This document summarizes and compares four detection algorithms for spatial modulation with multiple active transmit antennas (MASM): 1) Maximum likelihood (ML) detection, which performs an exhaustive search over all possible symbol vectors; 2) A decorrelator-based suboptimal detection method; 3) An ordered block minimum mean square error (OB-MMSE) detection method; 4) A proposed simplified ML detection method based on symbol cancellation and multi-level subset searching. Through simulations, the proposed detector is found to perform the same as ML detection down to bit error rates of 10-6, with lower complexity than ML detection and OB-MMSE for most configurations.

1. 1
A SimplifiedML Detectionfor Spatial ModulationwithMultiple Active Transmit
Antennas
Lloyd Blackbeard 1, Hongjun Xu1 and Fengfan Yang2
School of Engineering
1 University of KwaZulu-Natal, Durban, 4041, Republic of South Africa
2 Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China
email: xuh@ukzn.ac.za
Abstract: Spatial modulation (SM) with multiple active transmit antennas (MASM) is a scheme
capable of higher spectral efficiency than conventional spatial modulation. In this paper, the
authors simulate MASM with optimal maximum likelihood (ML) detection, a decorrelator based
detector, an ordered block minimum mean square error detector (OB-MMSE) and with a
proposed simplified maximum likelihood detector. In simulations, the proposed detector
performs the same as ML detection down to bit error rates of 10−6
for three considered MASM
configurations, whilst simulations for the sub-optimal detectors are shown to perform worse
than the proposed detector simulations. The complexity of the four detectors is considered,
showing that the proposed simplified ML detector is less complex than ML detection and less
complex than the average complexity of OB-MMSE, for all but the lowest spectral efficiency
configuration. The proposed detector has a fixed complexity, contrary to OB-MMSE, which has
a variable complexity.
Index Terms— Bit Error Rate (BER), Spatial Modulation (SM), Multiple-Input-Multiple-Output
(MIMO), 𝑴-ary Quadrature Amplitude Modulation (𝑴-QAM), Maximum Ratio Combining
(MRC), Maximum Likelihood (ML)

2. 2
I. Introduction
Multiple-input multiple-output (MIMO) schemes can provide greater bandwidth efficiency than
traditional single-input single-output (SISO) schemes. A benchmark for MIMO schemes is
vertical Bell Laboratories layered space-time V-BLAST [1], also known as spatial multiplexing
(SMX). In [1], a number of transmit antennas simultaneously transmit M-ary quadrature
amplitude modulation (MQAM) symbols and thus V-BLAST suffers from inter-channel
interference (ICI), inter-antenna coupling (IAC) and also requires antenna synchronization.
Unfortunately, optimal maximum likelihood (ML) detection of V-BLAST is of a high complexity
and sub-optimal methods require the number of receive antennas to equal or outnumber the
number of transmit antennas.
Spatial modulation (SM), another MIMO scheme proposed in [2], intrinsically avoids ICI and IAC
and also does not require antenna synchronization. In [2], a transmit antenna is selected from
an array to transmit an MQAM symbol, with selections of both transmit antenna and MQAM
symbol conveying data. Space-shift keying (SSK) [3] is a simplified version of SM that uses on-
off-keying (OOK) in place of MQAM.
SM has been developed further since its inception: [4] and [5] convey one set of data by
antennas which convey real symbols and another set by antennas which convey imaginary
symbols, [6] improves performance by allowing the number of active transmit antennas to
change, [7] incorporates a property of MQAM constellations in a sub-optimal detector, [8]
features a detector using compressed sensing, [9] offers a low-complexity near-optimal
detector by feeding a sub-optimal detector into an optimal one, [10] combines trellis coding
with SM, [11] allows spatial constellations whose sizes are not powers of two and [12]
combines space-time block codes with SM.
Wang et al propose SM with multiple active transmit antennas (MASM) [14], in which there are
multiple active transmit antennas over the single active transmit antenna in conventional SM
[2]. In MASM, a group of active antennas transmit MQAM symbols and thus data is carried both

3. 3
via the MQAM symbol and the group selection. Let 𝑁𝑡, 𝑁 𝑝 and 𝑁 𝛾 be the numbers of transmit
antennas, active transmit antennas and antenna groups respectively and M be the size of the
MQAM constellation. MASM has a spectral efficiency of log2 𝑁 𝛾 + 𝑁 𝑝 𝑙𝑜𝑔2 𝑀, where log2 𝑁 𝛾 ≤
⌊𝑙𝑜𝑔2 (
𝑁𝑡
𝑁 𝑝
)⌋, which is usually larger than SM at 𝑙𝑜𝑔2 𝑀𝑁𝑡.
[14] presents a suboptimal detector for MASM based on decorrelation but did not explore
maximum likelihood (ML) detection, nor analyze ML performance. This motivates the authors
to, in this work, simulate MASM with ML detection and analyze the ML BER performance of
MASM systems.
The ML MASM detector requires an exhaustive search among all possible 𝑁 𝛾 𝑀 𝑁 𝑝 MASM
symbols to choose the most probable estimate. The detection complexity increases rapidly with
𝑁 𝑝 and M. For large M the complexity is extremely high. Although low-complexity, close to
optimal techniques such as sphere decoding exist for SM [15-16], such has not been applied to
MASM. In addition, to the best of the authors’ knowledge at the time of writing, only [17] has
been proposed to deal with the high complexity of MASM. However, in our simulations, [17] is
unable to achieve ML performance and has a significantly fluctuating complexity at different
signal to noise ratios (SNR). This motivates the development of a low-complexity, ML
performing detection scheme for MASM with a fixed complexity. In this paper, the authors
propose a simplified ML detection scheme for MASM based on the symbol cancellation method
in [1] and the multi-level subset searching method of [18]. In simulations, the proposed
technique is found to perform the same as the ML detection down to bit error rates in the
order of 10−6
for three considered configurations of MASM.
The paper is organized as follows: a transmission model of MASM is presented in section 2,
with models of the signal, channel, noise and received signal. ML based optimal detection,
suboptimal detection based on decorrelation [14], OB-MMSE detection [17] and the proposed
simplified ML detection schemes are described in section 3. A lower bound for the performance

4. 4
of ML detection is derived in section 4. Section 5 presents a complexity analysis of the four
considered detectors. Simulation and complexity results are shown in section 6, and finally,
concluding remarks are made in section 7.
The following notation convention is used in this work unless otherwise specified:
(⋅)−1
,(⋅) 𝑇
,(⋅) 𝐻
,(⋅)†
, 𝐸[⋅],| ⋅ | and |⋅| 𝐹 refer to 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 𝑖 𝑡ℎ
row, 𝑗 𝑡ℎ
column entry in the corresponding matrix and subscript (⋅)𝑖 denotes the 𝑖 𝑡ℎ
entry in the
corresponding vector or the 𝑖 𝑡ℎ
column in the corresponding matrix.
II. SystemModel
[Fig. 1 Here]
The MASM transmission scheme is described in [14]. For convenience, it is described again
here. We consider a MIMO environment with 𝑁𝑡 transmit and 𝑁𝑟 receive antennas. In Fig. 1, 𝑁 𝑝
groups of log2 𝑀 bits are taken from the input bitstream and each is mapped to an MQAM
symbol for each of the 𝑁 𝑝 ≤ 𝑁𝑡 active transmit antennas. Another log2 𝑁 𝛾 bits are used to
select one of 𝑁 𝛾 antenna groups that prescribe which of the 𝑁𝑡 transmit antennas are active. A
symbol to antenna mapper maps each MQAM symbol, respectively, to its designated transmit
antenna in the selected antenna group. This mapping creates a transmit MASM symbol vector 𝒙
which is transmitted across the wireless fading channel 𝑯 to 𝑁𝑟 receive antenna, producing a
received signal vector 𝒚 after additive white Gaussian noise (AWGN) is added. The Gray coded
MQAM alphabet 𝒮 of size 𝑀 has symbols 𝑠𝑖, 𝑖ϵ[1: 𝑀] that are normalized so that 𝐸[| 𝑠𝑖|2] =
1/𝑁 𝑝. The set of antenna groups Γ has antenna group vectors 𝛾𝑐
1×𝑁𝑡
𝑐ϵ[1: 𝑁 𝛾] with ones in
positions corresponding to active antennas and zeroes elsewhere. For example, if a black/white
dot represents 1/0, then the antenna group selected in Fig. 1 is [1,0,1,0,0], meaning that the
first and second antennas are active. We also define for a given 𝛾𝑐 , the numbers 𝑙 𝑘, 𝑘ϵ[1: 𝑁 𝑝],
corresponding to the index of the 𝑘 𝑡ℎ
active antenna – note that the antenna group which 𝑙 𝑘

5. 5
belongs to is implicit. In Fig. 1, 𝑙1 = 1 and 𝑙2 = 3. Each 𝛾𝑐 has an associated rotation angle
which is applied to the symbols in the transmit vector1.
The set of MASM symbol vectors 𝒜 has transmit symbol vectors 𝒂 𝑞
𝑁𝑡 ×1
, 𝑞ϵ[1: 𝑀 𝑁 𝑝 𝑁 𝛾], each
with 𝑁 𝑝 entries 𝑎 𝑞𝑘, 𝑘𝜖[1: 𝑁 𝑝], taken from 𝒮 in positions corresponding to the active antennas
and zeros elsewhere. For example, Fig. 1 would have 𝒙 = 𝒂 𝑞 = [𝑎 𝑞1,0, 𝑎 𝑞2, 0,0], 𝑎 𝑞1, 𝑎 𝑞2ϵ𝒮. 𝒙
is that 𝒂 𝑞 chosen for transmission with 𝑥 𝑘 = 𝑎 𝑞𝑘. The received signal is given by
𝒚 = 𝑯𝒙 + 𝒏 (1)
where 𝑯 𝑁𝑟×𝑁𝑡 is the complex MIMO channel matrix with entries ℎ𝑖𝑗 corresponding to the
complex channel gain between the 𝑗 𝑡ℎ
transmit antenna and the 𝑖 𝑡ℎ
receive antenna, 𝑗 ∈
[1: 𝑁𝑡], 𝑖 ∈ [1: 𝑁𝑟]. 𝑯 has independent and identically distributed (i.i.d.) entries with complex
Gaussian distributions ℎ𝑖𝑗~ℂ𝒩(0;1), ∀𝑖, 𝑗. 𝒏 𝑁𝑟 ×1
is the noise vector with i.i.d. complex
Gaussian distributions 𝑛𝑖~ℂ𝒩(0; 𝜎2),∀𝑖, where 𝜎2
is the variance of the noise describing the
signal-to-noise ratio (SNR).
III. DetectionAlgorithms
In this section, we describe, for MASM, the ML detector, the decorrelator-based detector [14],
the OB-MMSE detector [17] and then present the proposed simplified ML detector. It is
assumed that full channel knowledge is available at the receiver.
A. Maximum Likelihood (ML) Detection
The ML MASM detector fully exploits the advantages of MASM by performing a joint detection
of transmit antenna group and MQAM symbols. The ML estimation of 𝒙 can be expressed as:
𝒙̃ = argmax
𝒂 𝑞ϵ𝒜
𝑃(𝒚|𝒂 𝑞 , 𝑯) (2)
(2) is developed further into the following form [19]:
1
It was found when simulating the MASM system with ML detection that the rotation of the MQAM
constellationbydifferentanglesforeachantennagroup as described in [14] made no difference to the
performance. Since the simplified decoding method proposed in this paper is based on maximum
likelihood, the rotation step is omitted for simplification.

6. 6
𝒙̃ = arg min
𝒂 𝑞ϵ𝒜
‖𝒚 − 𝑯𝒂 𝑞‖
𝐹
𝟐
(3)
For an MASM system with 𝑁 𝑝 active transmit antennas, the ML detector requires an exhaustive
search among all possible 𝑁 𝛾 𝑀 𝑁 𝑝 symbol vectors to choose only one of them. Clearly, the
detection complexity increases rapidly with 𝑁 𝑝 and M.
B. Decorrelator-Based Sub-OptimalDetection [14]
Since the ML detector for MASM has a very high detection complexity, [14] proposes a
decorrelator-based sub-optimal detector. The detector uses a zero-forcing (ZF) detector 𝑻 [19]
to estimate both the transmit antenna group and the MQAM symbols. The detector 𝑻, the
pseudo-inverse of the channel matrix, and subsequent antenna detection are given by:
𝑻 = 𝑯†
= ( 𝑯 𝐻
𝑯)−1
𝑯 𝐻
(4)
{𝑙̂1, …, 𝑙̂ 𝑁 𝑝
} = 𝑠𝑜𝑟𝑡 (argmax
𝑘
( 𝑻𝒚) 𝑘) (5)
where, in (5), the arg 𝑚𝑎𝑥 function returns the largest 𝑁 𝑝 entries, as opposed to the single
maximum entry, 𝑠𝑜𝑟𝑡(⋅) arranges the input vector in ascending order and 𝑙̂ 𝑘 is the estimated
index of the 𝑘 𝑡ℎ
transmit antenna. The active antennas can be estimated in such a way because
the inactive antennas transmit 0 and thus are expected to be the minimum entries in 𝑻𝒚 [14].
In [14], no procedure is described to deal with the selection of an invalid antenna group, thus,
we create our own solution. We say that an invalid antenna group is selected when ∄ 𝛾𝑐 with
{𝑙1, …, 𝑙 𝑁 𝑝
} = {𝑙̂1, …, 𝑙̂ 𝑁 𝑝
}. Let us define 𝒃 as a vector with 𝑁𝑡 entries, containing the entries of
𝑻𝒚 sorted in descending order. If the first 𝑁 𝑝 entries of 𝒃 result in an invalid group, we select
another group from 𝒃 in lexicographic order. If the second selection is also invalid, we continue
in lexicographic order until a valid group is found. Note that the procedure is repeated
(including the first selection) a maximum of (
𝑁𝑡
𝑁 𝑝
) − 𝑁 𝛾 + 1 times.
Assuming the estimate is correct, we estimate the transmitted MQAM symbols:
{𝒙̂1, … , 𝒙̂ 𝑁 𝑝
} = 𝑄( 𝑻𝒚) (6)

7. 7
where 𝑄(⋅) is the MQAM slicing function applied to each entry in the input vector [1]. For (6),
we choose the entries in 𝑻𝒚 that correspond to the estimated active antennas. From
{𝑙̂1, …, 𝑙̂ 𝑁 𝑝
} and {𝒙̂1,… , 𝒙̂ 𝑁𝒑
} , we create an estimate 𝒙̂ of 𝒙.
C. Ordered Block Minimum Mean SquareError Detection (OB-MMSE) [17]
The OB-MMSE detector is another method to deal with the high complexity of ML detection, in
which the focus is to estimate the most likely transmitted antenna groups in descending order,
which is apt for the configurations of 𝑁𝑡 = 16, 32 considered in [17]. OB-MMSE begins by
creating a variable 𝑧𝑖 for each transmit antenna 𝑖 ∈ 1: 𝑁𝑡 by multiplying the pseudo-inverse of
each channel column 𝒉𝑖 by the received signal vector:
𝑧𝑖 = ( 𝒉𝑖)†
𝒚 (7)
A sorted list 𝒋 of most likely antenna groups is then obtained using:
𝑤𝑐 = ∑ 𝑧𝑙 𝑘
𝑙 𝑘 ∈ 𝛾𝑐
𝑁 𝑝
𝑘=1
∀𝑐 (8)
[𝑗1, 𝑗2, … , 𝑗 𝑁𝛾
] = arg 𝑠𝑜𝑟𝑡( 𝒘) (9)
A minimum mean square error (MMSE) detector is applied to each antenna group in the sorted
list, in turn, using iteration number 𝑞, beginning with the most likely, until a stopping criterion is
met. The MMSE detector and stopping criterion are written:
𝒔̃ 𝑞 = 𝑄 ((𝑯 𝛾 𝑗 𝑞
𝐻
𝑯 𝛾 𝑗 𝑞
+ 𝜎2
𝑰) 𝑯 𝛾 𝑗 𝑞
𝐻
𝒚) (10)
‖𝒚 − 𝑯 𝛾 𝑗 𝑞
𝒔̃ 𝑞‖ < 𝑉𝑡ℎ (11)
Where 𝑉𝑡ℎ = 2𝑁𝑟 𝜎2
is the threshold distance and 𝑯 𝛾 𝑗 𝑞
is a matrix containing the columns of 𝑯
corresponding to 𝛾𝑗 𝑞
. If (11) is satisfied, the OB-MMSE detector constructs an estimate of the
transmitted MASM symbol using the current iteration of 𝒔̃ 𝑞 and 𝛾𝑗 𝑞
. If all 𝑁 𝛾 groups are
explored and the stopping criterion is still not met, the detector falls back to ML detection as in
(3).

8. 8
D. Simplified ML-Based Detection
Compared to ML detection, the decorrelator-based detector has a negligibly small complexity.
However, in simulations, the bit error rate (BER) performance of the decorrelator-based
detector is far worse than ML detection. In addition, again in simulations, the BER performance
of OB-MMSE detection is worse than ML and has a complexity which fluctuates widely with
SNR. Therefore, we propose a detector of simpler complexity than ML that can achieve ML
performance. We achieve lower complexity in the proposed scheme by reducing the set of
MASM symbol vectors that are evaluated by an ML detector - specifically, we create a subset
𝒳′
⊂ 𝒜 that contains only probable MASM symbols. The ML detector (3) is revised to:
𝒙̃ = arg min
𝒂 𝑞ϵ𝒳′
‖𝒚 − 𝑯𝒂 𝑞‖
𝐹
2
(12)
For the MASM configurations in this paper, and, more generally, for systems that do not have
𝑁𝑡 ≫ 1, we have 𝑁 𝛾 ≪ 𝑀 𝑁 𝑝. Thus, a reduction in antenna group candidates is not deemed a
priority. In light of this, 𝒳′
is created by sequentially considering each antenna group 𝛾𝑐 ∈ Γ a
priori. In each consideration, we use the successive interference cancellation (SIC) detector of
[1] to find an estimate 𝒙̂ 𝑐 of 𝒙 which contains entries 𝑥̂ 𝑐𝑘 for the 𝑘 𝑡ℎ
active antenna. To
improve the reliability of the SIC detector, we use the multi-level subset searching method of
[15] to find those symbols in 𝒮 which are adjacent to 𝑥̂ 𝑐𝑘 (neighbours) as alternative estimates.
Finally, we create 𝒳′ containing those 𝒂 𝑞 which are allowed by the estimates and alternative
estimates.
The SIC detector operates as follows: for each 𝛾𝑐 ϵΓ, we associate an 𝑁𝑟 × 𝑁 𝑝 transmission
matrix 𝑯 𝛾 𝑐
containing only those columns in 𝑯 corresponding to 𝛾𝑐 . Execution is performed
∀𝛾𝑐 to output estimates 𝒙̂ 𝑐 𝜖𝒜 of the transmit vector 𝒙. The pseudo-code of the proposed SIC
detection is given in (13):
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑖𝑠𝑎𝑡𝑖𝑜𝑛
𝑐 ← 0
𝑂𝑢𝑡𝑒𝑟 𝑅𝑒𝑐𝑢𝑟𝑠𝑖𝑜𝑛
(13a)

9. 9
𝑖 ← 0
𝑯1 ← 𝑯 𝛾 𝑐
𝒚1 ← 𝒚
𝐼𝑛𝑛𝑒𝑟 𝑅𝑒𝑐𝑢𝑟𝑠𝑖𝑜𝑛
(13b)
(13c)
(13d)
𝑖 ← 𝑖 + 1
𝑮𝑖 ← ( 𝑯𝑖
𝐻
𝑯𝑖)−1
𝑯𝑖
𝐻
𝑡𝑖 ← argmin
𝑘
‖( 𝑮𝑖) 𝑘‖2
𝑧𝑖 ← ( 𝑮𝑖) 𝑡 𝑖
𝑦𝑖
𝑥̂ 𝑐𝑖 ← 𝑄(𝑧 𝑡 𝑖
)
𝑯𝑖+1 ← ( 𝑯𝑖) 𝑡 𝑖
𝒚 𝑖+1 ← 𝒚 − 𝒉𝑡 𝑖
𝑥̂ 𝑐𝑖
(13e)
(13f)
(13g)
(13h)
(13i)
(13j)
(13k)
𝑈𝑛𝑡𝑖𝑙 𝑖 > 𝑁 𝑝
𝑈𝑛𝑡𝑖𝑙 𝑐 > 𝑁 𝛾
𝐸𝑛𝑑
where the scalar 𝑥̂ 𝑐𝑖 refers to the 𝑖 𝑡ℎ
symbol to be used in the construction of 𝒙̂ 𝑐 and 𝑖
indicates the 𝑖 𝑡ℎ
recursion for a given 𝛾𝑐 . Note that trivial steps, not included in the pseudo-
code, are needed for proper ordering and distribution in the construction of 𝒙̂ 𝑐. ( 𝑯𝑖) 𝑡 𝑖
is the
matrix 𝑯𝑖 with column 𝑡𝑖 removed, ( 𝑮𝑖) 𝑡 𝑖
is the 𝑡𝑖
𝑡ℎ
row of 𝑮𝑖 and 𝑄(⋅) is an MQAM slicing
function. After (13), we are presented with 𝑁 𝛾 SIC estimates 𝒙̂ 𝑐, one estimate per antenna
group.
As discussed, the next step is to find the subset of MQAM symbols lying adjacent to 𝑥̂ 𝑐𝑘, ∀𝑐, 𝑘.
We define the set of neighbours to an MQAM symbol 𝑠𝑖 as (here, 𝑖 refers to the 𝑖 𝑡ℎ
MQAM
symbol in 𝒮, not the 𝑖 𝑡ℎ
iteration of the SIC detector):

10. 10
𝒮𝑠𝑖
= { 𝑠 𝑚| ‖ 𝑠 𝑚 − 𝑠𝑖 ‖2
< 𝑑2
∀𝑠 𝑚 ∈ 𝒮}, 𝑚 ∈ [1: 𝑀] (14)
where 𝑑 is the radius within which the neighbours lie. In this paper, 𝑑 is the distance between
two MQAM symbols lying diagonally adjacent in 𝒮. We expand 𝒙̂ 𝑐 to include neighbours and
the resultant set 𝒳𝑐 is:
𝒳𝑐 = {𝒂 𝑞 | 𝑎 𝑞𝑘 ∈ 𝒮 𝑥̂ 𝑐𝑘
, 𝒂 𝑞 ∈ 𝛾𝑐 , 𝒂 𝑞 ∈ 𝒜} (15)
where 𝒂 𝑞 ∈ 𝛾𝑐 means that 𝒂 𝑞 lies in the same antenna group as 𝒙̂ 𝑐. Finally, the reduced set
upon which (12) is performed is given by:
𝒳′
= {𝒳1, … , 𝒳 𝑁𝛾
} (16)
IV. Theoretical Performance Analysis of MASM withML Detection
Considering that the final step of the proposed detector is ML detection and assuming that the
reduced set of MASM symbols to be searched by the final step includes the transmitted symbol,
the performance bound for ML detection is applicable to the proposed scheme. It is shown in
Fig. 2, the simulation results, that this assumption is correct. Thus, we derive an asymptotic
performance bound for MQAM MASM with ML detection in i.i.d. Rayleigh flat fading channel
conditions.
We note that ML detection performs a joint detection of MQAM symbols and transmit antenna
group. In order to derive a closed form BER expression we simplify the analysis by decoupling
transmit antenna group detection and symbol detection performance as in [9]. In the process of
doing this, we assume perfect MQAM symbol detection when deriving antenna group detection
and vice-versa. The overall bit error probability is bounded by
𝑝𝑒 ≥ 𝑝 𝑎 + 𝑝 𝑑 − 𝑝 𝑎 𝑝 𝑑 (17)
where 𝑝 𝑎 is the bit error probability (BEP) for antenna group detection, 𝑝 𝑑 is the BEP for MQAM
symbol detection and 𝑝𝑒 is the overall BEP for the whole system.
In the following two subsections, we derive 𝑝 𝑎 and 𝑝𝑒.

11. 11
A. Antenna Group Detection
Noting (3) and the assumption of correct symbol detection, we can say that an antenna group
detection error will occur when:
‖𝒚 − ∑ 𝒉𝑙 𝑘
𝑥 𝑘
𝑁 𝑝
𝑘=1
‖
𝐹
2
> ‖𝒚 − ∑ 𝒉 𝑙̂ 𝑘
𝑥 𝑘
𝑁 𝑝
𝑘=1
‖
𝐹
2
(18)
where 𝑙, 𝑙̂ ϵΓ, 𝒉𝑙 𝑘
is the channel vector in the 𝑙 𝑘
𝑡ℎ
column of 𝑯, corresponding to the 𝑘 𝑡ℎ
active
transmit antenna, 𝑙 ≠ 𝑙̂ and 𝑥 𝑘 is the 𝑘 𝑡ℎ
MQAM symbol in 𝒙. As such and similarly, 𝑙̂ 𝑘
corresponds to the 𝑘 𝑡ℎ
active transmit antenna in an incorrect antenna group.
For simplicity, we further assume that only one antenna is decoded incorrectly and, the BEP for
antenna group detection based on (18) is given by (intermediate steps can be found in the Equ.
(36) in [9])
𝑝 𝑎 = 𝑃 (‖ 𝒏‖ 𝐹
2
> ‖(𝒉𝑗 − 𝒉𝑗̂)𝑥 + 𝒏‖
𝐹
2
| 𝑯 ) (19)
where 𝑗, 𝑗̂ ∈ [1: 𝑁𝑡], 𝑗 ≠ 𝑗̂.
This problem is similar to the case in Equ.(14) in [9] which has solution Equ.(19) in [9] and is
written in (20):
𝑝 𝑎 ≤ ∑ ∑ 𝑁( 𝑗,𝑗̂)
𝑁𝑡
⋅ ∑
𝜇 𝛼
𝑁 𝑟 ∑ (
𝑁𝑟 −1+𝑤
𝑤
)[1−𝜇 𝛼] 𝑤𝑁 𝑟−1
𝑤=0
𝑀
𝑀
𝑞=1
𝑁𝑡
𝑗̂=1
𝑁𝑡
𝑗=1 (20)
where, 𝜇 𝛼 =
1
2
(1 − √
𝜎 𝛼
2
1+𝜎 𝛼
2 ), 𝜎𝛼
2
=
𝑝
2
|𝑥𝑗|
2
, 𝑁( 𝑗, 𝑗̂) is the number of bits in error between
transmit antenna index j and estimated transmit antenna index 𝑗̂ and 𝑝 is the SNR per active
antenna. Note that the total power must be divided among all active antennas.
The solution Equ.(19) in [9] is for a single active transmit antenna and [9] has not dealt with the
case of multiple active transmit antennas. Since MASM has multiple active transmit antennas,
we adapt (20) to cater to MASM. We observe that the first factor in (20), the double
summation, can be written as 𝐸[ 𝑁( 𝑗, 𝑗̂)] 𝑁𝑡, which will be generalized to 𝑁̅𝑎−𝑏𝑖𝑡𝑠 𝑁̅Γ−1. Here
𝑁̅𝑎−𝑏𝑖𝑡𝑠 is the average number of bits in error when a single active antenna is mistaken, and
𝑁̅Γ−1 is the average number of incorrect positions that can be occupied by the single active
antenna when the positions of the other 𝑁 𝑝 − 1 active antenna(s) are fixed. Similar to the

12. 12
discussion in Appendix A of [14], we see that when finding the antenna error probability for
MASM, we must take into account the probability of any of the active antennas being mistaken.
Thus, we multiply our expression for 𝑝 𝑎 by 𝑁 𝑝.
Let us define 𝑁𝑐(𝑙 𝑘,𝑙 𝑘̂
′
) as a function which returns the number of bits in error when, for the 𝑐 𝑡ℎ
antenna group, the 𝑘 𝑡ℎ
active antenna in position 𝑙 𝑘 is mistaken for the 𝑘̂ 𝑡ℎ
inactive antenna in
position 𝑙 𝑘̂
′
. Using these definitions, we write 𝑁̅𝑎−𝑏𝑖𝑡𝑠 𝑁̅Γ−1 as ∑ ∑ ∑
𝑁𝑐( 𝑙 𝑘,𝑙 𝑘̂
′ )
𝑁𝛾 𝑁 𝑝
𝑁𝑡 −𝑁 𝑝
𝑘̂=1
𝑁 𝑝
𝑘=1
𝑁𝛾
𝑐=1
. Note
that 𝑁𝑐(𝑙 𝑘, 𝑙 𝑘̂
′
) returns zero if an invalid 𝛾 is described. Also note that 𝑁̅𝑎−𝑏𝑖𝑡𝑠 𝑁̅Γ−1 is dependent
on the selection of the antenna set.
Finally, we can write the BEP of antenna group detection as:
𝑝 𝑎 ≤ ∑ ∑ ∑
𝑁𝑐( 𝑙 𝑘,𝑙 𝑘̂
′ )
𝑁𝛾
⋅ ∑
𝜇 𝛼
𝑁 𝑟 ∑ (
𝑁𝑟 −1+𝑤
𝑤
)[1−𝜇 𝛼 ] 𝑤𝑁 𝑟−1
𝑤=0
𝑀
𝑀
𝑞=1
𝑁𝑡 −𝑁 𝑝
𝑘̂ =1
𝑁 𝑝
𝑘=1
𝑁𝛾
𝑐=1
(21)
B. MQAM SymbolError probability
Similar to (18) and under the assumption of perfect antenna group detection, we write that an
MQAM symbol error will occur when:
‖ 𝒚 − 𝑯𝒙‖ 𝐹
2
> ‖ 𝒚 − 𝑯𝒙′‖ 𝐹
2
(22)
where 𝒙 ∈ 𝒜 is the correct MASM symbol and 𝒙′
∈ 𝒜 is incorrect, but from the same antenna
group as 𝒙. For simplicity, we further assume that 𝒙 differs from 𝒙′
by only one MQAM symbol,
giving the BEP for MQAM symbol detection, based on (22):
𝑝 𝑑 = 𝑃(‖ 𝒏‖ 𝐹
2
> ‖ 𝒉 𝑘( 𝑥 𝑘 − 𝑥 𝑘
′ ) + 𝒏‖ 𝐹
2
| 𝑯) (23)
This probability is equivalent to the symbol error rate 𝑆𝐸𝑅 for MQAM with 𝑁𝑟 receivers and ML
detection as given in [9], which is the following numerically integrated solution:
𝑆𝐸𝑅 =
𝑎
𝑐
{
1
2
(
2
𝑏𝑝+2
)
𝑁𝑟
−
𝑎
2
(
1
𝑏𝑝+1
)
𝑁𝑟
+ (1 − 𝑎) ∑ (
𝑆𝑖
𝑏𝑝+𝑆𝑖
)
𝑁𝑟
+ ∑ (
𝑆𝑖
𝑏𝑝+𝑆𝑖
)
𝑁𝑟
2𝑐−1
𝑖=𝑐
𝑐−1
𝑖=1 } (24)
where 𝑎 = 1 −
1
√ 𝑀
, 𝑏 =
3
𝑀−1
, 𝑚 = log2 𝑀, 𝑆𝑖 = 2sin2
𝜃𝑖, 𝜃𝑖 =
𝑖𝜋
4𝑐
, 𝑐 is the number of
summations and 𝑝 is the SNR per antenna. [9] shows that for 𝑐 > 10, there is a 0.0015dB,
0.0025dB and 0.0029dB error between the simulated and theoretical results for each of 4, 16
and 64 MQAM constellations respectively.

13. 13
Since gray mapping is used, we assume a symbol error results in a single bit error at high SNR.
Therefore, we write:
𝑝 𝑑 ≈ 𝑆𝐸𝑅/ log2 𝑀 (25)
V. Complexity Analysis
In this section, we analyze the computational complexity of the ML detector, the decorrelator-
based detector [14], the OB-MMSE detector [17] and the proposed detector. As in [9], the
computational complexity is in terms of complex multiplications and additions.
A. Computational Complexity of ML Detection
There are 𝑁 𝛾 𝑀 𝑁 𝑝 MASM symbols to be evaluated. Let us note the ML metric (3): we see that
for each evaluated symbol, we need 𝑁𝑟 𝑁𝑝 complex multiplications in the matrix multiplication
𝑯𝒂 𝑞 and 𝑁𝑟 complex multiplications in finding the Frobenius norm. The multiplication 𝑯𝒂 𝑞
requires 𝑁𝑟(𝑁 𝑝 − 1) complex additions, the subtraction of this product from 𝒚 requires 𝑁𝑟
complex additions and finding the Frobenius norm requires 𝑁𝑟 𝑁 𝑝 + 𝑁𝑟 − 1 complex additions.
We write these as:
𝛿 𝑀𝐿−𝑚𝑢𝑙𝑡 = 𝑁 𝛾 𝑀 𝑁 𝑝 𝑁𝑟(𝑁𝑝 + 1) (26)
𝛿 𝑀𝐿−𝑎𝑑𝑑 = 𝑁 𝛾 𝑀 𝑁 𝑝 (𝑁𝑟 𝑁 𝑝 + 𝑁𝑟 + 1) (27)
Let us note that the number of complex multiplications and additions here reduce to the case in
Table 1 of [20] if we: i) choose a square channel matrix; ii) choose 𝑁𝑡 − 𝑁 𝑝 = 0.
B. Computational Complexity of Decorrelator-Based Detection [14]
Note that the matrix inverse operation in [14] uses Gaussian elimination, whilst [20], which will
be used for our analysis, uses 𝐿𝐷𝐿 𝐻
decomposition. We begin with the computation of the
pseudo-inverse for (4). There are three sub-steps: matrix multiplication of the Hermitian of the
channel matrix with the channel matrix, the inverse operation and the multiplication of the
inverse matrix with the Hermitian of the channel matrix. These sub-steps are adapted from 𝛿1
in Section V-D. 𝑻 is then multiplied by the received signal vector 𝒚 for (5) and the complexity for

14. 14
this step is adapted from 𝛿3 in Section V-D. A zero complexity slicing function 𝑄(⋅) is assumed
[14].
The complexity for multiplication and addition are respectively given by:
𝛿 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙−𝑚𝑢𝑙𝑡 =
1
2
(3𝑁𝑟 𝑁𝑡
2
+ 𝑁𝑡
3
+ 3𝑁𝑟 𝑁𝑡 + 𝑁𝑡
2
− 2𝑁𝑡) (28)
𝛿 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙−𝑎𝑑𝑑 =
1
2
(3𝑁𝑟 𝑁𝑡
2
+ 𝑁𝑡
3
+ 𝑁𝑟 𝑁𝑡 − 2𝑁𝑡
2
− 3𝑁𝑡) (29)
C. Computational Complexity of OB-MMSE Detection [17]
The complexity of OB-MMSE can be broken into five steps: finding 𝑧𝑖, finding 𝑤𝑖, performing
MMSE, determining if the stopping criterion is met and performing ML decoding should no
suitable candidates be found. These steps have complexities represented here as
𝛿 𝑧, 𝛿 𝑤, 𝛿 𝑀𝑀𝑆𝐸, 𝛿 𝑠𝑡𝑜𝑝 and 𝛿 𝑀𝐿, with relevant suffixes in the subscripts to denote multiplication or
addition. We find in [17] that the ML and stopping criterion steps are not considered, thus for
fair comparison, the complexity of OB-MMSE is recalculated and found to be comprised of the
following:
𝛿 𝑧−𝑚𝑢𝑙𝑡 = 2𝑁𝑡( 𝑁𝑟 + 1) (30)
𝛿 𝑧−𝑎𝑑𝑑 = 𝑁𝑡( 𝑁𝑟 − 1) (31)
𝛿 𝑤−𝑚𝑢𝑙𝑡 = 0 (32)
𝛿 𝑤−𝑎𝑑𝑑 = 𝑁 𝛾(𝑁 𝑝 − 1) (33)
𝛿 𝑀𝑀𝑆𝐸−𝑚𝑢𝑙𝑡 = (
1
2
𝑁 𝑝
3
+
1
2
𝑁 𝑝
2 (3𝑁𝑟 + 1) +
3
2
𝑁 𝑝 𝑁𝑟) 𝜌 𝑀𝑀𝑆𝐸 (34)
𝛿 𝑀𝑀𝑆𝐸−𝑎𝑑𝑑 = (
1
2
𝑁 𝑝
3
+
1
2
𝑁 𝑝
2(3𝑁𝑟 − 2) +
1
2
𝑁 𝑝( 𝑁𝑟 − 1)) 𝜌 𝑀𝑀𝑆𝐸 (35)
𝛿 𝑠𝑡𝑜𝑝−𝑚𝑢𝑙𝑡 = 𝑁𝑟(𝑁 𝑝 + 1)𝜌 𝑀𝑀𝑆𝐸 (36)
𝛿 𝑠𝑡𝑜𝑝−𝑎𝑑𝑑 = (𝑁𝑟 𝑁 𝑝 + 𝑁𝑟 + 1)𝜌 𝑀𝑀𝑆𝐸 (37)
𝛿 𝑀𝐿−𝑚𝑢𝑙𝑡 = 𝑁 𝛾 𝑀 𝑁 𝑝 𝑁𝑟(𝑁 𝑝 + 1)𝜌 𝑀𝐿 (38)
𝛿 𝑀𝐿−𝑎𝑑𝑑 = 𝑁 𝛾 𝑀 𝑁 𝑝 (𝑁𝑟 𝑁 𝑝 + 𝑁𝑟 + 1)𝜌 𝑀𝐿 (39)
Where 𝜌 𝑀𝑀𝑆𝐸 , 𝜌 𝑀𝐿 are the average number of executions of the MMSE and ML step in each
frame, respectively, found via simulation.

15. 15
D. Computational Complexity of the Proposed Simplified ML detector
We tackle the complexity analysis of the proposed detector in four steps:
1) The first step is the calculation of the ZF receiver 𝑮 (13f), which is further broken down
into three sub-steps: The multiplication of the Hermitian of the nulled channel matrix by
itself 𝑯𝑖
𝐻
𝑯𝑖, the calculation of the matrix inverse and the multiplication of said inverse
by the Hermitian of the nulled channel matrix.
In sub-step 1, the result contains 𝑁 𝑝
2
entries, each requiring 𝑁𝑟 complex multiplications
and 𝑁𝑟 − 1 complex additions. Noting symmetry, the complexity is reduced in light of
there being 𝑁 𝑝 unique entries and
𝑁 𝑝
2
−𝑁 𝑝
2
symmetrical entries.
Thus, we write 𝛿 𝑎−𝑏 for this sub-step, where 𝑎 and 𝑏 indicate the step and sub-step
number and the suffix in the subscript denotes multiplication or addition:
𝛿1−1−𝑚𝑢𝑙𝑡 = 𝑁𝑟 (
𝑁 𝑝
2
+𝑁 𝑝
2
) (40)
𝛿1−1−𝑎𝑑𝑑 = ( 𝑁𝑟 − 1)(
𝑁 𝑝
2
+𝑁 𝑝
2
) (41)
In sub-step 2, we use 𝐿𝐷𝐿 𝐻
decomposition as in [17] to give:
𝛿1−2−𝑚𝑢𝑙𝑡 =
1
2
𝑁 𝑝
3
+
1
2
𝑁 𝑝
2
− 𝑁 𝑝 (42)
𝛿1−2−𝑎𝑑𝑑 =
1
2
𝑁 𝑝
3
−
1
2
𝑁 𝑝
2
(43)
For sub-step 3, the result contains 𝑁𝑟 𝑁 𝑝 entries, each requiring 𝑁 𝑝 multiplications and
𝑁 𝑝 − 1 additions:
𝛿1−3−𝑚𝑢𝑙𝑡 = 𝑁𝑟 𝑁 𝑝
2
(44)
𝛿1−3−𝑎𝑑𝑑 = 𝑁𝑟 𝑁 𝑝(𝑁𝑝 − 1) (45)
2) In the second step, we compute the symbol which has the largest post-detection SNR by
evaluating the ZF receiver 𝑮. Each of the 𝑁𝑟 𝑁 𝑝 entries in 𝑮 is squared and each
subsequent row is summed before selecting the largest entry in the resulting column.
Thus, we write:

16. 16
𝛿2−𝑚𝑢𝑙𝑡 = 𝑁𝑟 𝑁𝑝 (46)
𝛿2−𝑎𝑑𝑑 = ( 𝑁𝑟 − 1) 𝑁 𝑝 (47)
3) Step three multiplies a row of 𝑮 by 𝒚 𝒊. It is trivial that the complexity of this step is
written:
𝛿3−𝑚𝑢𝑙𝑡 = 𝑁𝑟 (48)
𝛿3−𝑎𝑑𝑑 = ( 𝑁𝑟 − 1 ) (49)
4) Step four is the preparation for the next iterative step in which the contribution to 𝒚𝑖 by
the detected symbol in the current step is subtracted from 𝒚𝑖. It is trivial to write:
𝛿4−𝑚𝑢𝑙𝑡 = 𝑁𝑟 (50)
𝛿4−𝑎𝑑𝑑 = 𝑁𝑟 (51)
We now develop the complexity analysis which began with the four steps, noting:
1) We assume a zero complexity slicing function 𝑄(⋅).
2) It is clear that the complexity for each recursive step reduces as the recursion number
increases, since the number of columns in 𝑯𝑖 decreases as 𝑖 increases. We account for
this by replacing 𝑁 𝑝 in iterated steps with 𝑁𝑖ϵ[1: 𝑁 𝑝].
3) Step four, being a preparation, need not be executed on the final iterative step.
Once the symbol has been estimated, we proceed with ML decoding the reduced set 𝒳′. The
complexity of this step is dependent on the average number of neighbours to a given symbol in
the MQAM constellation 𝑁̅𝑠 – it can be shown that this number is:
𝑁̅𝑠 =
4𝑁𝑐𝑜𝑟𝑛𝑒𝑟𝑠 +4𝑁 𝑠𝑖𝑑𝑒𝑠 (√ 𝑀−2)+𝑁 𝑚 𝑖𝑑𝑑𝑙𝑒𝑠 (√ 𝑀−2)
2
𝑀
(52)
where 𝑁𝑐𝑜𝑟𝑛𝑒𝑟𝑠 , 𝑁𝑠𝑖𝑑𝑒𝑠 and 𝑁 𝑚𝑖𝑑𝑑𝑙𝑒𝑠 refer to the number of neighbours to a constellation point
(including itself) if the constellation point lies in the corner, on the side or in the middle of the
MQAM constellation respectively. These numbers, in turn, depend on 𝑑, the radius of a circle
encompassing the neighbour set of MQAM symbols with its centre at the estimated symbol.
Thus, we can write the total complexities for the simplified detection algorithm as:

17. 17
𝛿 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑−𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠
= 𝑁 𝛾 (𝑁̅𝑠
𝑁 𝑝
𝑁𝑟(𝑁𝑝 + 1)
+
1
2
∑ [ 𝑁𝑖
3
+ 𝑁𝑖
2 (3𝑁𝑟 + 1) + 𝑁𝑖(3𝑁𝑟 − 2) + 2𝑁𝑟] + 𝑁𝑟(𝑁 𝑝 − 1)
𝑁 𝑝
𝑁𝑖 =1
)
(53)
𝛿 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑−𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑠
= 𝑁 𝛾 (𝑁̅𝑠
𝑁 𝑝
(𝑁𝑟 𝑁 𝑝 + 𝑁𝑟 + 1)
+
1
2
∑ [ 𝑁𝑖
3
+ 𝑁𝑖
2(3𝑁𝑟 − 2) + 𝑁𝑖( 𝑁𝑟 − 3) + 2𝑁𝑟 − 2]
𝑁 𝑝
𝑁𝑖 =1
+ 𝑁𝑟(𝑁 𝑝 − 1))
(54)
VI. Results andDiscussion
In this section, simulation and complexity results are presented to compare the ML detector,
the decorrelator-based detector [14], the OB-MMSE detector [17] and the proposed detector.
In addition, the results of the theoretical bound are also presented. The three MASM
configurations considered for simulations are taken from [14] and are shown in Table 1.
The simulation environment regarding fading and noise is described in Section II and the
following parameters are assumed for simulation: a Gray coded M-QAM constellation; full
channel knowledge at the receiver and a total transmit power that is the same for all
configurations.
A. BER Simulation Results
[Fig. 2 here]

18. 18
Fig. 2 shows the BER simulation results for the ML detector, the proposed detector, the
decorrelator-based detector [14] and the OB-MMSE detector [17]. We see that the proposed
detector performs with a negligible difference to ML detection for all MASM configurations
considered down to the order of 10−6
, whilst the OB-MMSE detector exhibits a ~1dB drop in
performance from ML detection for all configurations down to the order of 10−5
. The
performance of the decorrelator-based detector is much worse than ML detection, with a gap
at a BER of 10−5
of approximately 14dB for the 6bits/s/Hz and 10bits/s/Hz schemes
respectively and very large for 15bits/s/Hz (assuming no error floor exists, in which case, the
gap is infinite). We can conclude from this that for the configurations visited, both the proposed
detector and the OB-MMSE detector perform aptly whilst the decorrelator-based detector
performs badly.
Fig. 3 shows the analytical bounds and ML detection simulation BERs for each MASM
configuration. The analytical bounds predict well the ML detection BER of MASM over the
entire range of considered SNR for each of the three MASM configurations. Since the proposed
detector is ML based and exhibits an almost identical performance to ML detection, we can say
that the theoretical bounds apply to the proposed detector too, if we assume the candidate list
includes the transmitted symbol. We see from Fig. 2 that this assumption is valid.
[Fig. 3 here]
B. Complexity Analysis Results
In this section, we combine the results in (26-27), (28-29), (30-39), (53-54) to create figures of
merit measured in floating operations per second (FLOPS) for each of the detectors
respectively. We assume a complex multiplication and a complex addition require 6 and 2
FLOPS respectively as in [20]. The results are tabulated, plotted and compared for the decoding
schemes in question at the three spectral efficiencies simulated. The spectral efficiencies of
6bits/s/Hz, 10bits/s/Hz and 15bits/s/Hz are shown in Figures 4, 5 and 6.

19. 19
From Figures 4, 5 and 6, we see that the decorrelator-based detector [14] exhibits significantly
lower complexity than the other detectors for all three configurations. However, Figure 1 shows
that the BER performance for said detector is significantly lacking in comparison to ML
detection. We also see for all configurations that ML detection has the highest complexity in all
but the 6bits/s/Hz configuration, where the proposed method has a slightly higher complexity.
The focus of comparison is thus between the OB-MMSE and proposed detectors.
For OB-MMSE, we derive two figures of merit: the first is the complexity plotted for each SNR
used in simulations and the second is the complexity averaged over all SNRs. For 6bits/s/Hz, the
complexity is relatively stable over the SNR range, however, for 10bits/s/Hz and 15bits/s/Hz, we
see a variation of about 10 times.
In the case of 6bits/s/Hz, we see that the OB-MMSE detector has a lower average complexity,
by about 10 times, than both the ML detector and the proposed detector. This fact, combined
with stability of complexity with SNR, makes OB-MMSE a good decoding candidate for the
6bits/s/Hz configuration. It is obvious that for the 4QAM (6bits/s/Hz) case, in which the
neighbour set is identical to the full MQAM set, no reduction in complexity can be achieved
using the proposed detector due to the SIC overhead.
At 10bits/s/Hz, the proposed detector has a slightly lower complexity than the average OB-
MMSE complexity, although, at high SNR, the OB-MMSE complexity is significantly lower than
the proposed method. Taking into account the instability of OB-MMSE over the SNR range and
the lower complexity of the proposed detector versus the average complexity of OB-MMSE, the
proposed detector is has merit.
At 15bits/s/Hz, the proposed decoder complexity is significantly lower than the average OB-
MMSE detector complexity and thus, notwithstanding the instability of OB-MMSE complexity in
this configuration, the proposed detector is advantageous in comparison.

20. 20
Analysis of (53) and (54) shows that the SIC overhead of the proposed detector accounts for
25%, 12% and 4% of the total complexity for 6, 10 and 15bits/s/Hz respectively. Further, we see
that the modulation size is not included and that complexity increases linearly with the number
of antenna groups. Thus, the ML step is significant in all configurations. With the significance of
the ML step in mind, greater complexity savings can be expected for MQAM constellations
larger than the largest 16QAM constellation considered in this work if 𝑑, the radius encircling
neighbouring MQAM points, is kept constant.
[Fig. 4 here]
[Fig. 5 here]
[Fig. 6 here]
VII. Conclusion
Higher spectral efficiency transmission schemes such as MASM perform well when the
exhaustive search method of optimal (ML) detection is used. A decorrelator-based detector
[14] and OB-MMSE detector [17] have been proposed to cater to this problem, however, when
applied to the MASM configurations in this paper, the decorrelator-based detector exhibits
poor BER performance and the OB-MMSE detector performs slightly worse than ML, with a
complexity that varies with SNR.
We propose a simplified ML detection scheme based on the symbol cancellation method in [1]
and multi-level subset searching method [18] that, in simulation, achieves the performance of
optimal detection down to bit error rates in the order of 10−6
. The proposed detector makes
an estimate of a transmitted symbol for each antenna group by assuming, in turn, each antenna
group a priori. The estimates are expanded to include the adjacent neighbours in the MQAM
constellation and the resultant reduced set of possible transmitted MASM symbols is forwarded
to an ML detector. Complexity analysis shows that for the higher two spectral efficiencies
considered, the proposed detector is favourable in comparison to the sub-optimal methods,
whilst at the lowest spectral efficiency, OB-MMSE is favourable. In all configurations
considered, the decorrelator-based detector [14] exhibits poor BER performance.

21. 21
In the work, we also found a bound for the ML detector performance, which was shown to
aptly predict the simulation BER.
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23. 23
FIGURE CAPTIONS
Fig. 1 Block diagram of MASM transmission with pseudo-example
Fig. 2 Performance of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and
ML detectors for 3 different MASM configurations
Fig. 3 Performance of ML detection and theoretical performance bound for 3 different MASM
configurations
Fig. 4 Complexities of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and
ML detectors at 6bits/s/Hz over the simulated SNR range
Fig. 5 Complexities of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and
ML detectors at 10bits/s/Hz over the simulated SNR range
Fig. 6 Complexities of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and
ML detectors at 15bits/s/Hz over the simulated SNR range

24. 24
FIGURES
Fig. 1
Fig. 2

25. 25
Fig. 3
Fig. 4
0 5 10 15 20 25
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
BER(bits/bit)
6bits/s/Hz ML
10bits/s/Hz ML
15bits/s/Hz ML
15 bits/s/Hz Theory
10 bits/s/Hz Theory
6 bits/s/Hz Theory
1.E+02
1.E+03
1.E+04
0 2 4 6 8 10 12 14 16
FLOPS
Signal to Noise Ratio (dB)
6 bits/s/Hz
OB-MMSE [17]
Average OB-MMSE [17]
Decorrelator [14]
Proposed
ML

26. 26
Fig. 5
Fig.6
1.E+03
1.E+04
1.E+05
0 2 4 6 8 10 12 14 16 18 20 22
FLOPS
Signal to Noise Ratio (dB)
10 bits/s/Hz
OB-MMSE [17]
Average OB-MMSE [17]
Decorrelator [14]
Proposed
ML
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
0 2 4 6 8 10 12 14 16 18 20 22 24
FLOPS
Signal to Noise Ratio (dB)
15 bits/s/Hz
OB-MMSE [17]
Average OB-MMSE [17]
Decorrelator [14]
Proposed
ML

27. 27
TABLE CAPTIONS
Table 1: Simulation Parameters
Table 1
6bits/s/Hz 10bits/s/Hz 15bits/s/Hz
𝑁 𝑝 2 2 3
𝑁𝑡 4 4 5
𝑁𝑟 5 5 5
𝑁 𝛾 4 4 8
𝑀 4 16 16