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arXiv:1202.5187v1 [cs.IT] 23 Feb 2012
Sphere Decoding for Spatial Modulation Systems
with Arbitrary Nt
Rakshith Rajashekar and K.V.S. Hari
Department of Electrical Communication Engineering
Indian Institute of Science, Bangalore 560012
{ rakshithmr, hari}@ece.iisc.ernet.in
Abstract—Recently, three Sphere Decoding (SD) algorithms
were proposed for Spatial Modulation (SM) scheme which
focus on reducing the transmit-, receive-, and both transmit
and receive-search spaces at the receiver and were termed as
Receiver-centric SD (Rx-SD), Transmitter-centric SD (Tx-SD),
and Combined SD (C-SD) detectors, respectively. The Tx-SD
detector was proposed for systems with Nt≤Nr, where Nt
and Nrare the number of transmit and receive antennas of the
system. In this paper, we show that the existing Tx-SD detector is
not limited to systems with Nt≤Nrbut can be used with systems
Nr< Nt≤2Nr−1as well. We refer to this detector as the
Extended Tx-SD (E-Tx-SD) detector. Further, we propose an E-
Tx-SD based detection scheme for SM systems with arbitrary Nt
by exploiting the Inter-Channel Interference (ICI) free property
of the SM systems. We show with our simulation results that
the proposed detectors are ML-optimal and offer significantly
reduced complexity.
Index Terms—Sphere decoding, ML decoding, spatial modu-
lation, space-time shift keying, complexity.
I. INTRODUCTION
Spatial Modulation (SM) [1], [2] is a recently devel-
oped low-complexity Multiple-Input Multiple-Output (MIMO)
scheme that exploits the channel for information transmission
in an unprecedented fashion. Specifically, the information
bitstream is divided into blocks of length log2(NtM)bits, and
in each block, log2(M)bits select a symbol sfrom M-ary
signal set (such as M-QAM or -PSK), and log2(Nt)bits select
an antenna out of Nttransmit antennas for the transmission
of the symbol s. The throughput achieved by this scheme
is R= log2(NtM)bpcu. Thus, the SM scheme achieves
an increase in spectral efficiency of log2Ntbits over single-
antenna systems with a marginal increase in the complexity
since it still needs only one RF chain at the transmitter.
However, in order to achieve high throughputs either Ntor
M, or both need to be increased which renders this scheme
suitable for low and moderately high spectral efficiencies.
Recently, three specially tailored Sphere Decoding (SD)
detectors were proposed for SM systems in [3] which were
termed as Receiver-centric SD (Rx-SD) [4], Transmitter-
centric SD (Tx-SD), and Combined-SD (C-SD) detectors. It
was shown in [3] that the Rx-SD detector is suitable for SM
systems with large Nr, the number of receive antennas, and
C-SD detector is suitable for systems operating at relatively
The financial support of the DST, India is gratefully acknowledged.
high spectral efficiencies i.e., with large Ntor M, or both. But,
the applicability of the existing Tx-SD detector and hence, the
C-SD detector is limited to systems with Nt≤Nr, which is
due to the zero diagonal entries in the Rmatrix of the QR
decomposition associated with the underdetermined channel
matrix. This problem is inherent to any MIMO system with
underdetermined channel matrix that employs SD detector at
the receiver. In [5], this problem was addressed by transform-
ing the underdetermined channel matrix into a full-column
rank matrix and applying standard SD on it. But, this SD
which was termed as λ-Generalized SD (λ-GSD) detector is
asymptotically Maximum Likelihood (ML) optimal with SNR
for non-constant modulus signal sets such as M-QAM. Thus,
at low Signal-to-Noise Ratios (SNR) the performance of the
λ-GSD detector cannot be expected to be near-ML. The GSD
of [6] is also not ML-optimal for non-constant modulus signal
sets under all SNR conditions like the λ-GSD detector, and
also the complexity offered by the GSD of [6] is significantly
higher than that of the λ-GSD [5]. In this paper, we do not
take these approaches, instead, we show that the existing Tx-
SD detector of [4] is not limited to SM systems with Nt≤Nr
but is applicable to systems with Nr< Nt≤2Nr−1as
well, we refer to this detector as the Extended Tx-SD (E-Tx-
SD) detector, and further, we propose an ML-optimal detection
scheme termed as Generalized Tx-SD (G-Tx-SD) detector for
SM systems with arbitrary Nt.
II. SYSTEM MODEL
We consider a MIMO system having Nttransmit as well as
Nrreceive antennas and a quasi-static, frequency-flat fading
channel, yielding:
y=Hx +n,(1)
where x∈CNt×1is the transmitted vector, y∈CNr×1is
the received vector, H∈CNr×Ntis the channel matrix, and
n∈CNr×1is the noise vector. The entries of the channel
matrix and the noise vector are from circularly symmetric
complex-valued Gaussian distributions C N (0,1) and CN (0,
σ2), respectively, where σ2
2is the noise variance per dimen-
sion.
A. Spatial Modulation
In SM scheme [1], we have
x= [0,...,0
|{z }
l−1
, s, 0,...,0
|{z }
Nt−l
]T∈CNt×1,(2)
where sis a complex symbol from the signal set Swith
|S|=M. Throughout this paper we assume Sto be a lattice
constellation such as QAM. Thus, for an SM system, eq.(1)
becomes
y=Hxl,s +n,(3)
where l∈L={i}Nt
i=1 and the subscript scaptures the
dependence of xon the signal set S. Assuming perfect channel
state information and ML decoding at the receiver, we have
(ˆ
l, ˆs)ML = arg min
l∈L,s∈Sky−Hxl,sk2
2,(4)
= arg min
l∈L,s∈S(Nr
X
i=1
|yi−hl,is|2),(5)
where yiand hl,i are the ith and the (l, i)th entry of the
received vector yand the channel matrix H, respectively.
The computational complexity in terms of number of real
multiplications involved in computing ML solution of eq.(5)
is given by
CNt×Nr
ML = 8M NtNr,(6)
since, 8 real multiplications are required in computing |yi−
hl,is|2for any legitimate (s, l, i). If Nt≥Nr, then from the
well known division algorithm we have unique non-negative
integers qand r(0≤r < Nr) such that Nt=qNr+r. Thus,
from eq.(6) we can write
CNt×Nr
ML =C(q Nr+r)×Nr
ML =qCNr×Nr
ML +Cr×Nr
ML .(7)
B. Review of Tx-SD detector [4]
The complex-valued system in eq.(3) can be expressed in
terms of real variables as
ℜ(y)
ℑ(y)
|{z }
¯
y
=ℜ(H)−ℑ(H)
ℑ(H)ℜ(H)
|{z }
¯
H
ℜ(xl,s)
ℑ(xl,s)
|{z }
¯
xl,s
+ℜ(n)
ℑ(n)
|{z }
¯
n
,
(8)
where, ℜ(·)and ℑ(·)represent the real and imaginary parts,
¯
yand ¯
n∈R2Nr×1,¯
H∈R2Nr×2Nt, and ¯
xl,s ∈R2Nt×1.
For Nt≤Nr, the Tx-SD detector [4] is given by
(ˆ
l, ˆs)T x−SD = arg min
(l,s)∈ΘR
k¯
y−¯
H¯
xl,sk2
2,(9)
where, ΘR=(l, s)|l∈L, s ∈S, and k¯
y−¯
H¯
xl,sk2
2≤R2
with Ras the initial search radius of the sphere decoder.
From [4] and [7], we have R=αNrσ2, where α
is a parameter chosen to maximize the probability
of detection. By expressing ¯
Hin terms of its QR
decomposition we have k¯
y−¯
H¯
xl,sk2
2≤R2equivalent
to k¯
z−¯
R¯
xl,sk2
2≤R2
Qwhere, ¯
Ris an upper triangular
matrix given by ¯
R1 (2Nt×2Nt)
0(2Nr−2Nt×2Nt),¯
z=¯
QT
1¯
ywith
Fig. 1. Pictorial representation of the two level SD tree associated with
each of the elements of xl,s. The dependence of ¯xion ¯xi+Ntfor each iis
indicated by grouping their branches in the same block.
¯
Q=¯
Q1 (2Nr×2Nt)¯
Q2 (2Nr×2Nr−2Nt)such that ¯
H=¯
Q¯
R,
and R2
Q=R2− k ¯
QT
2¯
yk2. Thus, we have
ΘR=(l, s)|l∈L, s ∈S, and k¯
z−¯
R1¯
xl,sk2
2≤R2
Q.
(10)
Since ¯
R1is upper triangular and ¯
xl,s has only two non-zero
elements (from eq.(2) and eq.(8)) the elements of ΘRare given
by −RQ+ ¯zi
¯ri,i
≤¯xi≤RQ+ ¯zi
¯ri,i
,(11)
for Nt+ 1 ≤i≤2Ntand
−RQ+ ¯zi−¯ri,i+Nt¯xi+Nt
¯ri,i
≤¯xi≤RQ+ ¯zi−¯ri,i+Nt¯xi+Nt
¯ri,i
,
(12)
for 1≤i≤Nt, where, ¯xiand ¯ziare the ith entries of ¯
xl,s
and ¯
zrespectively, and ¯ri,j is the (i, j )th entry of the upper
triangular matrix ¯
R1. From eqs. (11) and (12) we have the SM
specific SD tree shown in the Fig.(1) where Nindicates the
number of signal points in the real and imaginary dimensions
of the constellation. For example, in a 64-QAM constellation,
M= 64 and N= 8.
III. PROPOSED DETECTOR FOR SM SYSTEMS WITH
ARBITRARY Nt
The Tx-SD detector discussed in the previous section was
proposed for SM systems with Nt≤Nr. In this section, we
show that the existing Tx-SD detector is applicable to systems
with Nr< Nt≤2Nr−1as well, and further extend it to
systems with arbitrary Nt.
A. Tx-SD detector in SM systems with Nr< Nt≤2Nr−1
P roposition 1 : The Tx-SD detector [4] originally pro-
posed for SM systems with Nt≤Nris applicable to systems
with Nr< Nt≤2Nr−1as well.
Proof: Consider an SM system with
Nt> Nr. The QR decomposition of ¯
His
given by ¯
Q¯
Rwhere ¯
Q=¯
Q1 (2Nr×2Nr)and
¯
R=¯
R1 (2Nr×2Nr)¯
R2 (2Nr×2Nt−2Nr)where ¯
R1is
an upper triangular matrix. Recall that only two elements are
non-zero in ¯
xl,s and are apart by Nt−1zero elements as
shown below.
¯
xl,s = [0,...,0
|{z }
l−1
,ℜ(s),0,...,0
| {z }
Nt−1
,ℑ(s),0,...,0
|{z }
Nt−l
]T∈R2Nt×1.
(13)
Considering l=Ntand some s∈Swe have,
¯
xNt,s = [0,...,0
|{z }
Nt−1
,ℜ(s),0,...,0
| {z }
Nt−1
,ℑ(s)]T∈R2Nt×1,(14)
and the last element of ¯
p=¯
R¯
xNt,s is given by ¯p2Nr=
¯
R(2Nr,:)¯
xNt,s. It is easy to see that ¯
R(2Nr,:) has first 2Nr−
1elements zero since ¯
R1is upper triangular, and if the number
of non-zero elements 2Nt−2Nr+ 1 in ¯
R(2Nr,:) is less than
or equal to Nt= (Nt−1) + 1, the number of zeros between
ℜ(s)and ℑ(s)plus one non-zero element ℑ(s)in ¯
xNt,s, we
see that ¯p2Nrdepends only on ℑ(s). Since this is true for all
{¯pi}2Nr
i=2 and any legitimate pair (l, s), we have the condition
2Nt−2Nr+ 1 ≤Nt≡Nt≤2Nr−1for independent
detection of imaginary components in ¯
xl,s. This results in the
following intervals analogous to those in eq.(11),
−RQ+ ¯z2Nr−i+1
¯r2Nr−i+1,2Nt−i+1
≤¯x2Nt−i+1 ≤RQ+ ¯z2Nr−i+1
¯r2Nr−i+1,2Nt−i+1
,(15)
for 1≤i≤Nt. Proceeding in the lines similar to that of
[4] for real components in ¯
xl,s we get the intervals of eq.(16)
given in the next page. From eq.(16) we observe that ¯
x, With
Nt=Nr, it can be checked that the intervals of eq.(15) and
eq.(16) reduce to those of eq.(11) and eq.(12), respectively.
We refer to this Tx-SD detector as the Extended Tx-SD (E-
Tx-SD) detector in the rest of the paper.
B. Tx-SD detector for SM systems with arbitrary Nt
We have shown in the previous subsection that the Tx-SD
detector of [4] can be used in SM systems with Nr≤Nt≤
2Nr−1. In this subsection we propose a SD detection scheme
for arbitrary Ntby partitioning the antenna search space into
disjoint subsets each of size 2Nr−1and running E-Tx-SD
decoders sequentially.
Consider an SM system with Nt>2Nr−1 = N′. From
the division algorithm we have unique non-negative integers
qand rsuch that Nt=qN ′+r, where 0≤r < N ′. For the
ease of presentation we assume r= 0 and hence Nt=qN ′
for now, and later generalize our results for non-zero r. Let L,
the set of antenna indices, be partitioned into disjoint subsets
Lk={i}kN ′
i=(k−1)N′+1 so that L=Sq
k=1 Lkand |Lk|=N′
for all 1≤k≤q.
Let J(l, s)represent the ML metric PNr
i=1 |yi−hl,is|2of
eq.(5). Then, the ML solution in terms of J(l , s)is (ˆ
l, ˆs)ML =
arg minl∈L,s∈SJ(l, s)and we have
min
l∈L,s∈SJ(l, s)a
= min
l∈L{min
s∈SJ(l, s)},(17)
b
= min "q
[
k=1 min
l∈Lk
{min
s∈SJ(l, s)}#,(18)
= min "q
[
k=1
Θ(k)#,(19)
where Θ(k) = minl∈Lk{mins∈SJ(l, s)}. In the above,
(a) follows from the assumption that the antenna index and
the transmitted symbol are encoded by independent sets of
bits, and (b) follows directly from the fact that Lpartitions
into Lk’s. It is straightforward that each of the Θ(k)’s can
be obtained by running E-Tx-SD detector discussed in the
previous subsection. The sphere radius Rkfor all 1≤k≤q
is taken as R=αNrσ2. Thus, we have
Θ(k) = (l, s)|l∈Lk, s ∈S, and k¯
zk−¯
Rk¯
xl,sk2
2≤R2
Q,k,
(20)
where, ¯
Qk¯
Rk=¯
H(:,[Lk]),R2
Q,k =R2and ¯
zk=¯
QT
k¯
y
for all 1≤k≤q. Equations (15) and (16) directly give the
elements of Θ(k)for all 1≤k≤q.
Now, for a system with Nt=qN′+rand r6= 0, it is
straightforward that there is an additional set Θ(q+ 1) in
eq.(19) which is same as eq.(20) with Lq+1 =L\Sq
k=1 Lk.
Thus, the E-Tx-SD based solution for systems with arbitrary
Ntis given by
(ˆ
l, ˆs)E−T x−SD = arg min
(l,s)∈Sq+1
k=1 Θ(k)
k¯
y−¯
H¯
xl,sk2
2.(21)
This solution is referred to as the Generalized Tx-SD (G-Tx-
SD) detector. We note here that the partitions Lkconsidered
here are of size N′= 2Nr−1, it is straightforward that the
detector in eq.(21) can be run over partitions of any size, for
example N′=Nras well.
From eq.(7) it is clear that the ML complexity scales linearly
with the number of transmit antennas. As the proposed G-
Tx-SD detector runs over partitioned channel blocks, it is
obvious that the complexity of the proposed detector can be
expected to be lesser than that of the ML detector, since the
complexity of the individual Tx-SD/E-Tx-SD detector is much
lesser than that of the ML detection [4]. We note here that the
proposed E-Tx-SD and G-Tx-SD detectors are not limited to
SM scheme alone, but are applicable to any system with the
ICI-free property, for example, the Space-Time Shift Keying
[8]. The C-SD detector proposed in [4] uses both the Tx-SD
and the Rx-SD detectors in order to reduce the overall search
complexity. It is straightforward that the proposed detectors
can be further extended by incorporating Rx-SD detector for
reduction in the receive search space complexity as well.
IV. SIMULATION RESULTS
Consider an SM system with Nr= 4 and Nt= 7,
employing 16- and 64-QAM signal sets. Let NM L =MNt
denote the number of search points in the ML detection, and
E[NE−T x−SD ]denote the expected number of points that lie
inside the hypersphere of radius Runder E-Tx-SD detection.
in this paper, we use the metric E[NE−T x−SD ]/NML to
measure the reduction in search space with respect to the
ML detection. From Fig.(2), it is clear that the Symbol Error
Rate (SER) performance of the ML and the E-Tx-SD detectors
overlap for both the modulation schemes considered, and from
−RQ+ ¯zNt−i+1 −¯rNt−i+1,2Nt−i+1 ¯x2Nt−i+1
¯rNt−i+1,Nt−i+1
≤¯xNt−i+1 ≤RQ+ ¯zNt−i+1 −¯rNt−i+1,2Nt−i+1 ¯x2Nt−i+1
¯rNt−i+1,Nt−i+1 for 1≤i≤Nt
(16)
4 6 8 10 12 14 16 18 20 22 24
10−4
10−3
10−2
10−1
100
SNR (dB)
SER
SER comparison of ML and E−Tx−SD detectors
ML (M=16)
ML (M=64)
E−Tx−SD (M=16)
E−Tx−SD (M=64)
M = 64
M = 16
Fig. 2. SER curves of ML and E-Tx-SD detectors in an SM system with
Nr= 4,Nt= 7, employing 16- and 64-QAM signal sets.
Fig.(3) we see that the expected number of nodes that lie
inside the sphere reduces significantly with increase in SNR.
Specifically, at an SNR of 20 dB, we see a reduction of about
90% and 70% for 16- and 64-QAM signal sets, respectively.
Fig.(4) shows the SER performance and the reduction in
complexity due to the G-Tx-SD detector in an SM system
with Nr= 4,Nt= 8, and 16-QAM signal set. We observe
from Fig.(4)(a) that the SER curves of the proposed and the
ML detector overlap, and from Fig.(4)(b) that there is about
85% reduction in complexity with respect to ML detection at
an SNR of about 18 dB. The partitions considered here are of
size |L1|=|L2|= 4. The reduction in complexity involved in
precomputations such as QR decomposition, etc., by using the
E-Tx-SD detector with different |Li|’s instead of the Tx-SD
detector with equal |Li|’s is left for our future study.
V. CONCLUSION
We have shown that the existing Tx-SD detector is not
limited to SM systems with Nt≤Nrbut is applicable to
systems with Nr< Nt≤2Nr−1as well. Further, we
have proposed a generalized Tx-SD detector for SM systems
with arbitrary Ntby exploiting the ICI-free nature of the
system, and have shown with our simulation results that the
proposed detectors give ML performance with significantly
reduced complexity.
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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1
SNR (dB)
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