On Design of Improper Signaling for SER Minimization in K

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On Design of Improper Signaling for SER Minimization in K
On Design of Improper Signaling for SER
Minimization in K -User Interference Channel
Hieu Duy Nguyen† , Rui Zhang†‡ , and Sumei Sun†
† Institute
for Infocomm Research, Singapore
Department, National University of Singapore, Singapore
Emails: {nguyendh, sunsm}@i2r.a-star.edu.sg, [email protected]
‡ ECE
Abstract—The rate maximization for the K-user interference
channels (ICs) has been investigated extensively in the literature.
However, the dual problem of minimizing the error probability
with given signal constellations and/or data rates of the users
is less exploited. In this paper, by utilizing the additional
degrees of freedom attained from the improper signaling (versus
the conventional proper signaling), we optimize the precoding
matrices for the K-user single-input single-output (SISO) ICs
to achieve minimal transmission symbol error rate (SER). Compared to conventional proper signaling as well as other stateof-the-art improper signaling designs, our proposed improper
signaling scheme is shown to achieve notable SER improvement in
SISO-ICs by simulations. Our study provides another viewpoint
for optimizing transmissions in ICs and further justifies the
practical benefit of improper signaling in interference-limited
communication systems.
I. I NTRODUCTION
Interference channel (IC) is a fundamental model in multiuser wireless communication and has been extensively studied to date. However, complete characterization of the capacity
region of the IC corrupted by additive Gaussian noise, is in
general still open, even for the simplest two-user case [1]. The
recent advance in a technique so-called interference alignment
(IA) has motivated numerous studies on characterizing the rate
performance of ICs under the high signal-to-noise ratio (SNR)
regime. With the aid of IA, the maximum achievable rates in
terms of degree of freedom (DoF) have been obtained for various IC models to provide useful insights on designing optimal
transmission schemes for interference-limited communication
systems (see, e.g., [2] and the references therein).
Another notable advancement is the use of improper Gaussian signaling (IGS) for ICs. Note that for conventional
systems employing proper Gaussian signals, the real and
imaginary parts of the transmitted signal have equal power
and are independent zero-mean Gaussian random variables [3].
In contrast, a complex Gaussian symbol is called improper if
its real and imaginary parts have unequal power and/or are
correlated. Improper signals have been studied in applications
such as detection and estimation [3], [4]. Studies of IGS in
communication systems, however, only appeared recently. This
may be due to the fact that proper Gaussian signaling (PGS)
has been known to be capacity optimal for the Gaussian pointto-point, multiple-access, and broadcast channels; and as a
result, it was presumably deemed to be optimal for ICs as
well. In [5], it has been shown that IGS can further improve
the achievable rates of the 3-user IC in high-SNR regime.
Inspired by this work, subsequent studies have investigated ICs
with IGS in finite-SNR regime. For instance, the achievable
rate region and minimal signal-to-interference-plus-noise ratio
(SINR) maximization for the K-user IC have been characterized in [6]-[9]. These works have reported a significant
rate improvement of IGS over conventional PGS in terms of
achievable rate under finite SNR.
It comes to our attention that most of the existing works
on ICs have focused on investigating the rate performance.
However, a dual problem for ICs, which minimizes the transmission error probability with given users signal constellations
and/or rates is less exploited. Notice that this problem may be
more practically sensible for the scenarios when the users have
their desired quality-of-service (QoS) in terms of data rate and
error performance to be met. It is worth noting that in [10],
the authors considered the problem of minimizing the mean
squared error (MSE) in ICs as an indirect way to minimize the
error rate. Although MSE is a meaningful criterion in practice,
minimizing the MSEs in ICs does not necessarily lead to the
error probability minimization. Moreover, [11] has studied the
error performance for ICs based on IA. However, it is restricted
only to the case of three-user ICs with at least two antennas at
each node, in which each of the three user links can achieve
at least one DoF. In contrast to the above prior works, in
this paper, we study the problem of minimizing the users’
symbol error rates (SERs) explicitly in the K-user single-input
single-output (SISO) IC by applying improper signaling over
finite signal constellations. We are motivated by the results
that IGS can provide rate gains for ICs over the conventional
PGS [5]-[9]. It is thus expected that the additional degrees of
freedom provided by improper signaling can also be exploited
to improve the SER performance in ICs, even with practical
(non-Gaussian) modulation schemes.
Specifically, we investigate the K-user SISO IC with given
signal constellations and transmission rates of the users assuming perfect channel state information (CSI). With improper
signaling, we first derive a closed-form expression for the pairwise error probability (PEP) in decoding each user’s signal
under a mild assumption on the Gaussian distribution of the
interference that is treated as additional noise at each receiver.
Based on this result, we then propose an efficient algorithm to
jointly optimize the precoding matrices at all transmitters to
minimize the maximum PEP/SER among all users. Through
simulations, we show that a significant SER improvement
can be achieved by the proposed algorithm in SISO-IC as
compared to the conventional proper signaling as well as other
state-of-the-art improper signaling designs.
h21
n2
hK 1
II. S YSTEM M ODEL
We consider a K-user SISO IC as shown in Fig. 1, where
the received complex baseband signals are expressed as
yk = hk1 x1 + hk2 x2 + · · · + hkK xK + nk ,
h11
(1)
hKK
nK
Fig. 1: The K-user SISO IC.
jθkl
where hkl = |hkl |e , k, l ∈ {1, . . . , K}, is the complex
coefficient for the channel from transmitter l to receiver k;
xk is the transmitted symbol for user k; and nk is the
additive white Gaussian noise (AWGN) at receiver k, which
is assumed to be a circularly symmetric complex Gaussian
(CSCG) random variable (RV) with zero mean and variance
of σk2 , denoted by nk ∼ CN (0, σk2 ), k = 1, . . . , K. The
transmit power of user k is assumed to be limited by Pk , i.e.,
E[||xk ||2 ] ≤ Pk . For convenience, we define the SNR of user
k as SNRk = Pk /σk2 . At the receiver side, each user aims to
decode the desired signal with minimum error probability. In
this study, we assume that each receiver employs the practical
single-user detection and thus the interference from all other
users is treated as additional noise. However, different from
the conventional setup where proper signaling is assumed, we
consider the use of more general improper signals. We first
define the propriety and impropriety for a complex RV as
follows.
Definition 2.1: Given a zero-mean complex RV
α = αR + jαI and
its real covariance matrix C α =
E [αR αI ]T [αR αI ] , the RV α is called proper if its
covariance matrix C α is a scaled identity matrix in the form
of C α = pI with p > 0, which means that the real and
imaginary parts αR and αI are uncorrelated and have the
equal variance of p. Otherwise, we call α improper.
It is worth noting that in practical communication systems,
modulation schemes such as PSK (e.g., QPSK, 8PSK) and
square QAM (e.g., 16QAM, 64QAM) all have proper signal
constellations. For the convenience of our analysis in the
sequel, we use the following equivalent real-valued representation of the complex-valued system in (1), which is essentially
a K-user 2 × 2 MIMO IC with all real entries given by
cos θkk − sin θkk xkR
ykR
= |hkk |
ykI
xkI
sin θkk
cos θkk
{z
} | {z }
| {z }
|
,y k
,J (θkk )
,xk
K
X
cos θkl − sin θkl xlR
n
(2)
|hkl |
+
+ kR
sin θkl
cos θkl
xlI
nkI
l=1,l6=k
{z
} | {z } | {z }
|
nk
,J (θkl )
,xl
for k = 1, . . . , K, where “R” and “I” denote the real and
imaginary parts, respectively; and nkR and nkI are independent and identically distributed (i.i.d.) real Gaussian RVs with
zero mean and variance of σk2 /2, denoted as N (0, σk2 /2).
Consider a normalized proper constellation for each user k,
represented by a pair of real symbols
inidk = [dkR dkI ]T with
h
the covariance matrix C dk = E dk dTk = I. For example, the
symbol
set for a normalized constellation from
QPSK is {dk }
= [1 1]T , [1 − 1]T , [−1 − 1]T , [−1 1]T . The improper
symbol xk transmitted by user k can then be obtained via the
following transformation
xkR
ak,12 dkR
a
= k,11
,
(3)
xkI
ak,21 ak,22 dkI
{z
}
|
,A k
or equivalently xk = Ak dk . Here Ak is referred to as
the precoding matrix, which is also called the widely linear
processing [3], [4]. Specifically, Ak can also be regarded as
a rotation and scaling matrix applied over any proper signal
constellation of user k to obtain improper signals. Furthermore, the
o power constraint for user k is re-expressed
n transmit
T
as T r Ak Ak = a2k,11 + a2k,12 + a2k,21 + a2k,22 ≤ Pk .
From (2), the real system model of the K-user IC with
improper signaling can be expressed in the following form,
y k = |hkk |J (θkk )Ak dk +
K
X
|hkl |J (θkl )Al dl + nk .
l=1,l6=k
(4)
Without loss of generality, we assume that each user k
applies the decoding matrix at the receiver in the form of
J (θkk )Rk . The signal after applying the decodng matrix is
given by
r k = RTk J T (θkk )y k = |hkk |RTk Ak dk
+
K
X
|hkl |RTk J (φkl )Al dl + RTk ñk ,
(5)
l=1,l6=k
where φkl = θkl − θkk , k, l ∈ {1, . . . , K} and l 6= k; and ñk
= J T (θkk )nk , [ñkR ñkI ]T with ñkR and ñkI being i.i.d.
Gaussian RVs each distributed as N (0, σk2 /2), k = 1, . . . , K.
PK
Denote wk = l=1,l6=k |hkl | J (φkl ) Al dl + nk as the
effective noise at the receiver of user k, which includes both
additive noise and interference. Then the post-processed signal
in (5) can be re-expressed as
r k = |hkk |RTk Ak dk + RTk wk .
(6)
In this paper, we consider the use of the whitening filter for
−1/2
the effective noise as the decoding matrix, i.e., Rk = W k
where W k is defined as
W k = E wk wTk
=
K
X
σk2
|hkl |2 J (φkl )Al ATl J T (φkl ).
I+
2
(7)
l=1,l6=k
Considering the general Mk -ary (Mk > 1) constellations for
dk of each user k, the (exact) SER expression is difficult to
be obtained even assuming the conventional proper signalling,
i.e., Ak is a scaled identity matrix for all k’s. Therefore, in
this paper, we approximate the SER with improper signalling
by an upper bound, which is obtained by applying the union
bound as follows:
X X
1
SERk ≤
P r{dk → d˜k |dk }P r{dk }
Mk − 1
dk d˜k 6=dk
X X
1
P r{dk → d˜k |dk }, (8)
=
Mk (Mk − 1)
dk d˜k 6=dk
where P r{dk → d˜k |dk } is the so-called PEP when dk is
erroneously decoded as d˜k with d˜k 6= dk conditional on that
dk is transmitted for user k; and in (8) we have assumed that
all constellation symbols are selected with equal probability.
In the rest of this paper, we consider the above SER upper
bound as our performance metric.
To derive the PEP, we need to make one further assumption
that the interference-plus-noise term wk in (6) is Gaussian
distributed. This means that for deriving the PEP of user k,
we need to assume that dl ∼ N (0, I), ∀ l ∈ {1, . . . , K},
l 6= k, i.e., dl is a Gaussian random vector with zero mean
and identity covariance matrix, despite of the practical Ml -ary
modulation used. Under the above Gaussian assumption for
the interference and hence the interference-plus-noise, after the
application of the whitening filter, the interference-plus-noise
RTk wk in (6) becomes a Gaussian random vector with zero
mean and identity covariance matrix. Under this assumption,
the optimal maximum likelihood (ML) detector for user k
can be shown to be equivalent to an Euclidean-distance based
detector: for each symbol transmitted by user k, its receiver
finds the constellation symbol dˆk which gives the smallest
distance of ||r k − |hkk |RTk Ak dˆk ||2 and declares it as the
transmitted symbol. Thereby, we are able to obtain a closedform expression for the PEP as shown in the following lemma,
while the accuracy of the above approximation approach for
the PEP will be validated later in Section V by simulations.
Lemma 2.1: Assuming the interference-plus-noise vector
wk is Gaussian distributed and the Euclidean-distance based
detector is used, the PEP at the receiver of user k is given by
P r{dk → d˜k |dk }
q


˜
|hkk | (dk − d˜k )T ATk W −1
k Ak (dk − dk )
,
= Q
2
(9)
where Q(x) ,
√1
2π
R∞
x
exp(−u2 /2)du.
III. M AXIMUM SER M INIMIZATION
In this section, we aim to minimize the maximum SER given
in (8) among all K users with improper signalling. This is
achieved by jointly optimizing the precoding matrices Ak ’s of
all users to minimize the maximum PEP given in (9) among
all different signal constellation pairs for each user k, and also
over all k’s. Given a set of transmit power constraints for the
users, Pk ’s, the optimization problem is thus formulated as
s,
min.
K s
Ak k=1
s.t. P r{dk → d˜k |dk } ≤ s,
T r(Ak ATk ) ≤ Pk ,
∀ d˜k 6= dk , k = 1, . . . , K.
(10)
Using (9), problem (10) is equivalent to the following
problem
t,
max.
K t
Ak k=1
˜
s.t. |hkk |2 (dk − d˜k )T ATk W −1
k Ak (dk − dk ) ≥ t,
T r(Ak ATk ) ≤ Pk ,
∀ d˜k 6= dk , k = 1, . . . , K.
(11)
Note that in the above problem, the first constraint for each
user k in fact corresponds to Dk = Mk (M2 k −1) number of
constraints, which are for all the different signal constellation
pairs, d˜k 6= dk . We denote F k as a Dk × 2 matrix which
consists of all the vectors dk − d˜k , ∀ d˜k 6= dk . For example, the corresponding F k for the normalized QPSK signal
constellation is
T
0 2 2 2 2
0
F k,QP SK =
.
(12)
2 2 0 0 −2 −2
Then the first set of constraints in (11) for each user k are
equivalent to
h
i
T
|hkk |2 F k AT W −1
≥ t, i = 1, . . . , Dk , (13)
k AF k
ii
where [Q]ii denotes the (i, i)-th element of a matrix Q. Hence,
we can express problem (11) equivalently as
t,
max.
K t
Ak k=1
h
i
T
s.t. |hkk |2 F k ATk W −1
A
F
≥ t,
k
k
k
ii
T r(Ak ATk ) ≤ Pk ,
i = 1, . . . , Dk , k = 1, . . . , K.
(14)
Problem (14) can be shown to be non-convex, and thus it is
in general difficult to find its optimal solution. Therefore, in the
following, we propose an efficient algorithm that is guaranteed
to find at least a locally optimal solution for problem
(14).
K
First, we introduce a set of auxiliary variables B k k=1
TABLE I: M INMAX -PEP A LGORITHM
and reformulate problem (14) as the following optimization
problem
t,
Ak
Kmin.
k=1
,
Bk
K
1.
t
2.
k=1
s.t. [Gk ]ii ≤ t,
T r(Ak ATk ) ≤ Pk ,
i = 1, . . . , Dk , k = 1, . . . , K.
(15)
In (15), we have
K
X
Gk =
|hkl |2 B Tk J (φkl )Al ATl J T (φkl )B l
− |hkk |B Tk Ak F Tk − |hkk |F k ATk B k +
p
Pk /2I, k = 1, . . . , K.
2
PK
σk
2
l=1,l6=k |hkl |
2 I +
−1
×J (φkl )Al ATl J T (φkl )
, k = 1, . . . , K.
K
Update Ak k=1 by solving problem (18).
K
Repeat steps 3 and 4 until both Ak k=1 and
K
B k k=1 converge within the prescribed accuracy.
Set B Tk = |hkk | F k ATk
IV. B ENCHMARK S CHEMES
l=1,l6=k
σk2
3.
4.
Initialize Ak =
B Tk B k , (16)
In this section, we present alternative approaches in the
existing
literature
and furthermore,
the
Kprecoding and decoding ma Kto optimize
and
R
trices,
i.e.,
A
k k=1
k k=1 , for the K-user SISO-IC
K
X
σk2
T
with
proper/improper
signalling.
It is worth pointing out that,
2
2
2
[Gk ]ii =
|hkl | ||bk,i J (φkl )al,1 ||
||bk,i || +
2
unlike
our
proposed
Minmax-PEP
algorithm, these benchmark
l=1,l6=k
schemes
might
not
be
originally
introduced
for minimizing the
K
X
T
T
T
2
2
SER
(or
PEP)
in
ICs
explicitly.
+
|hkl | ||bk,i J (φkl )al,2 || − 2|hkk |T r f k,i bk,i Ak ,
2
l=1,l6=k
(17)
where bk,i is the i-th column of the matrix B k ; ak,1 and ak,2
are the first and second column of the matrix Ak ; and f k,i is
the i-th row of the matrix F k , k = 1, . . . , K.
Since problem (15) can be shown
be convex over each
to
K
K
of the two sets Ak k=1 and B k k=1 when one of them
is given as fixed, we can apply the technique of alternating
optimization
Specifically,
Kto solve the problem
iteratively.
K
given Ak k=1 , the solution of B k k=1 can be obtained
by setting the gradient of [Gk ]ii given in (17) with respect
to bk,i as zero, i.e., ∂ [Gk ]ii /∂bk,i = 0. On the other hand,
K
K
optimizing Ak k=1 with given B k k=1 can be obtained
by solving the following convex problem by using primal-dual
interior point method, via, e.g., CVX [12].
t,
min.
K t
Ak k=1
s.t.
A. Proper Signaling with Power Control
For the K-user SISO IC with conventional proper signaling,
the
p pkprecoding and decoding matrices are reduced to Ak =
2 I and Rk = I, where pk ≤ Pk . Considering the use
of ML or Euclidean-distance based detection at each receiver,
user k detects that the symbol dˆk is transmitted if it gives
the smallest distance of ||r k − pk |h2 kk | dˆk ||2 among all signal
symbols. In this case, we search over all user power allocations
(p1 , . . . , pK ) subject to pk ≤ Pk and find the best (p1 , . . . , pK )
which achieves the minimum of max SERk , k = 1, . . . , K. We
refer to this scheme as Proper Signalling with Power Control
(PS-PC).
B. MSE-based Schemes
K
X
σk2
|hkl |2 ||bTk,i J (φkl )al,1 ||2
||bk,i ||2 +
2
l=1,l6=k
+
K
X
|hkl |2 ||bTk,i J (φkl )al,2 ||2
l=1,l6=k
− 2|hkk |T r f Tk,i bTk,i Ak ≤ t,
T r(Ak ATk ) ≤ Pk ,
k = 1, . . . , K, i = 1, . . . , Dk ,
(18)
To summarize, the proposed algorithm to solve problem
(15) is given in Table I, which is referred to as p
MinmaxPEP. Note that in Table I, we have initialized Ak = Pk /2I,
k = 1, . . . , K, i.e., assuming all users to employ conventional
proper signalling initially.
Although there has been no existing study on directly minimizing the SERs for ICs with improper signalling, minimizing
the MSEs has been considered as an alternative approach (see,
e.g., [10]). Following [10], we first define the MSE matrix for
r k as
E k = E (r k − dk )(r k − dk )T
K
2
X
T σk
|hkl |2 J (φkl )Al ATl J T (φkl )
I+
= Rk
2
l=1,l6=k
2
+ |hkk | Ak ATk Rk − |hkk |RTk Ak − |hkk |ATk Rk + I,
where we have used the following assumptions: E[dl dTk ] =
0; E[dl nTk ] = 0; and E[dk dTk ] = I. The minimization of
the sum of MSEs and the maximum per-stream MSE can be
expressed as the following optimization problems in (19) and
(20), respectively [10].
T r {E k }
k=1
T r(Ak ATk )
−1
10
≤ Pk , k = 1, . . . , K.
(19)
−0.2
10
2
s.t.
K
X
max(SER ,SER )
Kmin.
K
Ak k=1 , Rk k=1
0
10
s.t.
max max
k=1,...,K i=1,2
T r(Ak ATk )
[E k ]i,i
−2
10
1
Kmin.
K
Ak k=1 , Rk k=1
≤ Pk , k = 1, . . . , K.
(20)
The algorithms for solving the above two problems have
been given in [10] and are thus omitted here for brevity.
We thus denote the algorithms to solve (19) and (20) as
Minsum-MSE and Minmax-MSE, respectively. Note that the
convergence of these algorithms to (at least) a local optimum
of (19) or (20) is guaranteed. With the obtained precoding and
decoding matrices applied, the receiver of each user k finds
the nearest constellation symbol dˆk to r k given in (6), i.e.,
which gives the smallest distance of ||r k − dˆk ||2 , and then
declares it as the transmitted symbol.
−0.6
10
−3
10
−5
−4
10
−5
10
−5
0
PS−PC
Minsum−MSE
Minmax−MSE
MinIL−IA
MaxSINR−IA
Minmax−PEP
0
RTk J (φkl )Al = 0,
rank(RTk Ak ) = 1.
(21)
in which k, l ∈ {1, . . . , K}, l 6= k. In (21), the first and
second set of equations are for nullifying the interference and
enforcing the desired signal to span exactly one dimension
at each receiver, respectively. In this paper, we consider two
well-established IA-based schemes, i.e., interference leakage
minimization and SINR maximization [13], denoted as MinILIA and MaxSINR-IA, respectively.
Note that the above IA-based algorithms will yield the
precoding and decoding matrices simplified as 2 × 1 transmit
and receive beamforming vectors, denoted by {v k }K
k=1 and
,
respectively;
and
thus
each
user
needs
to
transmit
{uk }K
k=1
with only one-dimensional (1D) signal constellations. Therefore, for a fair comparison with other considered improper
signalling schemes in this paper which use two-dimensional
(2D) signal constellations, the transmitted symbols of IAbased schemes are assumed to be drawn from the 1D PAM
modulation with the same number of symbols and average
transmit power as other 2D modulation schemes. For example,
the corresponding constellation for the normalized QPSK is
5
10
15
20
SNR (dB)
Fig. 2: Comparison of the minimized maximum SER over
SNR in a two-user IC.
C. IA-based Schemes
It is known that the maximum sum-rate DoF of the equivalent K-user 2×2 real IC given in (2) with improper signalling
is 2 when K = 2, which is achieved when each of the
two users sends one data stream, based on the principle of
zero-forcing (ZF) [2]. In this subsection, we consider another
possible strategy to minimize the SER by applying IA-based
precoder and decoder designs for the K-user SISO-IC with
improper signalling. Specifically, we aim to find two sets of
K
K
{Ak }k=1 and {Rk }k=1 such that [13]
5
the normalized 4PAM, i.e.,
n √
√
√
√ o
−3 √25 , − √25 , √25 , 3 √25 .
−1
−1
1
1
−−→
,
,
,
1
−1
−1
1
V. N UMERICAL RESULTS
In this section, we present simulation results to compare
the SER performance of the proposed scheme with other
benchmark schemes in the K-user SISO-IC. First, Fig. 2
shows the results for the minimized maximum SER of the
users over SNR in a two-user IC with the channel coefficients
given by
1.9310e−j2.0228 0.7732ej0.5865
.
0.9249ej3.0213 2.3742ej0.2089
Here, we assume that both users apply 8PSK modulation.
Therefore, the equivalent constellation for IA-based schemes
is 8PAM. The SNRs of the two users are assumed to be
equal and are varied from 0 to 20 dB. We observe that
the proper signaling with power control (PS-PC) and MSEbased schemes result in saturated SERs for both users over
SNR. This is because under such schemes, both users’ signals
span two dimensions each, and thus the SER performance
is interference-limited with increasing SNR. In contrast, the
maximum SERs by the proposed Minmax-PEP and IA-based
schemes decrease over SNR. Similar to the IA-based schemes,
as the SNR increases, the precoding matrices A1 and A2
obtained from Minmax-PEP gradually converge to rank-1
matrices, i.e., each user’s precoded signal spans over only one
dimension although the original signal dk is drawn from a 2D
constellation (8-PSK). As a result, orthogonal transmissions
of the two users are achieved at each receiver and the system
SER is not limited by the interference as SNR increases. It is
VI. C ONCLUSION
0
10
In this paper, we study the problem of minimizing SERs
for the K-user SISO IC with improper signalling applied over
practical signal modulations. We propose an efficient algorithm
to jointly optimize the precoding matrices of users to directly
minimize the maximum SER, by exploiting the given signal
constellations and the additional degrees of freedom provided
by improper signalling. Several benchmark schemes based on
conventional proper signalling as well as other state-of-theart improper signaling designs are compared; and it is shown
by simulations that our proposed improper signalling scheme
achieves competitive or improved SER performances in SISOICs. Our study provides a different view on transmission
optimization for ICs in contrast to the large body of existing
works on the rate optimization of ICs.
−1
10
−0.1
max(SER1,SER2,SER3)
10
−2
10
−3
10
−0.7
10
−5
0
5
10
−4
10
−5
10
−6
10
−5
PS−PC
Minsum−MSE
Minmax−MSE
MinIL−IA
MaxSINR−IA
Minmax−PEP
0
5
R EFERENCES
10
15
20
25
30
SNR (dB)
Fig. 3: Comparison of the minimized maximum SER over
SNR in a three-user IC.
also observed that Minmax-PEP performs better than IA-based
schemes when SNR is less than 0 dB; however, as SNR further
increases, the SER performance of Minmax-PEP becomes
worse than that of IA-based schemes, which is mainly due
to the use of suboptimal noise whitening filters for MinmaxPEP in order to derive the closed-form PEP expression given in
(9). Thus, it is interesting to jointly optimize the decoding and
precoding matrices to further improve the SER performance
in our proposed improper signaling scheme, which will be left
to our future work.
Next, Fig. 3 compares the minimized maximum SER in a
three-user IC given by


1.9310e−j2.0228 0.7732ej0.5865
0.9766ej1.1907
 0.9249ej3.0213
2.3742ej0.2089 0.3009e−j1.5307  .
−j0.4282
1.7628e
0.3127e−j1.4959 2.1935ej1.7364
Here users (1, 2, 3) apply (QPSK, 8PSK, 8PSK) modulations, respectively. The equivalent constellations for IAbased schemes are hence (4PAM, 8PAM, 8PAM). The users’
SNRs are set to be equal and are varied from 0 to 30 dB.
It is observed that Minmax-PEP achieves substantial SER
improvement over all other schemes. Note that in this case,
the IA-based schemes cannot reduce SER over SNR, since
there is no feasible IA solution to (21) with given 1D signal
constellations for all the three users.
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