Transceiver design using linear precoding in a ... - Semantic Scholar
www.ietdl.org Published in IET Communications Received on 7th January 2010 Revised on 19th April 2010 doi: 10.1049/iet-com.2010.0015
ISSN 1751-8628
Transceiver design using linear precoding in a multiuser multiple-input multiple-output system with limited feedback M.N. Islam R. Adve Electrical and Computer Engineering, University of Toronto, Toronto, Canada E-mail: [email protected]
Abstract: The authors investigate quantisation and feedback of channel state information in a multiuser (MU) multiple-input multiple-output (MIMO) system. Each user may receive multiple data streams. The authors design minimises the sum mean squared error (SMSE) while accounting for the imperfections in channel state information (CSI) at the transmitter. This study makes three contributions: first, the authors provide an end-toend SMSE transceiver design that incorporates receiver combining, feedback policy and transmit precoder design with channel uncertainty. This enables the proposed transceiver to outperform the previously derived limited feedback MU linear transceivers. Second, the authors remove dimensionality constraints on the MIMO system, for the scenario with multiple data streams per user, using a combination of maximum expected signal combining and minimum MSE receiver. This makes each user’s feedback independent of the others and the resulting feedback overhead scales linearly with the number of data streams instead of the number of receiving antennas. Finally, the authors analyse the SMSE of the proposed algorithm at high signal-to-noise ratio (SNR) and large number of transmit antennas. As an aside, the authors show analytically why the bit error rate, in the high SNR regime, increases if quantisation error is ignored.
1
Introduction
The advantages of spatial diversity and multiplexing has led to the investigation of multiuser (MU) multiple-input single-output (MISO) and multiple-input multiple-output (MIMO) wireless communication systems [1]. Spatial diversity can increase system reliability as well as the spectral efficiency of MU systems. However, limitations caused by interference and channel fading remain a concern in MU systems. These can be mitigated by precoding the signals before transmission, in turn requiring channel state information at the transmitter (CSIT). This paper focuses on linear transceiver design in the downlink of MU MISO [2, 3] and MIMO systems [4, 5], a single base station (BS) communicating with multiple receivers. Each user may receive multiple data streams; the figure of merit here is the sum mean squared error (SMSE) across all data streams. In a frequency division duplexing system, channel information needs to be estimated at the receiver and sent IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
back to the BS after quantisation. Recent works suggest that this might be required in broadband time division duplex systems as well [6]. In general, providing accurate CSIT and reducing feedback overhead are important considerations in a linear transceiver design. Our work assumes perfect channel estimation at the receiver end with zero delay error-less feedback and focuses on quantising channel state information (CSI). In the available literature, scalar quantisation [7 – 10], vector quantisation (VQ ) [11, 12] and matrix quantisation [13, 14] have all been used to quantise CSI. It is now well established in the single user, single data stream, case that projecting the MIMO channel to an appropriate vector downlink channel yields better performance than full channel scalar quantisation with same feedback overhead [15]. This has led to considerable research in VQ , which reduces the feedback overhead by allocating bits in the proper vector downlink channel. In VQ , to send B feedback bits as the channel index to the BS, each user 27
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www.ietdl.org needs a codebook with 2B code vectors. Grassmannian line packing [16], VQ using MSE as the optimality criterion [17] and random vector quantisation [18] have been the most popular approaches for the MU case. In this paper, we investigate VQ , based on the mean squared inner product (MSIP) criterion [19] as the feedback method. In a MU-MISO system, users can feed back the channel vectors using VQ. However, in the MIMO case, one needs to combine the receive antennas to convert the MIMO channel to the effective vector downlink MISO channel. Since the receivers cannot cooperate, the quantisation scheme of each user must be independent from the others. Projecting the MIMO channel to the direction of its maximum eigen vector (MET) [12] is the optimal solution at low signal-to-noise ratio (SNR). Jindal [11, 20] proposed quantisation-based computing (QBC) which chooses the effective vector downlink channel to produce the least quantisation error; this is optimal at high SNR. Trivellato et al. [21] proposed maximum expected signal combining (MESC) to maximise the signal to interference plus noise ratio (SINR). Their scheme outperforms both QBC and MET; MESC converges to MET and QBC in the low and high SNR region, respectively, [21]. All these schemes discussed so far assume that each user receives a single data stream. However, multiple data streams per user complicate the feedback process, requiring linearly independent information for each stream. In this paper, we extend the MESC algorithm to the case with multiple data streams per user. Most of the relevant works in the limited feedback MU literature suffer from dimensionality constraints. With M transmit antennas, N total receive antennas and L data streams in total, either M ≥ N [18] or N ¼ L [22]. To the best of our knowledge, only authors in [7 – 9] avoid these constraints. However, by using scalar quantisation, the feedback overhead in these systems scale linearly with MN [7, 8] and (M 2 − 1) [9], respectively. Owing to the formulation of our MESC receive combining, the feedback overhead in our proposed system scales only with M × L (where L is the total number of receive data streams). Since, with linear precoding L ≤ M and L ≤ N, the proposed transceiver allows significant performance improvements with lower feedback rate. Finally, previous works [17] have shown, by simulation, that if the quantisation error is ignored, the MSE increases at high SNR. Here we investigate why this is true theoretically. The overall contributions of this paper are therefore the following: 1. We provide an end-to-end SMSE transceiver design that eliminates the dimensionality constraint and tie the feedback overhead to the number of data streams which is always less 28
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than or equal to the number of both transmit and receive antennas. 2. We extend and make the MESC receiver flexible by allowing multiple data streams per user scenario. 3. We show the error floor effect in terms of SMSE in MU broadcast systems. Previous works on this area focused on the ceiling effect in terms of capacity [11, 18] and SINR [17]. As an aside, we show why SMSE and bit error rate (BER) increase instead of converging to a constant if quantisation error is not considered. Preliminary results from this paper have appeared before [23, 24]. The new developments are as follows. First, the receive combining scheme proposed in this paper, namely MESC for multiple data streams per user, holds the benefits of eigen-based combining (EBC) and QBC at low and high SNR, respectively. In [23], we used EBC, suboptimal at high SNR, for all SNR. Second, the relation of the proposed algorithm to the existing transceivers and the analysis of the quantisation error were not presented in [23, 24]. Finally, we use a different proof in this manuscript to analyse the asymptotic SMSE of the proposed system. The remainder of this paper is organised as follows. Section 2 describes the system model and reviews the transceiver design problem with full channel knowledge. Section 3 reviews the proposed quantisation method and shows the linear precoder design. Section 4 illustrates the two-step receiver design process and gives the overall algorithm. We analyse our proposed transceiver in Section 5. After providing the numerical simulation results in Section 6, we draw our conclusions in Section 7. Notation: Lower case x denotes scalar whereas lower case bold face, h means column vector. Upper case boldface, V denotes matrix whereas uppercase normal font N represents constant entry. The superscripts (.)T and (.)H denote the transpose and conjugate transpose operators, respectively. tr [.] denotes the trace operator. I is reserved for the identity matrix whereas 1 represents the column with all one vector. diag(.) denotes the diagonal matrix where the diagonal entries contain the bracketed terms. · 1 denotes the L1 norm of the vector. E(.) denotes statistical expectation.
2
System model
We consider both MU-MISO and MU-MIMO systems in our design. We first describe the MU-MIMO system model. Then we show that, with our approach, transceiver design in the MU-MIMO system is very similar to that in a MU-MISO system.
2.1 MIMO system model Consider a single BS equipped with M transmit antennas communicating with K independent users. User k has Nk IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org antennas and receives Lk data streams. Let L = k Lk , N = k Nk . To ensure resolvability, we assume L ≤ M and Lk ≤ Nk . The ith data stream is processed by a unit norm linear precoding vector ui with the global precoder U = [u1 , u2 , . . . , uL ]. Let p = [p1 , p2 , . . . , pL ]T be the powers allocated to the L data streams and define the downlink power matrix P = diag( p). p1 ≤ Pmax where Pmax is the total available power. The data vector x = [x1 , . . . , xL ]T = [xT1 , xT2 , . . . , xTK ]T , includes all L data streams to the K users. The Nk × M block fading channel, HH k , between the BS and user is assumed to be flat. The global channel matrix is H H , with H = [H 1 , . . . , H k ]. User k receives √ H yDL k = H k U P x + nk
(1)
where nk represents the zero mean additive white Gaussian noise at the receiver with E[nnH ] = s2 I Nk . We also assume, E[xxH ] = I L . To estimate its own transmitted symbols, from yDL k , user k forms xˆ k =
√ xˆ = V H H H U P x + V H n √ = F H U Px + V H n
(2)
The MSE of the ith data stream is given by = E[( xi − xi )( xi − xi )H ] eDL i
(3)
The SMSE minimisation problem is
p,U
UL
y
=
eDL subject to p1 ≤ Pmax , ui = vi = 1 i
i=1
L
f j qj x j + n
(5)
j=1
where n = [nT1 , nT2 , . . . , nTK ]T and to facilitate our analysis, we define the M × L matrix F ¼ HV with F = [ f 1 , . . . , f L ]. The vectors f 1 , . . . , f L are the effective M × 1 MISO channels for the individual data streams.
L
In designing the precoder U, it is computationally efficient to use a virtual dual uplink [4]. In this uplink the transmit powers are q = [q1 , . . . , qL ]T for the L data streams, while the matrices U and V remain the same as before. The global virtual uplink power allocation matrix Q is defined as, Q ¼ diag(q) where q1 ≤ Pmax . So the received data in the BS in the virtual uplink is given by
DL VH k yk
Here V k is the Nk × Lk decoder vector for user k. Fig. 1 shows the block diagram of the proposed system in the downlink. Let V be the N × L block diagonal global decoder matrix, V = diag(V 1 , . . . , V K ). Overall
min
Figure 2 Block diagram of MU-MIMO uplink with channel and decoder combined as single block
(4)
Therefore data stream i is decoded as xˆ UL i
=
uH i
L
f j qj x j + n
(6)
j=1
Fig. 2 shows the proposed system model in the virtual uplink. As (6) and Fig. 2 show, the system has become an effective MU single input multiple output system in the virtual uplink. Uplink – downlink duality states that the same MSEs can be achieved in the uplink and the downlink with the same matrices U and V and the same power constraint. A recent result shows that at the optimal solution, p ¼ q [25]. With perfect channel knowledge, the transmitter iterates between V and Q and converges to the the optimum solution using a convex optimisation problem formulation [4]. Then the precoder finds the optimal U using the minimum mean square error (MMSE) solution [4]. The downlink power allocation is then set equal to Q [25].
3 MSIP quantisation and linear precoder design 3.1 MSIP quantisation
Figure 1 Block diagram of MU-MIMO downlink IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
We assume that the receivers have perfect CSI using training. For the purposes of quantisation only, the ith user chooses the quantised codevectors, fˆ i , . . . , fˆ L that would maximise i the SINR of its receiving data streams. Choosing the best quantised codevector is described in detail in the receiver design section. Each user has a codebook consisting of 2B unit norm vectors w ˆ 1, . . . , w ˆ 2B . Each user feeds back B bits per data stream to the BS. The receivers individually 29
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www.ietdl.org normalise and then quantise each of the Lk effective channels using the chordal distance [13] fˆ i = arg
2
min
w[w ˆ 1 ,...,w ˆ 2B
sin (/( f i , w)) ˆ
2
C(·)
(8)
Here, fˆ = C( f ) is the quantised effective channel and k·,·l denotes the inner product. Overall, we consider the following channel model at the BS for precoder design. f i + fi f i =
F = F + F
or
√ uMSE f i qi = J −1 i
(7)
The use of chordal distance over the Euclidean distance leads to a higher inner product between the original and quantised channels [19]. Here, we only quantise the direction of the effective channel and this direction can lie anywhere on the M-dimensional complex unit-norm sphere. Therefore we generate the quantisation codebook as a VQ problem using the MSIP optimality criterion; the details of MSIP VQ codebook generation can be found in [19]. Each user at first generates a large set of random unit norm Mdimensional complex vectors f and finds the quantiser codebook C to maximise the MSIP ˆ 2B ) = max E| , f , C( f ) . | (w ˆ 1, . . . , w
virtual uplink by [26]
(9)
F comprises L unit-norm effective channel vectors with the original channel directions. F denotes the L quantised feedback unit norm vectors. F denotes the error in the quantisation. We assume that the quantisation error matrix F has M×L independent identically distributed elements with zero mean and a variance of s2E /M. s2E is the quantisation error variance associated with each quantised F is assumed to be independent of x, n and F. vector fˆ i . Since s2E depends on receiver combining, the details regarding the expected value of this term in our proposed algorithm will be clarified in the receiver design and analysis section.
(10)
2
H s J = FQ F + s2 I M + E (q1 + · · · + qL )I M M
(11)
So eUL,MSE =1− i
√ H −1 √ q i f i J f i qi
(12)
Therefore the uplink SMSE is (see (13)) As F is fixed, the SMSE expression is a function of uplink power allocation Q.
Proposition 1: The optimisation problem for power allocation
q + · · · + qL 2 sE tr( J −1 ) Qopt = min s2 + 1 Q M
(14)
subject to tr[Q] ≤ Pmax , qk ≥ 0 for all k is convex in Q.
Proof: Ding [27] shows that SMSE remains a non-increasing function of SNR if channel uncertainty isequal and all available power is used because tr(Q) = Li=1 qi = Pmax makes the term within the brackets a constant. J is a positive definite matrix and therefore the optimisation problem is convex in J [28]. Since J is linear in Q, it can be readily proved that the problem is convex in Q. A The power allocation problem is therefore convex given F. In the next section we discuss the solution for V (equivalently F ). This section represents the core contribution of the paper.
4
Receiver design
It should be noted that the channel vector of each user is already in the form of f i in the MU-MISO case. Therefore in this case, we just quantise the normalised channel using chordal distance and return the corresponding index to the BS.
We propose a two-step receiver design. For the purposes of quantisation only, each user uses a MESC receiver and chooses the quantised codevectors that would maximise the SINR of their data streams. However, the users implement MMSE receivers while receiving the actual data. This is unlike the single MMSE solution in [2, 4] but allows for the channel feedback to be independent of the other users’ actions.
3.2 Linear precoder design
4.1 Receive combining with MESC
The optimum U in the SMSE minimisation problem of the system model proposed in (9) and (6) has been solved in the
Before proceeding with the analysis, let us clarify the relation between F and HV that will be used interchangebly in this
SMSEUL =
L
eUL,MSE = i
i=1
L i=1
2
=L−M + s +
⎡ −1 ⎤ L 2 L H H H s q √ −1 √ ⎦ FQ F + s2 + E i=1 i I M 1− qi f i J f i qi = L − tr⎣ FQ F M i=1
s2E
L
k=1 qk
M
tr[ J −1 ]
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(13)
IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org section. We have f i = H k vi where H k is the channel of the kth user receiving the ith data stream and vi is the decoding vector used for the ith stream; ui and uj are the precoding vectors of the ith and jth data stream. Now, using our quantisation policy in (7), we define the quantisation angle ui [ [0, p/2] as H cos ui = | f i fˆ i |
(15)
assumes that the vector downlink channels for all other users’ stream are mutually orthogonal to its own channels. We also assume that noise variance, signal power, quantisation error variance in the BS and total number of data streams sent by the BS are known to each of the users. Therefore because of the construction of (18), each user can find the expected value of f H i ui and ui even without co-operating with other users. Therefore by seperating the intra-user streams from inter-user streams, (17) takes the following form
Here, f i represents the unit norm effective vector downlink channel, that is, f = f /f . f is the effective MISO channel of the data stream. Since, the receivers know the quantisation angle exactly, we can use this information to improve the expected SINR. As in [21], define the quantisation error as
SINRDL = i
f i f i ) fi fi = fi −(
(16)
Here, f i 2 = sin2 ui . Now, in the downlink, the SINR of the ith stream is = SINR DL i
H 2 i ui |
s2 +
(P/L)| f H 2 j[L,j=i (P/L)| f i uj |
Now
(17)
Here, ui is normalised such that ui = 1. Since the users do not cooperate, the quantisation and feedback methods implemented by the users need to be independent of each other. Based on the fact that, the effective MISO channels of different users are statistically independent of each other, we assume different user’s quantised channels to be H mutually orthogonal, that is, fˆ fˆ = 0 where i and j j
indicate data streams that are being received by two different users. Following this assumption, it can be easily H verified from (18) that fˆ u = 0. i
2 2 |f H i uj | = f i
j[L, jLk
H
H
H
||( f i f i ) f i uj + f i uj ||2
j[L, jLk
(20) = f i 2
H
|| f i uj ||2
(21)
j[L, jLk
In (17) equal power allocation was assumed to simpify the receiver combining analysis. Here, ui and uj follow the form in (10). Now using (10) and the matrix inversion lemma [29] (see (18))
i
(19)
j[L, jLk
H
H 2 (P/L)|vH i H k ui | H 2 s2 + j[Lk , ji (P/L)|vH i H k uj | 2 + (P/L)| f H i uj |
j
Since each user knows the inner product of different code vectors in its codebook, the assumption of orthogonality is not valid for two different streams of the same user. Therefore in our proposed algorithm, each user uses its known codevectors, that is, the effective channels of its data streams, as a set of column vectors f in the F matrix and
f i 2 = f i 2
H
|| f i uj ||2
(22)
j[L, jLk
L − Lk M −1
(23)
=
L − Lk H ( f i 2 − ( f H i f i )( f i f i )) M −1
(24)
=
L − Lk H H H v (H k (I − fˆ i fˆ i )H k )vi M −1 i
(25)
= f i 2 sin2 ui
Here, (20) is obtained by taking out the norm of f i and using
H (16). Equation (21) follows since f i uj = 0 for mutually orthogonal reported channels from different users. Equation (22) was obtained by setting f = f / f . Equation (23) i
i
i
was derived using the analysis of [21]. In the presence of large number of codevectors, ui is very small which leads to H f i ≃ 0. Therefore, the unit vectors f i and uj are both fi identically distributed in the (M 2 1)-dimensional plane H
f i uj 2 follows a beta orthogonal to f i . This implies, distribution with parameter (1, M 2 1) and has expected value 1/(M 2 1) [21]. The factor of (L − Lk ) arises since the kth user receives Lk data streams and therefore (L − Lk )
−1 H s2E 1 1 √ ˆ Pmax I + FQF f i qi = 2 1− 2 ui = s + 2 2 M s + (sE /M)Pmax s + (sE /M)Pmax −1 1 H H √ fˆ i q i × Fˆ Q−1 + 2 Fˆ Fˆ Fˆ 2 s + (sE /M)Pmax 2
IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
(18)
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www.ietdl.org data streams are mutually orthogonal to the ith data stream. Equation (24) was obtained using the quantisation angle definition of (15). In (25), we again use f i = H k vi . Using the results of (25) in (19) and defining ⎛
⎞
P L − Lk ⎝ ⎠ (I − f i f H u uH + Bi = H H i ) H k (26) M −1 L k j[L , ji j j k
second stream of the kth user. So
BLk
2
P H L−2 H H H uLk uLk + (I − w ˆ Lk w = ˆ Lk ) H k 1 1 2 2 L k M −1 (32)
H H P vH Lk2 L H k uLk2 uLk2 H k vLk2 SINR DL = (33) Lk2 s2 + vH Lk BLk vLk 2
(w ˆ Lk , vLk ) =
Equation (19) takes the following form
2
2
max
(vLk =1,w ˆ Lk [W ) 2
= SINR DL i
H H P vH i L H k ui ui H k vi s2 + vH i B i vi
(27)
in (27) is a Owing to the structure of ui and Bi , SINR DL i function of vi and fˆ j ∀j [ Lk . Each of these fˆ j vectors are in the codebook C which consists of w ˆ 1, . . . , w ˆ 2B codevectors. Therefore the linear decoding vector vj and w ˆ j ∀j [ Lk should be chosen jointly as (w ˆ j , vj )∀j [ Lk =
Lk
max
vj =1,w ˆ j [W
SINR DL j
(28)
j=1
Here, vj is a complex Nk -dimensional vector. Joint optimisation for all the data streams of a particular user in (28) will lead to a computational complexity proportional to (2B )Lk . One sub-optimal solution to reduce complexity is to find the optimum decoding vector and quantised channel one data stream at a time. This simplified algorithm is given below: 1. First, assume that intra-user streams are orthogonal and find the vector downlink channel of the first stream. Therefore maximising (27) becomes an optimisation problem of w ˆ Lk and vLk where Lk1 denotes the first 1 1 data stream of the kth user. So BLk = 1
P H L−1 Hk (I − w ˆ Lk w ˆH ) Hk Lk1 1 L M −1
SINR DL Lk =
H H P vH Lk1 L H k uLk1 uLk1 H k
1
(w ˆ Lk , vLk ) = 1
1
s2
+
(vLk =1,w ˆ Lk [W ) 1
vLk
1
vH Lk1 BLk1 vLk1
max
(29)
SINR DL Lk 1
(30)
(31)
1
2. Once the quantised channel of the first stream is chosen, the user assumes it to be a non-orthogonal channel for the second stream’s vector downlink channel. However, vector downlink channels for the other streams of the same user are still considered to be orthogonal to both first and second stream’s channel. Thus maximising (27) again becomes an optimisation problem with variable vLk and 2 w ˆ Lk for the present data stream where Lk2 denotes the 2
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2
2
SINR DL Lk 2
(34)
2
3. For the third data stream of the kth user B Lk = 3
P H H (u uH + uLk uH Lk2 2 L k Lk1 Lk1 L−3 (I − w ˆ Lk w ˆH + Lk3 ) H k 3 M −1
(35)
The other equations take the forms of (29) – (35). The same policy continues upto the last stream of the kth user. Note that as we increase the number of data streams in the reported non-orthogonal channels part, number of the increases and components in the summation term uj uH j (L − Lk )/(M − 1) decreases. This follows the reasonings provided in deriving (20) – (25). With this algorithm, the SINR expression for a particular data stream remains a function of only its decoding vector and its quantised channel. This leads to a computational complexity of Lk × 2B in finding the channels of Lk data streams. Now (26) and (27) can be thought as a general form of all the data stream’s SINR expressions. In (26) and ˆ i. (27), both f i and u depend on the chosen codevector w For any particular w ˆ i , the linear decoding vector that maximises (27) can be obtained by the MMSE detector, √ u [21]. Then vi = (s2 I + Bi )−1 (P/L)H H k i SINR DL = i
P H u H (s2 I + Bi )−1 H H k ui L i k
(36)
The user finds the value of SINR DL for every w ˆ i using (36) i and chooses the w ˆ i , as the quantised channel fˆ i , that maximises SINR DL i . It is worth emphasising that, to our knowledge, this is the first receive combining scheme that considers signal power, inter-user and intra-user interference while accounting for multiple data streams per user.
4.2 Receiver design for data processing As mentioned earlier, MESC is for quantisation purposes only. The BS determines p and U based on the quantised F. However, for mutually non-orthogonal reported channels and a finite number of users, using MMSE receivers for data processing provide better results than IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org MESC receivers [21]. Therefore for the data H 2 −1 H vi = (H H k U PU H k + s I ) H k ui pi
5
5.1 Relation to the existing algorithms (37)
which can be normalised to make vi = 1. H H k is the MIMO channel of the kth user receiving the ith data stream. ui and pi respresent the designed precoder and allocated power for the ith stream. Note that the MMSE receiver cannot be implemented at the time of channel quantisation since the precoder matrix U was not designed at that time. The implementation of the decoder mentioned in (37) requires infinite training symbols. Therefore from a practical point of view, the BS either sends a finite number of dedicated symbols [30] or uses limited feedforward [31] to convey the post-processing information to the receivers. However, in our simulations, we restrict ourselves to the case where the users can estimate the effective channels of their data streams.
4.3 Overall algorithm Using the developments in Sections 3, 4.1 and 4.2, the steps of the proposed overall algorithm for SMSE minimisation in the MU system are as follows: 1. Send common pilots to the users in the system so that each user can estimate its own channel. 2. Each user generates a separate codebook of 2B unit norm vectors using MSIP VQ in off-line. In the MUMIMO case, each user converts its estimated MIMO channel to effective MISO channels using the MESC algorithm proposed in Section 4.1 and sends the codebook indexes of the effective channels to the BS. In a MUMISO system, each user quantises its own channel and the BS assumes V ¼ I. 3. Virtual uplink power allocation: Qopt = minQ (s2 + (s2E Pmax /M))tr( J −1 ), such that tr(Q) ≤ Pmax . This is a convex optimisation problem. Here, J follows (11). √ 4. Uplink beamforming: ui = J −1 fˆ i qi , ui = 1. 5. Downlink power allocation P ¼ Q. 6. Send dedicated pilot symbols for each of the data streams. Thereafter implement the MMSE downlink decoders using (37). vi = 1 for the ith data stream. The algorithm above results in a precoder U, decoder V and power allocation p. Note that the solution is sub-optimal because U and p are designed using MESC, not MMSE. This is the price paid for the feedback to be independent across users. IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
Analysis and discussion
As the proposed receive combining technique maximises the expected SINR of the data streams at the user end, it is equivalent to the MESC algorithm in the case of one data stream per user of [21] which was designed for the zero forcing (ZF) precoder. To illustrate this, let Lk = 1. Since intra-user interference is not present, all the quantised effective channels in F are assumed to be mutually orthogonal. Using this in (18) we obtain ui =
1
ˆ √ qi −
f s2 + s2E Pmax i
ˆ × F Q−1 +
1 (s2 + s2E Pmax )2
1 I s2 + s2E Pmax
−1
√ [1, 0, . .. , 0]T qi / fˆ i (38)
Equation (38) follows since (Q−1 + (1/(s2 + s2E Pmax ))I )−1 is a diagonal matrix. Since ui = 1, ui = fˆ i in this scenario. Using this in (19) we find SINR DL i =
H vH ˆ iw ˆH i ((P/L)H k w i H k )vi H H 2 s + vi ((P/L)H k ((L − 1)/(M − 1)(I − w ˆ iw ˆH i ))H k )vi
(39) This is the exact same expression obtained in [21] as the MESC combiner with noise variance s2 = 1. Trivellato et al. [21] has shown that this algorithm takes the form of MET combining at low SNR and QBC at high SNR. Thus MESC combining of [21] considers signal power and inter-user intereference while choosing the code vector. Since we are considering multiple data streams to each user, our proposed SINR expression in (19) considers signal power, inter-user and intra-user interference altogether. Thus our proposed algorithm is a generalised form for MESC combining with multiple data streams.
5.2 Quantisation error analysis Owing to the structure of the receive combining, the quantisation error in the quantised feedback effective MISO channel varies from low to high SNR. Thus, the variance of f˜ i varies, too. In the following, we give a brief explanation of the quantisation error variance in the high and low SNR scenarios.
5.2.1 Quantisation error at low SNR: In the low2 SNR region, we can assume, s2 ≫ j[L, j=i (P/L)| f H i uj | in (17). Therefore the proposed scheme leads to maximising signal power and the quantisation problem can 33
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www.ietdl.org be formulated as finding the decoding vector that would maximise the signal power and then finding the quantised code vector that is closest to the newly formed vector downlink MISO channel. Owing to the formulation of the MSIP approach, the error variance of quantisation error, s2E , is measured in terms of the angle spread between the original and quantised vectors. In [19], the quantisation error of f˜ was given in the following form
s2E = E[ sin2 (k(f i , f i ))] ≤ 2−B/(M−1)
(40)
power so that it converged to the form of (41) at high SNR (30 dB). Investigating the expected quantisation error at the intermediate SNR remains an open research problem. In summary, the quantisation error of the proposed algorithm ranges between 2−B/(M−1) and i × 2−B/(M−Nk ) . Note that, in most of the cases, both these error variances are lower than the errors in VQ MSE of [17] (which quantises both magnitude and direction) with s2E ≥ 2−B/M . Therefore the proposed algorithm quantises the channel directions more precisely than the previously proposed VQ MSE feedback policy.
Since we are only quantising the direction, not magnitude, this error variance denotes the angular spread of the quantised effective MISO channel.
5.3 SMSE analysis
5.2.2 Quantisation error at high SNR: In Section
In the absence of quantisation error, the SMSE of the precoder with perfect CSIT is [4]
5.1, we have shown that our proposed algorithm is equivalent to QBC at high SNR for one data stream per user. Simulation results in Section 6 will show the simulation of the convergence of this algorithm to QBC for multiple data streams per user. Therefore we analyse the high-SNR quantisation error of our receiving combining scheme using the concepts of QBC. When each user receives one data stream, QBC chooses the codevector with the least quantisation error and thus converts a MIMO channel into an effective MISO channel [11]. The quantisation error in this case is upper bounded by 2−B/(M−Nk ) [11]. Using the same notion, for a multiple data stream per user scenario, the effective MISO channel of the ith stream of a particular user can be chosen to generate the ith least quantisation error with respect to its original MIMO channel. The expected quantisation error of the ith data stream (in terms of error tolerance) of the kth user in this method satisfies [11, 32]
s2E ≤ i × 2−B/(M−Nk )
(41)
Note that the quantisation method described in the previous passage can lead to intra-user interference because of the correlation of two codevectors of a particular codebook. Our proposed algorithm avoids this scenario by incorporating the intra-user interference in receiver combining. However, the codevectors chosen for two different streams of a user vary with time and become mutually statistically independent in the long term of multiple channel realisations. Therefore we hypothesise that the quantisation error of our algorithm matches with that given by (41) at high SNR. The proposed receive combining scheme incorporates both an increase in signal power and a reduction in (intra and inter user) interference. The trade-off between these two depends on the SNR. Owing to the adaptive nature of this method, the expected quantisation errors for intermediate SNR cases are very hard to derive. In our simulations, we assumed the quantisation error to take the form of (40) at low SNR (0 dB) and changed this value linearly with transmitted 34
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SMSE = L − M + s2 tr[(FQF H + s2 I M )−1 ] −1 Pmax H = L − M + tr FF + I M (42) Ls2 In (42), we assumed Q = (Pmax /L)I L , that is, equal power allocation for simplicity of the analysis. At very high SNR, the SMSE approaches zero in (42) as tr((Pmax /Ls2 )FF H + I M )−1 is a decreasing function of SNR. However, with quantisation error, if the original precoder [4] is used L H −1 H −2 s2E P q f J fi SMSE = 1 − qi f i J f i + M max i i i=1 (43) H H f i J −1 f i and (s2E /M) where J = FQ F + s2 I M . Both qi
H Pmax qi f i J −2 f i increase with SNR. Since the former term is a linear-over-affine function and the latter is a quadraticover-quadratic function of Pmax , at high SNR the latter term dominates and SMSE increases with SNR, which explains the results of [17].
In our proposed algorithm
2 s ˆ Fˆ H SMSE = L − M + s2 + E Pmax tr FQ M −1 s2E 2 P + s + I M max M =L−M
P H Fˆ Fˆ + I M + tr 2 max L s + s 2E /M Pmax
(44)
−1
(45) In (45), we again assumed equal power allocation for analysis IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org simplicity. (Pmax )/(L(s2 + (s2E /M)Pmax )) is a non-increasing function of Pmax . Thus the proposed precoder makes sure that SMSE does not increase with SNR at high SNR region. Fig. 3 illustrates all these effects. Since, the increase in SMSE is most apparent in MU-MISO systems, the simulations use a MU-MISO system with independent channel realizations where M ¼ 5, Lk = 1∀k and B ¼ 10 bits per data stream. We use uncoded quadrature phase shift keying (QPSK) for data transfer. The proposed algorithm clearly stabilises the SMSE at high SNR.
H H antennas, fˆ i fˆ j = 0 when i = j. Therefore Fˆ Fˆ = I M . H Equation (49) follows since the ith diagonal entry of Fˆ Fˆ L ˆ 2 ˆ is given by j=1 f ij , f i is an M × 1 column vector and
f i 2 = 1. Therefore using the law of large numbers, as 2 M 1, fˆ ij 1/M. Using (50) in (45), we obtain SMSEM 1 = L − M +
5.3.1 Asymptotic SMSE analysis with large transmit antennas: At high SNR, s2 /Pmax ≃ 0.
=
M(M + Ls2E − L) M + Ls2E
L2 s2E M + Ls2E
(51)
Therefore the third term in (45) takes the following form tr
M ˆ ˆH FF +IM Ls2E
−1
⎡
−1 ⎤ 2 LsE H H I + Fˆ Fˆ = tr⎣I M − Fˆ Fˆ ⎦ M M
⎡
≃ tr⎣I M − Fˆ
= tr I M − ≃ tr I M − =
(46) ⎤
−1 Ls2E H IM 1+ Fˆ ⎦ M
M H Fˆ Fˆ 2 M + LsE
M L I M + Ls2E M M
M(M + Ls2E − L) M + Ls2E
(47) (48)
(49)
To avoid an error floor, the receivers have to decrease the quantisation error proportionately to the SNR. This condition can be met by increasing feedback bit with varying power so that (2−B/(M−1) /Pmax ) remains constant. This relation of feedback bits and varying power was at first noticed in terms of sum-rate in [18].
(50)
Here, (46) was obtained using matrix inversion lemma [29]. Equation (47) follows because we assumed fˆ i 2 = 1 in our BS channel model and because of the law of large numbers, with the presence of high number of transmit
Figure 3 Comparing the SMSE of the traditional and proposed precoder (M ¼ 5, K ¼ 5, N k ¼ 1, L k ¼ 1 ∀k, B ¼ 10, QPSK) IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
Equation (51) has been approximated using our simplistic BS channel model, where we assumed the effective channels of the data streams to be unit norm. However, both (45) and (51) suggest that, at very high SNR, SMSE becomes saturated. This leads to the following result. For a fixed quantisation error, the SMSE of a MU system is lower bounded by a fixed value which does not depend on SNR. We call this the error floor effect of MU broadcast systems. This is similar to the ceiling effect, in terms of capacity and SINR, seen previously in limited feedback literature [17, 18].
6
Numerical simulations
In this section, we compare our proposed scheme with the leading feedback schemes in the literature. Since our proposed algorithm uses channel resources to know the post-processing information of U and P, we use an MMSE receiver to simulate the other existing algorithms. This preserves the fairness of the comparisons since the performance of the system always improves with an MMSE receiver for mutually non-orthogonal channels [21]. Unless specified, all transmissions use QPSK. As mentioned before, our proposed transceiver for MUMIMO systems can be readily generalised to MU-MISO systems. In Fig. 4, we compare the performance of the proposed algorithm to the available precoders in a limited feedback MU-MISO system. The system uses M ¼ 4, K ¼ 4, Lk = 1∀k and B ¼ 10. The proposed algorithm performs better than the MMSE precoder [17] by using MSIP quantisation and convexity of the power allocation problem. The traditional SMSE transceiver, which ignores quantisation error, performs well at low SNR but begins to worsen at an SNR of 15 dB. Thus the proposed transceiver improves over the state-of-the-art in MU-MISO precoders based on limited feedback. To the best of our knowledge, coordinated beamforming is one of the very few existing 35
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Figure 4 Comparison with available MU-MISO precoding techniques (M ¼ 4, K ¼ 4, Lk ¼ 1 ∀k, B ¼ 10, QPSK) linear transceivers that avoids the dimensionality constraint in the MU-MIMO with multiple data stream per user scenario. In Fig. 5, we compare the proposed algorithm with coordinated beamforming. Here M ¼ 4, K ¼ 2, Nk = 4, Lk = 1∀k and B ¼ 15. Since coordinated beamforming implements joint transceiver design, it performs better than the proposed algorithm with full CSIT. However, coordinated beamforming needs at least (M 2 − 1) bits for ˆH ˆ H /H ˆ 2F ). To create Fig. 5, we used the feedback of (H 15 bits feedback overhead per data stream in a MUMIMO system with four transmit antennas. This means only 1 bit is available per unique scalar entry of ˆ 2F ), introducing large quantisation error. The ˆH ˆ H /H (H eigen structure of the channel therefore gets mangled at the BS [15], leading to loss of performance. On the other hand, since our proposed algorithm expends 15 bits to quantise the 4 × 1 vector, the quantisation error of the fed back vector always remains less than or equal to 2−B/(M−1) = 0.03125. Thus, the proposed algorithm performs very close to its full CSIT curve and outperforms coordinated beamforming [9] with limited feedback.
Figure 5 Comparison with the coordinated beamforming (M ¼ 4, K ¼ 2, Nk ¼ 4, Lk ¼ 1 ∀k, B ¼ 15, QPSK) 36
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Figure 6 Comparison with available MU-MIMO VQ precoding techniques (M ¼ 4, N1 ¼ N 2 ¼ 2, K ¼ 3, N3 ¼ 3, Lk ¼ 1 ∀k, B ¼ 15, QPSK) In Fig. 6, we compare our proposed scheme with other VQ combining limited feedback MU-MIMO transceivers. In this example, M ¼ 4, K ¼ 2, N1 = N2 = 2, N3 = 3, Lk = 1 ∀k and B ¼ 15. Since to the best of our knowledge, existing VQ combining MU-MIMO schemes have not dealt with multiple data streams per user, we stick with one data stream per user in this comparison. The proposed scheme outperforms the QBC [11] and MET [12] approaches because of the use of SMSE precoder, adaptive receive combining and optimal power allocation. Although our algorithm outperforms [21] up to 20 dB, Boccardi’s MESC [21] seems to converge at a lower error floor than the proposed algorithm. This happens because in our proposed algorithm the actual quantisation error variance is not known at the BS. Owing the adaptive quantisation policy of the proposed algorithm, the quantisation error variance changes from low to high SNR; since we only quantise the direction of the effective channels, the norm of the quantisation error is not available at the BS. The quantisation error in MESC case [21] also changes from low to high SNR but the BS does need this knowledge because of the use of a ZF precoder. Our proposed transceiver adds to the literature by allowing multiple data streams per user. Fig. 7 shows the comparison of the transceiver’s performance to other possible methods to transmit multiple data streams per user. In Fig. 7, EBC projects the MIMO channel to its dominant eigenvectors to create effective MISO channels [23] and QBC chooses the set of codevectors that will generate least quantisation error as effective MISO channels [11]. The proposed transceiver approaches EBC at low SNR and QBC at high SNR. Thus the proposed algorithm retains the advantages of both EBC and QBC by providing a trade-off between signal power, intra-user and inter-user interference. IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org the channel magnitude will also be important. An extension of the present work will be the optimisation of feedback bits among the channel magnitude and direction information, such as done in a different context in [33]. Our present work can also be extended to a slowly time varying channel. The temporal correlation between different blocks can be used to reduce the amount of feedback overhead.
8
References
[1] TSE D., VISWANATH P.: ‘Fundamentals of wireless communications’ (Cambridge University, Cambridge, UK, 2005) Figure 7 Different receive combining techniques with multiple data streams per user (M ¼ 4, K ¼ 2, N k ¼ 3, Lk ¼ 2∀k, B ¼ 12, BPSK)
[2] SHI S., SCHUBERT M.: ‘MMSE transmit optimization for multi-user multi-antenna systems’. Proc. IEEE ICASSP’2005, March 2005, vol. 3, pp. 409– 412
7
[3] SCHUBERT M. , BOCHE H.: ‘Solution of the multiuser downlink beamforming problem with individual SINR constraints’, IEEE Trans. Veh. Technol., 2004, 53, pp. 18– 28
Conclusions
In this paper, we proposed linear transceiver design in the downlink of a MU-MISO and MU-MIMO system (with multiple data streams per user) using SMSE precoder at the BS, MSIP VQ as the feedback algorithm and MMSE decoder at the receivers. However, to encode the channel information, the receivers use MESC first. In the MUMISO, the individual users send back the indexes of their quantised channels to the BS. In the MU-MIMO scenario, the users convert their MIMO channels to effective vector downlink MISO channels to maximise the expected SINR and then send the indexes of these quantised MISO channels. The BS uses the quantisation error of MSIP in the SMSE precoder design and finds the downlink precoder and power allocation vector using a convex optimisation problem. The proposed system was shown to outperform the previously existing linear transceivers in the MU scenario for limited feedback, while also allowing for multiple data streams per user. One possible extention of the present work will be the detailed analysis of the expected quantisation error variance in the intermediate SNR range. The way the receivers find the trade-off between signal power increase, and intra-user and inter-user interference reduction will give an insight to analyse this problem. In this work, only shape feedback is sent to the BS. This is reasonable if the average channel magnitudes of all users are equal. In this case, the alignment of precoding vector with channel direction is more important than the power allocation. However, this assumption may remain valid only in a small scale fading scenario. In a more practical scenario, different users will be located at different distances from the BS and therefore average channel magnitude will be different. In that case, bit allocation in IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
[4] KHACHAN A.M., TENENBAUM A.J., ADVE R.S.: ‘Linear processing for the downlink in multiuser MIMO systems with multiple data streams’. Proc. IEEE ICC’2006, June 2006, vol. 9, pp. 4113 – 4118 [5] TENENBAUM A., ADVE R.S.: ‘Joint multiuser transmit-receive optimization using linear processing’. Proc. IEEE ICC 2004, Paris, France, June 2004, pp. 588 – 592 [6] HAARTSEN J.C.: ‘Impact of non-reciprocal channel conditions in broadband TDD systems’. Proc. IEEE PIMRC 2008, September 2008, pp. 1 – 5 [7] DHARAMDIAL N., ADVE R.S.: ‘Efficient feedback for precoder design in single- and multi-user MIMO systems’. Conf. on Information Sciences and Systems, March 2005 [8] KHACHAN A.M.: ‘Linear processing for the downlink in multiuser MIMO systems’. MS thesis, University of Toronto, 2006 [9] CHAE C., MAZZARESE D., JINDAL N., HEATH R.W. JR.: ‘Coordinated beamforming with limited feedback in the MIMO broadcast channel’, IEEE J. Sel. Areas Commun., 2008, 26, pp. 1505 – 1515 [10] NARULA A., LOPEZ M.J., TROTT M.D., WORNELL G.W.: ‘Efficient use of side information in multiple-antenna data transmission over fading channels’, IEEE J. Sel. Areas Commun., 1998, 16, pp. 1423 – 1436 [11] JINDAL N.: ‘Antenna combining for the MIMO downlink channel’, IEEE Trans. Wirel. Commun., 2008, 7, pp. 3834 – 3844 37
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www.ietdl.org [12] BOCCARDI F., HUANG H., TRIVELLATO M.: ‘Mutliuser eigenmode transmission for MIMO broadcast channels with limited feedback’. Proc. IEEE SPAWC 2007, June 2007, pp. 1 – 5 [13] CONWAY J.H. , HARDIN R.H., SLOANE N.J.A.: ‘Packing lines, planes, etc. packings in grassmannian spaces’, Exp. Math., 1996, 5, pp. 139 – 159 [14] RAVINDRAN N., JINDAL N.: ‘MIMO broadcast channels with block diagonalization and finite rate feedback’. Proc. IEEE ICASSP 2007, April 2007, pp. III-13 – III-16 [15] LOVE D.J., HEATH R.W.: ‘Feedback techniques for MIMO channels’ in TSOULOS G. (ED.) : ‘MIMO system technology for wireless communications’ (Taylor and Francis Group, Boca Raton, FL, 2006), pp. 113– 146 [16] LOVE D.J., HEATH R.W. JR., STROHMER T.: ‘Grassmannian beamforming for multiple-input multiple-output wireless systems’, IEEE Trans. Inf. Theory, 2003, 49, pp. 2735 – 2747 [17] DABBAGH A.D. , LOVE D.J.: ‘Multiple antenna MMSE based downlink precoding with quantized feedback or channel mismatch’, IEEE Trans. Commun., 2008, 7, pp. 1859 – 1868 [18] JINDAL N.: ‘MIMO broadcast channels with finite-rate feedback’, IEEE Trans. Commun., 2006, 52, pp. 5045 – 5060 [19] ROH J.C., RAO B.D.: ‘Transmit beamforming in multipleantenna system with finite rate feedback: a VQ-based approach’, IEEE Trans. Inf. Theory, 2006, 52, pp. 1101 – 1112
[23] ISLAM M.N. , ADVE R.S.: ‘Linear transceiver design in a multiuser MIMO system with quantized channel state information’. Proc. IEEE ICASSP 2010, March 2010, pp. 3410 – 3413 [24] ISLAM M.N., ADVE R.S.: ‘SMSE precoder design in a multiuser MISO system with limited feedback’. Proc. 25th Biennial Symp. on Communications 2010, May 2010, pp. 352–356 [25] TENENBAUM A.J., ADVE R.S.: ‘Minimizing sum-MSE implies identical downlink and dual uplink power allocations’, To be published [26] SHENOUDA M.B., DAVIDSON T.N.: ‘On the design of linear transceivers for multiuser systems with channel uncertainty’, IEEE J. Sel. Areas Commun., 2008, 26, pp. 1015 – 1024 [27] DING M.: ‘Multiple-input multiple-output system design with imperfect channel knowledge’. PhD thesis, Queen’s University, 2008 [28] BOYD S., VANDENBERGHE L.: ‘Convex optimization’ (Cambridge University, Cambridge, UK, 2004) [29] STRANG G.: ‘Introduction to linear algebra’ (WelleslyCambridge Press, Wellesly, MA, USA, 2009) [30] KOBAYASHI M., CAIRE G., JINDAL N.: ‘How much training and feedback are needed in MIMO broadcast channels?’. Proc. IEEE ISIT 2008, July 2008, pp. 2663 – 2667
[20] JINDAL N.: ‘A feedback reduction technique for MIMO broadcast channels’. Proc. IEEE ISIT 2006, July 2006, pp. 2699 – 2703
[31] CHAE C., MAZZARESE D., INOUE T., HEATH R.W.: ‘Coordinated beamforming for the multiuser mimo broadcast channel with limited feedforward’, IEEE Trans. Signal Process., 2008, 56, pp. 6044– 6056
[21] TRIVELLATO M., HUANG H., BOCCARDI F.: ‘Antenna combining and codebook design for MIMO broadcast channel with limited feedback’. Proc. Asilomar Conf. on Signals and Systems, November 2007, pp. 302 – 308
‘Order statistics and records’ in (EDS.) : ‘Handbook of beta distribution and its applications’ (Marcel Dekker, Inc., New York, 2004), pp. 89 – 96
[22] RAVINDRAN N., JINDAL N.: ‘Limited feedback-based block diagonalization for the MIMO broadcast channel’, IEEE J. Sel. Areas Commun., 2008, 26, pp. 1473 – 1482
[33] KHOSHNEVIS B., YU W.: ‘Limited feedback multi-antenna quantization codebook design part ii: multiuser channels’, IEEE Trans. Signal Process., To be published
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[32]
GUPTA A., NADARAJAH S.:
GUPTA
A. ,
NADARA JAH
S.
IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
ISSN 1751-8628
Transceiver design using linear precoding in a multiuser multiple-input multiple-output system with limited feedback M.N. Islam R. Adve Electrical and Computer Engineering, University of Toronto, Toronto, Canada E-mail: [email protected]
Abstract: The authors investigate quantisation and feedback of channel state information in a multiuser (MU) multiple-input multiple-output (MIMO) system. Each user may receive multiple data streams. The authors design minimises the sum mean squared error (SMSE) while accounting for the imperfections in channel state information (CSI) at the transmitter. This study makes three contributions: first, the authors provide an end-toend SMSE transceiver design that incorporates receiver combining, feedback policy and transmit precoder design with channel uncertainty. This enables the proposed transceiver to outperform the previously derived limited feedback MU linear transceivers. Second, the authors remove dimensionality constraints on the MIMO system, for the scenario with multiple data streams per user, using a combination of maximum expected signal combining and minimum MSE receiver. This makes each user’s feedback independent of the others and the resulting feedback overhead scales linearly with the number of data streams instead of the number of receiving antennas. Finally, the authors analyse the SMSE of the proposed algorithm at high signal-to-noise ratio (SNR) and large number of transmit antennas. As an aside, the authors show analytically why the bit error rate, in the high SNR regime, increases if quantisation error is ignored.
1
Introduction
The advantages of spatial diversity and multiplexing has led to the investigation of multiuser (MU) multiple-input single-output (MISO) and multiple-input multiple-output (MIMO) wireless communication systems [1]. Spatial diversity can increase system reliability as well as the spectral efficiency of MU systems. However, limitations caused by interference and channel fading remain a concern in MU systems. These can be mitigated by precoding the signals before transmission, in turn requiring channel state information at the transmitter (CSIT). This paper focuses on linear transceiver design in the downlink of MU MISO [2, 3] and MIMO systems [4, 5], a single base station (BS) communicating with multiple receivers. Each user may receive multiple data streams; the figure of merit here is the sum mean squared error (SMSE) across all data streams. In a frequency division duplexing system, channel information needs to be estimated at the receiver and sent IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
back to the BS after quantisation. Recent works suggest that this might be required in broadband time division duplex systems as well [6]. In general, providing accurate CSIT and reducing feedback overhead are important considerations in a linear transceiver design. Our work assumes perfect channel estimation at the receiver end with zero delay error-less feedback and focuses on quantising channel state information (CSI). In the available literature, scalar quantisation [7 – 10], vector quantisation (VQ ) [11, 12] and matrix quantisation [13, 14] have all been used to quantise CSI. It is now well established in the single user, single data stream, case that projecting the MIMO channel to an appropriate vector downlink channel yields better performance than full channel scalar quantisation with same feedback overhead [15]. This has led to considerable research in VQ , which reduces the feedback overhead by allocating bits in the proper vector downlink channel. In VQ , to send B feedback bits as the channel index to the BS, each user 27
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www.ietdl.org needs a codebook with 2B code vectors. Grassmannian line packing [16], VQ using MSE as the optimality criterion [17] and random vector quantisation [18] have been the most popular approaches for the MU case. In this paper, we investigate VQ , based on the mean squared inner product (MSIP) criterion [19] as the feedback method. In a MU-MISO system, users can feed back the channel vectors using VQ. However, in the MIMO case, one needs to combine the receive antennas to convert the MIMO channel to the effective vector downlink MISO channel. Since the receivers cannot cooperate, the quantisation scheme of each user must be independent from the others. Projecting the MIMO channel to the direction of its maximum eigen vector (MET) [12] is the optimal solution at low signal-to-noise ratio (SNR). Jindal [11, 20] proposed quantisation-based computing (QBC) which chooses the effective vector downlink channel to produce the least quantisation error; this is optimal at high SNR. Trivellato et al. [21] proposed maximum expected signal combining (MESC) to maximise the signal to interference plus noise ratio (SINR). Their scheme outperforms both QBC and MET; MESC converges to MET and QBC in the low and high SNR region, respectively, [21]. All these schemes discussed so far assume that each user receives a single data stream. However, multiple data streams per user complicate the feedback process, requiring linearly independent information for each stream. In this paper, we extend the MESC algorithm to the case with multiple data streams per user. Most of the relevant works in the limited feedback MU literature suffer from dimensionality constraints. With M transmit antennas, N total receive antennas and L data streams in total, either M ≥ N [18] or N ¼ L [22]. To the best of our knowledge, only authors in [7 – 9] avoid these constraints. However, by using scalar quantisation, the feedback overhead in these systems scale linearly with MN [7, 8] and (M 2 − 1) [9], respectively. Owing to the formulation of our MESC receive combining, the feedback overhead in our proposed system scales only with M × L (where L is the total number of receive data streams). Since, with linear precoding L ≤ M and L ≤ N, the proposed transceiver allows significant performance improvements with lower feedback rate. Finally, previous works [17] have shown, by simulation, that if the quantisation error is ignored, the MSE increases at high SNR. Here we investigate why this is true theoretically. The overall contributions of this paper are therefore the following: 1. We provide an end-to-end SMSE transceiver design that eliminates the dimensionality constraint and tie the feedback overhead to the number of data streams which is always less 28
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than or equal to the number of both transmit and receive antennas. 2. We extend and make the MESC receiver flexible by allowing multiple data streams per user scenario. 3. We show the error floor effect in terms of SMSE in MU broadcast systems. Previous works on this area focused on the ceiling effect in terms of capacity [11, 18] and SINR [17]. As an aside, we show why SMSE and bit error rate (BER) increase instead of converging to a constant if quantisation error is not considered. Preliminary results from this paper have appeared before [23, 24]. The new developments are as follows. First, the receive combining scheme proposed in this paper, namely MESC for multiple data streams per user, holds the benefits of eigen-based combining (EBC) and QBC at low and high SNR, respectively. In [23], we used EBC, suboptimal at high SNR, for all SNR. Second, the relation of the proposed algorithm to the existing transceivers and the analysis of the quantisation error were not presented in [23, 24]. Finally, we use a different proof in this manuscript to analyse the asymptotic SMSE of the proposed system. The remainder of this paper is organised as follows. Section 2 describes the system model and reviews the transceiver design problem with full channel knowledge. Section 3 reviews the proposed quantisation method and shows the linear precoder design. Section 4 illustrates the two-step receiver design process and gives the overall algorithm. We analyse our proposed transceiver in Section 5. After providing the numerical simulation results in Section 6, we draw our conclusions in Section 7. Notation: Lower case x denotes scalar whereas lower case bold face, h means column vector. Upper case boldface, V denotes matrix whereas uppercase normal font N represents constant entry. The superscripts (.)T and (.)H denote the transpose and conjugate transpose operators, respectively. tr [.] denotes the trace operator. I is reserved for the identity matrix whereas 1 represents the column with all one vector. diag(.) denotes the diagonal matrix where the diagonal entries contain the bracketed terms. · 1 denotes the L1 norm of the vector. E(.) denotes statistical expectation.
2
System model
We consider both MU-MISO and MU-MIMO systems in our design. We first describe the MU-MIMO system model. Then we show that, with our approach, transceiver design in the MU-MIMO system is very similar to that in a MU-MISO system.
2.1 MIMO system model Consider a single BS equipped with M transmit antennas communicating with K independent users. User k has Nk IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org antennas and receives Lk data streams. Let L = k Lk , N = k Nk . To ensure resolvability, we assume L ≤ M and Lk ≤ Nk . The ith data stream is processed by a unit norm linear precoding vector ui with the global precoder U = [u1 , u2 , . . . , uL ]. Let p = [p1 , p2 , . . . , pL ]T be the powers allocated to the L data streams and define the downlink power matrix P = diag( p). p1 ≤ Pmax where Pmax is the total available power. The data vector x = [x1 , . . . , xL ]T = [xT1 , xT2 , . . . , xTK ]T , includes all L data streams to the K users. The Nk × M block fading channel, HH k , between the BS and user is assumed to be flat. The global channel matrix is H H , with H = [H 1 , . . . , H k ]. User k receives √ H yDL k = H k U P x + nk
(1)
where nk represents the zero mean additive white Gaussian noise at the receiver with E[nnH ] = s2 I Nk . We also assume, E[xxH ] = I L . To estimate its own transmitted symbols, from yDL k , user k forms xˆ k =
√ xˆ = V H H H U P x + V H n √ = F H U Px + V H n
(2)
The MSE of the ith data stream is given by = E[( xi − xi )( xi − xi )H ] eDL i
(3)
The SMSE minimisation problem is
p,U
UL
y
=
eDL subject to p1 ≤ Pmax , ui = vi = 1 i
i=1
L
f j qj x j + n
(5)
j=1
where n = [nT1 , nT2 , . . . , nTK ]T and to facilitate our analysis, we define the M × L matrix F ¼ HV with F = [ f 1 , . . . , f L ]. The vectors f 1 , . . . , f L are the effective M × 1 MISO channels for the individual data streams.
L
In designing the precoder U, it is computationally efficient to use a virtual dual uplink [4]. In this uplink the transmit powers are q = [q1 , . . . , qL ]T for the L data streams, while the matrices U and V remain the same as before. The global virtual uplink power allocation matrix Q is defined as, Q ¼ diag(q) where q1 ≤ Pmax . So the received data in the BS in the virtual uplink is given by
DL VH k yk
Here V k is the Nk × Lk decoder vector for user k. Fig. 1 shows the block diagram of the proposed system in the downlink. Let V be the N × L block diagonal global decoder matrix, V = diag(V 1 , . . . , V K ). Overall
min
Figure 2 Block diagram of MU-MIMO uplink with channel and decoder combined as single block
(4)
Therefore data stream i is decoded as xˆ UL i
=
uH i
L
f j qj x j + n
(6)
j=1
Fig. 2 shows the proposed system model in the virtual uplink. As (6) and Fig. 2 show, the system has become an effective MU single input multiple output system in the virtual uplink. Uplink – downlink duality states that the same MSEs can be achieved in the uplink and the downlink with the same matrices U and V and the same power constraint. A recent result shows that at the optimal solution, p ¼ q [25]. With perfect channel knowledge, the transmitter iterates between V and Q and converges to the the optimum solution using a convex optimisation problem formulation [4]. Then the precoder finds the optimal U using the minimum mean square error (MMSE) solution [4]. The downlink power allocation is then set equal to Q [25].
3 MSIP quantisation and linear precoder design 3.1 MSIP quantisation
Figure 1 Block diagram of MU-MIMO downlink IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
We assume that the receivers have perfect CSI using training. For the purposes of quantisation only, the ith user chooses the quantised codevectors, fˆ i , . . . , fˆ L that would maximise i the SINR of its receiving data streams. Choosing the best quantised codevector is described in detail in the receiver design section. Each user has a codebook consisting of 2B unit norm vectors w ˆ 1, . . . , w ˆ 2B . Each user feeds back B bits per data stream to the BS. The receivers individually 29
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www.ietdl.org normalise and then quantise each of the Lk effective channels using the chordal distance [13] fˆ i = arg
2
min
w[w ˆ 1 ,...,w ˆ 2B
sin (/( f i , w)) ˆ
2
C(·)
(8)
Here, fˆ = C( f ) is the quantised effective channel and k·,·l denotes the inner product. Overall, we consider the following channel model at the BS for precoder design. f i + fi f i =
F = F + F
or
√ uMSE f i qi = J −1 i
(7)
The use of chordal distance over the Euclidean distance leads to a higher inner product between the original and quantised channels [19]. Here, we only quantise the direction of the effective channel and this direction can lie anywhere on the M-dimensional complex unit-norm sphere. Therefore we generate the quantisation codebook as a VQ problem using the MSIP optimality criterion; the details of MSIP VQ codebook generation can be found in [19]. Each user at first generates a large set of random unit norm Mdimensional complex vectors f and finds the quantiser codebook C to maximise the MSIP ˆ 2B ) = max E| , f , C( f ) . | (w ˆ 1, . . . , w
virtual uplink by [26]
(9)
F comprises L unit-norm effective channel vectors with the original channel directions. F denotes the L quantised feedback unit norm vectors. F denotes the error in the quantisation. We assume that the quantisation error matrix F has M×L independent identically distributed elements with zero mean and a variance of s2E /M. s2E is the quantisation error variance associated with each quantised F is assumed to be independent of x, n and F. vector fˆ i . Since s2E depends on receiver combining, the details regarding the expected value of this term in our proposed algorithm will be clarified in the receiver design and analysis section.
(10)
2
H s J = FQ F + s2 I M + E (q1 + · · · + qL )I M M
(11)
So eUL,MSE =1− i
√ H −1 √ q i f i J f i qi
(12)
Therefore the uplink SMSE is (see (13)) As F is fixed, the SMSE expression is a function of uplink power allocation Q.
Proposition 1: The optimisation problem for power allocation
q + · · · + qL 2 sE tr( J −1 ) Qopt = min s2 + 1 Q M
(14)
subject to tr[Q] ≤ Pmax , qk ≥ 0 for all k is convex in Q.
Proof: Ding [27] shows that SMSE remains a non-increasing function of SNR if channel uncertainty isequal and all available power is used because tr(Q) = Li=1 qi = Pmax makes the term within the brackets a constant. J is a positive definite matrix and therefore the optimisation problem is convex in J [28]. Since J is linear in Q, it can be readily proved that the problem is convex in Q. A The power allocation problem is therefore convex given F. In the next section we discuss the solution for V (equivalently F ). This section represents the core contribution of the paper.
4
Receiver design
It should be noted that the channel vector of each user is already in the form of f i in the MU-MISO case. Therefore in this case, we just quantise the normalised channel using chordal distance and return the corresponding index to the BS.
We propose a two-step receiver design. For the purposes of quantisation only, each user uses a MESC receiver and chooses the quantised codevectors that would maximise the SINR of their data streams. However, the users implement MMSE receivers while receiving the actual data. This is unlike the single MMSE solution in [2, 4] but allows for the channel feedback to be independent of the other users’ actions.
3.2 Linear precoder design
4.1 Receive combining with MESC
The optimum U in the SMSE minimisation problem of the system model proposed in (9) and (6) has been solved in the
Before proceeding with the analysis, let us clarify the relation between F and HV that will be used interchangebly in this
SMSEUL =
L
eUL,MSE = i
i=1
L i=1
2
=L−M + s +
⎡ −1 ⎤ L 2 L H H H s q √ −1 √ ⎦ FQ F + s2 + E i=1 i I M 1− qi f i J f i qi = L − tr⎣ FQ F M i=1
s2E
L
k=1 qk
M
tr[ J −1 ]
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(13)
IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org section. We have f i = H k vi where H k is the channel of the kth user receiving the ith data stream and vi is the decoding vector used for the ith stream; ui and uj are the precoding vectors of the ith and jth data stream. Now, using our quantisation policy in (7), we define the quantisation angle ui [ [0, p/2] as H cos ui = | f i fˆ i |
(15)
assumes that the vector downlink channels for all other users’ stream are mutually orthogonal to its own channels. We also assume that noise variance, signal power, quantisation error variance in the BS and total number of data streams sent by the BS are known to each of the users. Therefore because of the construction of (18), each user can find the expected value of f H i ui and ui even without co-operating with other users. Therefore by seperating the intra-user streams from inter-user streams, (17) takes the following form
Here, f i represents the unit norm effective vector downlink channel, that is, f = f /f . f is the effective MISO channel of the data stream. Since, the receivers know the quantisation angle exactly, we can use this information to improve the expected SINR. As in [21], define the quantisation error as
SINRDL = i
f i f i ) fi fi = fi −(
(16)
Here, f i 2 = sin2 ui . Now, in the downlink, the SINR of the ith stream is = SINR DL i
H 2 i ui |
s2 +
(P/L)| f H 2 j[L,j=i (P/L)| f i uj |
Now
(17)
Here, ui is normalised such that ui = 1. Since the users do not cooperate, the quantisation and feedback methods implemented by the users need to be independent of each other. Based on the fact that, the effective MISO channels of different users are statistically independent of each other, we assume different user’s quantised channels to be H mutually orthogonal, that is, fˆ fˆ = 0 where i and j j
indicate data streams that are being received by two different users. Following this assumption, it can be easily H verified from (18) that fˆ u = 0. i
2 2 |f H i uj | = f i
j[L, jLk
H
H
H
||( f i f i ) f i uj + f i uj ||2
j[L, jLk
(20) = f i 2
H
|| f i uj ||2
(21)
j[L, jLk
In (17) equal power allocation was assumed to simpify the receiver combining analysis. Here, ui and uj follow the form in (10). Now using (10) and the matrix inversion lemma [29] (see (18))
i
(19)
j[L, jLk
H
H 2 (P/L)|vH i H k ui | H 2 s2 + j[Lk , ji (P/L)|vH i H k uj | 2 + (P/L)| f H i uj |
j
Since each user knows the inner product of different code vectors in its codebook, the assumption of orthogonality is not valid for two different streams of the same user. Therefore in our proposed algorithm, each user uses its known codevectors, that is, the effective channels of its data streams, as a set of column vectors f in the F matrix and
f i 2 = f i 2
H
|| f i uj ||2
(22)
j[L, jLk
L − Lk M −1
(23)
=
L − Lk H ( f i 2 − ( f H i f i )( f i f i )) M −1
(24)
=
L − Lk H H H v (H k (I − fˆ i fˆ i )H k )vi M −1 i
(25)
= f i 2 sin2 ui
Here, (20) is obtained by taking out the norm of f i and using
H (16). Equation (21) follows since f i uj = 0 for mutually orthogonal reported channels from different users. Equation (22) was obtained by setting f = f / f . Equation (23) i
i
i
was derived using the analysis of [21]. In the presence of large number of codevectors, ui is very small which leads to H f i ≃ 0. Therefore, the unit vectors f i and uj are both fi identically distributed in the (M 2 1)-dimensional plane H
f i uj 2 follows a beta orthogonal to f i . This implies, distribution with parameter (1, M 2 1) and has expected value 1/(M 2 1) [21]. The factor of (L − Lk ) arises since the kth user receives Lk data streams and therefore (L − Lk )
−1 H s2E 1 1 √ ˆ Pmax I + FQF f i qi = 2 1− 2 ui = s + 2 2 M s + (sE /M)Pmax s + (sE /M)Pmax −1 1 H H √ fˆ i q i × Fˆ Q−1 + 2 Fˆ Fˆ Fˆ 2 s + (sE /M)Pmax 2
IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
(18)
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www.ietdl.org data streams are mutually orthogonal to the ith data stream. Equation (24) was obtained using the quantisation angle definition of (15). In (25), we again use f i = H k vi . Using the results of (25) in (19) and defining ⎛
⎞
P L − Lk ⎝ ⎠ (I − f i f H u uH + Bi = H H i ) H k (26) M −1 L k j[L , ji j j k
second stream of the kth user. So
BLk
2
P H L−2 H H H uLk uLk + (I − w ˆ Lk w = ˆ Lk ) H k 1 1 2 2 L k M −1 (32)
H H P vH Lk2 L H k uLk2 uLk2 H k vLk2 SINR DL = (33) Lk2 s2 + vH Lk BLk vLk 2
(w ˆ Lk , vLk ) =
Equation (19) takes the following form
2
2
max
(vLk =1,w ˆ Lk [W ) 2
= SINR DL i
H H P vH i L H k ui ui H k vi s2 + vH i B i vi
(27)
in (27) is a Owing to the structure of ui and Bi , SINR DL i function of vi and fˆ j ∀j [ Lk . Each of these fˆ j vectors are in the codebook C which consists of w ˆ 1, . . . , w ˆ 2B codevectors. Therefore the linear decoding vector vj and w ˆ j ∀j [ Lk should be chosen jointly as (w ˆ j , vj )∀j [ Lk =
Lk
max
vj =1,w ˆ j [W
SINR DL j
(28)
j=1
Here, vj is a complex Nk -dimensional vector. Joint optimisation for all the data streams of a particular user in (28) will lead to a computational complexity proportional to (2B )Lk . One sub-optimal solution to reduce complexity is to find the optimum decoding vector and quantised channel one data stream at a time. This simplified algorithm is given below: 1. First, assume that intra-user streams are orthogonal and find the vector downlink channel of the first stream. Therefore maximising (27) becomes an optimisation problem of w ˆ Lk and vLk where Lk1 denotes the first 1 1 data stream of the kth user. So BLk = 1
P H L−1 Hk (I − w ˆ Lk w ˆH ) Hk Lk1 1 L M −1
SINR DL Lk =
H H P vH Lk1 L H k uLk1 uLk1 H k
1
(w ˆ Lk , vLk ) = 1
1
s2
+
(vLk =1,w ˆ Lk [W ) 1
vLk
1
vH Lk1 BLk1 vLk1
max
(29)
SINR DL Lk 1
(30)
(31)
1
2. Once the quantised channel of the first stream is chosen, the user assumes it to be a non-orthogonal channel for the second stream’s vector downlink channel. However, vector downlink channels for the other streams of the same user are still considered to be orthogonal to both first and second stream’s channel. Thus maximising (27) again becomes an optimisation problem with variable vLk and 2 w ˆ Lk for the present data stream where Lk2 denotes the 2
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2
2
SINR DL Lk 2
(34)
2
3. For the third data stream of the kth user B Lk = 3
P H H (u uH + uLk uH Lk2 2 L k Lk1 Lk1 L−3 (I − w ˆ Lk w ˆH + Lk3 ) H k 3 M −1
(35)
The other equations take the forms of (29) – (35). The same policy continues upto the last stream of the kth user. Note that as we increase the number of data streams in the reported non-orthogonal channels part, number of the increases and components in the summation term uj uH j (L − Lk )/(M − 1) decreases. This follows the reasonings provided in deriving (20) – (25). With this algorithm, the SINR expression for a particular data stream remains a function of only its decoding vector and its quantised channel. This leads to a computational complexity of Lk × 2B in finding the channels of Lk data streams. Now (26) and (27) can be thought as a general form of all the data stream’s SINR expressions. In (26) and ˆ i. (27), both f i and u depend on the chosen codevector w For any particular w ˆ i , the linear decoding vector that maximises (27) can be obtained by the MMSE detector, √ u [21]. Then vi = (s2 I + Bi )−1 (P/L)H H k i SINR DL = i
P H u H (s2 I + Bi )−1 H H k ui L i k
(36)
The user finds the value of SINR DL for every w ˆ i using (36) i and chooses the w ˆ i , as the quantised channel fˆ i , that maximises SINR DL i . It is worth emphasising that, to our knowledge, this is the first receive combining scheme that considers signal power, inter-user and intra-user interference while accounting for multiple data streams per user.
4.2 Receiver design for data processing As mentioned earlier, MESC is for quantisation purposes only. The BS determines p and U based on the quantised F. However, for mutually non-orthogonal reported channels and a finite number of users, using MMSE receivers for data processing provide better results than IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org MESC receivers [21]. Therefore for the data H 2 −1 H vi = (H H k U PU H k + s I ) H k ui pi
5
5.1 Relation to the existing algorithms (37)
which can be normalised to make vi = 1. H H k is the MIMO channel of the kth user receiving the ith data stream. ui and pi respresent the designed precoder and allocated power for the ith stream. Note that the MMSE receiver cannot be implemented at the time of channel quantisation since the precoder matrix U was not designed at that time. The implementation of the decoder mentioned in (37) requires infinite training symbols. Therefore from a practical point of view, the BS either sends a finite number of dedicated symbols [30] or uses limited feedforward [31] to convey the post-processing information to the receivers. However, in our simulations, we restrict ourselves to the case where the users can estimate the effective channels of their data streams.
4.3 Overall algorithm Using the developments in Sections 3, 4.1 and 4.2, the steps of the proposed overall algorithm for SMSE minimisation in the MU system are as follows: 1. Send common pilots to the users in the system so that each user can estimate its own channel. 2. Each user generates a separate codebook of 2B unit norm vectors using MSIP VQ in off-line. In the MUMIMO case, each user converts its estimated MIMO channel to effective MISO channels using the MESC algorithm proposed in Section 4.1 and sends the codebook indexes of the effective channels to the BS. In a MUMISO system, each user quantises its own channel and the BS assumes V ¼ I. 3. Virtual uplink power allocation: Qopt = minQ (s2 + (s2E Pmax /M))tr( J −1 ), such that tr(Q) ≤ Pmax . This is a convex optimisation problem. Here, J follows (11). √ 4. Uplink beamforming: ui = J −1 fˆ i qi , ui = 1. 5. Downlink power allocation P ¼ Q. 6. Send dedicated pilot symbols for each of the data streams. Thereafter implement the MMSE downlink decoders using (37). vi = 1 for the ith data stream. The algorithm above results in a precoder U, decoder V and power allocation p. Note that the solution is sub-optimal because U and p are designed using MESC, not MMSE. This is the price paid for the feedback to be independent across users. IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
Analysis and discussion
As the proposed receive combining technique maximises the expected SINR of the data streams at the user end, it is equivalent to the MESC algorithm in the case of one data stream per user of [21] which was designed for the zero forcing (ZF) precoder. To illustrate this, let Lk = 1. Since intra-user interference is not present, all the quantised effective channels in F are assumed to be mutually orthogonal. Using this in (18) we obtain ui =
1
ˆ √ qi −
f s2 + s2E Pmax i
ˆ × F Q−1 +
1 (s2 + s2E Pmax )2
1 I s2 + s2E Pmax
−1
√ [1, 0, . .. , 0]T qi / fˆ i (38)
Equation (38) follows since (Q−1 + (1/(s2 + s2E Pmax ))I )−1 is a diagonal matrix. Since ui = 1, ui = fˆ i in this scenario. Using this in (19) we find SINR DL i =
H vH ˆ iw ˆH i ((P/L)H k w i H k )vi H H 2 s + vi ((P/L)H k ((L − 1)/(M − 1)(I − w ˆ iw ˆH i ))H k )vi
(39) This is the exact same expression obtained in [21] as the MESC combiner with noise variance s2 = 1. Trivellato et al. [21] has shown that this algorithm takes the form of MET combining at low SNR and QBC at high SNR. Thus MESC combining of [21] considers signal power and inter-user intereference while choosing the code vector. Since we are considering multiple data streams to each user, our proposed SINR expression in (19) considers signal power, inter-user and intra-user interference altogether. Thus our proposed algorithm is a generalised form for MESC combining with multiple data streams.
5.2 Quantisation error analysis Owing to the structure of the receive combining, the quantisation error in the quantised feedback effective MISO channel varies from low to high SNR. Thus, the variance of f˜ i varies, too. In the following, we give a brief explanation of the quantisation error variance in the high and low SNR scenarios.
5.2.1 Quantisation error at low SNR: In the low2 SNR region, we can assume, s2 ≫ j[L, j=i (P/L)| f H i uj | in (17). Therefore the proposed scheme leads to maximising signal power and the quantisation problem can 33
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www.ietdl.org be formulated as finding the decoding vector that would maximise the signal power and then finding the quantised code vector that is closest to the newly formed vector downlink MISO channel. Owing to the formulation of the MSIP approach, the error variance of quantisation error, s2E , is measured in terms of the angle spread between the original and quantised vectors. In [19], the quantisation error of f˜ was given in the following form
s2E = E[ sin2 (k(f i , f i ))] ≤ 2−B/(M−1)
(40)
power so that it converged to the form of (41) at high SNR (30 dB). Investigating the expected quantisation error at the intermediate SNR remains an open research problem. In summary, the quantisation error of the proposed algorithm ranges between 2−B/(M−1) and i × 2−B/(M−Nk ) . Note that, in most of the cases, both these error variances are lower than the errors in VQ MSE of [17] (which quantises both magnitude and direction) with s2E ≥ 2−B/M . Therefore the proposed algorithm quantises the channel directions more precisely than the previously proposed VQ MSE feedback policy.
Since we are only quantising the direction, not magnitude, this error variance denotes the angular spread of the quantised effective MISO channel.
5.3 SMSE analysis
5.2.2 Quantisation error at high SNR: In Section
In the absence of quantisation error, the SMSE of the precoder with perfect CSIT is [4]
5.1, we have shown that our proposed algorithm is equivalent to QBC at high SNR for one data stream per user. Simulation results in Section 6 will show the simulation of the convergence of this algorithm to QBC for multiple data streams per user. Therefore we analyse the high-SNR quantisation error of our receiving combining scheme using the concepts of QBC. When each user receives one data stream, QBC chooses the codevector with the least quantisation error and thus converts a MIMO channel into an effective MISO channel [11]. The quantisation error in this case is upper bounded by 2−B/(M−Nk ) [11]. Using the same notion, for a multiple data stream per user scenario, the effective MISO channel of the ith stream of a particular user can be chosen to generate the ith least quantisation error with respect to its original MIMO channel. The expected quantisation error of the ith data stream (in terms of error tolerance) of the kth user in this method satisfies [11, 32]
s2E ≤ i × 2−B/(M−Nk )
(41)
Note that the quantisation method described in the previous passage can lead to intra-user interference because of the correlation of two codevectors of a particular codebook. Our proposed algorithm avoids this scenario by incorporating the intra-user interference in receiver combining. However, the codevectors chosen for two different streams of a user vary with time and become mutually statistically independent in the long term of multiple channel realisations. Therefore we hypothesise that the quantisation error of our algorithm matches with that given by (41) at high SNR. The proposed receive combining scheme incorporates both an increase in signal power and a reduction in (intra and inter user) interference. The trade-off between these two depends on the SNR. Owing to the adaptive nature of this method, the expected quantisation errors for intermediate SNR cases are very hard to derive. In our simulations, we assumed the quantisation error to take the form of (40) at low SNR (0 dB) and changed this value linearly with transmitted 34
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SMSE = L − M + s2 tr[(FQF H + s2 I M )−1 ] −1 Pmax H = L − M + tr FF + I M (42) Ls2 In (42), we assumed Q = (Pmax /L)I L , that is, equal power allocation for simplicity of the analysis. At very high SNR, the SMSE approaches zero in (42) as tr((Pmax /Ls2 )FF H + I M )−1 is a decreasing function of SNR. However, with quantisation error, if the original precoder [4] is used L H −1 H −2 s2E P q f J fi SMSE = 1 − qi f i J f i + M max i i i=1 (43) H H f i J −1 f i and (s2E /M) where J = FQ F + s2 I M . Both qi
H Pmax qi f i J −2 f i increase with SNR. Since the former term is a linear-over-affine function and the latter is a quadraticover-quadratic function of Pmax , at high SNR the latter term dominates and SMSE increases with SNR, which explains the results of [17].
In our proposed algorithm
2 s ˆ Fˆ H SMSE = L − M + s2 + E Pmax tr FQ M −1 s2E 2 P + s + I M max M =L−M
P H Fˆ Fˆ + I M + tr 2 max L s + s 2E /M Pmax
(44)
−1
(45) In (45), we again assumed equal power allocation for analysis IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org simplicity. (Pmax )/(L(s2 + (s2E /M)Pmax )) is a non-increasing function of Pmax . Thus the proposed precoder makes sure that SMSE does not increase with SNR at high SNR region. Fig. 3 illustrates all these effects. Since, the increase in SMSE is most apparent in MU-MISO systems, the simulations use a MU-MISO system with independent channel realizations where M ¼ 5, Lk = 1∀k and B ¼ 10 bits per data stream. We use uncoded quadrature phase shift keying (QPSK) for data transfer. The proposed algorithm clearly stabilises the SMSE at high SNR.
H H antennas, fˆ i fˆ j = 0 when i = j. Therefore Fˆ Fˆ = I M . H Equation (49) follows since the ith diagonal entry of Fˆ Fˆ L ˆ 2 ˆ is given by j=1 f ij , f i is an M × 1 column vector and
f i 2 = 1. Therefore using the law of large numbers, as 2 M 1, fˆ ij 1/M. Using (50) in (45), we obtain SMSEM 1 = L − M +
5.3.1 Asymptotic SMSE analysis with large transmit antennas: At high SNR, s2 /Pmax ≃ 0.
=
M(M + Ls2E − L) M + Ls2E
L2 s2E M + Ls2E
(51)
Therefore the third term in (45) takes the following form tr
M ˆ ˆH FF +IM Ls2E
−1
⎡
−1 ⎤ 2 LsE H H I + Fˆ Fˆ = tr⎣I M − Fˆ Fˆ ⎦ M M
⎡
≃ tr⎣I M − Fˆ
= tr I M − ≃ tr I M − =
(46) ⎤
−1 Ls2E H IM 1+ Fˆ ⎦ M
M H Fˆ Fˆ 2 M + LsE
M L I M + Ls2E M M
M(M + Ls2E − L) M + Ls2E
(47) (48)
(49)
To avoid an error floor, the receivers have to decrease the quantisation error proportionately to the SNR. This condition can be met by increasing feedback bit with varying power so that (2−B/(M−1) /Pmax ) remains constant. This relation of feedback bits and varying power was at first noticed in terms of sum-rate in [18].
(50)
Here, (46) was obtained using matrix inversion lemma [29]. Equation (47) follows because we assumed fˆ i 2 = 1 in our BS channel model and because of the law of large numbers, with the presence of high number of transmit
Figure 3 Comparing the SMSE of the traditional and proposed precoder (M ¼ 5, K ¼ 5, N k ¼ 1, L k ¼ 1 ∀k, B ¼ 10, QPSK) IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
Equation (51) has been approximated using our simplistic BS channel model, where we assumed the effective channels of the data streams to be unit norm. However, both (45) and (51) suggest that, at very high SNR, SMSE becomes saturated. This leads to the following result. For a fixed quantisation error, the SMSE of a MU system is lower bounded by a fixed value which does not depend on SNR. We call this the error floor effect of MU broadcast systems. This is similar to the ceiling effect, in terms of capacity and SINR, seen previously in limited feedback literature [17, 18].
6
Numerical simulations
In this section, we compare our proposed scheme with the leading feedback schemes in the literature. Since our proposed algorithm uses channel resources to know the post-processing information of U and P, we use an MMSE receiver to simulate the other existing algorithms. This preserves the fairness of the comparisons since the performance of the system always improves with an MMSE receiver for mutually non-orthogonal channels [21]. Unless specified, all transmissions use QPSK. As mentioned before, our proposed transceiver for MUMIMO systems can be readily generalised to MU-MISO systems. In Fig. 4, we compare the performance of the proposed algorithm to the available precoders in a limited feedback MU-MISO system. The system uses M ¼ 4, K ¼ 4, Lk = 1∀k and B ¼ 10. The proposed algorithm performs better than the MMSE precoder [17] by using MSIP quantisation and convexity of the power allocation problem. The traditional SMSE transceiver, which ignores quantisation error, performs well at low SNR but begins to worsen at an SNR of 15 dB. Thus the proposed transceiver improves over the state-of-the-art in MU-MISO precoders based on limited feedback. To the best of our knowledge, coordinated beamforming is one of the very few existing 35
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Figure 4 Comparison with available MU-MISO precoding techniques (M ¼ 4, K ¼ 4, Lk ¼ 1 ∀k, B ¼ 10, QPSK) linear transceivers that avoids the dimensionality constraint in the MU-MIMO with multiple data stream per user scenario. In Fig. 5, we compare the proposed algorithm with coordinated beamforming. Here M ¼ 4, K ¼ 2, Nk = 4, Lk = 1∀k and B ¼ 15. Since coordinated beamforming implements joint transceiver design, it performs better than the proposed algorithm with full CSIT. However, coordinated beamforming needs at least (M 2 − 1) bits for ˆH ˆ H /H ˆ 2F ). To create Fig. 5, we used the feedback of (H 15 bits feedback overhead per data stream in a MUMIMO system with four transmit antennas. This means only 1 bit is available per unique scalar entry of ˆ 2F ), introducing large quantisation error. The ˆH ˆ H /H (H eigen structure of the channel therefore gets mangled at the BS [15], leading to loss of performance. On the other hand, since our proposed algorithm expends 15 bits to quantise the 4 × 1 vector, the quantisation error of the fed back vector always remains less than or equal to 2−B/(M−1) = 0.03125. Thus, the proposed algorithm performs very close to its full CSIT curve and outperforms coordinated beamforming [9] with limited feedback.
Figure 5 Comparison with the coordinated beamforming (M ¼ 4, K ¼ 2, Nk ¼ 4, Lk ¼ 1 ∀k, B ¼ 15, QPSK) 36
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Figure 6 Comparison with available MU-MIMO VQ precoding techniques (M ¼ 4, N1 ¼ N 2 ¼ 2, K ¼ 3, N3 ¼ 3, Lk ¼ 1 ∀k, B ¼ 15, QPSK) In Fig. 6, we compare our proposed scheme with other VQ combining limited feedback MU-MIMO transceivers. In this example, M ¼ 4, K ¼ 2, N1 = N2 = 2, N3 = 3, Lk = 1 ∀k and B ¼ 15. Since to the best of our knowledge, existing VQ combining MU-MIMO schemes have not dealt with multiple data streams per user, we stick with one data stream per user in this comparison. The proposed scheme outperforms the QBC [11] and MET [12] approaches because of the use of SMSE precoder, adaptive receive combining and optimal power allocation. Although our algorithm outperforms [21] up to 20 dB, Boccardi’s MESC [21] seems to converge at a lower error floor than the proposed algorithm. This happens because in our proposed algorithm the actual quantisation error variance is not known at the BS. Owing the adaptive quantisation policy of the proposed algorithm, the quantisation error variance changes from low to high SNR; since we only quantise the direction of the effective channels, the norm of the quantisation error is not available at the BS. The quantisation error in MESC case [21] also changes from low to high SNR but the BS does need this knowledge because of the use of a ZF precoder. Our proposed transceiver adds to the literature by allowing multiple data streams per user. Fig. 7 shows the comparison of the transceiver’s performance to other possible methods to transmit multiple data streams per user. In Fig. 7, EBC projects the MIMO channel to its dominant eigenvectors to create effective MISO channels [23] and QBC chooses the set of codevectors that will generate least quantisation error as effective MISO channels [11]. The proposed transceiver approaches EBC at low SNR and QBC at high SNR. Thus the proposed algorithm retains the advantages of both EBC and QBC by providing a trade-off between signal power, intra-user and inter-user interference. IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
www.ietdl.org the channel magnitude will also be important. An extension of the present work will be the optimisation of feedback bits among the channel magnitude and direction information, such as done in a different context in [33]. Our present work can also be extended to a slowly time varying channel. The temporal correlation between different blocks can be used to reduce the amount of feedback overhead.
8
References
[1] TSE D., VISWANATH P.: ‘Fundamentals of wireless communications’ (Cambridge University, Cambridge, UK, 2005) Figure 7 Different receive combining techniques with multiple data streams per user (M ¼ 4, K ¼ 2, N k ¼ 3, Lk ¼ 2∀k, B ¼ 12, BPSK)
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[3] SCHUBERT M. , BOCHE H.: ‘Solution of the multiuser downlink beamforming problem with individual SINR constraints’, IEEE Trans. Veh. Technol., 2004, 53, pp. 18– 28
Conclusions
In this paper, we proposed linear transceiver design in the downlink of a MU-MISO and MU-MIMO system (with multiple data streams per user) using SMSE precoder at the BS, MSIP VQ as the feedback algorithm and MMSE decoder at the receivers. However, to encode the channel information, the receivers use MESC first. In the MUMISO, the individual users send back the indexes of their quantised channels to the BS. In the MU-MIMO scenario, the users convert their MIMO channels to effective vector downlink MISO channels to maximise the expected SINR and then send the indexes of these quantised MISO channels. The BS uses the quantisation error of MSIP in the SMSE precoder design and finds the downlink precoder and power allocation vector using a convex optimisation problem. The proposed system was shown to outperform the previously existing linear transceivers in the MU scenario for limited feedback, while also allowing for multiple data streams per user. One possible extention of the present work will be the detailed analysis of the expected quantisation error variance in the intermediate SNR range. The way the receivers find the trade-off between signal power increase, and intra-user and inter-user interference reduction will give an insight to analyse this problem. In this work, only shape feedback is sent to the BS. This is reasonable if the average channel magnitudes of all users are equal. In this case, the alignment of precoding vector with channel direction is more important than the power allocation. However, this assumption may remain valid only in a small scale fading scenario. In a more practical scenario, different users will be located at different distances from the BS and therefore average channel magnitude will be different. In that case, bit allocation in IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
[4] KHACHAN A.M., TENENBAUM A.J., ADVE R.S.: ‘Linear processing for the downlink in multiuser MIMO systems with multiple data streams’. Proc. IEEE ICC’2006, June 2006, vol. 9, pp. 4113 – 4118 [5] TENENBAUM A., ADVE R.S.: ‘Joint multiuser transmit-receive optimization using linear processing’. Proc. IEEE ICC 2004, Paris, France, June 2004, pp. 588 – 592 [6] HAARTSEN J.C.: ‘Impact of non-reciprocal channel conditions in broadband TDD systems’. Proc. IEEE PIMRC 2008, September 2008, pp. 1 – 5 [7] DHARAMDIAL N., ADVE R.S.: ‘Efficient feedback for precoder design in single- and multi-user MIMO systems’. Conf. on Information Sciences and Systems, March 2005 [8] KHACHAN A.M.: ‘Linear processing for the downlink in multiuser MIMO systems’. MS thesis, University of Toronto, 2006 [9] CHAE C., MAZZARESE D., JINDAL N., HEATH R.W. JR.: ‘Coordinated beamforming with limited feedback in the MIMO broadcast channel’, IEEE J. Sel. Areas Commun., 2008, 26, pp. 1505 – 1515 [10] NARULA A., LOPEZ M.J., TROTT M.D., WORNELL G.W.: ‘Efficient use of side information in multiple-antenna data transmission over fading channels’, IEEE J. Sel. Areas Commun., 1998, 16, pp. 1423 – 1436 [11] JINDAL N.: ‘Antenna combining for the MIMO downlink channel’, IEEE Trans. Wirel. Commun., 2008, 7, pp. 3834 – 3844 37
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[32]
GUPTA A., NADARAJAH S.:
GUPTA
A. ,
NADARA JAH
S.
IET Commun., 2011, Vol. 5, Iss. 1, pp. 27– 38 doi: 10.1049/iet-com.2010.0015
