A Low-Complexity Detection Scheme for Differential Spatial Modulation

Abstract

Differential spatial modulation (DSM), which does not require the channel state information at the receiver, is an attractive alternative to its coherent counterpart. The optimal maximum-likelihood (ML) detector of the DSM system employs the classic block-by-block method for jointly detecting the activated antenna matrix (AM) and the modulation symbols, resulting in high computational complexity. In this letter, a low-complexity near-ML detector, which operates on a symbol-by-symbol basis, is proposed for the DSM scheme. Specifically, in each block, the index of the activated transmit antenna and modulation symbol in each time slot are first obtained, and then, these antenna indices are utilized to simply determine the index of the activated AM. Simulation results show that the proposed algorithm is capable of offering almost the same performance as that of the ML detector with more than 90% reduction in complexity.

Full text

IEEE COMMUNICATIONS LETTERS 1
A Low-complexity Detection Scheme for Differential
Spatial Modulation
Lixia Xiao, Ping Yang, Xia Lei, Yue Xiao, Shiwen Fan, Shaoqian Li and Wei Xiang
Abstract—Differential spatial modulation (DSM),
which does not require the channel state information
(CSI) at the receiver, is an attractive alternative to
its coherent counterpart. The optimal maximum like-
lihood (ML) detector of the DSM system employs the
classic block-by-block method for jointly detecting the
activated antenna matrix (AM) and the modulation
symbols, resulting in a high computational complex-
ity. In this Letter, a low-complexity near ML detec-
tor, which operates on a symbol-by-symbol basis, is
proposed for the DSM scheme. Specifically, in each
block, the index of the activated transmit antenna (TA)
and modulation symbol in each time slot are firstly
obtained, and then these antenna indices (AIs) are
utilized to simply determine the index of the activated
AM. Simulation results show that the proposed algo-
rithm is capable of offering almost the same perfor-
mance as that of the ML detector with more than 90%
reduction in complexity.
Index Terms—Differential spatial modulation
(DSM), maximum likelihood (ML) detection, symbol-
based-symbol
I. Introduction
DIFFERENTIAL spatial modulation (DSM) [1]-[4]
is a novel multiple-input multiple-output (MIMO)
wireless transmission technique, which relies on a single
radio-frequency (RF) transmit structure without the need
of the channel state information (CSI). It is an attractive
alternative to the coherent spatial modulation (SM) [5]-[8],
which is considered as a promising transmission technique
for large-scale MIMO systems in terms of both theoretical
researches and practical implementations [9]-[10].
In DSM, one out of Qantenna matrices (AMs) is
activated to dispense Ntsymbols to Nttransmit antennas
(TAs) in Nttime instants. Therefore, high-data transmis-
sion is attainable in comparison with the differential space-
time shift keying (DSTSK) scheme [11], where only a single
symbol is transmitted in a space-time block.
This work was supported in part by the National Science Foun-
dation of China under Grant number 61471090, the National
Basic Research Program of China under Grant 2013CB329001,
the Fundamental Research Funds for the Central Universities
(No.ZYGX2013J112), the Program for New Century Excellent Tal-
ents in University (NCET-11-0058), and the open research fund of
National Mobile Communications Research Laboratory, Southeast
University (No. 2013D05). L. Xiao, P. Yang, X. Lei, Y. Xiao, S.
Fan and S. Li are with the National Key Laboratory of Science
and Technology on Communications, University of Electronic Sci-
ence and Technology of China, Chengdu 611731, China (e-mail:
xiaoyue@uestc.edu.cn and yang.ping@uestc.edu.cn). Y. Xiao is also
with the National Mobile Communications Research Laboratory,
Southeast University. W. Xiang is with the School of Mechani-
cal and Electrical Engineering, University of Southern Queensland,
Toowoomba, QLD 4350, Australia.
For the current DSM detector, it relies on the clas-
sic block-by-block-based maximum-likelihood (ML) de-
tection, where the AM indices and the amplitude and
phase modulation (APM) symbols are jointly detected
in a space-time block. Hence, the complexity of the ML
detector grows exponentially with the size of the con-
stellation set and the number of TAs. Note that some
low-complexity non-coherent detectors [11]-[13] have been
proposed for DSTSK. However, these detectors are not di-
rectly applicable to DSM, because multiple APM symbols
are transmitted in a single space-time block in the DSM
system. This is also the reason that the low-complexity
algorithms designed for coherent detection [14]-[16] are not
suitable for DSM systems.
Against this background, a novel low-complexity near
ML algorithm, which operates on a symbol-by-symbol ba-
sis, is proposed for DSM. The complexity of the proposed
detector is independent of the size of the APM symbols.
Specifically, the antenna index (AI) and the APM symbol
in each time slot are estimated firstly for each space-
time block. If the obtained NtAIs form a legitimate
AM, the output is taken as the final detection result.
Otherwise, we choose Pmost likely legitimate AMs for
further search. The selection principle is able to maximize
the number of identical elements between these AI vectors
and the original detected AIs. Furthermore, the proposed
algorithm is extended to large-scale DSM-MIMO system
with the aid of index mapping proposed in [4].
II. System Model
Consider a DSM system with Nttransmit and Nr
receive antennas. For the DSM scheme, one out of QAMs
is activated to disperse Ntsymbols to NtTAs in Nttime
slots. In the DSM system, there are total Nt!AMs. In fact,
only Q= 2blog2(Nt!)cAMs are permitted for conveying
information bits, where b·c is the floor operator. For each
(Nt×Nt)-element full-rank AM, there is only one nonzero
element in each column. Hence, each AM Aq(q∈(1, Q))
corresponds to one unique AI vector Lq= (l1
q, l2
q, ..., lNt
q),
where lj
q,(j∈(1, Nt)) is the activated index of the j-th
column of Aq. Therefore, QAMs correspond to an AI set
with Q vectors, i.e., L= (L1, ..., Lq, ..., LQ).
For each group of Nttime slots, the information bits
of length Bare divided into two parts: 1) B1= log2(Q)
bits are used to map one of the AMs Aq; and 2) B2=
log2(L)Ntbits are modulated to NtL-PSK symbols that
are transmitted by the activated AM Aq. As a result, the
k-th space-time block signal is expressed as
Xk=Aqdiag[s1, s2..., sNt],(1)

IEEE COMMUNICATIONS LETTERS 2
where diag[·]returns a square diagonal matrix with the
elements of vector and sj, j ∈(1, Nt)denotes the L-PSK
symbol. Differential encoding of DSM can be expressed as
Sk=Sk−1Xk,(2)
where S0is the (Nt×Nt)-element identity matrix.
At the receiver, the received k-th space-time block Yk∈
CNr×Ntis modeled as
Yk=HkSk+nk,(3)
where Hk∈CNr×Ntand nk∈CNr×Ntdenote the channel
matrix and the noise matrix, whose elements obey the
complex Gaussian distributions CN (0,1) and CN 0, σ2,
respectively. Assuming that channel coefficients remain
approximately unchanged over Nttime slots, i.e., Hk−1≈
Hk, the received signal Ykcan be represented as
Yk=Yk−1Xk+Nk,(4)
where Nk=nk−nk−1Xk. Then the classic block-by-block
ML detector can be expressed as
(ˆ
Xk)ML = arg min
Xk∈χ
kYk−Yk−1Xkk2,(5)
where χis the set of DSM transmit vectors with a size of
QLNt, which is shown in Table I. As can be seen, the sizes
of Qand χincrease exponentially with Land Nt.
TABLE I
Sizes of Qand χwith different system setups.
(Nt,L) (4,4) (6,4) (6,8) (8,4) (8,8) (16,4) (16,8)
Size
of Q242929215 215 244 244
Size
of χ212 221 227 231 239 276 292
III. Proposed Detection Algorithm for DSM
A. Proposed Detection Algorithm
In this section, a novel low-complexity symbol-by-
symbol detector is proposed for DSM to attain near-
optimal performance with reduced complexity. Specifical-
ly, the AI of the TA and the symbol in each time slot are
estimated in each time-space block. If the obtained AIs
forms a legitimate AM, they alongside the symbol vector
are taken as the final results. Otherwise, we choose Pmost
likely legitimate AMs for further search.
Firstly, the NtAIs and the corresponding symbol vector
are estimated. According to Eq.(4), the receiver signal
Yi
k∈CNr×1,(i∈(1, Nt)) can be obtained as
Yi
k=Yk−1Xi
k+Ni
k,(6)
where Yi
k,Xi
kand Ni
kdenote the i-th column of Yk,
Xkand Nk, respectively. Since there is only one nonzero
element in Xi
k, the activated index and the symbol can be
estimated by the HL-ML detector as [16]
(ˆ
li,ˆsi)ML = arg min
li∈(1,...,Nt)
(|yli
k−ˆsi|2− |yli
k|2)kYli
k−1k2,(7)
where
yli
k=(Yli
k−1)HYi
k
(Yli
k−1)HYli
k−1
,ˆsi=Q(yli
k),(8)
where Qis defined as the digital demodulation function.
After NtAIs are estimated by Eq.(7), we can obtain the
AI vector ˆ
Lk= (ˆ
l1, ..., ˆ
lNt)and the symbol vector ˆsk=
(ˆs1, ..., ˆsNt). Then the number of the identical elements
between ˆ
Lkand L= (L1, ..., Lq, ..., LQ)can be given as
N= [N1, ..., Nq, ..., NQ],(9)
where Nqis number of overlapped elements between ˆ
Lk
and Lq. Then the element in Nis sorted in descending
order as ˆ
N=hˆ
N1, ..., ˆ
Nq, ..., ˆ
NQi, and the corresponding
index order is given by
m= [m1, ..., mq, ..., mQ],(10)
where m1and mQdenote the indices of ˆ
N1and ˆ
NQ,
respectively. If the condition of ˆ
N1=Ntis satisfied, the
obtained ˆ
Lkis considered as a legitimate solution, and
thus the achieved (m1,ˆsk)is taken as the final output.
If ˆ
N1< Nt,ˆ
Lkis considered as a illegitimate solution.
Moreover, the largest element in Nmay not be unique,
and we define QMas the number of the largest elements
in N. In this case, we choose first P(P>QM)legitimate
AMs in Eq.(10) for detection, which can be expressed as
(ˆq) = arg min
q∈(m1,...,mP)
kYk−Yk−1Aqdiag(ˆsq)k2,(11)
where ˆsq= (s1
q, s2
q, ..., sNt
q)is obtained using Eq.(8).
B. Extended Version for Large-scale DSM-MIMO
As shown in Table I, the size of Qgrows exponentially
as the number of TAs increases. This makes the design
of the AMs to become a challenge. To tackle this issue,
an effective method of index mapping principle for DSM
was introduced in [4]. In this subsection, we extend the
proposed algorithm in Section III-A to large-scale DSM
systems by using the mapping method. The extension to
large-scale DSM focuses mainly on searching QMlegiti-
mate AMs, which are most likely to be the obtained ˆ
Lk.
1) Index mapping of large-scale DSM-MIMO: As shown
in [4], given a length of B1= log2(Q)bits denoted by a bi-
nary sequence [a(m)
1, ..., a(m)
B1], they are firstly transformed
into a decimal integer m∈(0, Q −1). Then, the number
mis expressed in factorial representation as
m=b(m)
1(Nt−1)! + · · · +b(m)
Nt0!,(12)
where b(m)= (b(m)
1, ..., b(m)
Nt)denotes the corresponding
factorial sequence. In [4], the sequence b(m)is mapped
to a permutation vector Lq= (l1
q, l2
q, ..., lNt
q). Specifically,
we first have l1
q=θb(m)
1
, where θ= [1,2, ..., Nt]. Then,
the element θb(m)
1
is removed from the set θ, so that we
have l2
q=θb(m)
2
. After that, θb(m)
2
is removed from θ. This
process is repeated until lNt
qis obtained.
2) Index de-mapping of large-scale DSM-MIMO: For a
legitimate AI vector Lq, it is de-mapped to a factorial
sequence b(m)= (b(m)
1, ..., b(m)
Nt)in [4]. Specifically, ˆ
lNtis
firstly added to ˜
θ, which is initialized as an empty vector
and we have b(m)
Nt= 0. Then the term ˆ
lNt−1is added to the

IEEE COMMUNICATIONS LETTERS 3
set ˜
θ, which is then sorted in ascending order. After that,
the index of ˆ
lNt−1in the ordered ˜
θis obtained as `Nt−1
and we have b(k)
Nt−1=`Nt−1−1. This process is repeated
until b(m)
1is obtained. Finally, the obtained vector b(m)
can be transformed to an integer ˆm.
3) Extension of the proposed algorithm: The proposed
algorithm in Section III-A is extended to large-scale DSM-
MIMO systems with the aid of index mapping and de-
mapping, as shown in Fig. 1. As can be seen from the fig-
ure, one first checks whether the estimated ˆ
Lkis legitimate
through index mapping and de-mapping. When ˆ
Lkfalls
within the legitimated set, it is mapped to a corresponding
AM. If the resultant AM is a full-rank matrix, and the
de-mapped ˆmsatisfies that ˆm∈(0, Q −1), ( ˆm,ˆsk)is
taken as the final solution. Otherwise, the obtained ˆ
Lkis
regarded as illegitimate, and one modifies some elements
in ˆ
Lkfollowing the principle of maximizing the number of
identical elements between the modified AI vector and the
obtained ˆ
Lk. Finally, all the legitimate AMs are selected
for detection using Eq.(11).
start
AM is
full-rank
yes
no
ˆ
(0, 1)
m Q
∈ −
end
ˆ
k
L
yes
no
1
ˆ
ˆ
( ,..., )
ˆ
t
k N
l l
=L
Index de-
mapping
Detection based on
Eq. (11)
Change
Fig. 1. The process of the extended detector for large-scale DSM.
In Fig. 1, if the AM is full rank and the de-mapped
number satisfies ˆm > (Q−1), the most likely erroneous
AI is found according to the index de-mapping factorial se-
quence b(m), and then amended to other legitimate values.
According to Eq.(12), the inequality of b(m)
j(Nt−j)! 6
m−(b(m)
1(Nt−1)!+, ..., +b(m)
j−1(Nt−j+ 1)!) should hold
for the estimated b(m)
j. Otherwise, the index ˆ
ljin vector
ˆ
Lkmay be incorrect. In this case, we exchange ˆ
ljwith
ˆ
lj+1, ..., ˆ
lNt, respectively, to obtain Nt−jchanged AI
vectors, corresponding to Nt−jpossible AMs.
On the other hand, if the AM is not a full-rank matrix,
this implies that some elements in ˆ
Lkare identical, and
hence some AIs are not included in the set ˆ
Lk. In this case,
we replace these identical elements in ˆ
Lkwith these non-
included AIs to make the AM to be a full-rank matrix.
Assume that there are tgroups of repeated AIs and ˆnt
non-included AIs. These ˆntnon-included AIs are able to
form ˆnt!combinations.
For these elements of tgroups, we keep only one element
for each, the rest elements are substituted by the remain-
ing ˆntindices. Assuming that each group has ni(i∈(1, t))
identical elements, there are Qt
i=1 niˆnt!possible AMs, and
all the legitimate AMs are selected for detection. For
example, letting the number of the TA be Nt=16 and
ˆ
Lk= (2,1,4,3,6,5,1,8,13,16,9,12,4,14,7,10), we have
t=2→ {1,1},{4,4}, n1=2, n2=2.(13)
Then, according to ˆ
Lk, the non-included ˆntAIs are
ˆnt=2→ {11,15}.(14)
Hence, there are 8possible AMs given as
ˆ
L1
k= (2,1,4,3,6,5,11,8,13,16,9,12,15,14,7,10)
ˆ
L2
k= (2,1,4,3,6,5,15,8,13,16,9,12,11,14,7,10)
ˆ
L3
k= (2,1,11,3,6,5,15,8,13,16,9,12,4,14,7,10)
ˆ
L4
k= (2,1,15,3,6,5,11,8,13,16,9,12,4,14,7,10)
ˆ
L5
k= (2,15,4,3,6,5,1,8,13,16,9,12,11,14,7,10)
ˆ
L6
k= (2,11,4,3,6,5,1,8,13,16,9,12,15,14,7,10)
ˆ
L7
k= (2,11,15,3,6,5,1,8,13,16,9,12,4,14,7,10)
ˆ
L8
k= (2,15,11,3,6,5,1,8,13,16,9,12,4,14,7,10)
(15)
In summary, denote by ˆ
QMthe number of legitimate
AMs corresponding to the changed AI vectors in the
large-scale DSM-MIMO detection. ˆ
QMmay be larger than
QMobtained by Eq.(10), which imposes extra complexity.
However, this extension will still be attractive for large-
scale DSM-MIMO detection, as shown in Section IV.
C. Complexity Analysis
The complexities of the proposed algorithm and the ML
detector are evaluated in terms of the number of real-
valued multiplications, which can be counted as
Cproposed =C1γ1+γ2(C1+ (2NrNt+ 4NrNt2)P)
CML = (4NrNt+ 2NrNt)2blog2(Nt!)cLNt,(16)
where C1= (6Nr+ 11)Nt+ (4Nr+ 11)Nt(Nt−1) is the
complexity of the proposed detector with legitimate ˆ
Lk,γ1
and γ2(γ1+γ2= 1) denote the percentages of legitimate
and illegitimate ˆ
Lk, respectively. As shown in Eq.(16), the
complexity of the proposed detector is independent of the
size of APM symbols, and dominated by γ1,γ2and P.
IV. Simulation Results
In this section, computer simulation are carried out over
Rayleigh flat fading channels. Fig. 2 plots the bit error
rate (BER) of the DSM system with Nt= 6 and Nt= 16
using the proposed detector. For comparative purposes,
the curves of ML detector of DSM and SM with perfect
CSI are added to Fig. 2(a) and Fig. 2(b), respectively.
Furthermore, the BER performance and the complexity
of the proposed algorithm are dominated by γ1, as shown
in Fig. 3(a). These values of γ1are statistic simulation
results and equivalent to r1/r, where r1is the number
of legitimate ˆ
Lkand ris the total number of legitimate
and illegitimate ˆ
Lk. Finally, the complexity comparison
between the proposed detector and the ML detector is
shown in Fig. 3(b).

IEEE COMMUNICATIONS LETTERS 4
As can be seen from Fig. 3(a), γ1increases with the
SNR. Specifically, γ1is able to reach 100% at the SNRs of
12 dB and 6 dB with Nt= 6 and Nt= 16, respectively.
When the condition γ1= 100% is met, the performance
of the proposed detector is the same as that of the ML
detector. Meanwhile, γ2decreases as SNR increases, and
approaches zero at high SNRs. As a result, the complexity
of the proposed detector drops as the SNR increases, and
becomes a constant at high SNRs, which is shown in
Fig. 3(b). Furthermore, although the complexity of the
proposed detector increases as the parameter Pincreases
at low SNRs, the proposed detector with P=Qis still
able to achieve more than 90% complexity reduction over
the ML detector.
At low SNRs, the value of γ1is not equal to 100%,
implying that there are inaccurate AIs in ˆ
Lk. Since the
proposed detectors with P=QMand P=ˆ
QMare both
designed based on the same location maximization criteri-
on, the accurate AM may be contained in the selected QM
or ˆ
QMAMs. This is the main reason that the proposed
detectors with P=QMand P=ˆ
QMhave negligible
performance losses compared with the ML detector at low
SNRs, which can be seen from Fig. 2(a). As expected, the
proposed detector with P=Qalways performs as well as
the ML detector.
As for large-scale DSM detection with Nt= 16, where
the ML detector for DSM is computational intractable,
the proposed algorithm with P=ˆ
QMstill performs well,
and has only about 2dB performance loss compared to
coherent ML detection.
0 5 10 15 20
SM/ML, 4PSK
SM/ML, 8PSK
DSM, 4PSK,
ˆ
M
P Q
=
DSM, 8PSK,
ˆ
M
P Q
=
8PSK
4PSK
0 5 10 15 20
BER
10
-6
10
-5
10
-4
10
-2
10
0
10
-1
10
-3
proposed,
ML
proposed,
ˆ
M
P Q
=
M
P Q
=
proposed,
P Q
=
(a) (b)
Fig. 2. BER performance of DSM with different antennas configu-
rations: a) Nt= 6,Nr= 6; b) Nt= 16,Nr=16.
V. Conclusion
This Letter proposed a low-complexity detector based
on symbol-by-symbol detection for the DSM system. Com-
pared to the classic block-by-block based ML detector, the
proposed detector is capable of achieving near the same
BER performance with significantly reduced complexity.
In future, the impact of potential bandwidth expansion
[17] of DSM system will be investigated, so as to improve
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0.9
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M
P Q
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