IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 7, JULY 2009 1939
An Efficient Greedy Scheduler for Zero-Forcing Dirty-Paper Coding
Jisheng Dai, Student Member, IEEE, Chunqi Chang, Member, IEEE, Zhongfu Ye,
and Yeung Sam Hung, Senior Member, IEEE
Abstract—Inthispaper,anefficient greedy scheduler for zero-
forcing dirty-paper coding (ZF-DPC), which can be incorporated
in complex Householder QR factorization of the channel matrix,
is proposed. The ratio of the complexity of the proposed scheduler
to the complexity of the channel matrix factorization required
by ZF-DPC is O(M
−1
), while such ratio for the original greedy
scheduler is O(M ),whereM is the number of transmitters.
Therefore, the new scheduler reduces the overhead of scheduling
from being the bottleneck of ZF-DPC to being negligible.
Index Terms—Efficient greedy scheduler, multiple-input
multiple-output (MIMO), vector Gaussian broadcast channel
(GBC), zero-forcing dirty-paper coding (ZF-DPC), QR factor-
ization by Householder transformation.
I. INTRODUCTION
C
OMPARED to traditional single-input single-output
communication systems, multiple-input multiple-output
(MIMO) systems support greater data rate and higher re-
liability over wireless links [1], [2]. Recen tly, there has
been tremendous interest in MIMO multiuser systems with
multiple-a ntenna (vector) Gaussian broadcast channel (GBC),
for which the transmitters can cooperate in encoding their
signals but the receivers (users) are constrained to decode their
signals independently.
It is well known that dirty-paper coding (DPC) [3], which
encodes signals at the transmitter w ith known interference an d
constrained transmitting power, achieves the capacity region
of the multiple antenna GBC [4]–[6] for the case of flat-
fading channels with perfect chan nel state information (CSI)
at all transmitters. A practical coding strategy called zero-
forcing dirty-paper coding (ZF-DPC) is proposed in [7] to
approach the capacity. It is noticed in [7] that the sum rate
(or throughput) of the channels with ZF-DPC can be affected
by user ordering, and the maximal sum rate can be achieved
when the number of ordered users is equal to the rank of the
channel matrix. Hence, there is a need to find the optimal user
ordering that approaches the maximal sum rate. A sub-optimal
user ordering algorithm (scheduler) called greedy scheduling
Paper approv e d by A. Lozano, the Editor for Wi reless Network Access
and Performance of the IEEE Communications Society. Manuscript received
December 17, 2007; revised July 18, 2008 and October 7, 2008.
J. Dai and Z. Ye are with the Department of Electronic Engineering and
Information Science, University of Science and Technology of China, Hefei,
Anhui, 230027, P.R. China (e-mail: jisheng.dai@ieee.org; yezf@ustc.edu.cn).
Z. Ye is also with the National Mobile Communications Research Laboratory ,
Southeast University, P. R. China.
C. Chang and Y. S. Hung are with the Department of Electrical and
Electronic Engineering, The Univ e rsity of Hong Kong, Pokfulam Road, Hong
Kong (e-mail: {cqchang, yshung}@eee.hku.hk).
This work is supported by the open issues of National Mobile Communi-
cations Research Laboratory, Southeast University (No. A200509).
Digital Object Identifier 10.1109/TCOMM.2009.07.070162
is proposed in [8]. The performance of this scheduler and its
slightly modified versions are analyzed in [9].
However, the complexity of greedy scheduler proposed in
[8], [9] is very high if the number of transmit antennas is
large. As will be shown in this paper, the computing overhead
of th e scheduler is much larger than the computation of th e
channel matrix factorization itself required by ZF-DPC, the
ratio of the computational complexity of the overhead to that
of the channel matrix factorization being linearly proportional
to the number of selected users (which is chosen equal to the
number of transmitters). Therefore, fast scheduling algorithms
are d esired to reduce the computational load of the scheduler.
In this paper, an efficient algorithm for the greedy scheduler
[8], [9] is proposed, which renders the overhead of scheduling
in ZF-DPC to be negligible.
The rest of this paper is organized as follows. In Section II,
ZF-DPC and greedy ZF-DPC al gorithms are introduced. An
efficient greedy scheduler for near-optimal user ordering is
derived in Section III. Section IV evaluates the complexity of
different algorithms. Conclusions are drawn in Section V.
The following notations are used in this paper. The super-
scripts (·)
H
and . denote Hermitian operation a nd 2-n orm
of a vector, respectively. (·)
ij
denotes the (i, j)th element of
a matrix, and I denotes the identity matrix. A\B denotes the
set A excluding set B. C
n×m
is the n × m complex matrix
space.
II. ZF-DPC
AND GREEDY ZF-DPC ALGORITHM
A. ZF-DPC
Consider a N-user vector GBC with M (≤ N) trans-
mitters. The relationship between the received signals y =
[y
1
...y
N
]
T
and transmitted signals x =[x
1
...x
M
]
T
can be
written as
y =
⎡
⎢
⎣
h
H
1
.
.
.
h
H
N
⎤
⎥
⎦
x + n Hx + n, (1)
where h
H
i
∈ C
1×M
(i =1,...,N) is the channel vector
denoting the path gain from the transmit antennas to the
ith user, and n ∼ N(0,σ
2
n
I) is the cir cularly symmetric
complex Gaussian noise. The power-constrained dirty-paper
coding (DPC) can be implemented as follows. Assuming
rank(H)=K(≤ M ), we can perform a QR factorization
of (strictly speaking, the transpose of) H,sothat
H = GQ, (2)
where G ∈ C
N×K
is lower triangular and Q ∈ C
K×M
has
orthonormal rows.
0090-6778/09$25.00
c
2009 IEEE
1940 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 7, JULY 2009
We wish to transmit information to a number of users with
maximal sum rate (throughput) under the constraint of total
transmitting power P . It is shown in [7] that to achieve the
maximal sum rate we should transmit information to K users.
Let the information to be tran smitted to the u sers be u =
[u
1
...u
K
]
T
, then we can transform this information as x =
Q
H
u and transmit x through the channels of H.FromEq(1)
and (2) we have
y
k
= G
kk
u
k
+
j<k
G
kj
u
j
+ n
k
,k=1, 2, ··· ,K, (3)
while there is no information sent to users K +1,...,N.This
is called zero-forcing d irty-p aper coding (ZF-DPC) because
for the k-th user the signal of all other users with index greater
than k are forced to zero in the k-th channel. In this scheme, if
we encode the information in the order of u
1
,u
2
, ..., u
K
,then
for the k-th user the interference
j<k
G
kj
u
j
is noncausually
known at the transmitter, and according to the result of [6]
codewords can be designed in a way that the capacity of the
channel is the same as if
j<k
G
kj
u
j
,k =1, 2,...,K,were
not present.
The sum rate for this ZF-DPC program can be maximized
when the power is optimally allocated across the first K
channels. This maximal sum rate is determined in [7], [ 8]
as
R =
K
k=1
[log(ξd
k
)]
+
(4)
where d
k
= |G
kk
|
2
, [x]
+
max(x, 0),andξ is decided by
the following total power constraint
K
k=1
[ξ − 1/d
k
]
+
= P. (5)
B. Greedy ZF-DPC Algorithm
Notice that the sum rate determined by Eq (4) depends on
d
k
,k =1, 2, ··· ,K, which in turn are determined by the QR
factorization (2). If we permute the rows of the channel matrix
H by a permutation matr ix Π to ΠH, which is equivalent to
re-ordering the receivers (users), and perform ZF-DPC on the
permuted channel matrix ΠH, then we get a different sum
rate. This has been noted in [7]. Hence there is a need to find
an optimal user order, i.e. an optimal permutation matrix Π
o
,
so that ZF-DPC based on Π
o
H will give the maximal sum
rate.
Obviously it is not easy to solve this combinatorial op-
timization problem especially when N and K are large.
Therefore, in [8] a sub-optimal approach to th is optimal
user ordering (scheduling) problem called greedy ZF-DPC
scheduler is proposed. This algorithm finds the sub-optimal
permutation vector π =[π(1),...,π(K)] for the first K rows
of Π
o
, outlined as follows.
Greedy ZF-DPC Scheduler Algorithm
1) Initialization: Let K =0.
2) For k =1,...,M
Let ˜π
k
= {π(i)|i<k}.(˜π
1
= ∅)
Project all the h
H
j
,j ∈{1, ··· ,N}\˜π
k
, onto the
orthogonal complement of the subspace spanned
by {h
H
π(i)
|i<k}, and denote the 2-norm of these
projected vectors as γ
j
,j ∈{1,...,N}\˜π
k
.
Let k
∗
=argmax
j
γ
j
.
If γ
k
∗
=0
go to step 3
else
Let π(k)=k
∗
and K = K +1.
End
3) Output the scheduling represented by the permutation
vector π =[π(1),...,π(K)].
This algorithm also serves as the basis of the scheduler
employed in [9]–[11]. As we will see later, the computational
complexity o f this scheduler is of the order O(NM
3
) flops
(floating operations), while the computational complexity of
the factorization of the channel matrix H in Eq (2) is of
the order O(NM
2
). Therefore this scheduling algorithm is
the bottleneck in the computation of the ZF-DPC. In the
following, we propose an efficient scheduling algorithm to
substantially reduce its computational complexity.
III. E
FFICIENT GREEDY SCHEDULER FOR ZF-DPC
A. QR factorization of the channel matrix by Householder
transformation
Given two vectors a
H
, b
H
∈ C
1×m
with a = b,the
Householder transformation H(a
H
, b
H
) ∈ C
m×m
reflects a
H
to b
H
and is given by [12]
H(a
H
, b
H
)=I −
zz
H
a
H
z
= I +
zz
H
z
H
b
, (6)
where z = a − b. It can be readily checked that
a
H
H(a
H
, b
H
)=b
H
and H(a
H
, b
H
) is unitary.
The QR factorization of the channel matrix H in Eq (2) can
be implemented by means of the Householder transformation.
First we reflect h
H
1
to h
1
e
H
1
,wheree
H
1
=[10... 0],by
H(h
H
1
, h
1
e
H
1
),sothat
HH(h
H
1
, h
1
e
H
1
)=
h
1
0
× H
N−1
. (7)
Then we reflect
ˆ
h
H
1
,thefirst row of H
N−1
,to
ˆ
h
1
e
H
1
by
H(
ˆ
h
H
1
,
ˆ
h
1
e
H
1
),sothat
h
1
0
× H
N−1
1 0
0 H(
ˆ
h
H
1
,
ˆ
h
1
e
H
1
)
=
⎡
⎣
h
1
0 0
×
ˆ
h
1
0
××H
N−2
⎤
⎦
. (8)
Applying this procedure recursively to H
N−2
, we get after
at most M steps the QR factorization of H as in Eq (2).
Details of the Householder QR can be found in [13].
B. Efficient Greedy Scheduler
The greedy ZF-DPC as proposed in [8] consists of two
independent steps, namely the greedy scheduling and the
QR factorization. We note that if the QR factorization is
performed recursively by the Householder transformation de-
scribed above, then the scheduler can be incorporated very
DAI et al.: AN EFFICIENT GREEDY SCHEDULER FOR ZERO-FORCING DIRTY-PAPER CODING 1941
efficiently within the Householder QR factorization. The key
observations in our proposed method are:
• The subspace projection of step (2) of the greedy sched-
uler is automatically performed as part of the House-
holder QR and therefore to perform this scheduling
separately is a duplication of computational effort.
• The size o f the subspace projection problem gets progres-
sively smaller in the recursive procedure of the House-
holder QR factorization, and since the 2-norm of the
original projection is equal to the 2-norm of the projection
onto the space of progressively reduced dimension, the 2-
norm in turn can be inferred very efficiently in a recursive
manner.
According to the Greedy Scheduler Algorithm described
in the previous section, the first user selected is the one
whose channel has the maximal 2-norm, so we permute H
accordingly to Π
1
H with the first row associated to the
selected user, and we denote this channel as h
H
to simplify
the presentation. Applying Householder transformation to the
first row of Π
1
H gives
Π
1
H =
h 0
× H
N−1
Q
1
, (9)
where Q
H
1
is the unitary Householder transformation matrix
for the first row of Π
1
H.
We now proceed to select the (k+1)-th user by an induction
argument. Suppose the previous k users have already been
selected by the greedy scheduler, whereby H is permuted to
Π
k
H whose first k rows are associated with the selected k
users in an ordered way. Also assume that we have applied
Householder transformation to the first k rows of Π
k
H so
that
Π
k
H =
R0
XH
N−k
Q
k
, (10)
where R is a lower triangular matrix, and Q
k
is unitary since
Q
H
k
is a p roduct of k Householder transformation matrices.
Partitioning the matrices in Eq (10) accordingly, we have
ˆ
H
1
ˆ
H
2
=
R0
XH
N−k
Q
k,1
Q
k,2
. (11)
To select the (k +1)-th user in the greedy scheduling
algorithm, we need to project rows of
ˆ
H
2
onto the orthogonal
complement of the subspace spanned by rows of
ˆ
H
1
, and order
the users by the 2-norm of these projections. To show that the
greedy scheduling scheme can be incorporated into the QR
factorization, we need the following result.
Theorem 1: Let rows of
ˆ
H
p
2
be the pr ojection of cor re-
sponding rows of
ˆ
H
2
onto the orthogonal complement of
the subspace spanned by rows of
ˆ
H
1
,where
ˆ
H
1
and
ˆ
H
2
are
defined by Eq (11). Then the 2-norm of the rows of
ˆ
H
p
2
are
equal to the 2-norm of the corresponding rows of H
N−k
, i.e.,
diag{
ˆ
H
p
2
(
ˆ
H
p
2
)
H
} =diag{H
N−k
H
H
N−k
}.
Proof: Since
ˆ
H
1
= RQ
k,1
and R is nonsingular, the
row space of
ˆ
H
1
is the same as the row space of Q
k,1
.As
Q
k,2
Q
H
k,1
=0, the row space of Q
k,2
is just the orthogonal
complement of the row space of
ˆ
H
1
.Thenwehave
ˆ
H
p
2
=
ˆ
H
2
Q
H
k,2
Q
k,2
=(XQ
k,1
+ H
N−k
Q
k,2
)Q
H
k,2
Q
k,2
= H
N−k
Q
k,2
, (12)
ˆ
H
p
2
(
ˆ
H
p
2
)
H
= H
N−k
Q
k,2
Q
H
k,2
H
H
N−k
= H
N−k
H
H
N−k
. (13)
This proves that diag{
ˆ
H
p
2
(
ˆ
H
p
2
)
H
} =diag{H
N−k
H
H
N−k
}.
By Theorem 1, we can select the (k +1)-th user as the one
whose corresponding row in H
N−k
has the maximal 2-norm
among all the rows of H
N−k
, which is equivalent to Step 2
of the greedy ZF-DPC scheduler given in Section II-B. Then
we have the following algorithm of efficient ZF-DPC with
greedy scheduling.
Efficient ZF-DPC with Greedy Scheduling
1) Let K =0, H
N
= H,andG be a zero matrix with the
same size as H.
2) For k =1,...,M
Compute the 2-norm of the rows of H
N−(k−1)
.
Permute the rows of H
N−(k−1)
to H
∗
so that its
first row, denoted as h
H
, has the maximal 2-norm
among all its rows.
If h =0
go to step 3
else
Let H(h
H
, he
H
1
)=I −
h−he
1
h
H
(h−he
1
)
(h −
he
1
)
H
I − v
k
w
H
k
.
Perform the Householder transformation
H
∗
H(h
H
, he
H
1
)=
h 0
gH
N−k
. (14)
Define the kth column of G to be
⎡
⎣
0
(k−1)×1
h
g
⎤
⎦
.
Let K = K +1.
End
3) Output the lower triangular matrix G,andQ =
K
k=1
Q
k
,whereQ
k
=
I
k−1
0
0I− v
k
w
H
k
H
.We
note that Q needs not to be obtained explicitly since
it is used only for coding x = Q
H
u and this can be
calculated by using v
k
and w
k
, k =1,...,K, with
complexity of the same order as that of calculating Q
H
u
directly.
IV. C
OMPLEXITY EVA L UAT I O N
In practice, the number of users (receivers) N is always
much greater th an the number of transmitting antennas M,i.e.
N ≥ M. If the channels are not degenerate, which is assumed
in this sectio n, the rank of the channel matrix H is M so that
the number of selected users K = M. In the following, we
calculate the n umber of real multiplications and the n umber
of real additions separately, which are required by different
algorithm s. No te that one complex multiplication takes 4 real
1942 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 7, JULY 2009
multiplications and 2 real additions, and one complex additio n
needs 2 real additions. Unless otherwise specified, multiplica-
tions and additions refer to complex operations throughout this
section.
A. Complexity of Householder QR for channel matrix factor-
ization
For each iteration o f the QR factorization, we need to
calculate H
∗
H = H
∗
(I − v
k
w
H
k
)=H
∗
− (H
∗
v
k
)w
H
k
,
where the size of H
∗
is (N − k +1)× (M − k +1).To
calculate v
k
and w
k
we need 3(M −k+1) mu ltiplications and
2(M − k)+1 additions, and we need 2(N −k+1)(M −k +1)
multiplications an d (2M − 2k +1)(N − k +1) additions
for the remaining calculation. Therefore the number of real
multiplications needed in total is
C
×
QR
=4NM
2
−
4
3
M
3
+ O(NM), (15)
and the number of real additions is
C
+
QR
=4NM
2
−
4
3
M
3
+ O(NM). (16)
B. Complexity of Greedy ZF-DPC Scheduler
For each iteration, we need 2kM multiplications and
2kM − M − k additions for Gram-Schmidt orthogonalization,
kM(N −k +1) multiplications and kM(N −k)+M +k −N
additions for projection, and 2M (N − k) real multiplications
and (M − 1)(N − k)+M(N − k) real additions for 2-norm
calculation. Thus, the total number of real multiplications is
C
×
S
=2NM
3
−
4
3
M
4
+5M
3
+2NM
2
+ O(NM), (17)
and the total number of real additions is
C
+
S
=2NM
3
−
4
3
M
4
+4M
3
+2NM
2
+ O(NM). (18)
C. Complexity of Efficient Greedy Scheduler
In our efficient greedy scheduler, scheduling is performed
inside the QR factorization loop. The overhead of scheduling
is to compute the 2-norm of rows of H
N−k
at each iteration
k +1. From Eq (14) we have
H
∗
H
H
∗
=
h 0
gH
N−k
h g
H
0H
H
N−k
=
h
2
hg
H
hggg
H
+ H
N−k
H
H
N−k
. (19)
The diagonal of H
∗
H
H
∗
and diagonal of H
N−k
H
H
N−k
are
the squared 2-norm of rows of H
∗
and rows of H
N−k
,
respectively. So the squared 2-norm of i-th row of H
N−k
can
be calculated as
(H
N−k
H
H
N−k
)
ii
=(H
∗
H
H
∗
)
ii
−|g
i
|
2
, (20)
where g
i
is the i-th entry of g.Since(H
∗
H
H
∗
)
ii
has
been calculated in the previous step, only 2 real multipli-
cations and 2 real additions are needed for the calculation
of (H
N−k
H
H
N−k
)
ii
. The initial calculation of the squared
2-norm of rows of H requires 2NM real multiplications
and N(M − 1) + NM real additions, so the additional real
multiplications and real additions needed in our proposed
scheduling are
C
×
ES
=2NM +
M−1
k=1
2(N − k)+O(N )
=4MN − M
2
+ O(N), (21)
C
+
ES
= N (M − 1) + NM +
M−1
k=1
2(N − k)+O(N )
=4MN − M
2
+ O(N). (22)
Let α = M/N, which tends to 0 when N>>M,thenwe
have
C
ES
C
QR
=
1 − α/4
1 − α/3
M
−1
+ O(M
−2
), (23)
C
S
C
QR
=
1 − 2α/3
2 − 2α/3
M + O(M
0
). (24)
Note that the superscript (“ × ” or “+”) is dropped because
of the approximately equal number of multiplications and
additions in each algorithm.
By using our proposed efficient greedy scheduling algo-
rithm, we can thus reduce the overhead of scheduling from
being the bottle neck of ZF-DPC to being negligible.
V. C
ONCLUSIONS
Zero-forcing dirty paper coding (ZF-DPC) is used in vector
Gaussian broadcast channel. To maximize the sum rate, a
greedy scheduling is proposed in [8] to get a sub-optimal
user ordering. In this paper we propose an efficient algorithm
for this greedy scheduler by doing the scheduling within the
QR factorization of the channel matrix, which is a necessary
step in ZF-DPC. Thus, for a N receiver M transmitter
system the computational complexity of scheduling is reduced
from O(NM
3
) to O(NM), i.e., fr om being the bottleneck
of ZF-DPC to being negligible. Our alg orithm can also be
employed by related greedy schedulers described in [9]–[11],
but the computational complexity should be recalculated if the
number of selected users is 1 ≤ K<M.
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