This document discusses a vector precoding scheme for multi-user MIMO systems. It proposes using vector precoding to circumvent the channel inversion required for zero forcing precoding. The scheme develops a joint transmitter-receiver design where the transmitter precoder lies in the null space of other users' channels to eliminate multi-user interference. Simulation results show the proposed approach improves bit error rate performance by an order of magnitude compared to zero forcing, and increases MIMO broadcast channel capacity with lower complexity than inversion-based techniques.

1. Vector Precoding Scheme
for Multi-user MIMO Systems
Yogesh Nijsure, Charan Litchfield, Yifan Chen and Predrag B. Rapajic
Medway School of Engineering,
the University of Greenwich, UK,
Email: {y.nijsure, c.litchfield, y.chen, p.rapajic}@gre.ac.uk
Abstract— In this paper, the performance of Vector Precoding
in multiple input multiple output broadcast channels(MIMO BC)
is investigated and compared with other channel decomposition
techniques utilized for implementing zero forcing (ZF) precod-
ing. It is a known result that ZF precoding requires pseudo
inversion of the channel matrix, where this operation is only
optimum when the transmitter power is unconstrained. The
problem when the transmitter is subject to average or maximum
power constraints is well known, where results published have
indicated that ZF precoding approaches the maximum capacity
bound if the dimensionality of the system is greater than the
number of transmitter antennas. A vector precoding technique
for MIMO BC channels is investigated in this paper where
pseudo inversion is circumvented by employing joint co-operation
between transmitter and receiver for all users. This technique
adopts a time scheduling approach to service the users which
facilitates decentralized multi-user detection at the receiver. This
approach yeilds an improvement to the bit error rate probability
by approximately an order of magnitude as compared to the ZF
approach utilizing other channel decomposition techniques. The
scheme also enables an increase in the capacity of the MIMO BC,
with less computational complexity as compared to the techniques
employing Moore-Penrose pseudo inverse.
I. INTRODUCTION
Precoding uses the same idea as frequency equalization,
except that the fading is inverted at the transmitter instead of
at the receiver. The technique requires the transmitter to have
knowledge of the sub-channel flat fading gains, which must be
obtained through estimation. When the receiver has multiple
antennas, the transmit beamforming cannot simultaneously
maximize the signal level at all of the receive antenna and pre-
coding is used. Note that precoding requires knowledge of the
channel state information (CSI) at the transmitter. Precoding
is quite common on wireline multi-carrier systems like high
bit rate digital subscriber lines.There are two main problems
with precoding in the wireless setting. First, precoding is
basically channel inversion, and we know that inversion is
not power efficient in fading channels. In particular an infinite
amount of power is needed for channel inversion on a Rayleigh
fading channel. The other problem with precoding is the need
for accurate channel estimates at the transmitter, which are
difficult to obtain in a rapidly fading channel.
In multiuser communication scenario, diversity can be ex-
ploited through making appropriate choice among users with
independently faded channels [2]. In the literature, multiuser
scheduling has been considered in the context of channel
allocation for a space division multiple access/time division
multiple access network e.g.,[1],[3],[11] but mainly with the
downlink and the assumptions that users are equipped with
only one antenna or transmit only one data stream. As in-
dicated in [15] this approach raises two potential concerns.
First, a globally optimal allocation requires a thorough search
of all possible choices, and suboptimal or heuristic alternatives
induce complexity versus performance tradeoffs. Second, the
physical layer details are largely neglected: either the compati-
bility metric depends solely on the channel and is independent
of the underlying transceiver structures; or a conservative view
is taken that treats multiuser interference(MUI) as background
noise.
In this paper we adopt the vector precoding scheme for
multi-user multiple input multiple output (MU-MIMO) system
and adopt a time scheduling approach to service the users. The
key contributions of this paper are:
1) Developing a joint-transmitter receiver design for imple-
menting vector precoding for MU-MIMO systems.
2) Evaluation of multi-user channel capacity and Bit-error
rate performance.
The proposed research aims at addressing the issue of channel
inversion required for pre-coding and mitigation of multi-user
interference in MIMO systems. In this first section of this
paper we provide an introduction to the concept of vector
precoding for MIMO systems. In section II we describe the
proposed system model used in this paper. In section III we
analyze how vector precoding is used for our system. In
section IV we develop a multi-user MIMO system. In section
V we provide discussion on results. At the end of this paper
we provide a set of conclusions in section VI.
II. PROBLEM FORMULATION
This proposed scheme for precoding will adopt a joint effort
at both the transmitter and a receiver. This will use a Zero
forcing (ZF) like approach in order to mitigate multiuser inter-
ference. Let Hj be the jth
user subchannel and Xibe the user
i transmit vector. The fundamental idea of ZF solution[7],[13]
is that interference is removed by forcing Hj · Xi = 0 for i
not equal to j, which means that all the other users besides
the user of interest will be forced to have a zero contribution
by adopting this scheme, results in a constraint that the total
number of transmit antennas must always be greater than
number of receiver antennas. as in [16]. The transmitter matrix
for user j will not interfere with the signal at the output of the

2. Fig. 1. vector precoding
receivers for other users if it lies in the null space of the above
given channel vector. Let UjΣVj
H
represent the singular value
decomposition of the channel under consideration. where U
and V represent the left and right singular vectors respectively
and Σ represents the matrix of the singular values of the
decomposed channel. Let the received vector be represented
by Y , the channel matrix by H , the transmitted vector by
˜X. and ϑ represent the additive white gaussian noise and (.)H
represents the Hermitian transpose.
As indicated in Fig.1, the classical method utilizing the
channel decomposition approach can be described as follows.
The received signal can be represented as
Y = HX + ϑ.
Forming the singular value decomposition of the channel
Y = UΣV H
X + ϑ
By utilizing the the matrix of right singular vectors V of the
channel,pre-processing of the signal is achieved and matrix of
transmit vectors is formed
Let X = V ˜X
Y = UΣV H
V ˜X + ϑ
The received signal is partially whitened and post processing
at the receiver end is achieved by utilizing the matrix of left
singular vectors U.
UH
Y = UH
UΣ ˜X + UH
ϑ
Thus the received signal after post- processing can be repre-
sented as:
˜Y = Σ−1
Σ ˜X + Σ−1
UH
ϑ
˜Y = ˜X + υ
Extending the same concept to multi-user MIMO case:
Hj
˜V
(0)
j = [U
(1)
j U
(0)
j ]Σ[V
(0)
j V
(1)
j ] (1)
The transmitted signal X is subject to additive white
Gaussian noise (AWGN) n, and multipath propagation AWGN
channel H. The MU-MIMO system consists of nt transmitting
and nr receiving antennas. The channel matrix H is a (nr ×nt)
complex matrix, the received vector y is a nr dimensional
complex BPSK signal vector, the transmitted signal x is a nt
dimensional vector and n is the nt dimensional noise vector.
A BPSK modulation scheme is used in order to eliminate
modulation gain and simply show the performance advantage
of MU-MIMO. More advanced modulation scheme is expected
to offer extra gain in data rates but at the same time an
increased complexity.
III. MULTI-USER PRECODING
Multi user multiple-input multiple-output (MIMO) systems
provide high capacity with the benefits of space division
multiple access. The channel state information at the base
station (BS) or access point (AP) is very important since
it allows joint processing of all users signals which results
in a significant performance improvement and increased data
rates [1]. If the channel state information is available at the
BS/AP, it can be used to efficiently eliminate or suppress multi-
user interference (MUI) by beamforming or by using dirty-
paper codes. The precoding also allows us to perform most
of the complex processing at the BS/AP which results in a
simplification of users terminals. Linear precoding techniques
have an advantage in terms of computational complexity. [4]
Non-linear techniques have a higher computational complexity
and require some signaling overhead but can provide a better
performance than linear techniques.
The basic idea behind this solution is to utilize the right
singular vectors of the channel matrix in order to form the
precoding matrix.
Thus an optimal precoding matrix can be formed such that
all MUI is zero by choosing a precoding matrix Fi that lies
in the null space of the other users channel matrices. Thereby,
a MU MIMO downlink channel is decomposed into multiple
parallel independent single user MIMO channels [16]. Thus
we can define the zero MUI constraint forces Fi to lie in
the null space of Hi. From the singular value decomposition
(SVD) of Hi whose rank is Li. The proposed system chooses
the last right singular vectors Nt − Li where Nt is number of
transmitter antennas.
Thus the equivalent channel of user i after eliminating the
multi-user interference is identified. Each of these equivalent
single user MIMO channels has the same properties as a
conventional single user MIMO channel. As mentioned before,
by applying block diagonalization on the combined channel
matrix of all users the MU MIMO channel can be transformed
into a set of parallel single-user MIMO channels. However,
there is a capacity loss due to the nulling of overlapping
subspaces of different users. In [13], the authors propose a
successive precoding algorithm in order to define a simplified
solution of the power control problem. By allowing a certain
amount of interference, this algorithm reduces the capacity
loss due to the subspace nulling. In short, first calculate the
maximum capacity that an individual user can achieve. The
basic ideology behind this solution is to utilize the right
singular vectors of the channel matrix in order to form the
precoding matrix.
F = [F1F2F3....FK] ⊆ CNT ×R
(2)

3. Thus an optimal precoding matrix can be formed such that all
MUI is zero by choosing a precoding matrix Fi that lies in
the null space of the other users channel matrices. Thereby,
a MU MIMO downlink channel is decomposed into multiple
parallel independent single user MIMO channels [14], [12].
˜Hi = [HT
1 ....HT
i−1HT
i+1....HT
K]T
(3)
the zero MUI constraint forces Fi to lie in the null space of Hi.
From the singular value decomposition (SVD) of Hi whose
rank is Li. The proposed system chooses the last right singular
vectors MtLi where Mt is number of transmitter antennas.
Hi
˜V
(0)
i = UiΣ[V
(1)
i V
(0)
i ]H
(4)
IV. MULTI-USER MIMO
The proposed solution aims at identifying the user with
the smallest difference between its maximum capacity and its
capacity and generate its precoding matrix such that it lies
in the null space of the remaining users channel matrices.
Thereafter the new combined channel matrix without this users
channel matrix is formed. The proposed system repeats these
steps until the combined channel matrix is empty. The order
of the users in which they are precoded using zero forcing
precoding is the reverse of the order in which their precoding
matrices are generated. The capacity of a MIMO closed-loop
system, that is, perfect CSI at the transmitter, with worst-
case noise under a trace constraint (or worst-case interference)
equals the capacity of a MIMO open-loop system, that is,
no CSI at the transmitter, with white noise, that is, without
interference. The structure of the equivalent system is a single-
user MIMO system with uncorrelated noise and without CSI at
the transmitter [3]. The worst-case noise directions correspond
with the left eigenvectors of the channel matrix H. The optimal
transmit directions correspond with the right eigenvectors of
the channel matrix H. Both are independent of each other. The
power allocation is then the well-known waterfilling solution.
At the receiver each user utilizes the left singular matrix rows
or matrix U to decode the data that was transmitted. Thus the
signal processing at the user end has to be decentralized to
facilitate the successful operation of the proposed solution. It
is evident that a particular user has no idea about the channel
characteristics of different users in the network and a co-
operative scheme between the users cannot be implemented
for conveying the information about the left singular matrices
in between the users for decoding purposes. This is the issue
with the proposed system [6]. The computational complexity
involved in this SVD type of approach is O(nk2
) as compared
to the ZF approach involving Moore-Penrose pseudo inverse
has a greater computational complexity of O(n3
).
A. Performance analysis
The capacity of MU - MIMO downlinks is intimately
connected with a result as indicated in [6] called ”writing on
dirty paper” [6], which is briefly summarized here suppose X
represents a transmitted signal,W and Z are additive white
noise terms, so that the received signal is Y = X + W + Z.It
is shown in [6] that if W is known deterministically to the
transmitter , then the capacity of the communication channel
is same as a channel with only the second interference term:
Y = X + Z. Regardless of whether or not the receiver knows
W and independent of the statistics of W. When the users are
known at the transmitter , SDMA can be employed to increase
capacity [3]. In particular, the capacity of the channel for user
j is indicated as in [15]
Cj = maxXj log2 |I + (σ2
nI + Hj
˜Xj
˜Xj
∗
H∗
j )−1
HjXjX∗
j H∗
j |
(5)
where Rnj
is the covariance of the noise vector. The capacity
is thus the function of not only what modulation matrix
is chosen for the particular user of interest, but also those
chosen for all other co-channel users as well [9]. Viewing the
problem entirely form the perspective of receiver j , capacity
is maximised when, Hj
˜Xj = 0 Or in other words , when the
transmit matrix ˜Xj for all other then j lies in null space of
Hj. If this is done , then the capacity of user j is equal to the
waterfilling capacity of the channel matrix Hj [10]. note that
nt ≥ nr is necessary condition for achieving a requirement
not imposed in the blind transmitter case.
For the purpose of simulations the comparison was made
between the proposed vector precoding scheme and the zero
forcing approach. The systems considered for the simulation
were full rank systems. As seen from the simulations results
the proposed vector precoding scheme outperforms the zero
forcing precoding method even at significantly lower values
of SNR. The main reason for this can be due to the fact that
in traditional zero forcing approach the channel needs to be
inverted at the receiver and under such circumstances spectral
nulls are introduced in the process of reception. In the vector
precoding approach the channel doesnt need to be inverted
under the assumption that the transmitter has complete channel
state information. The vector precoding approach exploits the
orthogonal nature of the right singular matrix. In the case of a
multi-user MIMO system the proposed method adopts a time
division multiple access scheme in which each user is serviced
at a time. In this way the decentralization of users is achieved.
The proposed system is compared with different configurations
in MIMO adopting a zero forcing like approach. The problem
with sum capacity maximization in a multi-user channel is
that such an approach may result in one or two ”strong”
users large taking a dominant share of the available power,
potentially leaving weak users with little or no throughput
[7],[4]. Consequently, in practice, the dual problem is often of
more interest: i.e., minimize power output at the transmitter
subject to achieving a desired arbitrary rate for each user[4].
Assume Hj = AjBJ , where Aj is nR × Lj, Bj is Lj × nT ,
and Lj ≤ nRj
. Here , the condition HiXj = 0,i = j,
necessary to make the system block-diagonal, is equivalent
to BiMj = 0;i = j. Thus , we define the matrix ˜Bj
˜Bj = [BT
1 ...BT
j−1BT
j+1...BT
K]T
(6)
Let the SVD of
˜Bj = ˜UBj
˜ΣBj
[ ˜VBj
(1)
˜VBj
(0)
]∗
(7)

4. Fig. 2. error rate
, where ˜VBj
(0)
, corresponds to the right null space of ˜Bj.
The optimal modulation matrix for user j, subject to the
constraint that the inter-user interference is zero , is now
of the form ˜VBj
(0)
Xj,for some choice of transmit vectorsXj.
Substituting (6) and (7) in (5),the system capacity of the
approach in this case is thus of the form:
C = maxXj
,j=1,K
K
j=1
log2|I+1/σ2
nAjBj
˜VBj
XjX ∗
j
˜VBj
B∗
j A∗
j |
(8)
B. Algorithm
1) For j = 1,....K:
2) Compute ˜Vj
0
, the right null space of ˜HJ . Information
of the active users at the receiver side
3) Compute SVD
Hj
˜
V
(0)
j = Uj ·
Σ 0
0 0
· [V
(1)
j V
(0)
j ]∗
(9)
4) Use water filling on the diagonal elements of Σ to
determine the optimal power loading matrix Λ under
a total power constraint P
5) Xs = [ ˜V
(0)
1
˜V
(1)
1
˜V
(0)
2
˜V
(1)
2 ...... ˜V
(0)
k
˜V
(1)
k ]Λ1/2
6) Evaluate the received vector for the for the current sub-
channel conditions.
7) Post Processing: premultiply by the left matrix of left
singular vectors as obtained from SVD decomposition
of the current channel estimate.
8) Evaluate the bit error rate for the user of interest.
9) Repeat the process for each user and each user time slot.
10) End of algorithm.
At higher SNRs, the relatively small gap between channels
with and without channel information at the transmitter is suf-
ficiently small. However, this assumes that the channel is full
rank. When the channel is rank deficient, the gaps are larger,
and having complete, or even only partial channel information
Fig. 3. ergodic capacity
available can be advantageous. Multi-user capacity can be
taken to have different meanings. [8]It is possible to consider
the capacity of one particular user in the context of a system,
or to consider the sum capacity of all users in the system.
Under a single power constraint, it is possible to achieve a
variety of different combinations of rates for different users
by allocating resources differently to different users.
V. DISCUSSION ON RESULTS
Figure 2 indicates BER of a simple MU-MIMO system with
BPSK modulation over a channel,by adopting zero-forcing
precoding and geometric mean decomposition methods is
presented along with the comparison with our vector precoding
approach. Figure 3 shows the capacity of a MIMO system
is presented with a comparison of the ergodic capacity for
the transmitter with uniform power allocation ,transmitter
with CSI and the multiuser MIMO case with the proposed
approach. The joint transmitter and receiver scheme for im-
plementing the Multi-user downlink vector precoding scheme
was demonstrated in this paper. The simulation results indicate
a capacity improvement as well as the improvement in the
bit error rate performance compared with the zero forcing
approach. In the vector precoding approach the channel doesnt
need to be inverted under the assumption that the transmitter
has complete channel state information. Results show that
designing the precoders based on the standard pseudo-inverse
is optimal under the assumption of a total power constraint.
However, when more complex power constraints are involved,
e.g., individual total per antenna power constraints, the pseudo-
inverse is no longer sufficient and vector precoding provides
better performance. In general, finding the optimal inverse is
a difficult optimization problem which is highly dependent
on the specific design criterion. Such constraints may be
important in modern systems where multiple base stations,
each with multiple antennae, cooperatively transmit data to
the same users.

5. VI. CONCLUSION
In this paper we demonstrated the joint transmitter-receiver
design for multi-user precoding scenario. The BER perfor-
mance for such a system was evaluated and simulation results
suggest an improvement in BER performance of the system as
compared to the transmitter side precoding alone. This can be
mainly attributed to the fact that the proposed solution avoids
channel inversion usually required in the precoding process.
The capacity results also indicate a better mitigation of MUI in
the case of MIMO system. Thus, precoding with generalized
power constraints is an important problem in modern com-
munication systems and there are still many open questions.
More advanced linear precoding schemes should be addressed.
For example, it is well known that in low SNR conditions,
and under channel uncertainty, regularizing the pseudo-inverse
can considerably improve the performance. It is interesting to
examine this property in the context of generalized inverses.
Future work should also address the implications of the results
on non-linear schemes such as ZF DPC precoding. Another
extension of this work is to consider the well known duality
between receive and transmit processing. ZF decoding using
the pseudo-inverse (the decorrelator) is probably the most
common decoding algorithm. The results suggest that vector
precoding may outperform it under uncertainty conditions.
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