1. 1
Superimposed Spatial Modulation
L.M. Blackbeard, H. Xu
Abstract—In the quest for greater utilisation of wireless com-
munication channels, multiple input multiple output (MIMO)
systems have demonstrated promise in their utilisation of as-
sumed multiple uncorrelated channels. Spatial Modulation (SM)
[1], conveys information above the symbol index in the spatial
position of the transmitting antenna, selected from an array.
In this letter, a scheme is proposed which allows a doubling of
the information in spatial position, by superimposing two spatial
modulation systems which are uniquely identifiable because
they transmit orthogonal symbols. It is found that the system
outperforms conventional spatial modulation for all transmission
bit rates simulated.
I. INTRODUCTION
A. Background and Motivation
In higher throughput wireless communication links, multiple
input multiple output (MIMO) systems have been devised in
which more than one antenna is used at both or either the
transmit and receive side, in order to take advantage of realised
multiple uncorrelated channels and the resultant extension of
channel capacity.
V-BLAST [2] has demonstrated the potential of MIMO
systems, but suffers from high decoding complexity and a
requirement for a greater or equal number of receive antenna
as transmit antenna which may not be realisable on some plat-
forms. Space Shift Keying (SSK) [3] operates by transmitting
information bits in the spatial position of a single selected
transmit antenna: it offers simpler decoding complexity, the
use of a single receive antenna, avoids inter-channel interfer-
ence and does not require antenna synchronisation. Spatial
Modulation (SM), is similar to SSK, except that a signal
symbol is transmitted to convey extra information above the
spatial position of the transmit antenna.
Conventional spatial modulation has a spatial or antenna
constellation of size Nt, corresponding to each possible active
transmit antenna. In [4] and [5], SM is extended to allow
for an arbitrary number of active antenna, selected from a
combinational spatial constellation set of size 2 log2(Nt
N ) .
If it is assumed that correlation exists in the channels of
the transmit antennae, the less than full utilisation of the
combinational set can be used to an advantage by maximising
the minimum distance in the spatial set. Nevertheless, in [6]
a non-power-of-2 constellation is allowed for.
To the knowledge of the author, no SM scheme has been
proposed using the principle of superposition to allow for a
spatial constellation set of size N2
t .
B. Proposed Scheme and Contribution
In this letter, a new approach to spatial modulation is
proposed, in which one or two antenna(e) are active during any
symbol period, each conveying independent information in its
spatial position. It differs from schemes such as [4] and [5] in
that it approaches multiple active antenna from a superposition
point of view: that is, each transmitter is an independent
half-system operating on the same array of antennae and the
antenna index of one half-system may coincide with that of
the other. In doing this, the size of the antenna constellation
set is doubled, increasing spectral efficiency and improving
performance over conventional spatial modulation.
This approach requires a method to identify which half-
system carries more significant bits and in order to discern
each of the two half-systems, orthogonal components (real and
imaginary) of a quadrature symbol are separated so that each is
assigned to a half-system to be modulated. When the antenna
indices coincide, the original QAM symbol is transmitted on
one antenna, resembling conventional SM.
In this letter, the new scheme will be referred to as Super-
imposed Spatial Modulation (SSM).
II. SYSTEM MODEL
A. Notation Convention
In the mathematical description which ensues, the following
convention is utilised: bold faced upper/lower case letters
denote matrices/vectors respectively; (·)T
, (·)H
, E[·], |·| and
· F refer to transpose, Hermitian, expectation, Euclidean
norm and Frobenius norm operators respectively; regular text
letters with subscripts (·)ij denote the ith
row, jth
column
entry in the corresponding matrix and regular text letters with
subscript (·)i denote the ith
entry in the corresponding vector.
B. Concept of Transmission
Now is described the proposed Superimposed Spatial Mod-
ulation (SSM) scheme. As shown in Figure 1, a data stream
is mapped block wise into three parts: log2 M bits select
a symbol from a quadrature amplitude modulation (QAM)
Gray coded constellation set of size M, log2Nt bits select
an antenna from an array of Nt transmitters to transmit the
real component of the symbol and similarly, a further log2Nt
bits select an antenna to transmit the imaginary component
of the symbol: this results in a total spectral efficiency of
log2 M × N2
t bits/s/Hz.
Fig. 1. Block Diagram of the Proposed SSM Scheme

2. 2
C. Mathematical Model
With the concept of the proposed system in place, the
mathematical model is developed.
x ∈ CNt×1
refers to the SSM symbol vector which is to be
transmitted in a given symbol period. It has entries in positions
which correspond to the active transmit antennae, of which the
values are drawn from the and components of a symbol
in the given Gray coded MQAM constellation for the first
and second transmit antenna respectively. This means that the
entries resemble a pulse amplitude modulation (PAM) symbol
from a PAM constellation of order
√
M, except when the
antennae indices are identical, in which case the components
are added together to form the original QAM symbol. The rest
of the vector x is filled with zero entries.
H ∈ CNr×Nt
is the channel matrix which represents as-
sumed independent and identically distributed (i.i.d.) complex
channel gains between each transmit and receive antenna
pair in a Rayleigh fading environment. Each entry hij ∼
CN (0; 1) ∀i ∈ [1 : Nr] and ∀j ∈ [1 : Nt].
n ∈ CNr×1
is a vector with i.i.d entries ni ∼
CN 0; σ2
∀i ∈ [1 : Nr], it corresponds to the additive white
Gaussian noise (AWGN) on each respective receive antenna
with a variance σ2
determined by the signal to noise ratio.
With the above definitions in place, one can write the
received signal vector y as:
y = Hx + n (1)
As such, y ∈ CNr×1
.
D. Maximum Likelihood Decoding
To retrieve the transmitted SSM symbol x, maximum like-
lihood (ML) decoding is employed. That is; where the symbol
index i ∈ [1 : M], the antenna indices l , l ∈ [1 : Nt] and
the channel matrix H is assumed to be perfectly known at the
receiver; finding the solution to:
˜xi, ˜l , ˜l = arg max
xi,l ,l
P (y|xi, l , l , H) (2)
Where (˜·) refers to the maximum likelihood estimation of
transmitted parameter (·).
(2) can also be written as:
˜xi, ˜l , ˜l = arg min
xi,l ,l
y − hl [xi] − hl [xi] 2
F (3)
III. THEORETICAL PERFORMANCE ANALYSIS
In this section, an upper bound for SSM with ML detection
is derived. III-A will expand upon this.
A. The Union Bound
To arrive at an upper bound, we note that ML performs
a joint detection of symbol and antennae and simplify the
analysis by decoupling antenna and symbol detection as in
[7]. In the analysis of antenna detection, we assume perfect
symbol detection and in the analysis of symbol detection,
we assume perfect antenna detection. It is shown in IV
that this assumption is within a small margin of error. The
two decoupled results will be named Pa for antenna index
induced BER and Pd for symbol induced BER. Calling Pc
the probability of correct decoding and Pe the BER of the
whole system, one can write:
Pe 1 − Pc = 1 − (1 − Pa)(1 − Pd) (4)
B. Antenna Detection BER (Pa)
In the derivation of Pa, we assume no inter-channel inter-
ference between the two superimposed systems. The equation
for P˜a, the BER due to a single transmit antenna, is taken
from [7] after noting the halving of transmit power due to the
two concurrent transmitters. The equation follows:
P˜a
Nt
j=1
M
q=1
Nt
ˆj=1
N(j, ˆj)µNr
α
Nr−1
w=0
Nr−1+w
w [1 − µα]w
NtM log2 Nt
(5)
Where N(j, ˆj) is a function returning the representative
binary distance between the jth
and ˆjth
antennae, µα =
1
2 1 −
σ2
α
1+σ2
α
, σ2
α = p
4 |xq|2
and p is the SNR.
However, since there are two concurrent transmitters, the
effective antenna induced BER for P˜a 1 is written as:
Pa 1 − (1 − P˜a)2
≈ 2P˜a (6)
C. Symbol Detection BER (Pd)
First, we note, under the assumption of perfect antenna
detection and different antenna indices for each half-system
and in a MISO environment, that the received signal vector
can be written in scalar quantities as follows:
y = α1x + jα2x + n (7)
Where α1 and α2 are the fading coefficients after phase
removal for the real and imaginary channels respectively; x
and x are the real and imaginary components of the MQAM
symbol respectively; n ∈ CN(0; σ2
) is the AWGN noise on
the receive antenna with variance determined by SNR and
j =
√
−1. The symbol error rate (SER) of the MQAM symbol
is thus composed of the SERs of two orthogonal MPAM
symbols in i.i.d. Rayleigh fading environments with AWGN:
they shall be deemed Pd and Pd . The combined SER Psymbol
d
is then written as:
Psymbol
d = 1 − (1 − Pd )(1 − Pd ) (8)
Where M is the size of the MQAM constellation. This follows
the same procedure as for MQAM performance analysis when
the symbol is transmitted from a single antenna in an otherwise
identical environment [8]. Thus, Pd will be evaluated as if the
symbol were transmitted from a single antenna.
1) Symbol Error Rate for QAM: The symbol error rate for
QAM with Nr receivers and ML detection is given in [7] as
the following numerical solution:
Psymbol
d (p) =
a
c
1
2
2
bp + 2
Nr
−
a
2
1
bp + 1
Nr
+(1 − a)
c−1
i=1
Si
bp + Si
Nr
+
2c−1
i=c
Si
bp + Si
Nr
(9)

3. 3
Where a = 1 − 1/
√
M, b = 3/(M − 1), m = log2 M,
Si = 2 sin2
θi, θi = iπ/4n, c is the number of summations,
Nr is the number of receive antennae and p is the SNR. [7]
writes that for c > 10, there is a 0.0015 dB, 0.0025 dB and
0.0029 dB error between simulated and theoretical results for
each of 4, 16 and 64 QAM constellations respectively. Once
the SER is derived, one can approximate the BER at high
SNRs assuming a nearest neighbour approximation and Gray
mapping by:
Pd
Psymbol
d
m
(10)
Where m = log2 M is the number of bits per MQAM symbol.
IV. SIMULATION RESULTS
The performance of conventional SM and the proposed
SSM are simulated and plotted along with their respective
theoretical bounds. The simulations are written according to
the model described in II-C and it is assumed that perfect
channel state information (CSI) is known at the receiver for
a flat-fading Rayleigh environment; that power is allocated
(averaged over all symbols) equally to the active transmit
antennae and that communication is to a single user.
Fig. 2. Comparison of different schemes at 6 bits/s/Hz
At both 6 bits/s/Hz and 8 bits/s/Hz in the cases sim-
ulated, SSM shows a performance gain over conventional
SM: the gain is approximately 3 dB, 2 dB and 4 dB for
6 bits/s/Hz, 8 bits/s/Hz with Nt = 4 and 8 bits/s/Hz with
Nt = 8 respectively. The upper performance bounds are
within approximately a 1 dB margin from their corresponding
simulations.
V. CONCLUSION
In this letter, a new scheme referred to as SSM is proposed
to convey spatial information in the position of two superim-
posed active transmit antennae. The receiver is able to uniquely
identify each transmit antenna and thus the combined spatial
constellation set was extended to size N2
t . The simulation
results show that using SSM to increase spectral efficiency
performs better than increasing the size of the symbol con-
stellation in conventional SM.
Fig. 3. Comparison of different schemes at 8 bits/s/Hz
VI. REFERENCES
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