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Equivalent Expressions and Performance Analysis of SLNR Precoding Schemes: A Generalisation to Multi-Antenna Receivers

13 May 2013-IEEE Communications Letters (Institute of Electrical and Electronics Engineers (IEEE))-Vol. 17, Iss: 6, pp 1196-1199

TL;DR: It is shown that the SL NR scheme can be viewed as a generalised channel regularisation technique and the conditions for an equivalence between the SLNR, the Regularised Block Diagonalisation (RBD) and the Generalised MMSE Channel Inversion (GMI method 2) schemes are given.
Abstract: In this letter, equivalent expressions of transmit precoding solutions based on the maximum signal-to-leakage-plus-noise ratio (SLNR) are derived for multiuser MIMO systems with multi-antenna receivers. The performance of the SLNR precoding scheme is also analysed based on this equivalent form. Further, it is shown that the SLNR scheme can be viewed as a generalised channel regularisation technique and the conditions for an equivalence between the SLNR, the Regularised Block Diagonalisation (RBD) and the Generalised MMSE Channel Inversion (GMI method 2) schemes are given. Consequently, the performance analysis in this letter can be extended to the RBD and GMI schemes. This generalises the equivalence between the SLNR and MMSE schemes and its useful implications, from the case of single-antenna to multi-antenna receivers.
Topics: Precoding (55%)

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Patcharamaneepakorn, P., Doufexi, A., & Armour, S. M. D. (2013).
Equivalent expressions and performance analysis of SLNR precoding
schemes: a generalisation to multi-antenna receivers.
IEEE
Communications Letters
,
17
(6), 1196-1199.
https://doi.org/10.1109/LCOMM.2013.050313.130549
Peer reviewed version
Link to published version (if available):
10.1109/LCOMM.2013.050313.130549
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IEEE COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION 1
Equivalent Expressions and Performance Analysis of SLNR Precoding
Schemes: A Generalisation to Multi-antenna Receivers
Piya Patcharamaneepakorn, Angela Doufexi, Member, IEEE, and Simon Armour
Abstract—In this letter, equivalent expressions of transmit
precoding solutions based on the maximum signal-to-leakage-
plus-noise ratio (SLNR) are derived for multiuser MIMO systems
with multi-antenna receivers. The performance of the SLNR
precoding scheme is also analysed based on this equivalent form.
Further, it is shown that the SLNR scheme can be viewed as a
generalised channel regularisation technique and the conditions
for an equivalence between the SLNR, the Regularised Block
Diagonalisation (RBD) and the Generalised MMSE Channel
Inversion (GMI method 2) schemes are given. Consequently, the
performance analysis in this letter can be extended to the RBD
and GMI schemes. This generalises the equivalence between the
SLNR and MMSE schemes and its useful implications, from the
case of single-antenna to multi-antenna receivers.
Index Terms—Multiuser MIMO, linear precoding, SLNR,
RBD, GMI, equivalent forms, performance analysis
I. INTRODUCTION
I
N multiuser multiple-input multiple-output (MU-MIMO)
systems, linear precoding schemes are often considered for
practical implementations as they offer near-optimal perfor-
mance with relatively low computational complexity. Linear
precoding techniques, such as Zero-Forcing (ZF) [1] and
Minimum Mean Square Error (MMSE) [1], [2], are initially
developed for systems with single-antenna receivers. They
are later extended to the case of multi-antenna receivers,
e.g. ZF is developed into Block Diagonalisation (BD) [3]
precoding, and MMSE is extended to Regularised Block
Diagonalisation (RBD) [4] and Generalised MMSE Channel
Inversion (GMI) [5] schemes. A precoding scheme based on
the maximum signal-to-leakage-plus-noise ratio (SLNR) [6]
is another attractive technique which provides an alternative
approach to the signal-to-interference-plus-noise ratio (SINR)
maximisation problem and supports systems with both single-
antenna and multi-antenna receivers.
In this letter, equivalent solutions of the SLNR-based
precoding scheme with multi-antenna receivers are derived
and are shown to be a generalised form of the regularised
channel inversion, with regularisation factors for each user
inversely proportional to their average signal-to-noise ratios
(SNR) per data stream. Conditions for the equivalence between
the SLNR, RBD and GMI-2 are also presented. Moreover,
the performance of the SLNR scheme is analysed and can
be extended to the RBD and GMI-2 schemes, due to their
equivalence. This generalises the equivalence between the
SLNR and MMSE schemes proven in [7], [8] and establishes
the extension of its implications from the case of single-
antenna to multi-antenna receivers.
Manuscript received March 12, 2013. The associate editor coordinating the
review of this letter and approving it for publication was Z. Z. Lei.
The authors are with the Centre for Communications Research, Uni-
versity of Bristol, Bristol, BS8 1UB, UK (e-mail: {eezpp, A.Doufexi,
Simon.Armour}@bristol.ac.uk).
Digital Object Identifier 00.0000/LCOMM.0000.000000.000000
II. SYSTEM MODEL AND SLNR PRECODING SCHEMES
Consider a single-cell single-carrier downlink MU-MIMO
system with M transmit antennas at the base station (BS)
and K users, each with N
k
receive antennas. Each user ks
channel matrix H
k
C
N
k
×M
, assumed to have independent
and identically distributed (i.i.d.) entries, is known at the BS.
Each user k transmits B
k
r
k
= rank(H
k
) = min(N
k
, M)
data streams. The transmitted signal at the BS is expressed as
x = WAs =
P
k
W
k
A
k
s
k
. The vector s = [s
T
1
, s
T
2
, ..., s
T
K
]
T
denotes the overall data vector, where s
k
C
B
k
and
E{ss
H
} = I. W = [W
1
, W
2
, ..., W
K
] is the transmit pre-
coding matrix, where W
k
C
M×B
k
. A is the power normal-
isation matrix defined by A = blkdiag{A
1
, A
2
, ..., A
K
} with
A
k
= diag(a
k
) and a
k
= (a
k1
, a
k2
, ..., a
kB
k
)
T
R
B
k
, such
that the total transmission power
P
Tr(W
k
A
k
A
H
k
W
H
k
) =
P
P
k
= P . The additive Gaussian noise vector for each
user k, denoted as n
k
, has zero mean and covariance matrix
E{n
k
n
H
k
} = σ
2
k
I
N
k
. The user ks received signal is given by
y
k
= H
k
W
k
A
k
s
k
+ H
k
X
j6=k
W
j
A
j
s
j
+ n
k
. (1)
At user k, the receive processing can be decomposed as
G
k
= D
k
¯
G
k
, where
¯
G
k
C
B
k
×N
k
is the receive filter nor-
malised such that each row has unity norm and D
k
R
B
k
×B
k
is a diagonal matrix, wherein the diagonal entries represent the
norms of the associated rows of G
k
. Denoting the received
signal at the output of the receive filter as
ˆ
y
k
=
¯
G
k
y
k
, the
estimated data sequence,
ˆ
s
k
, can be written as
ˆ
s
k
= G
k
y
k
= D
k
ˆ
y
k
. (2)
A. The SLNR Precoding Scheme
For the SLNR scheme [6], the power normalisation is
usually assumed such that Tr
W
H
k
W
k
= B
k
and A
k
=
q
P
k
B
k
I
B
k
. Here, the SLNR maximisation criterion leads to the
following optimisation problem:
W
opt
k
= arg max
W
k
Tr
W
H
k
H
H
k
H
k
W
k
Tr
h
W
H
k
˜
H
H
k
˜
H
k
+ α
k
I
M
W
k
i
(3)
s.t. Tr
W
H
k
W
k
= B
k
with α
k
=
N
k
σ
2
k
P
k
and
˜
H
k
= [H
H
1
, ..., H
H
k1
, H
H
k+1
, ..., H
H
K
]
H
.
The solution to (3) can be given by [6]
W
k
= ρ
k
T
k
I
B
k
; 0
(M B
k
)×B
k
(4)
where the columns of T
k
C
M×M
defines the generalised
eigenspace of the pair
n
H
H
k
H
k
,
˜
H
H
k
˜
H
k
+ α
k
I
M
o
and ρ
k
is a power normalisation parameter, such that Tr
W
H
k
W
k
=
B
k
. The matched filter, given by
0000–0000/00$00.00
c
2013 IEEE

2 IEEE COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION
¯
G
k
= Ψ
k
W
H
k
H
H
k
(5)
where Ψ
k
R
B
k
×B
k
is a diagonal matrix chosen to normalise
each row to unity norm, is deployed as the receive filter [6].
III. EQUIVALENT EXPRESSIONS OF SLNR-BASED
SOLUTIONS WITH MULTI-ANTENNA RECEIVERS
Lemma 1: Define the following SVD operations
˜
H
k
=
˜
U
k
˜
S
k
˜
V
H
k
, (6)
H
k
˜
V
k
˜
S
H
k
˜
S
k
+ α
k
I
M
1
2
=
¯
U
k
¯
S
k
¯
V
H
k
(7)
where
˜
U
k
,
˜
V
k
,
¯
U
k
, and
¯
V
k
are unitary matrices;
˜
S
k
and
¯
S
k
are (
P
j6=k
N
j
) ×M and N
k
×M (respectively) diagonal
matrices, with assumption that the singular values on the
diagonal entries of
¯
S
k
are sorted in decreasing order, i.e.
s
1
s
2
··· s
r
k
> 0; r
k
= rank(H
k
). Let
˘
S
k
= diag(s
1
, ··· , s
B
k
) and
˘
U
k
,
˘
V
k
denote submatrices
containing B
k
(B
k
r
k
) leading columns of
¯
U
k
and
¯
V
k
respectively, the solution (4) can be rewritten as:
W
k
= ρ
k
˜
V
k
˜
S
H
k
˜
S
k
+ α
k
I
M
1
2
˘
V
k
(8)
= ρ
k
˜
H
H
k
˜
H
k
+ α
k
I
M
1
H
H
k
˘
U
k
˘
S
1
k
. (9)
In addition, the normalised matched filter in (5) can be
expressed as
¯
G
k
=
˘
U
H
k
. (10)
Proof: From the definition of generalised eigenspaces,
there exists an invertible matrix T
k
C
M×M
such that
T
H
k
H
H
k
H
k
T
k
= Λ
k
(11)
T
H
k
˜
H
H
k
˜
H
k
+ α
k
I
M
T
k
= I
M
(12)
where Λ
k
= diag(λ
k,1
, ··· , λ
k,M
) with λ
k,1
λ
k,2
λ
k,r
k
> 0 and λ
k,r
k
+1
= ··· = λ
k,M
= 0. Using the SVD in
(6), (12) can be expressed as
T
H
k
˜
V
k
˜
S
H
k
˜
S
k
+ α
k
I
M
˜
V
H
k
T
k
= I
M
. (13)
It follows from (13) that T
k
=
˜
V
k
˜
S
H
k
˜
S
k
+ α
k
I
M
1
2
Q
k
where Q
k
C
M×M
is a unitary matrix. Let
˜
P
k
=
˜
V
k
˜
S
H
k
˜
S
k
+ α
k
I
M
1
2
, Q
k
can be determined by substi-
tuting T
k
into (11). Hence, (11) can be rewritten as
Q
H
k
˜
P
H
k
H
H
k
H
k
˜
P
k
Q
k
= Λ
k
. (14)
It can be clearly seen from (14) that the unitary matrix Q
k
diagonalises
˜
P
H
k
H
H
k
H
k
˜
P
k
; thus, Q
k
can be obtained by
eigenvalue decomposition of
˜
P
H
k
H
H
k
H
k
˜
P
k
, i.e. columns
of Q
k
contain eigenvectors associated to eigenvalues sorted
in decreasing order. Specifically, considering the SVD in (7),
it can be chosen such that Q
k
=
¯
V
k
. It also follows that
Λ
k
=
¯
S
H
k
¯
S
k
. Thus, T
k
can be expressed as
T
k
=
˜
V
k
˜
S
H
k
˜
S
k
+ α
k
I
M
1
2
¯
V
k
. (15)
Since the solution of the SLNR design only involves the
first B
k
columns of Q
k
, the solution (4) can be rewritten as
in (8). In addition, from (7), it can be shown that
˘
V
k
=
˜
P
H
k
H
H
k
˘
U
k
˘
S
1
k
=
˜
S
H
k
˜
S
k
+ α
k
I
M
1
2
˜
V
H
k
H
H
k
˘
U
k
˘
S
1
k
. It
follows that (9) is obtained by substituting
˘
V
k
into (8).
Furthermore, by considering submatrices of (7) and right
multiplication with
˘
V
k
, i.e. H
k
˜
P
k
˘
V
k
=
1
ρ
k
H
k
W
k
=
˘
U
k
˘
S
k
,
the receive matched filter can be obtained as W
H
k
H
H
k
=
ρ
k
˘
S
H
k
˘
U
H
k
. Accordingly, the (row) normalised matched filter
can be expressed as (10).
A. Equivalence between SLNR, RBD and GMI-2 schemes
As clearly seen from (8), (10), the SLNR precoding scheme
with multi-antenna receivers can be viewed as a regularised
channel inversion technique, similar to RBD [4], with differ-
ences in regularisation and power-normalisation parameters.
This conforms to the previous observations in [7], [8] for the
case of single-antenna receivers. Notice that, for N
k
= 1,
the precoding and decoding matrices in (9) and (10) can
be reduced to the solution given in [8]. Analogously, an
equivalence between SLNR and RBD schemes with multi-
antenna receivers can also be established as given in the
following theorem (only the proof outline is provided due to
the limited space).
Theorem 1: The precoding and decoding matrices of the
SLNR precoding scheme [6], obtained by (4) and (5) respec-
tively, are equivalent to those of RBD [4] and GMI-2 (GMI
method 2) [5] schemes under the assumption of the same
regularisation parameter and power normalisation procedure.
Proof: This theorem follows by applying [5, Theorem 1-
3] with the assumption of equal regularisation parameters, i.e.
α
1
= ··· = α
K
= α, and the same power normalisation, i.e.
ρ
1
= ··· = ρ
K
= ρ to the equivalent expressions of precoding
and decoding matrices in (9) and (10).
For N
k
M , the above assumptions can be met, for
instance, by imposing equal power allocation (EPA) and
full-eigenmode transmission constraints, i.e. choosing A
k
=
P
s
I
N
k
with B
k
= N
k
and P
s
= P/(
P
k
N
k
). This is
analogous to [8] for the case of single-antenna receivers.
Note that the equivalence between the SLNR and RBD
schemes has also been observed in [9] whereby the equiva-
lence has only been verified for the precoding matrices, by
back substitution of the RBD solution [e.g. in a form of
(8)] into the SLNR precoding design criteria (11) and (12).
The equivalence of decoding matrices, e.g. (5) and (10), has
not been provided in [9]. This paper, on the other hand,
obtains the equivalent expressions of the SLNR solutions (8)-
(10) by direct derivation of (11) and (12) and establishes the
equivalence of the decoding matrices in addition to that of the
precoding matrices as given in Lemma 1.
Theorem 1 implies the possible interchange of existing
algorithms and analysis among these schemes. For instance,
the matched filter given by (5) can be used instead of (10)
in RBD and GMI-2 schemes for reduced complexity. Existing
power allocation algorithms in RBD [4] can also be applied to
SLNR and GMI-2 schemes. Further, the performance analysis
given in the following can be extended to the RBD as well as
the GMI-2 schemes.

PATCHARAMANEEPAKORN et al.: EQUIVALENT EXPRESSIONS AND PERFORMANCE ANALYSIS OF SLNR PRECODING SCHEMES 3
IV. PERFORMANCE ANALYSIS OF THE SLNR SCHEME
A. SLNR analysis when
P
j6=k
N
j
< M
For
P
j6=k
N
j
< M , the precoding matrix in (9) can be
decomposed into two orthogonal subspaces, i.e.
W
k
=
1
ν
k
h
P
˜
H
k
H
H
k
+ α
k
P
//
˜
H
k
H
H
k
i
˘
U
k
˘
S
1
k
(16)
where P
˜
H
k
H
H
k
=
I
˜
H
H
k
˜
H
k
˜
H
H
k
1
˜
H
k
H
H
k
is an
orthogonal projection of H
H
k
into the null space of
˜
H
k
, i.e.
aligned with the BD solution, while the other part P
//
˜
H
k
H
H
k
=
˜
H
H
k
˜
H
k
˜
H
H
k
+ α
k
I
1
˜
H
k
˜
H
H
k
1
˜
H
k
H
H
k
leads to signal
leakage in the column space of
˜
H
k
, i.e. inter-user interference.
ν
k
is the power normalisation parameter. The inter-user inter-
ference is well-controlled as α
k
P
//
˜
H
k
0 at high SNR. How-
ever, it remains necessary to choose an appropriate number of
data streams to avoid a convergence to zero effective gain of
some data streams as suggested in the following theorem.
Theorem 2: For
P
j6=k
N
j
< M, a sufficient condition of
the number of data streams that ensures non-zero effective
gains at high SNR can be given by
B
k
min{M
X
j6=k
N
j
, N
k
}. (17)
Proof: At high SNR, α
k
P
//
˜
H
k
0, the precoding design
converges to the BD solution and leakage power converges
to zero. The effective channel of user k is thus interference-
free and has rank r = rank(H
k
W
k
) rank(H
k
P
˜
H
k
H
H
k
) =
min{M
P
j6=k
N
j
, N
k
}. Multiplexing excessive data streams
over this number involves choosing columns of
˘
U
k
in the null
space of H
k
P
˜
H
k
H
H
k
, potentially leading to zero-gain at high
SNR. Thus, it suffices to ensure non-zero effective gain for
each data stream if (17) is satisfied.
Note that substreams with effective gains converging to zero
account for irreducible BER and zero throughput at high SNR.
This results in an error floor of the average BER. The sum-
rate, however, still grows with SNR with a change of slope,
i.e. multiplexing gain reduces as substreams with zero-gain no
longer contribute to the sum throughput.
Theorem 3: Consider a case wherein the conditions in
Theorems 1 and 2 are satisfied, i.e. A
k
=
P
s
I
N
k
, B
k
= N
k
,
and
P
k
N
k
= M with N
k
P
j6=k
N
j
=
˜
N < M. The SINR
of the i
th
stream of user k can be approximated by
γ
SLN R
k,i
eig
i
(
˜
H
H
k
˜
H
k
+
σ
2
k
P
s
I
M
1
H
H
k
H
k
)
(18)
=
P
s
σ
2
k
eig
i
{B
k
+
k
} (19)
with B
k
= H
k
I
M
˜
H
H
k
˜
H
k
˜
H
H
k
1
˜
H
k
H
H
k
and
k
=
H
k
˜
H
H
k
˜
H
k
˜
H
H
k
1
˜
H
k
˜
H
H
k
+
σ
2
k
P
s
I
˜
N
1
˜
H
k
H
H
k
, and
eig
i
{·} denoting the i
th
largest eigenvalue of the argument.
Proof: Analogous to [8], it could be argued
that the interference-plus-noise covariance matrix can
be estimated by a leakage-plus-noise matrix, that is
P
j6=k
H
k
W
j
W
H
j
H
H
k
+
σ
2
k
P
s
I
N
k
P
j6=k
W
H
k
H
H
j
H
j
W
k
+
σ
2
k
P
s
W
H
k
W
k
= W
H
k
P
j6=k
H
H
j
H
j
+
σ
2
k
P
s
I
M
W
k
, which
can be shown to be generally tight when the interference
power is relatively small compared to the noise power, i.e. at
the asymptotic low and high SNR regimes
1
. Using the above
approximation, the SINR of the i
th
stream of user k can be
written as
γ
SLN R
k,i
=
G
k
H
k
W
k
W
H
k
H
H
k
G
H
k
ii
h
G
k
P
j6=k
H
k
W
j
W
H
j
H
H
k
+
σ
2
k
P
s
I
N
k
G
H
k
i
ii
W
H
k
H
H
k
H
k
W
k
W
H
k
H
H
k
H
k
W
k
ii
h
W
H
k
H
H
k
W
H
k
˜
H
H
j
˜
H
j
+
σ
2
k
P
s
I
M
W
k
H
k
W
k
i
ii
(a)
= λ
k,i
(20)
where the equality (a) follows from (11) and (12). Hence,
the SINR can be approximated by the corresponding eigen-
value. Note that (20) can be expressed as (18). By us-
ing eig{AB} = eig {BA}, (18) can be rewritten as
eig
i
H
k
˜
H
H
k
˜
H
k
+
σ
2
k
P
s
I
M
1
H
H
k
. (19) follows after ap-
plying a few matrix operations.
Although the approximation in Theorem 3 could not guar-
antee the tightness at moderate SNRs, it greatly simplifies
the SINR analysis and provides insights into the performance
with respect to the BD scheme as given below. Moreover,
reasonable accuracy can generally be observed by simulation.
Following [10], the SINR of BD can be expressed as
γ
BD
k,i
=
P
s
σ
2
k
eig
i
{B
k
}. Comparing to (19), it follows from [11,
4.3.1] that γ
SLN R
k,i
λ
SLN R
k,i
γ
BD
k,i
+ λ
N
k
(
P
s
σ
2
k
k
), with
the smallest eigenvalue of
P
s
σ
2
k
k
denoted as λ
N
k
(
P
s
σ
2
k
k
) =
eig
N
k
H
k
ˇ
V
k
ˇ
S
2
k
+
σ
2
k
P
s
I
˜
N
1
ˇ
V
H
k
H
H
k
.
ˇ
V
k
and
ˇ
S
k
are
the submatrices, corresponding to the
˜
N non-zero singu-
lar values, of
˜
V
k
and
˜
S
k
respectively. It follows that
λ
N
k
(
P
s
σ
2
k
k
) 0 (
P
s
σ
2
k
k
is non-negative definite Hermitian).
Further, λ
N
k
is a non-decreasing function of
P
s
σ
2
k
, it converges
to λ
N
k
,
(
P
s
σ
2
k
k
) = eig
N
k
H
k
ˇ
V
k
ˇ
S
2
k
ˇ
V
H
k
H
H
k
as
P
s
σ
2
k
.
This indicates the superiority of the SLNR scheme over BD.
Following [8], for Rayleigh fading channels, it can be shown
that the sum-rate of the SLNR scheme converges to that of
BD at high SNR, while there remains a non-vanishing gap of
BER performance as also shown in Section V.
B. SLNR analysis when
P
j6=k
N
j
M
For
P
j6=k
N
j
M, i.e.
˜
H
H
k
˜
H
k
is full rank, leakage power
from user k to user j generally increases with P
s
at high SNR,
as kH
j
W
k
A
k
k
2
F
P
s
kH
j
˜
H
H
k
˜
H
k
1
H
H
k
˘
U
k
˘
S
1
k
k
2
F
. In
general, this suggests a limit of performance in the high SNR
regime. However, from (7), the singular vectors (
˘
U
k
,
˘
V
k
)
1
This can be shown by first noticing that
σ
2
k
P
s
W
H
k
W
k
converges to
σ
2
k
P
s
I
N
k
at low and high SNR. Further, for i.i.d. channels, EkH
k
W
j
W
H
j
H
H
k
k
F
=
EkH
j
W
k
W
H
k
H
H
j
k
F
= EkW
H
k
H
H
j
H
j
W
k
k
F
. Since H
k
W
j
W
H
j
H
H
k
and W
H
k
H
H
j
H
j
W
k
are presumably small compared to
σ
2
k
P
s
I
N
k
, it fol-
lows that
P
j6=k
H
k
W
j
W
H
j
H
H
k
+
σ
2
k
P
s
I
N
k
P
j6=k
W
H
k
H
H
j
H
j
W
k
+
σ
2
k
P
s
I
N
k
P
j6=k
W
H
k
H
H
j
H
j
W
k
+
σ
2
k
P
s
W
H
k
W
k
.

4 IEEE COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION
−5 0 5 10 15 20 25 30 35
10
−4
10
−3
10
−2
10
−1
10
0
SNR (dB)
Average uncoded Bit Error Rate (BER)
Scn−1: 4x[2(2)x2], BD
Scn−1: 4x[2(2)x2], RBD
Scn−1: 4x[2(2)x2], GMI−2
Scn−1: 4x[2(2)x2], SLNR
Scn−2: 4x[3(2)x2], SLNR
Scn−3: 4x[4(2)x2], SLNR
Scn−4: 4x[2(1)x3], SLNR
Scn−5: 4x[2(2)x3], SLNR
Fig. 1. Average BER (QPSK), with configuration M × [N
k
(B
k
) × K].
are chosen in the directions which have direct and inverse
relationships with the singular values of H
k
and those of
˜
H
k
,
respectively. Severe interference can thus be alleviated when
M is large (high degree of freedom for transmit beamform-
ing design) and B
k
is small (only using data streams with
reasonably good designs, i.e. large singular values).
Additionally, with a specific case of K = 2, it can be shown
that the interference issue at high SNR can be completely
avoided as elaborated in the following theorem.
Theorem 4: For K = 2 and N
k
M, a necessary
condition for the convergence of inter-user interference to zero
at high SNR is given by
2
X
k=1
B
k
M. (21)
In fact, at high SNR, the beamforming designs of both users
are equivalent with eigenvalues sorted in reverse order.
Proof: For K = 2 with H
H
1
H
1
and H
H
2
H
2
full rank,
at asymptotically high SNR, (11) and (12) for user 1 can
be rewritten as
Λ
1
2
1
T
H
1
H
H
1
H
1
T
1
Λ
1
2
1
= I
M
and
Λ
1
2
1
T
H
1
H
H
2
H
2
T
1
Λ
1
2
1
= Λ
1
1
, respectively. It fol-
lows that T
2
= T
1
Λ
1
2
1
and Λ
2
= Λ
1
1
. Thus, Λ
1
and
Λ
2
are in reverse order. T
1
and T
2
are equivalent, in the
sense that column vectors (in reverse order) are aligned in the
same directions, i.e. there are M distinct independent beams
available for both users. Hence, zero inter-user interference is
plausible if (21) is satisfied.
V. SIMULATION RESULTS AND CONCLUSIONS
Figs. 1 and 2 show the sum-rate and BER performance for
various scenarios (Scn-1 to Scn-5) with EPA and independent
Rayleigh fading channels. The SNR is defined as P
2
. The
equivalence between the SLNR, RBD and GMI-2 schemes is
represented by Scn-1. For
P
j6=k
N
j
< M with the condition
given in Theorem 2 being not satisfied, an error floor and a
change of sum-rate slope at high SNR can be observed as
shown in Scn-2. For
P
j6=k
N
j
M, K = 2 (Scn-3), a limit
of performance at high SNR is shown to be avoided if the
condition in Theorem 4 holds. In contrast, for
P
j6=k
N
j
M,
K > 2 (Scn-4 and Scn-5), performance floors can be observed.
In this case, smaller B
k
results in less severe interference
−5 0 5 10 15 20 25 30 35 40
0
5
10
15
20
25
30
35
40
45
50
SNR (dB)
Average sum rate (bit/s/Hz)
Scn−1: 4x[2(2)x2], BD
Scn−1: 4x[2(2)x2], RBD
Scn−1: 4x[2(2)x2], GMI−2
Scn−1: 4x[2(2)x2], SLNR
Scn−1: 4x[2(2)x2], SLNR
(Sum−rate by Eq.18)
Scn−2: 4x[3(2)x2], SLNR
Scn−3: 4x[4(2)x2], SLNR
Scn−4: 4x[2(1)x3], SLNR
Scn−5: 4x[2(2)x3], SLNR
Scn−5: 4x[2(2)x3], SLNR
(Sum−rate by Eq.18)
Fig. 2. Average sum rate, with configuration M × [N
k
(B
k
) × K].
as analysed in Section IV-B. The approximation of sum-rate
given in Theorem 3 is also provided in Fig. 2. Notice that
the approximation is generally tight for the whole SNR range
when
P
j6=k
N
j
< M (Scn-1), whereas it is slightly loose for
high SNR when
P
j6=k
N
j
M (Scn-5) as the assumption of
small interference is no longer accurate.
In conclusion, this letter derived equivalent expressions of
SLNR-based precoding solutions and established the equiv-
alence between the SLNR, RBD, and GMI-2 precoding
schemes. With this equivalent form, the performance of the
SLNR scheme has been analysed. These analytic results can
be extended to the other schemes due to their equivalence.
This generalises [8] and its useful implications from the case
of single-antenna to multi-antenna receivers.
REFERENCES
[1] C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, “A Vector-
Perturbation Technique for Near-Capacity Multiantenna Multiuser Com-
munication - Part I: Channel Inversion and Regularization, IEEE Trans.
Commun., vol. 53, no. 1, pp. 195–202, Jan. 2005.
[2] M. Joham, W. Utschick, and J. A. Nossek, “Linear transmit processing in
MIMO communications systems, IEEE Trans. Signal Process., vol. 53,
no. 8, pp. 2700–2712, Aug. 2005.
[3] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-Forcing meth-
ods for Downlink Spatial Multiplexing in Multiuser MIMO channels,
IEEE Trans. Signal Process., vol. 52, no. 2, pp. 461–471, Feb. 2004.
[4] V. Stankovic and M. Haardt, “Generalized Design of Multi-User MIMO
Precoding Matrices, IEEE Trans. Wireless Commun., vol. 7, no. 3, pp.
953–961, Mar. 2008.
[5] H. Sung, S. Lee, and I. Lee, “Generalized Channel Inversion Methods
for Multiuser MIMO Systems, IEEE Trans. Commun., vol. 57, no. 11,
pp. 3489–3499, Nov. 2009.
[6] M. Sadek, A. Tarighat, and A. Sayed, “A Leakage-Based Precoding
Scheme for Downlink Multi-User MIMO Channels, IEEE Trans. Wire-
less Commun., vol. 6, no. 5, pp. 1711–1721, May 2007.
[7] F. Yuan and C. Yang. (2012, Feb.) Equivalence of SLNR Precoder and
RZF Precoder in Downlink MU-MIMO. ArXiv:1202.1888, Feb 2012.
[Online]. Available: http://arxiv.org/abs/1202.1888
[8] P. Patcharamaneepakorn, S. Armour, and A. Doufexi, “On the Equiv-
alence Between SLNR and MMSE Precoding Schemes with Single-
Antenna Receivers, IEEE Commun. Lett., vol. 16, no. 7, pp. 1034–1037,
Jul. 2012.
[9] K. Wang and X. Zhang, “On equivalence of SLNR-based precoding
and RBD precoding, Electron. Lett., vol. 48, no. 11, pp. 662–663, May
2012.
[10] R. Chen, J. G. Andrews, and J. Robert W. Hearth, “Multiuser Space-
Time Block Coded MIMO System with Downlink Precoding, in IEEE
Int. Conf. on Commun. (ICC), 2004, vol. 5, Jun. 2004, pp. 2589–2693.
[11] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge
University Press, 1985.
Citations
More filters

Proceedings ArticleDOI
Yavuz Yapici1, Sung Joon Maeng1, Ismail Guvenc1, Huaiyu Dai1  +1 moreInstitutions (2)
20 May 2019-
Abstract: Massive multiple-input multiple-output (MIMO) is a key technology for 5G wireless communications with a promise of significant capacity increase. The use of low-resolution data converters is crucial for massive MIMO to make the overall transmission as cost- and energy-efficient as possible. In this work, we consider a downlink millimeter-wave (mmWave) transmission scenario, where multiple users are served simultaneously by massive MIMO with one-bit digital-to-analog (D/A) converters. In particular, we propose a novel precoder design based on signal-to-leakage-plus-noise ratio (SLNR), which minimizes energy leakage into undesired users while taking into account impairments due to nonlinear one-bit quantization. We show that well-known regularized zero-forcing (RZF) precoder is a particular version of the proposed SLNR-based precoder, which is obtained when quantization impairments are totally ignored. Numerical results underscore significant performance improvements along with the proposed SLNR-based precoder as compared to either RZF or zero-forcing (ZF) precoders.

7 citations


Posted Content
Yavuz Yapici1, Sung Joon Maeng1, Ismail Guvenc1, Huaiyu Dai1  +1 moreInstitutions (2)
TL;DR: This work proposes a novel precoder design based on signal-to-leakage-plus-noise ratio (SLNR), which minimizes energy leakage into undesired users while taking into account impairments due to nonlinear one-bit quantization.
Abstract: Massive multiple-input multiple-output (MIMO) is a key technology for 5G wireless communications with a promise of significant capacity increase. The use of low-resolution data converters is crucial for massive MIMO to make the overall transmission as cost- and energy-efficient as possible. In this work, we consider a downlink millimeter-wave (mmWave) transmission scenario, where multiple users are served simultaneously by massive MIMO with one-bit digital-to-analog (D/A) converters. In particular, we propose a novel precoder design based on signal-to-leakage-plus-noise ratio (SLNR), which minimizes energy leakage into undesired users while taking into account impairments due to nonlinear one-bit quantization. We show that well-known regularized zero-forcing (RZF) precoder is a particular version of the proposed SLNR-based precoder, which is obtained when quantization impairments are totally ignored. Numerical results underscore significant performance improvements along with the proposed SLNR-based precoder as compared to either RZF or zero-forcing (ZF) precoders.

6 citations


Journal ArticleDOI
Yu Li1, Zufan Zhang1, Zufan Zhang2Institutions (2)
TL;DR: An algorithm which combines the triangular decomposition and signal to leakage and noise ratio (SLNR) (TD- SLNR) to suppress strong co-channel interference in multi-cell multiple input and multiple output (MIMO) heterogeneous networks is proposed.
Abstract: The heterogeneous network, contains a macro cell and a grid of low power nodes with the same frequencies, can improve the system capacity and spectrum efficiency. Configuring low-power nodes that share the same spectrum with macro cell to form heterogeneous networks makes it more likely to improve the system capacity and spectrum efficiency, but inevitably, strong co-channel interference is the main barrier to further improvement for heterogeneous networks. This paper proposes an algorithm which combines the triangular decomposition and signal to leakage and noise ratio (SLNR) (TD-SLNR) to suppress strong co-channel interference in multi-cell multiple input and multiple output (MIMO) heterogeneous networks. Firstly, the proposed algorithm can reduce the number of inter-cell interferences in half. As a result of triangular decomposition, an equivalent interference channel model is extracted to eliminate the rest of interferences using SLNR and interference suppression matrix. Theoretical analysis shows that the proposed algorithm provides a potential solution to suppress the co-channel interference with low complexity and reduce the computation complexity without adding extra interference suppression matrices and computation complexity at receivers. Furthermore, the simulation results show that TD-SLNR algorithm can improve system capacity and energy efficiency comparing with the traditional SLNR algorithm.

5 citations


Proceedings ArticleDOI
25 Nov 2013-
TL;DR: A modified definition of signal-to-leakage-plus-noise ratio (SLNR) as a criterion for linear transmit filter design in Multi-user MIMO systems and an iterative precoding algorithm is proposed and is studied with two linear receivers.
Abstract: This paper proposes a modified definition of signal-to-leakage-plus-noise ratio (SLNR) as a criterion for linear transmit filter design in Multi-user MIMO systems. The proposed metric incorporates receiver structures into the precoder design, which can potentially exploit unused receive signal subspaces in cases where the available eigenmodes are not fully transmitted. The improvement in terms of the degrees of freedom of the precoder is also elaborated. In addition, an iterative precoding algorithm is proposed and is studied with two linear receivers, namely the matched filter (MF) and the minimum mean square error (MMSE) receivers. Simulation results show that the proposed scheme outperforms the conventional SLNR scheme and can achieve further improvement with an MMSE receiver, at the expense of increased computational complexity and slow convergence rate. Moreover, the proposed scheme is shown to reduce its form to the conventional scheme when full-eigenmode transmission is assumed.

3 citations


Dissertation
21 Jun 2016-
TL;DR: This thesis investigates the application of an iterative leakage-based precoding algorithm to practical multiuser-MIMO systems and proposes several modifications to the aforementioned method which demonstrated improved performance under certain practical conditions.
Abstract: This thesis investigates the application of an iterative leakage-based precoding algorithm to practical multiuser-MIMO systems. We consider the effect of practical impairments including imperfect channel state information, transmit antenna correlation, and timevarying channels. Solutions are derived which improve performance of the algorithm with imperfect channel state information at the transmitter by leveraging knowledge of the second-order statistics of the error. From this work we draw a number of conclusions on how imperfect channel state information may impact the system design including the importance of interference suppression at the receiver and the selection of the number of co-scheduled users. We also demonstrate an efficient approach to improve the convergence of the algorithm when using interference-rejection-combining receivers. Finally, we conduct simulations of an LTE-A system employing the improved algorithm to show its utility for modern communication systems. Iterative Leakage-Based Precoding for Multiuser-MIMO Systems Eric Sollenberger General Audience Abstract This thesis investigates several aspects of a particular method by which multiple users can share radio resources within a wireless system i.e. they may operate on the same frequency and at the same time. This is a desirable capability in modern wireless systems because it improves the efficiency of radio spectrum usage. Radio spectrum has become a very expensive resource in recent years so achieving high efficiency is crucial. Our investigation led us to propose several modifications to the aforementioned method which demonstrated improved performance under certain practical conditions. We further demonstrated the effects of several common system impairments and provided insight into how these impairments effect the system design. Finally, we demonstrated that using this method provides significant gains when used for the latest cellular technology.

2 citations


Cites background from "Equivalent Expressions and Performa..."

  • ...It was shown in [18] that sufficient degrees of freedom exist for the SLNR scheme to drive the leakage power to zero at asymptotic high SNR if ∑ Mi j≠i < Nt is satisfied....

    [...]


References
More filters

Book ChapterDOI

[...]

01 Jan 2012-

123,310 citations


"Equivalent Expressions and Performa..." refers methods in this paper

  • ...Following [10], the SINR of BD can be expressed as γ k,i = Ps σ(2) k eigi {Bk}....

    [...]


Book
Roger A. Horn1, Charles R. Johnson2Institutions (2)
01 Jan 1985-
Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.

23,959 citations


MonographDOI
01 Jan 1985-

7,511 citations


Journal ArticleDOI
TL;DR: While the proposed algorithms are suboptimal, they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity.
Abstract: The use of space-division multiple access (SDMA) in the downlink of a multiuser multiple-input, multiple-output (MIMO) wireless communications network can provide a substantial gain in system throughput. The challenge in such multiuser systems is designing transmit vectors while considering the co-channel interference of other users. Typical optimization problems of interest include the capacity problem - maximizing the sum information rate subject to a power constraint-or the power control problem-minimizing transmitted power such that a certain quality-of-service metric for each user is met. Neither of these problems possess closed-form solutions for the general multiuser MIMO channel, but the imposition of certain constraints can lead to closed-form solutions. This paper presents two such constrained solutions. The first, referred to as "block-diagonalization," is a generalization of channel inversion when there are multiple antennas at each receiver. It is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high SNR. The second, known as "successive optimization," is an alternative method for solving the power minimization problem one user at a time, and it yields superior results in some (e.g., low SNR) situations. Both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined. In order to accommodate more general scenarios, we also propose a framework for coordinated transmitter-receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas. While the proposed algorithms are suboptimal, they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity.

3,117 citations


"Equivalent Expressions and Performa..." refers methods in this paper

  • ...ZF is developed into Block Diagonalisation (BD) [3] precoding, and MMSE is extended to Regularised Block Diagonalisation (RBD) [4] and Generalised MMSE Channel Inversion (GMI) [5] schemes....

    [...]


Journal ArticleDOI
TL;DR: A simple encoding algorithm is introduced that achieves near-capacity at sum rates of tens of bits/channel use and regularization is introduced to improve the condition of the inverse and maximize the signal-to-interference-plus-noise ratio at the receivers.
Abstract: Recent theoretical results describing the sum capacity when using multiple antennas to communicate with multiple users in a known rich scattering environment have not yet been followed with practical transmission schemes that achieve this capacity. We introduce a simple encoding algorithm that achieves near-capacity at sum rates of tens of bits/channel use. The algorithm is a variation on channel inversion that regularizes the inverse and uses a "sphere encoder" to perturb the data to reduce the power of the transmitted signal. This work is comprised of two parts. In this first part, we show that while the sum capacity grows linearly with the minimum of the number of antennas and users, the sum rate of channel inversion does not. This poor performance is due to the large spread in the singular values of the channel matrix. We introduce regularization to improve the condition of the inverse and maximize the signal-to-interference-plus-noise ratio at the receivers. Regularization enables linear growth and works especially well at low signal-to-noise ratios (SNRs), but as we show in the second part, an additional step is needed to achieve near-capacity performance at all SNRs.

1,687 citations


"Equivalent Expressions and Performa..." refers methods in this paper

  • ...This generalises the equivalence between the SLNR and MMSE schemes proven in [7], [8] and establishes the extension of its implications from the case of singleantenna to multi-antenna receivers....

    [...]

  • ...Available: http://arxiv.org/abs/1202.1888 [8] P. Patcharamaneepakorn, S. Armour, and A. Doufexi, “On theEquivalence Between SLNR and MMSE Precoding Schemes with SingleAntenna Receivers,”IEEE Commun....

    [...]

  • ...They are later extended to the case of multi-antenna receivers, e.g. ZF is developed into Block Diagonalisation (BD) [3] precoding, and MMSE is extended to Regularised Block Diagonalisation (RBD) [4] and Generalised MMSE Channel Inversion (GMI) [5] schemes....

    [...]

  • ...Linear precoding techniques, such as Zero-Forcing (ZF) [1] and Minimum Mean Square Error (MMSE) [1], [2], are initially developed for systems with single-antenna receivers....

    [...]


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