1. 1
Euclidean Distance Antenna Selection for Spatial
Modulation with Multiple Active Transmit Antennas
Lloyd Blackbeard, Hongjun Xu and Fengfan Yang
Abstract—
I. INTRODUCTION
Spatial Modulation (SM)[1] could be argued to be
known fairly ubiquitously in communications academia.
Much work has been done in improving its throughput,
bit error rate (BER) performance and detection complexity
[2][3][4][5][6][7][8][9][10][11].
In one such improvement, called Euclidean Distance An-
tenna Selection (EDAS)[9], 2n
n ∈ N transmit antennas are
selected from a larger set in a fashion that maximises the Eu-
clidean distance between any two symbols. In Multiple Active
Spatial Modulation (MASM)[11], more than one antenna is
selected to be active in each symbol period, usually allowing
for more information to be mapped to the spatial domain
and allowing the simultaneous transmission of unique symbols
from each active antenna, thereby improving throughput. To
the best of the authors’ knowledge, no system has been
implemented to improve the performance of MASM in a
fashion similar to EDAS. Thus, in this paper, EDAS will be
implemented for SM systems with multiple active antennas.
The remainder of the paper follows: in section II, a model
for MASM is presented that will be used in subsequent
sections; in section III, we implement the idea of EDAS
for MASM and summarise antecedent work with which the
proposed method will be compared; in section IV, we write
the computational complexity for all schemes implemented in
the paper and produce the analytical results; in section V, we
show simulation results in terms of bit error rate (BER) and
in section VI, concluding remarks are given.
Unless otherwise specified the notation convention follows:
(·)−1
, (·)T
, (·)H
, (·)†
, E[·], | · | and | · |F denote the matrix
inverse, transpose, Hermitian, Moore-Penrose pseudoinverse,
expectation, Euclidean norm and Frobenius norm operators
respectively; regular, bold face lower case, bold face upper
case and capital script/cursive text refer to scalars, vectors,
matrices and sets respectively; subscripts (·)ij denote the
ith
row, jth
column entry in the corresponding matrix and
subscript (·)i denotes the ith
entry in the corresponding vector
or the ith
column in the corresponding matrix.
II. SYSTEM MODEL
The following serves to summarise convential MASM. We
consider a MIMO environment with Nt transmit antennas and
Nr receive antennas. We observe that a maximum of NΓ =
2
log2(Nt
Np
) antenna groups can be defined for Np ≤ Nt active
transmit antennas1
. For each active antenna, we map log2M
bits from the input bitstream to an MQAM symbol selected
from an MQAM constellation of size M. A further log2NΓ
bits select which of the Nt antennas will be active, or rather,
which antenna group will be used in transmission.
Once mapping is complete for the symbol period, we have
the transmit symbol column vector x with MQAM symbols
in positions corresponding to the active antennas and zeros
elsewhere. This vector is transmitted over a channel matrix H
with entries CN(0, 1), which are independently and identically
distributed (i.i.d.). At the receive side, additive white Gaussian
noise is added in the form of a column vector n with i.i.d.
entries CN(0, σ2
), where σ2
is determined by the signal to
noise ratio (SNR). We can thus write the received signal vector
(RSV) y as:
y = Hx + n (1)
Detection on the receive side, in our case, is performed
using Maximum Likelihood (ML) detection, the explanation
of which follows.
Let us call the set of MASM symbols from which the
symbol x is chosen X. Detection is performed using an
exhaustive search over all X. We write the ML decision metric
as the following, where ˜x is the estimated symbol and xi is
an element of X:
˜x = arg min
xi
y − Hxi
2
F xi ∈ X (2)
III. PROPOSED ANTENNA GROUP SELECTION SCHEME
AND SIMPLIFICATIONS
We move to apply the work of [10] and [9], which are
proposed for improving the performance of conventional SM,
to MASM. In conventional SM, the number of active transmit
antennas is one, and therefore the spectral efficiency is given
by log2 MNt. By increasing the number of transmit antennas
to NT > Nt and choosing Nt antennas from the total NT
in an intelligent fashion, the bit error rate (BER) performance
can be improved [10][9]. We can intuit that this method is
indeed valid as a column in the channel matrix undergoing
fading can be replaced by another column that has a larger
norm.
Now, in MASM, if the number of transmit antennas is in-
creased from Nt to NT , the number of possible antenna groups
is increased from Nt
Np
to NT
Np
. Unlike the conventional SM
case, the question that must be asked in the MASM case is
1Fractional bit encoding can be used to overcome this limit [3].

2. 2
how to select those antenna groups that maximise the BER
performance.
A. Proposed Scheme - Euclidean Distance Antenna Group
Selection (EDAGS)
We take note in the MASM system described in II, that the
total number of possible antenna groups is NΨ = Nt
Np
and
that an integer power of two number of them NΓ, must be
selected for the construction of the symbol set X. However,
in contrast to conventional MASM, for the proposed scheme
and similar to conventional EDAS, we have a number of
transmit antennas greater than the minimum required for
NΓ = 2
log2(Nt
Np
) , written as NT ≥ Nt and thus, we have
NΨ = NT
Np
.
Now, in conventional MASM, the selection of NΓ from
NΨ is made arbitrarily, however, in the work presented in
this paper, we select those antenna groups which maximise
Euclidean distance between any two MASM symbols. We
call this method Euclidean Distance Antenna Group Selection
(EDAGS) and elucidate further with a mathematical descrip-
tion.
Let us note the ML metric (2) and rewrite it after substitut-
ing (1).
˜x = arg min
xi
Hx + n − Hxi
2
F xi ∈ X (3)
From (3), we can see that the performance of the system can
be improved by selecting a subset of MASM antenna groups
that maximises the minimum Euclidean distance between
symbol vectors.
Enumerating the NΨ
NΓ
ways of selecting valid antenna
groups as Ψ, we can write the selection as:
ΓED = arg max
Γ∈Ψ
{ min
γ1,γ2∈Γ
x1,x2∈S
x1=x2 iff γ1=γ2
Hγ1
x1 − Hγ2
x2
2
F } (4)
Where: Hγi has Np columns taken from those in the
channel matrix H which correspond to the active antennas
in the γi antenna group of Γ; xi is an Np × 1 vector from
the MNp
sized set S of all possible Np dimensional MQAM
symbols and ΓED is the resultant set of antenna groups.
B. Euclidean Distance Antenna Selection (EDAS)
Unfortunately, the complexity of the proposed EDAGS
scheme is very large. In order to reduce complexity, we apply
the results of [9] and [10] and apply them to the MASM
system. In order to do this, we find those Nt antennas of the
full NT antenna set that maximise the minimum Euclidean
distance between MASM symbols and subsequently create
antenna groups from the Nt antennas selected. To reiterate, this
is in contrast to selecting those antenna groups that maximise
the minimum Euclidean distance between symbols.
We obtain the equation for the EDAS method as described
when applied to MASM as:
IED = arg max
I∈I
{ min
x1=x2∈X
HI(x1 − x2) 2
F } (5)
Where I is the enumeration of each combination of NT
Nt
and HI has Nt columns described by I. Once IED is
found, we create the NΓ antenna groups required for MASM
transmission from the antennas described in IED.
C. Capacity Optimised Antenna Selection
In [9], a second scheme aside from EDAS is proposed that
is based upon the capacity of the channel matrix instead of the
Euclidean distance between symbols. The approach results in
drastically reduced complexity for the price of performance.
Being based upon capacity, the scheme is dubbed Capacity
Optimised Antenna Selection (COAS).
In this paper, we propose capacity optimisation by consid-
ering the equation for the capacity of a MIMO channel. This
capacity is given by:
C = EH log2 det INr +
ρ
η
HHH
(6)
Let us call the sum in the determinant function A, i.e.:
A = INr
+
ρ
η
HHH
(7)
Using QR decomposition, we can write A as the product of
a unitary, orthogonal matrix Q and an upper triangular matrix
R. The determinant function in (6) can thus be expressed as:
det INr
+
ρ
η
HHH
= det (QR) (8)
Using established properties of determinants, specifically,
the determinant of a product and the determinant of a unitary
matrix, we can further write:
det (QR) = det (Q) · det (R) = 1 · det (R) = det (R) (9)
Since R is upper triangular, its determinant is equal to the
product of its diagonal elements. This is written as:
det INr +
ρ
η
HHH
=
Nr
i=1
rii r ∈ R (10)
Now, since the objective of capacity optimised antenna
selection is to improve performance by selecting Nt antennas
from the larger set of NT antennas, the ensuing step, logically,
is to compute the upper triangular matrix in such a way
that the diagonal entries are monotonically decreasing. This
may be done intuitively by permuting the columns of the
channel matrix H and calculating the determinant for each
permutation. Once the diagonal is as desired, we select those
Nt antennas corresponding to the first Nt columns in the R
matrix.
Applying this method as is, we are presented with the
obstacle of very large computational complexity due to the
number of possible permutations, given as NT !. For example,
if NT = 6, there are 720 possible permutations. We are
able to drastically decrease the computational complexity by
altering the way in which QR decomposition is performed.
This method is presented in what follows.
α ≤ CSM ≤ α + log2(NSM ) (11)

3. 3
Where α = 1
NSM
NSM
i=1 log2(1 + ρ hi
2
). It is clear
from (11) that capacity is maximised if the Nt antennas
corresponding to the columns in the H with the largest norms
are chosen out of the NT columns.
Although COAS is intended for conventional SM, with a
single symbol being transmitted in each symbol period, we use
the scheme as a comparison to proposed schemes by creating
the NΓ antenna groups from those Nt that are selected.
D. Improved Capacity Optimised Antenna Selection
We note that COAS is intended to be implemented as part of
a conventional SM scheme. This can be readily seen from the
equation for capacity given in (11), which exhibits the norm
of a single column at a time. It is therefore advantageous to
improve upon COAS by considering the equation for capacity
that is applicable to simultaneous transmission from multiple
transmit antennas, since, discarding antenna group selection in
MASM, that which remains is spatial multiplexing (SMX).
We write the equation for the capacity of the MASM scheme
as:
α ≤ CMASM ≤ α log2(NΓ) (12)
Where, critically, α is given as:
We continue by analysing the term HHH
, that is, the chan-
nel matrix multiplied by its Hermitian or conjugate transpose.
If we apply QR decomposition to the channel matrix, the term
becomes:
HHH
= QRRH
QH
(13)
Where QQH
= I is a unitary matrix with orthogonal
columns and R is upper triangular. Trivially, the conjugate
transpose of an upper triangular matrix is a lower triangular
matrix, and thus we find RRH
as a diagonal matrix consisting
of the squares of the diagonal entries in the original upper
triangular matrix with unchanged positions.
Further, considering the intrinsic properties of the matrix Q
mentioned as unitary and orthogonal, we discard it in (6).
With such insight elucidated regarding the capacity of
the MASM system, it is readily observed that performing
QR decomposition in such a way that the columns of H
are permuted to produce an upper triangular matrix R with
monotonically decreasing entries on the diagonal is a simple
method to improve the capacity of the system.
Let us consider the ML metric (2). If we substitute the
received signal vector (1) into the ML metric and assume
x = xi, we can rewrite (2) as:
˜x = arg min
xi
H∆x + n 2
F xi ∈ X (14)
Where ∆x = x − xi. In the proposed method of this paper
named EDAGS, we maximise the minimum distance of (14)
(when x = xi) by considering simultaneously each antenna
and symbol combination. However, this exhaustive method is
computationally intensive. In order to reduce complexity, we
assume that the symbols resulting from ∆x are unknown. With
this assumption in mind, it is apparent that selecting those
transmit antennas which maximise the Euclidean distance
represented by H∆x + n is a method for minimizing the
BER of the system. Since, in this method, n is also unknown,
the objective is pursued in the following:
I = arg max
I∈I
HI∆x 2
F (15)
In COAS [9], as mentioned in Section III-C, we select those
columns in the channel matrix with the highest absolute norms
in order to maximise the capacity and thus the BER of the
system. If this method is used to solve for (15), we obtain a
fairly optimal result. However, the use of such a method would
assume that all columns in the channel matrix are orthogonal,
an assumption which is fairly accurate if the number of receive
antennas is large. However, in simulations, it is shown that
better performance can be achieved by pursuing (15) whilst
not assuming perfect orthogonality between every column in
H.
We thus propose a solution of (15) which assumes does
not assume perfect orthogonality between the columns in the
channel matrix, a method dubbed Sorted QR Decomposition
Based Antenna Selection (QRAS), which, to the best of the
authors’ knowledge, has not been offered elsewhere.
To support and give motivation for QRAS, we trace what
follows. To begin, we consider a MIMO system with Nt = 2.
The square of the triangle inequality gives:
h1 + h2
2
≤ h1
2
+ 2 h1 · h2 + h2
2
(16)
Since in uncorrelated flat-fading MIMO systems, column
vectors are pseudo-orthogonal, we can disregard the inner
product term 2 h1 · h2 and write a new triangle inequality
for the considered MIMO system:
h1 + h2
2
≤ h1
2
+ h2
2
(17)
Now, for the two column channel matrix in the considered
system, we decompose H∆x into the following:
H∆x = QR∆x = [q1 q2]
r11 r12
0 r22
∆x1
∆x2
(18)
Where, as is customary with QR decomposition, Q is
a unitary orthogonal matrix and R is an upper triangular
matrix. If we find the square of the Frobenius norm for the
decomposition in (18), we may begin by discarding Q as it is
unitary and write the remainder as:
H∆x 2
F = R∆x 2
F =
r11∆x1 + r12∆x2
r22∆x2
2
F (19)
Furthering (19), we are given:
H∆x 2
F = (r11∆x1 + r12∆x2)2
+ (r22∆x2)2
(20)
Nearing completion of the motivation, we consider two ex-
treme cases of (20): that in which both columns of the channel
matrix are parallel and that in which the columns of the chan-
nel matrix are completely orthogonal - over the comparison,
the amplitude of both vectors in the channel matrix remain
constant. In the first case, (20) is given by (r11∆x1+r12∆x2)2
and in the second, we have (r11∆x1)2
+ (r22∆x2)2

4. 4
Let us note that in a system with Nr receive antennas,
we can find Nr orthogonal vectors, provided no two column
vectors in the channel matrix are exactly parallel.
We decompose the channel matrix into a unitary orthogonal
matrix Q and an upper triangular matrix R. This decompo-
sition is performed for each permutation of the columns of
the channel matrix H until the diagonal of the R matrix is
monotonically decreasing. In this way, we find the optimal
antenna selection by choosing those columns with the greatest
orthogonal distance from other columns.
Note in QRAS that the number of times QR decomposition
must performed is upper bounded by NT ! and thus the
computational complexity rises with the factorial of NT .
Once again, for comparison to the proposed scheme, we
implement QRAS for MASM. After the implementation, we
are left with Nt chosen antennas, from which we construct
the NΓ antenna groups.
IV. COMPUTATIONAL COMPLEXITY FOR PROPOSED AND
ANTECEDENT SCHEMES
Drawing on the work in [], we quantify the computa-
tional complexity of the antecedent and proposed schemes
in floating point operations (FLOPs), where each multipli-
cation, division, addition and subtraction count as a single
FLOP. Each scheme was analysed by the author in the
same manner in order to ensure continuity in method and
for brevity, such is excluded in this paper as the deriva-
tions are trivial. Thus, formulae for each are simply stated:
For EDAS, we have NT
Nt
N2
t M2
[2NtNr + Nt + Nr − 1]; for
QRAS, 2NT !NrN2
T ; for COAS, 2NT Nr −NT and for EDAGS
(proposed),
(NT
Np
)
NΓ
N2
ΓM2Np
[4NpNr + Nr − 1].
Fig. 1: Computational Complexity for 4 Antenna Selection
Schemes in 2 MASM Configurations
Figure 1 shows the complexity of the proposed and an-
tecedent schemes in the configurations described in Table I.
The proposed scheme is clearly orders of magnitude more
complex than that of the schemes to which it is compared.
However, in the subsequent section, it can be seen that the
NT Nt Np Nr NΓ M
6bits/s/Hz 6 4 2 5 4 4
10bits/s/Hz 6 4 2 5 4 16
TABLE I: MASM Configurations
proposed outperforms those to which it is compared in terms
of bit error rate (BER) performance.
V. RESULTS
The proposed scheme was simulated in a Rayleigh flat
fading environment with AWGN for the configurations in
Table I.
For comparison, we include those methods explained in III.
Note that in all cases other than the proposed scheme, we
select the minimum number of transmit antennas Nt, needed
to attain NΓ (i.e. 4 out of NT = 6), whereas in the proposed
scheme, the antenna groups may include any of the NT = 6
transmit antennas - this is where the advantage of the proposed
scheme lies. The results are shown in Figure 3.
Fig. 2: Simulation Results for MASM 6bits/s/Hz
We can see from the simulation results that the proposed
scheme offers a significant performance gain over the other
schemes considered. However, intrinsically, the computational
complexity of the proposed method is far larger than any of
the schemes used for comparison. Thus, there is room for
further work in finding a simplified method of implementing
the proposed method.
VI. CONCLUSION
Building upon the work of [9] and [11], the authors propose
a method to select optimal antenna groups in MASM. The
proposed method is found to perform significantly better than
conventional EDAS and two other benchmarks considered.
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