Performance Analysis and Optimal Detection of Spatial Modulation

International Journal on Recent and Innovation Trends in Computing and Communication ISSN: 2321-8169
Volume: 5 Issue: 9 60 – 63
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60
IJRITCC | September 2017, Available @ http://www.ijritcc.org
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Performance Analysis and Optimal Detection of Spatial Modulation
Prof. K.S. Solanki
Ujjain Engineering College, Ujjain (M.P.)
khemsingh_solanki@rediffmail.com
Abhilasha Singh
Ujjain Engineering College, Ujjain (M.P.)
abhilasha_sng9@yahoo.com
Abstract—In this paper, we propose the optimal detector for spatial modulation. The new detector performs significant better than the original
(~4 dB gain), and we derive the closed form expression for the average bit error probability. The optimal detector of SM shows performance
gain (~1.5 −3 dB) over popular multiple antenna system, making it an excellent prospect for future wireless communication.
Keywords-Antenna modulation, spatial modulation, maximum likelihood detection, MIMO.
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I. INTRODUCTION
Multiple Input Multiple output (MIMO) scheme have been
proposed for wireless communication system to significantly
increase capacity, range and reliability when comparing with
convectional single antenna system. It provides increase in
data throughput and minimum probability of error without
additional frequency spectrum and transmission power. MIMO
systems can be categorized into Beam forming, spatial
multiplexing (SM) and diversity. Several MIMO techniques,
among which the Space Time Block Code (STBC) and spatial
multiplexing achieving diversity and multiplexing gain. The
spatial diversity gain can be exploited by STBC because of its
implementation simplicity and low decoding complexity. The
maximum likelihood (ML) decoder with linear complexity is
the main attraction of orthogonal STBC (OSTBC). Full -rate
full-diversity code for more than two transmit antennas with
linear complexity is proven impossible to be constructed.
Maximum multiplexing gain by simultaneous transmission
over all antennas can be achieved using V-BLAST scheme.
The joint ML decoding provides high capacity for the data
stream, but the complexity increases with the number of
streams. The error performance of the system can be
significantly reduced using linear sub-optimal decoders for V-
BLAST, such as linear minimum mean square error (MMSE),
successive cancellation, but the inter channel interference
(ICI) and Inter antenna interference (IAI) increases. Spatial
Modulation (SM) is a recently developed low-complexity
Multiple-Input Multiple-Output scheme that uses antenna
indices and a conventional signal set to convey information. It
has been shown that the Maximum-Likelihood (ML) detection
in an SM system involves joint detection of the transmit
antenna index and the transmitted symbol, and hence, the ML
search complexity grows linearly with the number of transmit
antennas and the size of the signal set. In this paper, we show
that the ML search complexity in an SM system becomes
independent of the constellation size when the signal set
employed is a square- or a rectangular-QAM. Further, we
show that Sphere Decoding (SD) algorithms become essential
in SM systems only when the number of transmit antennas is
large and not necessarily when the employed signal set is
large. We propose a novel sphere decoding detector whose
complexity is lesser than that of the existing detector and a
generalized detection scheme for SM systems with number of
transmit antennas. We support our claims with simulation
results that the proposed detectors are ML-optimal and offer a
significantly reduced complexity.
Organization: This paper is organized as follows. Section II
introduces the existing system and disadvantages of this
system. Section III introduces the basic system model. In
section IV we introduces the proposed system and its
advantages, In section V we derive the optimal detector and
provide a performance analysis for the SM system. Section VI
presents some simulation results and we conclude the paper in
section VII.
II. EXISTING SYSTEM AND ITS DISADVANTAGES
A. EXISTING SYSTEM
Trans-mission techniques designed for multiple input multiple
out-put (MIMO) systems, such as the Bell Laboratories
layered space-time (BLAST) architecture. Due to inter-
channel interference (ICI) caused by coupling multiple
symbols in time and space, maximum likelihood (ML)
detection increases exponentially in complexity with the
number of transmit antennas. Consequently, avoiding ICI
greatly reduces receiver complexity, and contributes in
attaining performance gains. The so-called spatial modulation
(SM), is an effective means to remove ICI and the need for
precise time synchronization amongst antennas. SM is a
pragmatic approach for transmitting information, where the
modulator uses well known amplitude/phase modulation
(APM) techniques such as phase shift keying (PSK) and
quadrature amplitude modulation, but also employs the

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antenna index to convey information. Only one antenna
remains active during transmission so that ICI is avoided. As
well, inter-antenna synchronization (IAS) during transmission
is no longer needed as in the case of Vertical-BLAST (V-
BLAST) [4], in which all antennas transmit symbols at the
same time. A sub-optimal detection method is presented and
only valid under some constrained assumptions about the
channel. For conventional channels, their detector fails and
even with their assumption, detection is not optimal. We
present the optimal detector for SM and show that the
detection is a joint optimization problem that cannot be
separated. We analyze the performance of the SM system and
derive a closed form expression for the bit error probability
when real constellations are used. As well, prior to this work,
SM’s advantages lied in removing ICI and IAS from the
communication systems, where gains in performance over
other schemes in the literature was not present. With optimal
SM however, we show that performance gains over maximum
ratio combining (MRC) and V-BLAST is observed, making
the use of SM in practical systems more attractive.
B. DISADVANTAGES
Inter-channel interference (ICI) caused by coupling multiple
symbols in time and space, maximum likelihood (ML)
detection increases exponentially in complexity with the
number of transmit antennas.
III. SPATIAL MODULATION
A. SYSTEM MODEL
The general system model consists of a MIMO wireless link
with 𝑁𝑡 transmit antennas and 𝑁𝑟 receive antennas. The
general system is shown in fig 1. The random sequence of
independent bits b enters the SM mapper, which groups B bits
and maps them to a constellation vector
𝑥 = 𝑥1 𝑥2 … …. 𝑥 𝑁𝑡
𝑇
where we assume power constraint
of unity (i.e. 𝐸𝑥 𝑥 𝐻
𝑥 = 1).
Where, [. ] 𝑇
represents transpose and 𝐸𝑥 [. ] represents
statistical expectation with respect to 𝑥.
In SM only one antenna is active at a time any other antenna is
deactivated and hence only one of the 𝑥𝑖 in 𝑥 is non-zero. The
signal is transmitted over an 𝑁𝑟 × 𝑁𝑡 wireless channel 𝐻 and
experiences an 𝑁𝑟 -dim additive white Gaussian noise 𝜂 =
𝜂1 𝜂2…………. 𝜂 𝑁 𝑟
𝑇
. The received can be expressed as
𝑦 = 𝜌𝐻𝑥 + 𝜂 (1)
Where, 𝜌 is the average signal to noise ratio (SNR) at each
receive antenna, and 𝐻 and ƞ have independent and identically
distributed (iid) entries according to𝒞𝒩 (0,1).
Fig. 1 General system model.
B. SM MODULATION
As mentioned earlier, SM utilizes the antenna index as an
another means to transmit information. The antenna combined
with symbol index make up the SM mapper. The mapper
collects 𝐵 = 𝑙𝑜𝑔2(𝑀𝑁𝑡) bits and maps them to a costellation
vector.
𝑥𝑗𝑞 ≜ 0 0 … … . 𝑥 𝑞 0 … … .0
𝑇
↑
𝑗 𝑡𝑕
𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
Where, j is the activated antenna and 𝑥 𝑞 is the 𝑞 𝑡𝑕
symbol
from the constellation 𝜒 𝑀 . Hence 𝑗 𝑡𝑕
antenna remains acive
during symbol transmission. For example in 3 bits/s/Hz
transmission with 𝑁𝑡 = 4 antennas, the information bits are
mapped to a ±1 binary PSK (BPSK) symbol, and transmitted
on one of the four available antennas. When 𝑥 𝑎 is transmitted
from the 𝑗 𝑡𝑕
antenna the output of the channel is expressed as,
𝑦 = 𝜌𝑕𝑗 𝑥 𝑎 + 𝜂
Where, 𝑕𝑗 is the 𝑗 𝑡𝑕
column of𝐻.
C. SM DETECTION
In [1], assuming constant modulus signaling such as PSK a
sub-optimal detection rule is given by,
𝑗 = 𝑎𝑟𝑔𝑗 max⁡|𝑕𝑗
𝐻
𝑦|
𝑞 = 𝑎𝑟𝑔𝑞 𝑚𝑎𝑥𝑅𝑒{(𝑕𝑗 𝑥 𝑞 ) 𝐻
𝑦}
Where, 𝑗 and 𝑞 is the estimated antenna and symbol index,
respectively. Since the mapping is one to one, the demapper
obtain an estimate of the transmitted bits by taking j and q as
inputs. However, this detector only works for transmission
over normalized channel.
IV. PROPOSED SYSTEM AND ITS ADVANTAGES
A. PROPOSED SYSTEM.
We propose the optimal detector for SM and show that the
detection is a joint optimization problem that cannot be
separated. We analyze the performance of the SM system and
derive a closed form expression for the bit error probability
when real constellations are used.SM’s advantages lied in
removing ICI and IAS from the communication systems,
where gains in performance over other schemes in the

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62
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literature was not present. We show that performance gains
over maximum ratio combining (MRC) and V-BLAST is
observed, making the use of SM in practical systems more
attractive. A MIMO wireless link with Nt transmit and Nr
receive antennas. A random sequence of independent bits b
enters the SM mapper. SM exploits the antenna index as an
additional means to transmit information. The antenna
combined with the symbol index make up the SM mapper. The
MRC scheme is essentially a single input multiple output
(SIMO) communication system usingAPM and employing an
ML receiver, where we use 8-QAM to achieve the spectral
efﬁciency requirement. V-BLAST using BPSK with Nt =4
antennas and ordered successive interference cancellation
(OSIC) using the minimum mean squared error (MMSE)
receiver is also compared.
B. ADVANTAGE
SM was inferior in terms of performance over V-BLAST and
MRC, and its advantages mainly lied in enabling simple
detection as well as removing the need for ICI and IAS.
V. OPTIMAL DETECTION AND PERFORMANCE
ANALYSIS
A. OPTIMAL DETECTION
Since the channel inputs are assumed equally likely, the
optimal detector is ML, which is given by,
𝑗 𝑀𝐿, 𝑞 𝑀𝐿 = 𝑎𝑟𝑔max
𝑗 ,𝑞
𝑝 𝑌(𝑦|𝑥𝑗𝑞 , 𝐻)
= argmin𝑗 ,𝑞 𝜌 𝑔𝑗𝑞 𝐹
2
− 2𝑅𝑒 𝑦 𝐻
𝑔𝑗𝑞 (2)
Where, 𝑔𝑗𝑞 = 𝑕𝑗 𝑥 𝑞 , 1 ≤ 𝑗 ≤ 𝑁𝑡 , 1 ≤ 𝑞 ≤ 𝑀, and
𝑝 𝑌 𝑦 𝑥𝑗𝑞 , 𝐻 = 𝜋−𝑁 𝑟 exp⁡(− 𝑦 − 𝜌𝐻𝑥𝑗𝑞 𝐹
2
) is PDF of y,
conditioned on 𝑥𝑗𝑞 and 𝐻. It can be seen that detection is a joint
optimization problem which cannot easily be separated. Even
with normalized channels and constant modulus signaling (i.e.
𝑔𝑗𝑞 𝐹
2
= 1), the detector reduces to,
𝑗 𝑀𝐿, 𝑞 𝑀𝐿 𝑃𝑆𝐾 = 𝑎𝑟𝑔max
𝑗 ,𝑞
𝑅𝑒 𝑦 𝐻
𝑔𝑗𝑞
B. PERFORMANCE ANALYSIS
The performance of SM system will be derived using well
known union bounding technique [5, P. 261-262]. The average
bit error rate (BER) in SM is union bounded as
𝑃𝑒,𝑏𝑖𝑡 ≤ 𝐸𝑥 𝑁 𝑞, 𝑞 𝑃(𝑥𝑗𝑞 → 𝑥𝑗 𝑞 )
𝑗 𝑞
=
𝑁 𝑞,𝑞 𝑃(𝑥 𝑗𝑞 →𝑥 𝑗 𝑞 )
𝑁𝑡 𝑀
𝑀
𝑞=1
𝑁𝑡
𝑗=1
𝑀
𝑞=1
𝑁𝑡
𝑗=1 (3)
Where, 𝑁(𝑞, 𝑞) is the number of bits in error between the
symbol 𝑥 𝑞 and 𝑥 𝑞 and 𝑃(𝑥𝑗𝑞 → 𝑥 𝑥 𝑞 ) denotes the pairwise error
probability (PEP) of deciding on the constellation vector 𝑥𝑗 𝑞
given that 𝑥𝑗𝑞 is transmitted. By simplifying (2) the PEP
conditioned on 𝐻 is given by,
P 𝑥𝑗𝑞 → 𝑥𝑗 𝑞 H = P 𝑑𝑗𝑞 > 𝑑𝑗 𝑞 H = Q( 𝜅)
Where, 𝑑𝑗𝑞 = ( 𝜌 𝑔𝑗𝑞 𝐹
2
− 2𝑅𝑒{𝑦 𝐻
𝑔𝑗𝑞 }) and 𝑄 𝑥 =
1
2𝜋
∞
𝑥
𝑒−𝑡2 2
𝑑𝑡. We define 𝜅 as
𝜅 ≜
𝜌
2𝑁 𝑟
𝑔𝑗𝑞 − 𝑔𝑗 𝑞 𝐹
2
= 𝐴 𝑛 + 𝑖𝐵(𝑛) 2𝑁 𝑟
𝑛=1 (4)
Where, 𝑖 = −1 and
𝐴 𝑛 =
𝜌
2𝑁𝑟
(𝑕 𝑛𝑗
𝑅
𝑥 𝑞
𝑅
− 𝑕 𝑛𝑗
𝐼
𝑥 𝑞
𝐼
− 𝑕 𝑛𝑗
𝑅
𝑥 𝑞
𝑅
+ 𝑕 𝑛𝑗
𝐼
𝑥 𝑞
𝐼
)
𝐵 𝑛 =
𝜌
2𝑁𝑟
(𝑕 𝑛𝑗
𝑅
𝑥 𝑞
𝐼
+ 𝑕 𝑛𝑗
𝐼
𝑥 𝑞
𝑅
− 𝑕 𝑛 𝑗
𝑅
𝑥 𝑞
𝐼
− 𝑕 𝑛 𝑗
𝐼
𝑥 𝑞
𝑅
)
The superscript R and I denote the real and imaginary
part,respectively, and 𝑕 𝑛𝑚 is the element of H in the
𝑛 𝑡𝑕
row,and 𝑚 𝑡𝑕
column. In this case, the performance can be
evaluated numerically. However, for symbols x drawn from a
real constellation XM , this independence is satisfied and (4)
reduces to 𝜅 = 𝛼 𝑛
22𝑁 𝑟
𝑛=1 where, 𝛼 𝑛 ~ 𝒩(0, 𝜍𝛼
2
) with 𝜍𝛼
2
=
𝜌( 𝑥 𝑞
2
+ 𝑥 𝑞
2
)
2𝑁 𝑟
Hence, 𝜅 is a chi-squared random variable with 𝑠 = 2𝑁𝑟
degrees of freedom and PDF 𝑝𝜅(𝜗) given in [5,p.41]. The PEP
can then be formulated as
𝑃 𝑥𝑗𝑞 → 𝑃𝑗 𝑞 = 𝐸𝜅 𝑃 𝑥𝑗𝑞 → 𝑥𝑗 𝑞 𝐻
= 𝑄 𝜗
∞
𝜗=0
𝑝𝜅 𝜗 𝑑𝜗
=
exp ⁡
−𝑡2
2
𝐹𝜅(𝑡2)𝑑𝑡
2𝜋
∞
𝑡=0
(5)
where the last line follows from a simple change of integration
order and𝐹𝜅 𝑦 = 𝑓𝜅(𝜗)
𝑦
𝜗=0
𝑑𝜗 is the chi-squared cumulative
distribution function (CDF). We use the expression for 𝐹𝜅(𝑦)
given in [5, p.42 Eq. (2.1-114)] and closed form integral
expression from [6,p.337, Eq. (3.326-2)] to simplify equation
(5) as
𝑃 𝑥𝑗𝑞 → 𝑥𝑗 𝑞 =
1 −
Г(𝑘′ )(2𝜍 𝛼
2)−𝑘
2𝜋𝑘!
𝜇 𝛼
2
−2𝑘′
𝑚−1
𝑘=0
2
Where, 𝜇 𝛼 =
𝜍 𝛼
2+1
𝜍 𝛼
2 , 𝑚 =
𝑠
2
= 𝑁𝑟 and 𝑘′
= 𝑘 +
1
2
using [6,
p.897 Eq. (8.339-2)], with some straightforward algebra, we
get the PEP expression as
𝑃 𝑥𝑗𝑞 → 𝑥𝑗 𝑞 =
𝜇 𝛼 − 2𝑘
𝑘 2𝜇 𝛼 𝜍 𝛼
−2𝑘
𝑁 𝑟−1
𝑘=0
2𝜇 𝛼
(6)
Plugging in (6) into (3), we obtain

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63
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𝑃𝑒,𝑏𝑖𝑡 ≤
N q,q 𝜇 𝛼 − 2𝑘
𝑘 2𝜇 𝛼 𝜍 𝛼
−2𝑘
𝑁 𝑟−1
𝑘=0
4M𝜇 𝛼
M
𝑞=1
M
𝑞=1 (7)
VI. SIMULATION RESULTS
We perform montecarlo simulation. We perform this
simulation for at least 105
channel realization and plot the
average BER performance versus 𝜌, the average SNR per
receive antenna. In all schemes 3bits/s/Hz transmission with
𝑁𝑡 = 4 antennas are assumed. In fig.2 dotted lines represents
constraint channel and solid line represents conventional
channel. We use 8-QAM with MRC scheme using APM and
employing ML-receiver. This scheme increases spectral
efficiency. V-BLAST using BPSK with 𝑁𝑡 = 3 antennas and
ordered successive interference constellation (OSCI) using the
minimum mean square error(MMSE) receiver is also
compared. SM with BPSK and 𝑁𝑡 = 4 antenna is shown for
sub-optimal [1] and optimal receivers along with the SM BER
bound of [7].
When simulation over conventional channels (solid line) are
performed. Higher gains are achieved.Mesleh’s detector fails
in achieving higher gain.
Fig. 2 BER performance of SM for 3 bits/s/Hz transmission.
VII. CONCLUSION
In this paper, optimal detector for SM is derived performance
gain of SM is observed over the detector in [1]. Closed form
expression for the average BER of SM is derived. Shown in
fig.2 optimal SM is better than V-BLAST and MRC. SM is an
excellent candidate future communication system.
REFERENCES
[1] R. Mesleh, H. Haas, C. W. Ahn, and S. Yun, ―Spatial
modulation – anew low complexity spectral efficiency
enhancing technique,‖ Conf. onComm. and Networking in
China, Oct. 2006.
[2] G. J. Foschini, ―Layered space-time architecture for wireless
communicationin a fading environment when using multiple
antennas,‖ Bell Labs.Tech. J., vol. 1, no. 2, pp. 41–59, 1996.
[3] R. Mesleh, ―Spatial modulation: a spatial multiplexing
technique forefficient wireless data transmission‖, PhD
thesis, Jacobs University, June2007.
[4] P. Wolniansky, G. Foschini, G. Golden, and R. Valenzuela,
―V-BLAST:an architecture for realizing very high data rates
over the rich-scatteringwireless channel,‖ in Proc. of
URSI’98.
[5] J.G. Proakis, Digital Communications, (4th ed.) McGraw-
Hill, New York,2001.
[6] I. S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series
and Products,(7th ed.) Elsevier Academic Press, 2007.
[7] R. Böhnke, D. Wübben, V. Kühn, and K. D. Kammeyer,
―Reducedcomplexity mmse detection for blast architectures,‖
in Proc. IEEE Globecom’03, San Francisco, California, USA,
Dec. 2003.

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