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 k’s
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 k’s 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
k−1
, 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/σ
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.
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![Fig. 2. Average sum rate, with configurationM × [Nk(Bk)×K].](/figures/fig-2-average-sum-rate-with-configurationm-x-nk-bk-xk-2tqs7hnc.webp)
![Fig. 1. Average BER (QPSK), with configurationM × [Nk(Bk)×K].](/figures/fig-1-average-ber-qpsk-with-configurationm-x-nk-bk-xk-2unka23c.webp)