A low complexity user grouping based
multiuser MISO downlink precoder
Saif Khan Mohammed and Erik G. Larsson
Linköping University Pre Print
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Saif Khan Mohammed and Erik G. Larsson, A low complexity user grouping based multiuser
MISO downlink precoder, 2011, accepted for IEEE GLOBECOM 2011.
Preprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-70205
A low complexity user grouping based multiuser
MISO downlink precoder
Saif Khan Mohammed, Member IEEE and Erik G. Larsson, Senior Member IEEE
Communication Systems Division, Dept. Electrical Engineering (ISY),
Link
¨
oping University, Sweden. E-mail: saif@isy.liu.se and erik.larsson@isy.liu.se.
Abstract—We consider low complexity precoding for the Mul-
tiple Input Single Output (MISO) Gaussian Broadcast channel
with N
t
antennas at the base station and N
u
single antenna
users in the downlink. Theoretical studies have suggested high
throughput communication with increasing spatial dimensions
i.e., min(N
t
, N
u
). Nevertheless, most modern communication
standards are unable to exploit the spatial dimension fully, since
they are restricted to orthogonal communication techniques like
TDMA/FDMA (Time/Frequency Division Multiplexed Access)
which are known to be sub-optimal. This restriction is mostly due
to the prohibitive complexity of optimal/near-optimal precoding
schemes. On the other hand low complexity techniques like Zero
Forcing (ZF) and MMSE have poor sum rate performance. In
this paper, we propose a novel low-complexity user grouping
based precoding scheme which schedules all users on the same
time-frequency resource (i.e., optimal utilization of resources).
The proposed precoder is analytically shown to achieve a sum
rate performance significantly better than the ZF precoder at
similar complexity. Through simulations, it is also observed to
achieve a significant fraction of the sum rate achieved by the
optimal schemes.
I. INTRODUCTION
Multiple-Input Multiple-Output (MIMO) technology holds
the key to very high throughput downlink communication in
fading wireless channels by exploiting the spatial dimension
[1]. However most modern wireless communication standards
support a maximum achievable spectral efficiency less than
10 bits/sec/Hz. This is because the multiple access scheme is
still TDMA or FDMA, in which each user communicates over
distinct frequency-time resource, i.e., orthogonal communica-
tion. The rate region and sum capacity of the Gaussian MIMO
broadcast channel (which models downlink communication
in modern wireless systems) is achieved by a scheme called
Dirty Paper Coding (DPC), in which all users share the same
frequency-time resource [2]. It is also known that orthogonal
access schemes (like TDMA, FDMA) are strictly sub-optimal
and achieve only a small fraction of the total sum capacity
[3]. However, TDMA and FDMA are still favored in practice,
due to less burden of obtaining channel state information
and also due to the high precoding complexity of optimal
precoders like DPC. Other near-optimal precoders like those
based on vector perturbation and lattice reduction [4] also
have prohibitive complexity. On the other hand low complexity
precoders, like ZF [5], MMSE are are known to achieve poor
sum rate performance especially in ill-conditioned channels.
This work was supported by the Swedish Foundation for Strategic Research
(SSF) and ELLIIT.
To keep the low-complexity benefit of the ZF precoder and
yet improve the overall sum rate, we propose a user grouping
based precoder. In the proposed precoder, the users are divided
into small groups of equal size. Downlink beamforming is
done in such a way that, at each receiver the interference from
the signal intended for users not in its group is nulled out.
However, there still remains interference from the signal of
users in the same group. This interference is pre-cancelled at
the transmitter, by performing dirty paper coding among the
users in the same group. With small groups (having only 2 or
3 users), dirty paper coding within each group is practically
feasible and is equivalent to dirty paper coding for a MISO
broadcast channel with small number of users [6], [7].
Inter-group interference pre-cancellation for a group of users
is achieved by choosing their beamforming vectors to lie in a
space orthogonal to the space spanned by the channel vectors
of the users in the other groups. One novel aspect of the pro-
posed precoder, is that we choose the beamforming vectors in
such a way that the effective channel matrix for each group is
lower triangular, which enables successive known interference
pre-cancellation within each group using DPC. With group
size greater than one and a per user power allocation same as
that of the ZF precoder, the proposed precoder is analytically
shown to achieve a sum rate greater than that achieved by
the ZF precoder. For a given grouping of users, the optimal
power allocation is given by the waterfilling scheme. Since
the achievable sum rate is observed to be sensitive towards the
chosen grouping of users, it is jointly optimized w.r.t. both the
per user power allocation as well as the grouping of users. This
optimization problem is inherently complex, and therefore we
propose near-optimal low-complexity solutions to it. Through
analysis and simulations, we show that indeed the proposed
user grouping based precoder achieves a sum rate significantly
greater than that achieved by the ZF precoder
2
, at similar com-
plexity. The low complexity attribute of the proposed precoder
could be an enabler for large MISO broadcast systems with
large N
t
and N
u
.
We also clarify that, the proposed precoder is entirely dif-
2
In this paper, the ZF precoder used as the benchmark precoder for
comparison, is based on the pseudo-inverse of the channel matrix, which
is one among all possible generalized inverses. In [8], it has been shown that
with a total transmit power constraint, among all possible generalized inverses,
the pseudo-inverse results in the maximum achievable sum rate. Since we only
consider a total transmit power constraint in this paper, comparison with the
pseudo-inverse based ZF precoder suffices and therefore we need not compare
with ZF precoder based on other generalized inverses.
ferent from the block diagonalization based precoder proposed
in [5], which considers a MIMO multiuser broadcast channel,
in which each user could have multiple receive antennas.
Beamforming vectors are chosen such that each user sees no
interference from the information intended for other users.
Hence, in the special case of MISO broadcast channel (which
we consider in this paper), the block diagonalization precoder
in [5] basically reduces to the ZF precoder. In addition to
this, the precoder that we propose performs beamforming in
groups of users and not separately for each user. Another
user pairing precoder has been proposed for the Gaussian
multiuser MIMO broadcast channel in [9]. However, in [9],
only 2 users share the same time-frequency resource, i.e., the
medium access is orthogonal in groups of 2 users, which
is a sub-optimal utilization of resources when compared to
the proposed precoder where all users share the same time-
frequency resource.
Notations: A
H
and A
T
represent conjugate transpose and
transpose of the matrix A respectively. For any arbitrary
complex number z, let ¯z and |z| denote its complex conjugate
and absolute value respectively. The complex and the real
fields are denoted by C and R respectively. Given a vector
x = (x
1
, x
2
, ··· , x
n
)
T
∈ C
n
, let kxk
∆
=
p
P
n
k=1
|x
k
|
2
. For
any positive integer n, n!
∆
= n.(n−1).(n−2). ··· .2.1. Further,
for any set S, |S| denotes the cardinality (size) of the set S.
II. SYSTEM MODEL
Let H = (h
1
, h
2
, ··· , h
N
u
)
H
, represent the N
u
× N
t
channel matrix between the base station and the N
u
users
3
(N
t
≥ N
u
). The channel vector from the transmitter to the
k-th user is denoted by h
H
k
∈ C
1×N
t
, with its i-th entry
¯
h
k,i
representing the channel gain from the i-th transmit antenna to
the receive antenna of the k-th user
4
. With channel knowledge
at the transmitter, the information symbols can be effectively
mapped onto the symbols to be transmitted from the N
t
transmit antennas. Let x = (x
1
, x
2
, ··· , x
N
t
)
T
∈ C
N
t
×1
represent the transmitted vector. The vector of received sym-
bols y = (y
1
, y
2
, ··· , y
N
u
)
T
∈ C
N
u
×1
(with y
k
denoting the
signal received by the k-th user) is then given by
y = Hx + n (1)
where n = (n
1
, n
2
, ···n
N
u
)
T
∈ C
N
u
×1
is the additive noise
vector with n
k
representing the noise at the k-th receiver.
Further, each entry of n is an i.i.d CN(0, 1) random variable.
Further, the transmitter is subject to an average transmit power
constraint given by
E[kxk
2
] = P
T
. (2)
Due to unit variance noise, we would refer to P
T
as the
transmit signal to receiver noise ratio (i.e., transmit SNR).
For the sake of clarity and conciseness we introduce the
following notations. Subsequently we shall refer to the k-th
3
Throughout the paper, H is assumed to be full rank.
4
Subsequently we shall also refer to the receiver at the k-th user as the
k-th receiver.
user by U
k
. In the proposed precoding scheme, the total set of
users S = {U
1
, U
2
, ···U
N
u
} is partitioned into N
g
= N
u
/g
disjoint groups of size
5
g. Let the i-th group of users be de-
noted by the ordered set S
i
= {U
i
1
, U
i
2
, ··· , U
i
g
}. Therefore,
S = ∪
N
g
i=1
S
i
, and S
i
∩ S
j
= φ, ∀i 6= j, where φ denotes the
null set. Also, let any arbitrary grouping of users be denoted
by the unordered set P =
n
S
1
, S
2
, ··· , S
N
g
o
. For example,
with N
u
= 4 and g = 2, one possible grouping of users is
given by P =
n
{U
1
, U
4
}, {U
2
, U
3
}
o
.
For notational purposes, let us denote the set of all possible
groupings of a set of N
u
users into groups of size g, by A
(g)
N
u
.
For example with N
u
= 4 users and g = 2
A
(2)
4
=
(
n
{U
1
, U
2
}, {U
3
, U
4
}
o
,
n
{U
2
, U
1
}, {U
3
, U
4
}
o
,
n
{U
1
, U
2
}, {U
4
, U
3
}
o
,
n
{U
2
, U
1
}, {U
4
, U
3
}
o
,
n
{U
1
, U
3
}, {U
2
, U
4
}
,
n
{U
3
, U
1
}, {U
2
, U
4
}
,
n
{U
1
, U
3
}, {U
4
, U
2
}
,
n
{U
3
, U
1
}, {U
4
, U
2
}
,
n
{U
1
, U
4
}, {U
3
, U
2
}
o
,
n
{U
4
, U
1
}, {U
3
, U
2
}
o
,
n
{U
1
, U
4
}, {U
2
, U
3
}
o
,
n
{U
4
, U
1
}, {U
2
, U
3
}
o
)
.
Let H[i] ∈ C
(N
u
−g)×N
t
denote the sub-matrix of H con-
sisting of only those rows which represent the channel vector
of users not in the set S
i
, and let G[i] ∈ C
g×N
t
denote the
sub-matrix containing the remaining rows of H. Specifically,
if S
i
= {U
i
1
, U
i
2
, ··· , U
i
g
} then
G[i]
∆
= (h
i
1
, h
i
2
, ··· , h
i
g
)
H
. (3)
Further let H
i
represent the subspace spanned by the rows of
H[i], and let H
⊥
i
be the subspace orthogonal to H
i
. Therefore,
P[i] = (I
N
t
− H[i]
H
(H[i]H[i]
H
)
−1
H[i]) ∈ C
N
t
×N
t
(4)
represents the projection matrix for the subspace H
⊥
i
.
III. ZF PRECODER AND THE MOTIVATION FOR GROUPING
USERS
One of the most simple and low complexity linear precoder
is the ZF precoder. For each user, the ZF precoder beamforms
the user’s information is a direction which is orthogonal to
the space spanned by the channel vectors of the remaining
N
u
− 1 users, resulting in no inter-user interference. Further,
for any given user, its effective channel gain is proportional
to the Euclidean length of the projection of its channel vector
onto the space orthogonal to the space spanned by the channel
vectors of remaining users. In case of ill-conditioned channels,
since the channel vectors of all the users are “highly” linearly
dependent, the effective channel gain of each user would be
small, implying low achievable rates. It would therefore be
ideal to keep the low-complexity benefit of the ZF precoder
and yet improve the overall sum rate, especially when the
channel is ill-conditioned.
By grouping users into groups of size larger than one,
beamforming can be done to nullify only inter-group inter-
ference. With small group size, intra-group interference can
then be pre-cancelled using practical DPC at the transmitter,
without any significant increase in the required transmit power.
5
The proposed precoder can be generalized to have groups of different size.
However, for simplicity, we only consider groups of equal size in this paper.
Therefore the effective channel gain for U
i
j
is the Euclidean
length of the projection of h
H
i
j
onto the space H
⊥
i
. On the
other hand, with the ZF precoder, the effective channel gain
is the Euclidean length of the projection of h
H
i
j
onto the
subspace orthogonal to all the rows of H except h
H
i
j
(We shall
subsequently denote this orthogonal subspace by H
⊥
i
j
). It is
noted that H
⊥
i
j
⊂ H
⊥
i
whenever g > 1. Since the projection of
a vector onto a subspace of some space G is of lesser Euclidean
length than its projection onto the space G, it follows that
the effective channel gain for U
i
j
is higher with the proposed
user grouping based precoder as compared to that with the ZF
precoder. This simple observation coupled with the availability
of practical low-complexity DPC for small systems, motivates
the proposed precoding scheme which is presented in the next
section in more detail.
IV. PROPOSED USER GROUPING BASED PRECODER
For the i-th group of users, let the QR decomposition of the
matrix F[i]
∆
= P[i]G[i]
H
be given by
F[i] = Q[i]R[i] (5)
where R[i] ∈ C
g×g
is an upper triangular matrix with
positive diagonal entries, and Q[i] ∈ C
N
t
×g
is a matrix with
orthonormal columns. From the above decomposition it is also
clear that, the g columns of Q[i] form an orthonormal basis
for the space H
⊥
i
. Further, for any k 6= i, we have
G[k]Q[i] = 0 , k 6= i. (6)
This is because, for any k 6= i, the rows of G[k] lie in H
i
.
Beamforming the information for the users in the i-th group
along the columns of Q[i] ensures that users in group i do not
observe any interference from other groups. The precoding
operation for the i-th group is therefore given by
x[i] = Q[i]W[i]u[i] (7)
where u[i]
∆
= (u
i
1
, u
i
2
, ··· , u
i
g
)
T
is the g × 1 vector of
auxiliary input symbols of the users in the i-th group S
i
.
Information is encoded over these auxiliary input symbols. The
auxiliary symbols are assumed to be i.i.d. Gaussian distributed
with mean 0 and variance 1. W[i] ∈ C
g×g
is an additional
linear precoder to optimize the sum rate achieved by the i-th
group of users. The transmitted vector is then given by
x =
N
g
X
i=1
x[i]. (8)
Let y[i]
∆
= (y
i
1
, y
i
2
, ··· , y
i
g
)
T
be the g × 1 vector of
symbols received by the users in the i-th group S
i
. Using
(1), (7) and (8), the received vector y[i] is given by
y[i] = G[i]
x[i] +
N
g
X
k=1,k6=i
x[k]
+ n[i]
= G[i]x[i] +
N
g
X
k=1,k6=i
G[i]Q[k]W[k]u[k] + n[i]
= G[i]x[i] + n[i] = G[i]Q[i]W[i]u[i] + n[i] (9)
where the last step follows from the application of (6). From
(9) it is clear that each group of users does not have any in-
terference from the other groups. Basically the original MISO
broadcast channel has been decomposed into N
g
parallel non-
interfering g-user MISO broadcast subchannels.
We next focus on the effective channel matrix for the i-th
group of users. From (9) it is again clear that the effective
channel matrix for the i-th group of users is given by
B[i]
∆
= G[i]Q[i]W[i]. (10)
In this paper, we restrict ourselves to diagonal W[i] =
diag(
√
p
i
1
,
√
p
i
2
, ···
√
p
i
g
), where p
i
j
is the power allocated
to the information symbol of the j-th user in the i-th group.
With diagonal W[i], the sum power constraint in (2) is
N
g
X
i=1
g
X
j=1
p
i
j
= P
T
. (11)
Subsequently, let p = (p
1
, p
2
, ··· , p
N
u
) denote the power
allocation vector, with p
i
being the power allocated to U
i
.
We next show that B[i] is actually a lower triangular matrix
and is equal to R[i]
H
W[i]. From the definitions of P[i] and
Q[i] in (4) and (5), it is clear that P[i] is the projection matrix
for the space spanned by the columns of Q[i] and therefore
P[i]Q[i] = Q[i]. (12)
Since F[i] = Q[ i] R[i] = P[i]G[i]
H
it follows that R[i] =
Q[i]
H
P[i]G[i]
H
, and hence using (12) and the fact that P[i]
is Hermitian, we have
Q[i]
H
G[i]
H
= R[i]. (13)
Combining (13) with (9) we have
y[i] = R[i]
H
W[i]u[i] + n[i]. (14)
From (14), the received signal at the j-th user in the i-th group
is given by
y
i
j
= R[i]
(j,j)
√
p
i
j
u
i
j
+
Interference term
z
}| {
(j−1)
X
k=1
R[i]
(k,j)
√
p
i
k
u
i
k
+ n
i
j
,
j = 1, 2, ···g (15)
where R[i]
(k,j)
denotes the entry of R[i] in the k-th row
and the j-th column. Due to the lower triangular structure
of the effective channel matrix for the i-th group, from (15),
we observe that the j-th user in the i-th group (i.e., U
i
j
) has
interference only from the symbols of the previous (j − 1)
users in the same group (i.e., U
i
1
, ···U
i
(j−1)
).
In the proposed coding scheme, we start with the first user
in the i-th group, and since it sees no interference from any
other users, we simply use an AWGN channel code with rate
r
i
1
= log
2
(1 + p
i
1
R[i]
2
(1,1)
) (16)
The second user, has an interference term with contribution
only from the first user. Since the transmitter knows the
transmitted symbol for the first user, it knows the interference
term for the second user, and can therefore perform known
interference pre-cancellation using the Dirty Paper Coding
scheme [10], [11], [12]. In a similar manner, for the j-th
user, the transmitter can perform Dirty Paper Coding for the
known interference term which has contributions only from
the previous (j −1) users. The rate achieved by the j-th user
in the i-th group is therefore given by
r
i
j
= log
2
(1 + p
i
j
R[i]
2
(j,j)
). (17)
For a given grouping of users P ∈ A
(g)
N
u
, total power constraint
P
T
, channel realization H and power allocation vector p, the
sum rate achieved is given by
r(H, P
T
, P, p)
∆
=
N
u
/g
X
i=1
g
X
j=1
r
i
j
(18)
where r
i
j
is given by (17). Maximization over p yields
r(H, P
T
, P)
∆
= max
p |
P
N
u
i=1
p
i
=P
T
, p
i
≥0
r(H, P
T
, P, p)(19)
In (19), the optimal power allocation for a given grouping of
users is given by the waterfilling scheme [13]. Subsequently,
for g = 1, we shall denote the optimal waterfilling power
allocation in (19), by p
∗
= ( p
∗
1
, p
∗
2
, ··· , p
∗
N
u
). The optimal
sum rate is achieved by jointly maximizing r(H, P
T
, P, p)
over both P and p and is given by
C(H, P
T
) = max
P∈A
(g)
N
u
,p |
P
N
u
i=1
p
i
=P
T
r(H, P
T
, P, p)
= max
P∈A
(g)
N
u
r(H, P
T
, P). (20)
This optimization problem is inherently complex due to its
combinatorial nature. It has been observed that the achievable
sum rate is sensitive to the chosen grouping of users. This
observation therefore motivates us to consider optimal/near-
optimal methods for solving (20).
For small N
u
, (20) can be solved simply by brute-force
enumeration of all possible groupings. However, for large N
u
,
the combinatorial nature of the problem makes it inherently
complex to solve by brute-force enumeration. Towards this
end, in Section V we propose low-complexity approximations
to the solution of (20).
Here we also note that, the ZF precoder is a special case
of the proposed user grouping scheme with g = 1, i.e., N
u
groups with one user per group. The other special case is for
g = N
u
, i.e., only one group consisting of all the N
u
users.
We shall refer to this as the ZF-DPC precoding scheme and
has been discussed in detail in [11] as the ZF-DP precoder.
Note that with g = N
u
, successive DPC has to be performed
for N
u
users, which can be prohibitive for large N
u
. Further,
the number of possible ordered groupings is N
u
! which is also
large for large N
u
.
V. PARTITIONING USERS INTO GROUPS
In this section, we consider the optimization problem in (20)
and propose low-complexity near-optimal solutions to it. We
firstly show that irrespective of the channel realization H, the
sum rate achieved by the proposed precoder with any arbitrary
grouping (g ≥ 2) and the ZF power allocation, is greater than
that achieved with g = 1.
Theorem 1: Let P ∈ A
g
N
u
= {S
1
, S
2
, ··· , S
N
g
}, S
k
=
{U
k
1
, U
k
2
, ··· , U
k
g
} be any arbitrary user grouping with g >
1. Then
r(H, P
T
, P, p
∗
) =
N
u
/g
X
k=1
g
X
j=1
log
2
(1 + p
∗
k
j
R[k]
2
(j,j)
) ≥ C
ZF
.
(21)
is satisfied for any channel realization H. C
ZF
is the sum rate
achieved by the ZF precoder.
We are unable to present the proof here due to lack of
space. We note that Theorem 1 holds for any arbitrary user
grouping. Therefore optimizing r(H, P
T
, P, p
∗
) w.r.t. P is
expected to achieve a sum rate significantly greater than C
ZF
.
We therefore propose the following optimization problem.
P
∗
= arg max
P∈A
g
N
u
N
u
/g
X
k=1
g
X
j=1
log
2
(1 + p
∗
k
j
R[k]
2
(j,j)
). (22)
In the next section, we propose a low-complexity near-optimal
solution to the problem in (22).
A. Generalized User Grouping Algorithm - GUGA
For a given H and P
T
, the optimization problem in (22) can
be shown to be equivalent to a weighted matching problem
(WMP) on g-uniform hypergraphs. This correspondence can
be used to solve (22). However, the general WMP is known
to be NP-hard, and is therefore not realizable in polynomial
time. Hence several polynomial time approximation algo-
rithms have been proposed, using standard linear and semi-
definite programming techniques [14]. However even these
approximations are too complex for application to symbol
by symbol precoding. We therefore propose a low-complexity
(polynomial time) user grouping algorithm to the optimization
problem in (22), which is shown to achieve a sum rate greater
than 1/g of the sum rate achieved with the optimal grouping
P = P
∗
(P
∗
is given by (22))
6
.
Before discussing the algorithm in detail, we define the
weight of any given group of users as follows. Given the
k-th group of g users, represented by the ordered set S
k
=
{U
k
1
, U
k
2
, ··· , U
k
g
}, we define its weight to be
7
W(S
k
)
∆
=
g
X
j=1
log
2
(1 + p
∗
k
j
R[k]
2
(j,j)
). (23)
The optimization problem in (22) is therefore expressed in
terms of the weight function as
P
∗
= arg max
P∈A
g
N
u
N
u
/g
X
k=1
W(S
k
). (24)
6
Since, g is typically small (usually 2,3,4), 1/g is a significant fraction of
1.
7
We reminder the reader that R[k] is implicitly dependent on the chosen
grouping.
