable more feedback bits are needed for MU-MIMO decomposition to perform well, which is caused by the detrimental impact of inter-user interference. As a result, if only very limited number of feedback bits is allowed single-user transmission is preferred. When the system allows much more feedback bits and can afford the complexity (as the number of feedback bits increases, the codebook size and search space increase exponentially), then MU-MIMO decomposition should be used. Although from this conclusion we may doubt the feasibility of multi-user MIMO schemes in practical wireless systems, there are still some possibilities of improving the performance. One aspect is that in a system with a large number of users, multi-user diversity in feedback quality can be exploited to improve the system performance with a fixed number of feedback bits. Since it is the residual inter-user interference that causes the major performance degradation, we may only allow ˆ V ˆ HHH exceeds a certain those users whose tr H k V k k k threshold to do feedback. In this way, inter-user interference can be better eliminated and a byproduct of this approach is that the overall feedback overhead is reduced. Also, the channel we considered in this paper is assumed to be uncorrelated, in practice however spatial correlation may exist. From [9] we know that in very highly correlated channels, the information of the spatial correlation matrix of the channel alone is sufficient for MU-MIMO techniques to provide much better performance than single-user transmission, therefore we may exploit the information contained in the correlation matrix in conjunction with those provided by limited feedback to improve the system performance for moderately correlated channels.
(
)
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