Low-Complexity Precoding for Sum Rate Maximization in Downlink Massive MIMO Systems

Abstract

We propose a precoding scheme to improve the downlink sum rate for a multicell massive multiple-input multiple-output (MIMO) system. We first present a lowcomplexity approach based on dirty paper coding and zeroforcing that combines a reduced form of QR decomposition and an orthogonal projection. We formulate a downlink sum rate optimization problem that takes both intracell and intercell interference into account, and then we use the convex conjugate to transform the problem into an unconstrained dual problem to find an optimal solution by applying a quasi-Newton algorithm with low complexity per iteration. We prove that the proposed algorithm exhibits faster convergence than other methods, and the numerical results verify that the proposed precoding design outperforms conventional precoding methods in multicell massive MIMO systems.

Full text

1
Low-Complexity Precoding for Sum Rate
Maximization in Downlink Massive MIMO Systems
Hieu V. Nguyen, Van-Dinh Nguyen, and Oh-Soon Shin
Abstract—We propose a precoding scheme to improve the
downlink sum rate for a multicell massive multiple-input
multiple-output (MIMO) system. We first present a low-
complexity approach based on dirty paper coding and zero-
forcing that combines a reduced form of QR decomposition
and an orthogonal projection. We formulate a downlink sum
rate optimization problem that takes both intracell and intercell
interference into account, and then we use the convex conjugate
to transform the problem into an unconstrained dual problem to
find an optimal solution by applying a quasi-Newton algorithm
with low complexity per iteration. We prove that the proposed
algorithm exhibits faster convergence than other methods, and
the numerical results verify that the proposed precoding design
outperforms conventional precoding methods in multicell massive
MIMO systems.
Index Terms—Dirty paper coding, massive MIMO, optimiza-
tion, precoding, zero-forcing.
I. INTRODUCTION
Massive multiple-input multiple-output (MIMO) has been
recently considered as a promising technology for next-
generation wireless systems [1], [2]. In particular, [2] showed
that the achievable rate of a massive MIMO system can nearly
reach the channel capacity even without intercell cooperation,
provided that a sufficiently large number of antennas are
employed at the base station (BS). As in traditional MIMO
systems, precoding at the BS plays an important role in
determining the downlink achievable rate in massive MIMO
systems. Most previous works have adopted linear precoding
schemes for downlink transmissions, including the maximum
ratio combining/maximum ratio transmission (MRC/MRT),
zero-forcing (ZF), and minimum mean square error (MMSE)
[3], [4], because linear precoding schemes can be implemented
with low computational complexity while also providing quite
good performance if the BS has a very large number of
antennas. In practice, however, the number of antennas is
limited, so linear precoding cannot fully exploit the capability
of massive antenna arrays.
To improve the throughput of a MIMO system, previous
works applied a combination of ZF and dirty paper coding
(DPC), referred to as successive zero-forcing dirty paper
coding (SZF-DPC), as originally proposed in [5] instead of
non-linear DPC with very high complexity in [6]. Accordingly,
recently published works optimize the downlink transmissions
[7], [8], and in particular, [8] designed a precoder based
on SZF-DPC to maximize the sum rate using an iterative
Newton’s method, which has a computational complexity
of O(KM 3), with Kas the number of users and Mas
the number antennas at the BS. This complexity might be
The authors are with the School of Electronic Engineering, Soongsil
University, Seoul 06978, Korea (email: {hieuvnguyen, nguyenvandinh, os-
shin}@ssu.ac.kr).
acceptable in a traditional MIMO system with only a few
antennas. However, the high level of complexity, the cube of
M, is likely a barrier to realize massive MIMO systems with
hundreds or thousands of antennas. Moreover, previous works
have only considered a single-cell system while ignoring
intercell interference. This motivates us to develop a low-
complexity precoding scheme suitable for multicell massive
MIMO systems.
In this letter, we design a low-complexity precoding method
for downlink transmission in multicell massive MIMO sys-
tems. This method is based on the SZF-DPC approach, and
we first formulate an achievable sum rate maximization prob-
lem with a sum power constraint. In particular, we exploit
economy-size QR decomposition and orthogonal projection
to eliminate both the intracell and intercell interference. As
a result, the computational complexity can be as low as
O(KM 2), which stands in contrast with the O(KM 3)com-
plexity of the scheme in [8]. We then derive a dual problem
by applying the convex conjugate to the original problem
to significantly reduce the number of variables. The optimal
solution of the dual problem is found using an iterative quasi-
Newton method we developed known as the Broyden-Fletcher-
Goldfarb-Shanno (BFGS) algorithm. Only two iterations are
required for the BFGS algorithm, no matter how the parame-
ters change. Interestingly, the complexity for each iteration is
O(1) when using the BFGS algorithm to compute a scalar
expression and O(K)when using the typical water-filling
method to solve a log-sum problem [9].
Notation: (·)Tand (·)Hrespectively denote the transpose
and the Hermitian transpose of a vector or matrix. (·)−1and
tr(·)are respectively the inverse and the trace of a matrix.
diag(x)denotes a diagonal matrix whose main-diagonal en-
tities are determined by a vector x.[x]istands for the i-th
element of vector x; and xy, where xand yare the same-
size vectors, denotes [x]i>[y]i,∀i.1Tand 1respectively
denote a row vector and a column vector with all elements
equal to 1.0Tand 0are similarly defined. The terms inf and
sup represent the infimum and the supremum, respectively.
II. SY ST EM MO DE L
We consider a multicell system comprised of Lcells with
each BS equipped with Mantennas serving Ksingle-antenna
users. The channel between the BS in the `-th cell and the k-
th user in the j-th cell is modeled as g`,jk =pβ`,j kh`,j k,
where β`,jk stands for the path-loss and shadowing, and
h`,jk ∈C1×Mrepresents the channel vector between M
antennas at the BS in the `-th cell and the k-th user in the j-th
cell. The entries in h`,jk are assumed to follow independent
and identically distributed (i.i.d.) CN(0,1). The BS of the `-
th cell broadcasts the information symbol vector s`to all K

2
users in the cell. Let s`=s`1s`2. . . s`K Tand assume that
Es`= 0 and Es`sH
`=IK.
Before transmitting the signal to the k-th user in the `-th
cell, the BS multiplies the information signal with a precoding
vector w`k ∈CM×1. The signal received at the k-th user in
the `-th cell is corrupted by intercell interference from the
signal of the BS of the j-th cell (j6=`). Therefore, the signal
received at the k-th user in the `-th cell can be written as
x`k =√pdg`,`kw`k s`k +√pdXL
j=1
j6=`XK
i=1 gj,`kwj isj i +z`k
+√pdXK
i=1
i<k
g`,`kw`i s`i +√pdXK
i=1
i>k
g`,`kw`i s`i,(1)
where pddenotes the transmit power at the BS and the
additive noise z`k is assumed to follow i.i.d. CN(0,1). The
precoding column vector w`k satisfies the power constraint
tr WH
`W`= 1 where W`,[w`1w`2···w`K ]∈CM×K.
As discussed in [7], [8], the intracell interference component
√pdPK
i=1,i<k g`,`kw`i s`i is considered to be noncausally
known to realize DPC precoding, and thus it is absolutely
eradicated from the received signal at the k-th user in the `-th
cell. Accordingly, the downlink achievable rate for all cells in
system is given as
RDPC =XL
`=1 XK
k=1 log21 + pd|g`,`kw`k |2
Iinter
`k +Iintra
`k + 1,(2)
where the intercell and the intracell interference terms are
defined as Iinter
`k ,pdPL
j=1,j6=`PK
i=1 |gj,`kwj i|2and Iintra
`k ,
pdPK
i=1,i>k |g`,`kw`i |2, respectively.
III. PROP OS ED PRECODING DESIGN
The sum rate in (2) is obviously nonconvex with respect
to the precoding vector w`k, and thus it is not easy to
solve a sum rate maximization problem. To deal with this,
the precoding matrix in `-th cell is designed to contain a
component orthogonal to the channel vectors from the BS in
the `-th cell to all users in the j-th cell (j6=`), so that the
signal from the `-th is eliminated at the users in j-th cell. The
orthogonal component is also incorporated into the precoding
design based on SZF-DPC to remove the intracell interference
as well. As a result, the sum rate maximization problem for
precoding design under the zero-forcing constraints and a sum
power constraint is derived from (2) as
maximize
w`k XK
k=1 log21 + pd|g`,`k w`k|2,(3a)
subject to g`,ji w`k = 0,∀j∈ IL\{`},and ∀i∈ IK,
(3b)
g`,`iw`k = 0,∀i < k, (3c)
tr WH
`W`≤1,(3d)
where INdenotes an index set defined as IN,
{1,2,·· · , N }, with Nas an arbitrary natural number. The
constraints (3b) and (3c) represent the conditions to respec-
tively eliminate the intercell and intracell interference while
(3d) denotes the sum power constraint at the `-th cell. Herein,
as compared to (2) the roles of the subscript iand kin
(3c) are interchanged to be convenient for expressing and
solving the problem. Note that (3) is concave and thus can
be solved using standard convex packages. However, the
computational complexity may be very large since the convex
tools cannot take the special features of the optimization
problem into account. This motivates us to develop an efficient,
low-complexity algorithm to solve (3) while preserving the
optimality of the solution.
Let G`,j =gH
`,j1···gH
`,jK H∈CK×M,∀j∈ IL.
Constraint (3b) can then be transformed into matrix form
as GH
−`W`=0, where G−`∈CM×(L−1)Kis defined as
G−`,GH
`,1·· · GH
`,(`−1)GH
`,(`+1) . . . GH
`,L. The constraint
(3b) can be reduced by constructing W`=PG⊥
−`
¯
W`,
where PG⊥
−`is an orthogonal projection onto G⊥
−`, which
is computed as PG⊥
−`=IM−G−`GH
−`G−`−1GH
−`. As a
result, constraint (3b) is always satisfied no matter how ¯
W`
has been chosen since GH
−`W`=GH
−`PG⊥
−`
¯
W`=0. Note
that each column of the matrix G−`and the product PG⊥
−`
¯
W`
correspond to g`,ji and w`k in (3b), respectively.
Now we define ¯
G`,` ,G`,`PG⊥
−`and express the QR
decomposition of ¯
G`,` as
¯
G`,` =Y`B`.(4)
By applying the Gram-Schmidt procedure to the rows of ¯
G`,`,
Y`∈CK×Kbecomes a lower triangular matrix, and B`∈
CK×Mhas Kpairwise orthogonal rows. It is implicit that
B`BH
`=IKbut BH
`B`6=IM. We construct ¯
W`=BH
`Ω`,
where Ω`= diagω`1··· ω`K , and ω`k is an optimization
variable. Accordingly, the precoding matrix is expressed as
W`=PG⊥
−`BH
`Ω`.(5)
As a result, constraint (3c) is always satisfied since G`,`W`=
¯
G`,`BH
`Ω`=Y`B`BH
`Ω`=Y`Ω`. Actually, the product
Y`Ω`is a lower triangular matrix where each upper diagonal
entry is equivalent to g`,`iw`k = 0 for ∀i<k. Meanwhile,
g`,`kw`k =Y`,k kω`k , where Y`,kk is the entry on the k-th row
and on the k-th column of Y`.
We define a Hermitian matrix as T`,B`PG⊥
−`BH
`.
From (5) and the relationship PH
G⊥
−`
=PG⊥
−`=P2
G⊥
−`
,
the left-hand side of the constraint equation (3d) can be
written as trWH
`W`= trT`Ω`ΩH
`. Let a vector c`∈
C1×Kcomprise the main diagonal entries of T`, and ¯ω`,
ω2
`1. . . ω2
`k . . . ω2
`K Tbe a vector of optimization variables.
We also define ¯y`,pdY2
`,11 . . . Y 2
`,kk . . . Y 2
`,KK . Then, the
optimization problem (3) becomes equivalent to
maximize
¯ω`0XK
k=1 log21 + ¯y`k¯ω`k(6a)
subject to c`¯ω`= 1.(6b)
Note that (6) can be easily solved with a K-length vector
variable ¯ω`using a water-filling algorithm. However, the
best water-filling implementation takes O(K)per iteration
[9]. Therefore, we propose an approach that reduces the
complexity per iteration and requires a fewer numbers of
iterations by first describing the following proposition.
Proposition 1: The Lagrangian Lassociated with problem
(6) can be defined as
L(Υ`, ν`) = f(Υ`−1) + ν`¯c`Υ`−ρ`,
where f(Υ`),−PK
k=1 log2(υ`k )and ν`is the dual
variable associated with problem (6). An alternative variable
υ`k ,1 + ¯y`k¯ω`k>1is introduced to define Υ`,
[υ`1υ`2. . . υ`K ]T1. Let ¯
Y`=diag¯y`−1,¯c`,c`¯
Y`,
and ρ`,c`¯
Y`1+ 1, then the Lagrange dual problem in (6)

3
Algorithm 1: Proposed algorithm to solve (7)
Input: Initial points ν`= 1, tolerance = 10−5, and step
size t= 1. Let φ=−∇ν`˜g(ν`),Θ=1,and φ0= 0.
Repeat:
1: Calculate φ=φ−φ0and ∆ν`= Θφ.
2: Exit the loop if kφk ≤ .
3: Update ν`,φ0and Θin the following order:
ν`←ν`+t∆ν`;φ0←φ+∇ν`˜g(ν`); and
Θ←result in (10)
Output: The dual optimal value ν`.
can be given as
max
ν`
g(ν`) = −ρ`ν`+ν`¯c`1+K+
K
X
k=1
log2ν`¯cT
`k.(7)
Finally, the solution for (6) can be found from the optimal
dual variable ν∗
`in (7) given as
¯ω`=¯
Y`diagν∗
`¯cT
`−11.(8)
Proof: See Appendix A.
Obviously, the maximization problem in (7) is concave. An
iterative algorithm can then be applied to find the solution.
We adopt the BFGS algorithm, which is a quasi-Newton
method, for the following reasons. (i) The BFGS algorithm
approximates the inverse of the Hessian matrix to lower the
complexity per iteration, and thus it is suitable for use in real-
time implementations [10]. (ii) By setting a special initial point
to the dual variable, the BFGS algorithm can find the optimal
solution with a limited number of iterations.
Let Θbe the approximation of the inverse of the Hessian
matrix. The gradient of the objective function ˜g(ν`) = −g(ν`)
is given as
∇ν`˜g(ν`) = ρ`−1T¯cT
`−K
ν`ln 2 .(9)
In the i-th iteration, the algorithm works as follows.
1) Calculate ∆ν(i)
`=−Θ(i)∇ν`˜gν(i)
`.
2) Stop if k∇ν`˜gν(i)
`k ≤ .
3) Update the optimal dual value as ν(i+1)
`=ν(i)
`+ ∆ν(i)
`.
4) Calculate φ0=∇ν`˜gν(i+1)
`− ∇ν`˜gν(i)
`.
5) Calculate Θ(i+1) given in [10] as
Θ(i+1) = Θ(i)−Θ(i)φ0∆ν(i)
`T+ ∆ν(i)
`(φ0)TΘ(i)
∆ν(i)
`Tφ0
+∆ν(i)
`Tφ0+ (φ0)TΘ(i)φ0∆ν(i)
`∆ν(i)
`T
∆ν(i)
`Tφ02.(10)
6) Go back to Step 1).
The above loop still requires the computation of gradients
for different values of ν`. To overcome this limitation, we
present a modified version of the original BFGS algorithm
in Algorithm 1. At the beginning, we set φ=−∇ν`˜g(ν`),
Θ=1, and φ0= 0. The update step calls for the gradient only
once, and the first step of the next iteration reuses the stored
values of φand φ0to compute −∇ν`˜g(ν`). This algorithm is
also applicable to the general case of multiple dual variables.
The dual problem in (7) is used to reduce the optimization
variable to a scalar ν`instead of a K-length vector ¯ω`in
(6). As a result, the complexity per iteration for the iterative
algorithm is only O(1). Furthermore, the proposed method can
200 300 400 500 600 700 800 900 1000
0.6
0.8
1
1.2
1.4
1.6
Number of Antennas at BS, M
(a)
Sum Throughput (Gbps)
MRT ZF MMSE Proposed DPC
0 4 8 12 16
0
0.2
0.4
0.6
0.8
1
pd= 15 dBm
Per-User Rate (bps/Hz)
(b)
Cumulative Distribution
0 4 8 12 16 20
0
0.2
0.4
0.6
0.8
1
pd= 30 dBm
Per-User Rate (bps/Hz)
(c)
Fig. 1. Comparison of the performance of the proposed design with those
of conventional schemes. (a) Sum rate per cell versus the number of antennas
M. (b) The cumulative distribution of per-user rate with pd= 15 dBm. (c)
The cumulative distribution of per-user rate with pd= 30 dBm.
2 4 6 8 10 12 14
10−5
10−4
10−3
10−2
10−1
100
101
102
Number of Iterations
Error Tolerance
K= 10
5 10 15 20
10−5
10−4
10−3
10−2
10−1
100
101
102
103
Number of Iterations
K= 20
CG Water-filling Newton’s method Proposed
Fig. 2. Convergence behavior for K= 10 and 20.
obtain an optimal solution within two iterations, no matter how
the other parameters change. The proof is given in Appendix
B. The total complexity of proposed method is determined
by computing PG⊥
`, QR decomposition in (4), and precoding
matrix in (5), taking (L+ 3)KM 2+ ((3(L−1)2+ 1)K2+
K+ 1)M+ (L−1)3K3flops. And thus, it is approximated
as O(KM 2), which is far less than O(KM 3)for single-cell
precoding based on Newton’s method in [8], especially for a
large M.
IV. NUMERICAL RES ULT S
We consider a hexagonal-cell system in which a center
cell is surrounded by 6 neighboring cells, L= 7. Each BS
is assumed to be located at the center of the cell, and the
bandwidth is set to 20 MHz with a power spectral density
of additive noise assumed to be −174 dBm/Hz. We consider
two values for the transmit power pdof 15 dBm and 30
dBm, according to 3GPP TR 36.942 v.9.0.1, with a 46-dBm
maximum transmit power for the 20 MHz bandwidth. We first
examine the case where the BS in each cell serves K= 10
users. Fig. 1(a) depicts the sum rate per cell versus the number
of antennas M. The proposed design is shown to provide better
performance than conventional precoding schemes, i.e., MRT,
ZF, and MMSE1. As compared to DPC in [6], the rate loss
when using our method significantly decreases as Mincreases,
which was proved in [11, Theorem 2]. Figs. 1(b) and 1(c) show
the cumulative distribution of the per-user rate with M= 500,
when pd= 15 dBm and 30 dBm, respectively. The proposed
design is seen to outperform the others, and the gain increases
with pd.
1Note that only linear precoding was considered in previous works in the
context of multicell massive MIMO systems. This is the reason why we
compare the sum throughput of the proposed scheme only with that of linear
precoding schemes.

4
TABLE I
COMPARISON OF CONVERGENCE AND COMPLEXITY
XXXXXXXXXX
X
Methods
Convergence &
Complexity Number of
Iterations
Complexity per
Iteration
Water-filling 8 O(K)
Conjugate gradient KO(1)
Newton’s 7 O(1)
BFGS 2 O(1)
Fig. 2 compares the convergence behavior of the proposed
algorithm with other three algorithms: water-filling [9], conju-
gate gradient (CG) [12], and Newton’s method [8] for different
values of K. We capture the most typical cases of ten thousand
random loops. Overall, the convergence rate of CG linearly
changes as the number of users, while that of the others
does not change with the number of users. The proposed
algorithm is shown to provide excellent convergence behavior;
it converges in two iterations as discussed in Section III.
In particular, although the complexity per iteration of the
proposed approach is comparable to Newton’s method and CG,
it converges in far less number of iterations, as summarized
in Table I.
V. CONCLUSIONS
This letter presents a precoding design based on the SZF-
DPC for a multicell massive MIMO system. We have derived a
dual problem associated with the sum rate maximization prob-
lem by using the QR decomposition and convex conjugate.
We developed a modified BFGS algorithm to find the optimal
solution with a reduced complexity, and numerical results
were presented here to verify that the proposed precoding
approach outperforms the previously proposed schemes while
also providing low complexity with fast convergence.
APPENDIX A
THE LAG RA NG E DUAL PROB LE M
First, let υ`k ,1 + ¯y`k¯ω`kbe an alternative variable,
then Υ`,[υ`1υ`2·· · υ`K ]T1since ¯ω`0. Therefore,
¯ω`and Υ`are related as
¯ω`=¯
Y`Υ`−1,(11)
where ¯
Y`=diag¯y`−1. By substituting (11) into (6), the
optimization problem is written as
minimize
Υ`1fΥ`=−XK
k=1 log2υ`k (12a)
subject to ¯c`Υ`=ρ`.(12b)
To derive a dual problem, we consider the Lagrangian asso-
ciated with (12) as LΥ`, ν`=f(Υ`−1)+ ν`(¯c`Υ`−ρ`),
where f(Υ`−1)instead of f(Υ`)is used so that constraint
Υ`1is included in the Lagrangian LΥ`, ν`. The La-
grange dual function associated with the optimization problem
(6) is expressed as
g(ν`) = inf
Υ`L(Υ`, ν`) = inf
Υ`f(Υ`−1) + ν`(¯c`Υ`−ρ`)
=−ρ`ν`−f∗−ν`¯cT
`+ν`¯c`1.(13)
Note that the convex conjugate of a shifted function
f(Υ`−1)is given by f∗−ν`¯cT
`−ν`¯c`1. As in
[13], the Legendre transform of −log (x)is given by
−(1 + log (−x∗)), where x∗belongs to the dual space of a
real vector space, which contains x. Accordingly, the convex
conjugate f∗(−ν`¯cT
`)is expressed as f∗(−ν`¯cT
`) = −K−
PK
k=1 log2ν`¯cT
`k. By applying this to (13), we obtain
the dual problem given in (7). By solving the dual problem,
we get the optimal dual variable ν∗
`. Then, the optimal
solution for problem (6) is computed by solving the dual
feasibility equation via the Karush-Kuhn-Tucker condition for
L(Υ`, ν`), which is given as ∇Υ`f(Υ`−1) + ν∗
`¯cT
`=0
where ∇Υ`f(Υ`)is the gradient of f(Υ`). The equivalent
equation is given as
υ`k =1
ln 2 ν∗
`¯cT
`k
+ 1.(14)
By substituting (14) into (11), the optimal solution ¯ω`of (6)
is derived as (8).
APPENDIX B
CONVERGENCE OF THE PROP OS ED ALGORITHM
With a scalar dual variable, the approximated inverse of the
Hessian Θis initially set to 1, which makes the first iteration
of the BFGS equivalent to a gradient descent. Interestingly,
the scalar ν`is simply set to ν(0)
`= 1. From (9), the gradient
of ˜g(ν(0)
`)is ∇ν`˜g(ν(0)
`) = ξ`−K
ln 2 , where ξ`=ρ`−1T¯
cT
`.
In the next iteration, the value of ν(1)
`is updated as ν(1)
`=
ν(0)
`− ∇ν˜g(ν(0)
`) = 1 −ξ`+K
ln 2 . Therefore, the gradient in
the second iteration becomes
∇ν˜g(ν(1)
`) = ξ`−1K−ξ`ln 2
ln 2 −ξ`ln 2 + K.(15)
With ρ`and c`defined in Proposition 1, ξ`= 1 is posed in
the optimization problem. Consequently, ∇ν˜g(ν(1)
`)in (15) is
equal to 0, which is an exit condition of the loop.
REFERENCES
[1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta,
O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and
challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30,
no. 1, pp. 40–60, Jan. 2013.
[2] T. L. Marzetta, “Noncooperative cellular wireless with unlimited num-
bers of base station antennas,” IEEE Trans. Wireless Commun., vol. 9,
no. 11, pp. 3590–3600, Nov. 2010.
[3] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot
contamination and precoding in multi-cell TDD systems,” IEEE Trans.
Wireless Commun., vol. 10, no. 8, pp. 2640–2651, Aug. 2011.
[4] J. Choi, “Massive MIMO with joint power control,” IEEE Wireless
Commun. Lett., vol. 3, no. 4, pp. 329–332, Aug. 2014.
[5] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna
Gaussian broadcast channel,” IEEE Trans. Inform. Theory, vol. 49, no. 7,
pp. 1691–1706, July 2003.
[6] N. Jindal, W. Rhee, S. Vishwanath, S. A. Jafar, and A. Goldsmith,
“Sum power iterative water-filling for multi-antenna Gaussian broadcast
channels,” IEEE Trans. Inform. Theory, vol. 51, no. 4, pp. 1570–1580,
Apr. 2005.
[7] A. D. Dabbagh and D. J. Love, “Precoding for multiple antenna Gaussian
broadcast channels with successive zero-forcing,” IEEE Trans. Signal
Process., vol. 55, no. 7, pp. 3837–3850, July 2007.
[8] L.-N. Tran, M. Juntti, M. Bengtsson, and B. Ottersten, “Beamformer
designs for MISO broadcast channels with zero-forcing dirty paper
coding,” IEEE Trans. Wireless Commun., vol. 12, no. 3, pp. 1173–1185,
Mar. 2013.
[9] S. Khakurel, C. Leung, and T. Le-Ngoc, “A generalized water-filling
algorithm with linear complexity and finite convergence time,” IEEE
Wireless Commun. Lett., vol. 3, no. 2, pp. 225–228, Apr. 2014.
[10] A. M. Nezhad, R. A. Shandiz, and A. E. Jahromi, “A particle swarm-
BFGS algorithm for nonlinear programming problems,” Comput. Oper.
Res., vol. 40, no. 4, pp. 963–972, Apr. 2013.
[11] J. Lee and N. Jindal, “High SNR analysis for MIMO broadcast channels:
Dirty paper coding versus linear precoding,” IEEE Trans. Inform.
Theory, vol. 53, no. 12, pp. 4787–4792, Dec 2007.
[12] S. D. Gray, “Multipath reduction using constant modulus conjugate
gradient techniques,” IEEE J. Select. Areas Commun., vol. 10, no. 8,
pp. 1300–1305, Oct. 1992.
[13] S. Boyd and L. Vandenberghe, Convex Optimization. New York, NY,
USA: Cambridge Univ. Press, 2004.