# Low-Complexity Improved-Throughput Generalised Spatial Modulation: Bit-to-
Symbol Mapping, Detection and Performance Analysis

Jiancheng An, Chao Xu, _Senior Member, IEEE_ , Yusha Liu, Lu Gan, and Lajos
Hanzo, _Fellow, IEEE_ This work was supported in part by the China Scholarship
Council. The work of L. Hanzo was supported in part by the Engineering and
Physical Sciences Research Council projects EP/Noo4558/1, EP/PO34284/1, and
COALESCE, in part by the Royal Society’s Global Challenges Research Fund
Grant, and in part by the European Research Council’s Advanced Fellow Grant
QuantCom.J. An and L. Gan are with the School of Information and Communication
Engineering, University of Electronic Science and Technology of China,
Chengdu, Sichuan 611731, China. (E-mail<EMAIL_ADDRESS>ganlu@uestc.edu.cn).C. Xu, Y. Liu and L. Hanzo are with the School of
Electronics and Computer Science, University of Southampton, SO17 1BJ, UK.
(E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>lh@ecs.soton.ac.uk).

###### Abstract

Low-complexity improved-throughput generalised spatial modulation (LCIT-GSM)
is proposed. More explicitly, in GSM, extra information bits are conveyed
implicitly by activating a fixed number $N_{a}$ out of $N_{t}$ transmit
antennas (TAs) at a time. As a result, GSM has the advantage of a reduced
number of radio-frequency (RF) chains and reduced inter-antenna interference
(IAI) at the cost of a lower throughput than its multiplexing-oriented full-RF
based counterparts. Variable-${N_{a}}$ GSM mitigates this throughput reduction
by incorporating all possible TA activation patterns associated with a
variable value $N_{a}$ ranging from $1$ to $N_{t}$ during a single channel-
use, which maximises the throughput of GSM but suffers a high complexity of
the mapping book design and demodulation. In order to mitigate the complexity,
_first of all_ , we propose two efficient schemes for mapping the information
bits to the TA activation patterns, which can be readily scaled to massive
MIMO setups. _Secondly_ , in the absence of IAI, we derive a pair of low-
complexity near-optimal detectors, one of them has a reduced search scope,
while the other benefits from a decoupled single-stream based signal detection
algorithm. _Finally_ , the performance of the proposed LCIT-GSM system is
characterised by the error probability upper bound (UB). Our Monte Carlo based
simulation results confirm the improved error performance of our proposed
scheme, despite its reduced signal detection complexity.

###### Index Terms:

Generalised spatial modulation, low-complexity improved-throughput generalised
spatial modulation, maximum likelihood detection, bitwise spatial mapping,
MIMO, antenna selection.

##  I Introduction

Spatial modulation (SM), constitutes a promising technique for the next-
generation multiple-input multiple-output (MIMO) system, which has attracted
substantial research interests [1, 2, 3]. The SM scheme strikes a flexible
trade-off between conventional MIMO techniques including Alamouti’s space-time
block code (STBC), and vertical Bell Laboratories layered space-time (V-BLAST)
system [4, 5, 6], which aim for retaining a diversity gain and a multiplexing
gain, respectively. More explicitly, for the SM scheme, the indices of the
transmit antenna (TA) are exploited as an extra dimension for transmitting
information besides the conventional amplitude-phase shift keying (APSK) such
as phase-shift keying (PSK) and quadrature amplitude modulation (QAM) [7, 3,
8]. Owing to the fact that only a single TA is activated in SM-MIMO at any
time instant, the inter-channel interference (ICI) and the inter-antenna
interference (IAI) of conventional MIMO techniques are mitigated, which
results in the simplified transceiver design and a reduced signal detection
complexity [2, 3]. Specifically, the MIMO power consumption is effectively
reduced by the SM philosophy, where only a single power amplifier is required
[9, 10, 11]. As a further advance, the SM scheme can be flexibly configured
for challenging communication scenarios, such as that of the downlink MIMO
systems having a rank-deficient channel matrix [8].

Inspired by the aforementioned advantages, the space shift keying (SSK)
concept was introduced in [12, 13] as a low-complexity implementation of SM,
where only the TA indices convey information. As a further advance, the
principle of SM has also been extended to the time domain, resulting in the
space-time shift keying (STSK) [14, 15, 16, 17], to the frequency domain
yielding the space-frequency shift keying (SFSK) [18] and to the time-
frequency domain creating the space-time-frequency shift keying (STFSK) [19,
20] to exploit all three degrees of freedom. As a result, the generalised
concept of index modulation (IM), has gained widespread interest in both
academia and industry [21, 22, 23, 24]. Furthermore, in [25, 26] and [27], the
IM philosophy was also extended to indoor optical wireless communication and
molecular communication, where visible light and molecules are the medium of
information transmission, respectively. Besides, both the analytical studies,
numerical simulations, as well as real-world experiments have verified that
SM-MIMOs have the inherent potential of outperforming many state-of-art MIMO
schemes, provided that a sufficiently high number of TAs is available at the
transmitter [28, 29].

TABLE I:  The contribution of the proposed LCIT-GSM scheme compared with other
GSM schemes.

Contributions | LCIT-GSM | [30] | [31] | [32] | [33] | [34] | [35] | [27] | [36] | [37] | [38]
---|---|---|---|---|---|---|---|---|---|---|---
Flexibility in the number of TAs | ✓ | ✓ | $\times$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Multiple active TAs | ✓ | ✓ | $\times$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Multiple streams∗ | $\times$ | ✓ | $\times$ | ✓ | ✓ | ✓ | ✓ | $\times$ | $\times$ | ✓ | $\times$
Variable number of active TAs | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | ✓ | ✓ | $\times$ | ✓
Bitwise spatial mapping | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | ✓ | $\times$ | $\times$ | $\times$
Joint spatial and classic symbol alphabet | ✓ | ✓ | ✓ | $\times$ | ✓ | $\times$ | ✓ | $\times$ | $\times$ | ✓ | $\times$
Low-complexity detectors | ✓ | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | ✓ | $\times$
Improved upper bound | ✓ | $\times$ | ✓ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$

  * •

[30]-2018: Single-RF and twin-RF SM for an arbitrary number of TAs;

  * •

[31]-2016: Joint spatial and classic symbol alphabet for SM;

  * •

[32]-2017: GSM aided mm-wave MIMO;

  * •

[33]-2016: Diagonal precoding aided GSM;

  * •

[34]-2019: Generalized quadrature spatial modulation;

  * •

[35]-2019: Generic signal shaping methods for the multiple-stream GSM;

  * •

[27]-2015: Depleted molecule shift keying;

  * •

[36]-2015: Variable active antenna SM;

  * •

[37]-2012: GSM with multiple active TAs;

  * •

[38]-2017: Variable-$N_{a}$ GSM.

∗The LCIT-GSM scheme considered in this paper is a single-stream GSM scheme
that improves the throughput by using more TA activation patterns in the
spatial domain while inherits the advantage of IAI-free. We note that the
LCIT-GSM scheme can also be combined with V-BLAST to improve throughput to a
greater extent, which, however, results in a variable-length code with serious
error propagation.

In order to improve the throughput of single-RF SM and SSK schemes,
generalised spatial modulation (GSM) was proposed in [39, 40], which activates
several TAs simultaneously. Furthermore, conventional full-RF diversity- and
multiplexing-oriented MIMO schemes have been combined with GSM [41, 37, 42,
43]. Specifically, in [41], the space-time block coding-spatial modulation
(STBC-SM) concept was proposed, where GSM was combined with STBC in order to
achieve a beneficial diversity gain at the cost of using a reduced number of
RF chains. Similarly, a novel multiplexing-oriented GSM scheme was proposed in
[37, 42, 43] that combines V-BLAST and GSM, where multiple TAs are activated
simultaneously in order to transmit independently modulated symbols. This
scheme imposed a reduced IAI compared to the full-RF V-BLAST scheme. Recently,
a new transmission scheme termed as generalised quadrature spatial modulation
(GQSM) was proposed in [34], which extends the GSM constellation to the in-
phase and quadrature dimensions [44]. The QSM mapping of [44] has then been
further improved in [45, 46, 47, 48] by increasing the system’s throughput
and/or reducing the bit error probability (BEP). However, all the
aforementioned GSM schemes – including the STBC-SM, SM+V-BLAST and GQSM –
always employ a fixed number $N_{a}$ of TAs, which inevitably results in the
well-known throughput limitation of the family of GSM schemes. In order to
eliminate this throughput limitation, the variable-$N_{a}$ GSM (VGSM) concept
was proposed for increasing the spatial constellation size by employing a
variable number $N_{a}$ of activated TAs [36, 49, 50].

Figure 1: “Spider diagram" comparing several space-time transmission schemes.
The hierarchy structure models ordered MIMO schemes in each category. From the
outer to the inner, the gain of the corresponding term becomes smaller for a
selected direction. Taking the throughput as an example, V-BLAST has the
highest throughput, followed by GSM+V-BLAST, GSM, GSM+STBC, SM and STBC.

Against this background, we propose a low-complexity improved-throughput
generalised spatial modulation (LCIT-GSM) scheme, which is inspired by the
novel bit-to-symbol mapping scheme of [27] in the context of molecular
communication. In order to reflect on these issues, first of all, the trade-
offs between the aforementioned state-of-the-art MIMO schemes are visualised
by the spider diagram of Fig. 1. The rationale of Fig. 1 is that considering
the throughput as an example, V-BLAST achieves the highest throughput,
followed by GSM+V-BLAST, GSM, GSM+STBC and STBC. Therefore, there are five
rings in Fig. 1, which represent the ordered position of the five main MIMO
families in each category. As summarised in Fig. 1, owing to its fixed number
of activated TAs, the GSM schemes generally suffer from a throughput loss.
Furthermore, the features of the new LCIT-GSM scheme using the improved
bitwise spatial mapping design of [27] are compared to the existing GSM
schemes in Table I, where the specific terminologies of the systems referenced
are listed below Table I for the readers’ convenience. More explicitly, the
contributions of the paper are as follows:

Figure 2: Schematic of the LCIT-GSM system model, where the number of
activated TAs is a variable that carries source information.

  * •

First of all, a LCIT-GSM scheme is proposed for improving the throughput by
activating a variable number $N_{a}$ of activated TAs, while reducing the
complexity of the mapping design. More specifically, we introduce three
spatial mapping arrangements that incorporate all the legitimate TA activation
patterns. The first two of them are motivated by the so-called depleted
molecule shift keying (MoSK) scheme of [27]. In order to accommodate the
special case of all-zeros in the bitstream that may result in no TA being
activated and hence no modulated symbols being transmitted, we conceive
specifically designed mapping arrangements for both SM and the classic APSK
modulation. This is in contrast to the MoSK regime of [27], where the all-zero
state of the nano-transmitter is directly exploited to convey information bits
in the context of SSK-based molecular communication. Furthermore, a look-up
table (LUT) based spatial mapping scheme similar to [36, 49] is also included
in order to demonstrate the advantages of the proposed low-complexity
detectors.

  * •

Secondly, in the absence of IAI in the LCIT-GSM mapping, we propose a pair of
low-complexity detectors based on the classical maximum likelihood detector
(MLD). Explicitly, a two-step near-maximum likelihood detector (TMLD) is
derived, which substantially reduces the size of the legitimate set to be
searched. Furthermore, a decoupling based maximum likelihood detector (DMLD)
is conceived for decoupling a two-dimensional joint search into a pair of one-
dimensional single-stream based searches, while maintaining the optimal
detection performance.

  * •

Thirdly, based on the prior work of _Haas et al._ [51], we derive a tighter
performance upper bound (UB) for the proposed LCIT-GSM system. Explicitly, the
improved performance bound is composed of three parts, which respectively
reflect the impacts of the LCIT-GSM scheme’s signal constellation diagram,
spatial constellation diagram and their interactions. Finally, our Monte Carlo
based simulation results confirm the improved spectral efficiency (SE),
reduced signal detection complexity and the improved BEP performance of our
proposed LCIT-GSM scheme compared to conventional GSM scheme [39] and VGSM
scheme [36, 49].

The rest of this paper is structured as follows. Section II presents the
improved LCIT-GSM system model and spatial mapping schemes. Following this,
our detection algorithms are conceived in Section III. Section IV analyses the
theoretical BEP and computational complexity of the LCIT-GSM system, while the
BEP performance of our LCIT-GSM systems is presented in Section V. Finally,
our conclusions are offered in Section VI.

_Notation:_ We use upper (lower) bold face letters to indicate matrices
(column vectors). ${\left(\cdot\right)^{T}}$, ${\left(\cdot\right)^{H}}$ and
${\left(\cdot\right)^{*}}$ represent the transpose, Hermitian transpose and
conjugate, respectively. $E\left\\{\cdot\right\\}$ stands for the expected
value. $\left\|\cdot\right\|$ is the Frobenius norm of a complex vector.
$\otimes$ represents the Kronecker product. We denote the $N\times N$ identity
matrix as ${{\bf{I}}_{N}}$. ${\bf{0}}$ and ${\bf{1}}$ denote an all-zero
vector and an all-one vector, respectively, with appropriate dimensions.
Furthermore, $\left\lfloor\cdot\right\rfloor$ and
$\left\lfloor\cdot\right\rceil$ represent the floor and rounding operation,
respectively. The $\log\left(\cdot\right)$ functions are base-2 by default.
$C_{n}^{k}$ denotes the number of combinations of taking $k$ out of $n$.
${{\mathcal{M}}_{\gamma}}\left(t\right)=E\left\\{{\exp\left({-\gamma
t}\right)}\right\\}$ is the moment generating function (MGF) of the random
variable $\gamma$.
$Q\left(x\right)=\frac{1}{\sqrt{2\pi}}{\int}_{x}^{+\infty}exp\left(\frac{-{t}^{2}}{2}\right)dt$
is the Q-function. $\angle\left(z\right)$ denotes the phase angle of a complex
number $z$. $\Re\left(z\right)$ and $\Im\left(z\right)$ denote the real and
imaginary part of a complex number $z$, respectively.

##  II System Model

###  II-A LCIT-GSM Transmission

The LCIT-GSM system model is depicted in Fig. 2. The incoming data bits are
first grouped into $m={m_{s}}+{m_{a}}$ bits. Following this, the grouped block
of bits ${{\bf{b}}^{T}}$ is split into $m_{s}$ spatial bits ${\bf{b}}_{s}^{T}$
and $m_{a}$ APSK modulated symbol bits ${\bf{b}}_{a}^{T}$, i.e.,
${{\bf{b}}^{T}}=\left[{\left.{{\bf{b}}_{s}^{T}}\right|{\bf{b}}_{a}^{T}}\right]$.
The first $m_{s}$ spatial bits are assigned to select the TA activation
pattern following the spatial mapping procedures, which will be detailed in
Section II-B. We note that only a reduced number of TAs are activated at any
time. The remaining $m_{a}$ bits are modulated using APSK, such as $M$-PSK or
$M$-QAM, where the modulation order is given by $M={2^{{m_{a}}}}$.

In contrast to the classic GSM, which activates a fixed number of TAs at each
slot, the number of activated TAs in the LCIT-GSM scheme is variable [36, 49].
This constitutes the most distinguished benefit of the LCIT-GSM design, where
the maximum throughput is pursued by utilizing all legitimate TA activation
patterns. The total number $N_{m}$ of all legitimate TA activation patterns is
given by:

${N_{m}}=\mathop{C_{{N_{t}}}^{0}}\limits_{----}+C_{{N_{t}}}^{1}+C_{{N_{t}}}^{2}+\cdots+C_{{N_{t}}}^{{N_{t}}}=2^{N_{t}},$
(1)

where the underlined $C_{{N_{t}}}^{0}$ corresponds to the spatial bit sequence
having all zeros.

Figure 3: The number $m_{s}$ of spatial bits versus the number $N_{t}$ of TAs
for SM, GSM, and LCIT-GSM, where we have a fixed number ${N_{a}}=2$ of
activated TAs for GSM.

However, it should be noted that the spatial bitstream with all zeros in LCIT-
GSM, i.e., ${\bf{b}}_{s}^{T}={\bf{0}}$ cannot transmit an APSK symbol carrying
$m_{a}$ bits, hence it must be avoided. This problem will be addressed in
Section II-B. If the case corresponding to ${\bf{b}}_{s}^{T}={\bf{0}}$ is
eliminated, only $\left({{N_{m}}-1}\right)$ TA activation patterns remain
legitimate. Bearing in mind that the number of TA activation patterns that
convey $m_{s}$ source bits must be a power of two, the maximum value of
${m_{s}}$ becomes111We note that $m_{s}$ is increased beyond the limit of (2)
by the first of our spatial mapping schemes proposed in Section II-B, namely
in DTAA-R, where we have $m_{s}=\log\left(2^{N_{t}}\right)=N_{t}$. The absence
of the all-zero activation pattern is compensated by the transformation of the
signal constellation, which ensures maintaining the maximum transmission rate
combined with the low complexity of the direct bit-to-antenna mapping scheme.
In this subsection, we assume that $\left({{N_{t}}-1}\right)$ spatial bits are
conveyed.

${m_{s}}=\left\lfloor{\log\left({{N_{m}}-1}\right)}\right\rfloor=\left\lfloor{\log\left({2^{N_{t}}-1}\right)}\right\rfloor={N_{t}}-1.$
(2)

In summary, the total number of source bits conveyed by the LCIT-GSM design is
given by

$m={m_{a}}+{m_{s}}=\log M+{N_{t}}-1,$ (3)

which is evidently higher than that of GSM given by $\log M+\left\lfloor\log
C_{{N_{t}}}^{{N_{a}}}\right\rfloor$, where $N_{a}$ is the number of RF chains.
Fig. 3 compares the number $m_{s}$ of spatial bits versus the number $N_{t}$
of TAs, for using GSM and LCIT-GSM, where the case of SM is portrayed as a
special case of GSM associated with ${N_{a}}=1$. It can be seen that when
$m_{s}=4$ spatial bits are transmitted, $N_{t}=4$ is needed for LCIT-GSM. For
the same throughput, the GSM scheme would require an increased value of
$N_{t}=6$, while the SM scheme requires a further increased value of
$N_{t}=16$, as demonstrated by Fig. 3.

_Remark 1:_ We note that, first of all, the number of spatial bits in LCIT-GSM
increases with the number of RF chains. By contrast, in GSM, the number of
spatial bits decreases when $N_{a}$ is increased beyond
$\left\lfloor\frac{N_{t}}{2}\right\rfloor$ [39]. Although we directly
introduce LCIT-GSM as having $N_{t}$ RF chains, hence exhibiting the maximum
throughput improvement, it must be mentioned that our system is also eminently
suitable for a limited number of RF chains. Secondly, we note that the same
APSK symbol is transmitted by all activated TAs in order to retain the IAI-
free feature. Finally, similar to the GSM scheme’s combination with STBC and
V-BLAST [52, 43], the proposed LCIT-GSM can also be combined with the classic
MIMO schemes for the sake of striking further trade-offs between throughput,
complexity and diversity, which is left for future work.

In the LCIT-GSM system, the transmit signal is modelled by a ${N_{t}}\times 1$
column vector $\bf{x}$, which is transmitted over an ${N_{t}}\times{N_{r}}$
MIMO Rayleigh flat fading wireless channel,
${\bf{H}}=[{{{\bf{h}}_{1}}}\quad{{{\bf{h}}_{2}}}\quad\cdots\quad{{{\bf{h}}_{{N_{t}}}}}]$,
where ${{\bf{h}}_{k^{\prime}}}$ is the $k^{\prime}$-th column of $\bf{H}$ that
represents the channel vector of the link spanning from the $k^{\prime}$-th TA
to all of $N_{r}$ receive antennas (RAs). The entries of ${\bf{H}}$ are
generated by complex independent and identically distributed (i.i.d.) Gaussian
random variables having zero-mean and unit-variance.

Figure 4: (a) DTAA-R mapping scheme: The $i$-th spatial bit controls the
on/off state of the $i$-th TA, where $1\leq i\leq 3$. For the special case of
’000’, all the three TAs are activated, while the $M$-PSK/QAM symbol is
rotated according to (7) or (8); (b) DTAA-D mapping scheme: The $i$-th spatial
bit controls the on/off state of the $i$-th TA, where $1\leq i\leq 3$. The
4-th TA is only activated when the spatial bitstream of ’000’ is encountered;
(c) LUT mapping scheme: The combination book selects the active TAs based on
the current spatial bitstream, which uses a mapping design similar to the VGSM
of [36, 49].

Finally, the signal ${\bf{y}}$ received at any symbol instant is given by

${\bf{y}}={\bf{Hx}}+{\bf{n}}={{\bf{h}}_{{{\mathcal{S}}_{k}}}}s+{\bf{n}},$ (4)

where $s\in M$-APSK is the modulated $M$-PSK/QAM symbol222We note that in the
first of the proposed bit-to-symbol mapping schemes detailed in Section II-B,
namely in DTAA-R, $s$ represents the rotated PSK/QAM symbol when the all-zero
spatial bitstream is encountered., while $\mathcal{S}_{k}$ is the $k$-th TA
activation combination associated with
$k\in\left\\{{1,2,\cdots,{2^{N_{t}-1}}}\right\\}$. The mapping mechanism of
$\left\\{{{\mathcal{S}_{1}},{\mathcal{S}_{2}},\cdots,{\mathcal{S}_{{2^{N_{t}-1}}}}}\right\\}$
will be detailed in Section II-B. Furthermore, the channel vector can be
expressed as

${{\bf{h}}_{{\mathcal{S}_{k}}}}=\sum\limits_{k\in{\mathcal{S}_{k}}}{{{\bf{h}}_{k}}},$
(5)

which is the summation of all channel vectors corresponding to the $k$-th TA
activation patterns. Moreover, ${\bf{n}}$ in (4) is an AWGN vector associated
with zero-mean and variance of $\sigma_{n}^{2}$ in each dimension at the
receiver.

###  II-B LCIT-GSM spatial mapping scheme

In this subsection, we will introduce three spatial mapping arrangements for
LCIT-GSM systems, which are: _1)_ direct TA activation relying on
constellation rotation (DTAA-R), _2)_ direct TA activation relying on a
dedicated TA (DTAA-D) and _3)_ LUT-based spatial mapping (LUT). More
specifically, DTAA-R and DTAA-D are the improved schemes of the bitwise
activation scheme of the MoSK [27], while LUT uses the same mapping design as
VGSM of [36, 49], which we include in order to share the low-complexity
detectors proposed in Section III.

1) DTAA-R: The motivation of the DTAA-R scheme is to establish a direct
mapping between the spatial bits and the TA activation pattern, which does not
require a large LUT. The simplest way of implementing this is to assign each
spatial bit to independently control the on/off state of the TAs, as portrayed
in Fig. 4 (a). Explicitly, the spatial bit-pattern ‘101’ activates the upper-
and lower-most TAs and the ‘0’ bit in the fourth position of the box is BPSK-
modulated onto both active TAs. The TA in the middle is deactivated. However,
as mentioned before, when all spatial bits of $b_{s}$ are zeros, no TAs are
activated to transmit the classic $M$-PSK/QAM symbol, which constitutes a
challenge in our DTAA-R design. In order to solve this problem, we propose the
following revised DTAA-R mapping rule:

$\left\\{{\begin{array}[]{*{20}{l}}{{{\bf{b}}_{s}}^{T}\neq{\bf{0}}\Leftrightarrow{\bf{x}}={\bf{b}}_{s}\times
s}\\\ {{{\bf{b}}_{s}}^{T}={\bf{0}}\Leftrightarrow{\bf{x}}={\bf{1}}\times
T\left(s\right)},\end{array}}\right.$ (6)

where ${\bf{1}}$ is a ${N_{t}}\times 1$ column vector of ones. In this way,
all TAs are activated, when the spatial bits are all zeros. Moreover, owing to
the fact that the TA activation pattern corresponding to
$\mathbf{b}_{s}^{T}=\mathbf{0}$ becomes exactly the same as in the case of
$\mathbf{b}_{s}^{T}=\mathbf{1}$, we propose to apply a transformation
$T\left(\cdot\right)$ to the APSK constellation.

More explicitly, for an $M$-PSK symbol, a simple phase rotation is applied to
the APSK symbol, when all the TAs are activated for
$\mathbf{b}_{s}^{T}=\mathbf{0}$, which can be expressed as:

$T_{PSK}\left(s\right)={e^{j\frac{\pi}{M}}}\cdot s.$ (7)

TABLE II:  LCIT-GSM mapping table for DTAA-R, DTAA-D, LUT, respectively, where
$\left({\cdot,\cdot}\right)$ indicates the indices of the active TAs, while
$N_{t}=\left\\{3,4\right\\}$, $m_{s}=3$ and $m_{a}=1$ are assumed∗.

Input Bits | DTAA-R with ${N_{t}}=3$ | DTAA-D with ${N_{t}}=4$ | LUT with ${N_{t}}=4$
---|---|---|---
TA Activation Pattern | Symbol | TA Activation Pattern | Symbol | TA Activation Pattern | Symbol
0000 | (1,2,3) | _i_ | (4) | -1 | (1) | -1
0001 | (1,2,3) | _-i_ | (4) | 1 | (1) | 1
0010 | (3) | -1 | (3) | -1 | (2) | -1
0011 | (3) | 1 | (3) | 1 | (2) | 1
0100 | (2) | -1 | (2) | -1 | (3) | -1
0101 | (2) | 1 | (2) | 1 | (3) | 1
0110 | (2,3) | -1 | (2,3) | -1 | (4) | -1
0111 | (2,3) | 1 | (2,3) | 1 | (4) | 1
1000 | (1) | -1 | (1) | -1 | (1,2) | -1
1001 | (1) | 1 | (1) | 1 | (1,2) | 1
1010 | (1,3) | -1 | (1,3) | -1 | (3,4) | -1
1011 | (1,3) | 1 | (1,3) | 1 | (3,4) | 1
1100 | (1,2) | -1 | (1,2) | -1 | (1,3) | -1
1101 | (1,2) | 1 | (1,2) | 1 | (1,3) | 1
1110 | (1,2,3) | -1 | (1,2,3) | -1 | (2,4) | -1
1111 | (1,2,3) | 1 | (1,2,3) | 1 | (2,4) | 1

∗Although we use a table here to describe the three mapping schemes, it is
worth noting that the proposed mapping schemes of DTAA-R and DTAA-D can be
readily extended to large-scale antenna arrays, which is challenging for LUT
with the same design as [36, 49].

By contrast, when $M$-QAM is employed instead of $M$-PSK, the phase rotation
of (7) no longer maximises the distance in the constellation. Instead, we
propose the following transformation:

${T_{QAM}}\left(s\right)={e^{j\frac{\pi}{{{M_{n}}}}}}\cdot s,$ (8)

where
${M_{n}}=\max\left\\{{count\left({\left|{{s_{i}}}\right|=\left|\chi\right|}\right)}\right\\}$
for $i\in[1,M]$, while $count\left(\cdot\right)$ represents the number of
occurrences of the event in parentheses, with $\left|\chi\right|$ representing
the set of magnitudes of all QAM constellation points. For example, when the
normalised 16QAM constellation is adopted, we have
$\left|\chi\right|=\left\\{\frac{1}{\sqrt{5}},1,\frac{3}{\sqrt{5}}\right\\}$,
yielding $a_{1}=count\left(\left|s_{i}\right|=\frac{1}{\sqrt{5}}\right)=4$,
$a_{2}=count\left(\left|s_{i}\right|=1\right)=8$ and
$a_{3}=count\left(\left|s_{i}\right|=\frac{3}{\sqrt{5}}\right)=4$. Therefore,
${M_{n}}=8$ is obtained according to (8). Following the same philosophy, we
have ${M_{n}}=12$ for 64QAM.

We note that although the special case of the all-zero spatial bitstream is
used for activating all TAs, other legitimate TA activation patterns that
contain at least one activated TA can also be readily chosen for the special
case in (6), provided that the modulated symbol for the chosen pattern is
rotated according to (7) and (8).

Again, the DTAA-R scheme is exemplified in Fig. 4 (a), where we have
${N_{t}}=3$ TAs and BPSK modulation. More explicitly, let us revisit Fig. 4
(a) a bit more formally, where the first $m_{s}=3$ bits are mapped to the TA
activation pattern, while the remaining $m_{a}=1$ bit modulates a BPSK symbol.
The full set of TA activation patterns is summarised in Table II. For example,
the bit sequence of $\mathbf{b}^{T}=[1\quad 0\quad 1\quad|0]$ results in the
TA activation pattern of $\left(1,3\right)$ and the BPSK symbol of $-1$, as
seen in Table II, which results in the transmitted signal vector of
$\mathbf{x}=[-1\quad 0\quad-1]^{T}$. Moreover, when the special case of the
all-zero spatial bitstream is encounted, the bit sequence of
$\mathbf{b}^{T}=[0\quad 0\quad 0\quad|1]$ is mapped to the TA activation
pattern of $\left(1,2,3\right)$ and to the rotated BPSK symbol $-i$, as shown
in Table II, which leads to the transmitted signal vector of
$\mathbf{x}=[-i\quad-i\quad-i]^{T}$ in Fig. 4 (a).

2) DTAA-D: Instead of applying a phase rotation to the APSK constellation, in
this section, we propose the DTAA-D mapping method, which assigns a dedicated
TA to be uniquely turned on only for the all-zero spatial bitstream. As a
result, the on/off states of the $\left(N_{t}-1\right)$ TAs are decided by the
$\left(N_{t}-1\right)$ non-all-zero spatial bits of Fig. 4 (b) and of Table II
without any ambiguity, hence eliminating the dependence of the DTAA-R scheme
on the constellation rotation of (7) and (8). More explicitly, the DTAA-D
mapping rule is defined by

$\left\\{{\begin{array}[]{*{20}{l}}{{{\bf{b}}_{s}^{T}}\neq{\bf{0}}\Leftrightarrow{\bf{x}}={{\left[{\begin{array}[]{*{20}{c}}{{\bf{b}}_{s}^{T}}&0\end{array}}\right]}^{T}}\times
s}\\\
{{{\bf{b}}_{s}^{T}}={\bf{0}}\Leftrightarrow{\bf{x}}={{\left[{\begin{array}[]{*{20}{c}}{{{\bf{0}}_{\left({{N_{t}}-1}\right)\times
1}^{T}}}&1\end{array}}\right]}^{T}}\times s}.\end{array}}\right.$ (9)

Therefore, the last TA of Fig. 4 (b) is activated for the special case of
${\bf{b}}_{s}^{T}={\bf{0}}$.

Again, the DTAA-D arrangement is exemplified in Fig. 4 (b), where the incoming
data bits are also divided into two parts: a spatial bit sequence
${{\bf{b}}_{s}^{T}}$ and a data bit sequence ${{\bf{b}}_{a}^{T}}$. We note
that ${N_{t}}=4$, $m_{s}=3$, and $m_{a}=1$ are assumed in Fig. 4 (b), and the
corresponding TA activation patterns are also summarised in Table II.
Specifically, based on Fig. 4 (b) and Table II, the information bit sequence
of ${\bf{b}}^{T}=[1\quad 0\quad 1\quad|0]$ is transmitted as
${\bf{x}}={[{-1}\quad 0\quad{-1}\quad 0]^{T}}$. However, based on (9), the bit
sequence of ${\bf{b}}^{T}=[0\quad 0\quad 0\quad|1]$ is mapped to
$\mathbf{x}=[0\quad 0\quad 0\quad 1]^{T}$, as seen in Table II, which avoids
the constellation rotation of $-i$ seen in the DTAA-R example. We will
demonstrate in Section V that as the benefit of eliminating the DTAA-R
scheme’s constellation rotation, the pairwise error probability is increased
by the DTAA-D scheme. Nonetheless, we note that this arrangement does not
compromise the LCIT-GSM throughput, since a total of $\left(N_{t}-1\right)$
spatial bits are transmitted by $N_{t}$ TAs, which is in line with (2).

3) LUT333We note that LUT uses the same mapping design as the VGSM of [36,
49], and we include it here in order to share the low complexity detectors
proposed in Section III.: The DTAA-D arrangement is capable of achieving the
full LCIT-GSM throughput without any constellation rotation, but
unfortunately, the distance between the legitimate TA activation patterns has
not been maximised. More explicitly, according to (1), there are a total of
$\left(2^{N_{t}}-1\right)$ legitimate TA activation patterns, where the case
of deactivating all TAs is eliminated. However, it can be seen in (2) that
only $2^{N_{t}-1}$ TA activation patterns are needed. Against this background,
the LUT arrangement is introduced to further optimize the DTAA-D scheme’s
spatial constellation, but suffer from the high complexity of the mapping
design [36, 49].

Specifically, for the case of $N_{t}=4$, there are a total of $2^{N_{t}}-1=15$
possible TA activation patterns, which is constituted by $C_{4}^{1}=4$,
$C_{4}^{2}=6$, $C_{4}^{3}=4$ and $C_{4}^{4}=1$ according to (1). Nonetheless,
only $2^{N_{t}-1}=8$ TA activation patterns are needed. Therefore, the first
rule of LUT is to prioritize the TA activation patterns with smaller number
$N_{a}$ of activated TAs. For example, the TA activation patterns that only
activate a single TA out of $N_{t}=4$ result in the transmitted signal vectors
of
$\mathbf{x}^{T}=\left\\{\left[s,0,0,0\right],\left[0,s,0,0\right],\left[0,0,s,0\right],\left[0,0,0,s\right]\right\\}$,
where the TA activation pattern always differs in the maximum possible number
of 4 positions. Following this, for the TA activation patterns associated with
the same $N_{a}$, the pairs that have the highest possible number of different
activated TA positions are chosen. For example, after admitting the
$C_{4}^{1}=4$ combinations associated with a single activated TA, we still
need $2^{N_{t}-1}-C_{4}^{1}=4$ patterns out of the $C_{4}^{2}=6$ combinations
associated with $N_{a}=2$ activated TAs. Hence, the TA activation pattern
pairs of $\left(1,2\right)$ and $\left(3,4\right)$ are chosen, which always
differ in the maximum possible number of 4 positions. Similarly, the pair of
$\left(1,3\right)$ and $\left(2,4\right)$ are subsequently chosen. Although
$\left(1,2\right)$ and $\left(1,3\right)$ now only differ in 3 activated TA
positions, we have ensured that the maximally separated pairs are used.

In summary, the LUT arrangement is exemplified in Fig. 4 (c) for the case of
$N_{t}=4$, $m_{t}=3$ and $m_{a}=1$, where the list of TA activation patterns
is also presented in Table II. The LUT mapping may be represented by

${{\bf{b}}_{s}^{T}}\Leftrightarrow{\bf{x}}=\mathrm{U}\left({{{\bf{b}}_{s}}}\right)\times
s,$ (10)

where $\mathrm{U}\left(\cdot\right)$ returns a column vector having entries of
$0$ or $1$, and the above LUT mapping rules ensure that the binary vectors
$\left\\{\mathrm{U}\left(\mathbf{b}_{s}\right)\right\\}_{\forall\mathbf{b}_{s}}$
have the maximum possible Hamming distance.

##  III Signal Detection

In this section, we proceed to devise low-complexity signal detectors for the
LCIT-GSM receiver. We first consider the optimal MLD in Section III-A,
followed by a pair of reduced-complexity detectors in Sections III-B and
III-C, respectively.

###  III-A Maximum Likelihood Detection

From the detector’s perspective, let us represent the tentative transmitted
signal vector as ${{\bf{x}}_{k,l}}$, where the subscript $k$ denotes the
$k$-th TA activation pattern444Note that the subscript $k$ represents the
$k$-th TA activation pattern, not the $k$-th TA in SM. The number of active
TAs is variable and determined by the $k$-th TA activation pattern., while the
subscript $l$ denotes the $l$-th $M$-PSK/QAM symbol. We assume that perfect
channel knowledge is available for the receiver. As a result, the MLD finds
the estimate of $\bf{x}$ by performing a full search over all legitimate
indices $k$ and $l$ formulated as [53]:

$\begin{split}\left[{\hat{k}_{MLD},\hat{l}_{MLD}}\right]&=\mathop{\arg\max}\limits_{k,l}{p_{\bf{y}}}\left({\left.{\bf{y}}\right|{{\bf{x}}_{k,l}},{\bf{H}}}\right)\\\
&=\mathop{\arg\min}\limits_{k,l}{\left\|{{\bf{y}}-{\bf{H}}{{\bf{x}}_{k,l}}}\right\|^{2}}\\\
&=\mathop{\arg\min}\limits_{k,l}\left\|\mathbf{y}-\mathbf{g}_{k}s_{l}\right\|^{2},\end{split}$
(11)

where the probability density function (PDF) of ${\bf{y}}$ conditioned on
${{{\bf{x}}_{k,l}}}$ and ${\bf{H}}$ is given by:

${p_{\bf{y}}}\left({\left.{\bf{y}}\right|{{\bf{x}}_{k,l}},{\bf{H}}}\right)=\frac{1}{{{{\left({\pi\sigma_{n}^{2}}\right)}^{{N_{r}}}}}}\exp\left({-\frac{{{{\left\|{{\bf{y}}-{\bf{H}}{{\bf{x}}_{k,l}}}\right\|}^{2}}}}{{\sigma_{n}^{2}}}}\right),$
(12)

${s_{l}}$ represents the $l$-th symbol of the constellation, and
$\mathbf{g}_{k}$ is the $k$-th column of the equivalent $N\times{N_{r}}$ LCIT-
GSM channel matrix $\mathbf{G}$ from the perspective of the APSK symbol, where
we have $N={2^{{N_{t}}}}$ for DTAA-R and $N={2^{{N_{t}}-1}}$ for DTAA-D and
LUT. In summary, the equivalent channel matrix $\mathbf{G}$ is defined as
follows

$\left\\{{\begin{array}[]{*{20}{l}}\mathbf{G}_{DTAA-R}=\begin{bmatrix}\mathbf{H}\mathbf{b}_{s,1}&\cdots&\mathbf{H}\mathbf{b}_{s,N-1}&\mathbf{H}\mathbf{1}e^{j\frac{\pi}{M_{n}}}\end{bmatrix}\\\
\mathbf{G}_{DTAA-D}=\begin{bmatrix}\mathbf{H}\begin{bmatrix}\mathbf{b}_{s,1}\\\
0\par\end{bmatrix}&\cdots&\mathbf{H}\begin{bmatrix}\mathbf{b}_{s,N-1}\\\
0\par\end{bmatrix}&\mathbf{H}\begin{bmatrix}\mathbf{0}\\\
1\par\end{bmatrix}\end{bmatrix}\\\
\mathbf{G}_{LUT}=\begin{bmatrix}\mathbf{H}\mathrm{U}\left(\mathbf{b}_{s,0}\right)&\mathbf{H}\mathrm{U}\left(\mathbf{b}_{s,1}\right)&\cdots&\mathbf{H}\mathrm{U}\left(\mathbf{b}_{s,N-1}\right)\end{bmatrix},\end{array}}\right.$
(13)

where $M_{n}=M$ for $M$-PSK, and ${\bf{b}}_{s,0}=\bf{0}$ denotes the spatial
bitstream with all zeros, while ${\bf{b}}_{s,1},\cdots,{\bf{b}}_{s,N-1}$
denote the other spatial bitstreams.

As a result, it can be readily seen from (11) that the LCIT-GSM system is now
equivalent to an SM system with $N$ TAs, where all of the three mapping
arrangements of DTAA-R, DTAA-D and LUT have the same detector structure.

###  III-B Two-stage near-Maximum Likelihood Detection

The MLD has to jointly search through $N$ TA activation patterns and $M$
symbols, hence its complexity order is given by
$\mathcal{O}\left({MN}\right)$, which grows exponentially both with the number
$N_{t}$ of TAs and number $m_{s}$ of modulated bits. In this section, we
conceive the low-complexity two-stage near-MLD (TMLD) scheme, which is similar
to the massive MIMO detector of [54] and to the sphere decoder of [55]. This
two-stage TMLD first detects the modulated symbol index $l$ from the reduced
search space as follows:

${\hat{l}}=\mathop{\arg\min}\limits_{l}\left\|\mathbf{y}-\mathbf{h}_{k^{\prime}}s_{l}\right\|^{2}.$
(14)

We note that the range of $k^{\prime}$ is given by
$k^{\prime}\in\left\\{{1,2,\cdots,{N_{t}}}\right\\}$, which is reduced from
$k\in\left\\{1,2,\cdots,N\right\\}$.

Then, the TMLD relaxes the range of $l$ appropriately based on the estimated
${\hat{l}}$, which means several $l$ values can be selected based on (14),
instead of a single ${\hat{l}}$, since a single one would unlikely to be the
globally optimal one. Therefore, TMLD defines the reduced candidate set of $l$
using ${\hat{l}}$ in (14) as

${\chi_{TMLD}}=\left\\{{\left.l\right|\frac{\left\|\mathbf{y}-\mathbf{h}_{k^{\prime}}s_{l}\right\|^{2}}{\left\|\mathbf{y}-\mathbf{h}_{k^{\prime}}s_{\hat{l}}\right\|^{2}}\leq
c}\right\\},$ (15)

where $c\geq 1$ is a constant that controls the size of the set
${\chi_{TMLD}}$.

Following this, the second stage of TMLD is performed based on (11) as:

$\left[{\hat{k}_{TMLD},\hat{l}_{TMLD}}\right]=\mathop{\arg\min}\limits_{k,l\in\chi_{TMLD}}\left\|\mathbf{y}-\mathbf{g}_{k}s_{l}\right\|^{2}.$
(16)

The MLD guarantees that the solution is optimal by searching through all the
elements in the available set. By contrast, TMLD searches for $k$ and $l$ in a
reduced region by relaxing the rough estimate of $l$, which is similar to the
sphere decoding philosophy of [55, 56]. When $c$ is infinite, TMLD becomes
equivalent to MLD.

###  III-C Decoupling Based Maximum Likelihood Detection

In this subsection, we propose the novel decoupling based MLD (DMLD), which
firstly obtains the optimum modulation indices for all TA activation pattern
candidates and then detects the optimal TA activation pattern with the aid of
the demodulated $M$-APSK symbol. More explicitly, for each tentative TA
combination index $k$, the LCIT-GSM detection in (11) can be simplified to

$\begin{split}{\hat{l}_{k}}&=\mathop{\arg\min}\limits_{l}\left\\{{\left\|{\mathbf{g}}_{k}\right\|}^{2}{s}_{l}-2\Re\left\\{{s}_{l}^{*}{\mathbf{g}}_{k}^{H}\mathbf{y}\right\\}\right\\}\\\
&=\mathop{\arg\min}\limits_{l}\left\\{{\left|{s}_{l}-{p}_{k}\right|}^{2}-{\left|{p}_{k}\right|}^{2}\right\\}\\\
&=\mathop{\arg\min}\limits_{l}\left\\{{\left|{s}_{l}-{p}_{k}\right|}^{2}\right\\},\end{split}$
(17)

where the demodulator’s decision variable is given by
${p}_{k}=\frac{{\mathbf{g}}_{k}^{H}\mathbf{y}}{{\left\|{\mathbf{g}}_{k}\right\|}^{2}}$.
In this way, the optimum modulated symbol index $\hat{l}_{k}$ associated with
all TA activation indices may be directly obtained by demapping $p_{k}$ to the
closest constellation point, assuming that a regular APSK constellation is
used.

When $M$-PSK is employed, the demodulation of (17) may be performed by
rounding the phase of the decision variable to the nearest $M$-PSK index as
[57]555In DMLD, we estimate $s$ instead of its index, since $s$ can be
directly obtained by the rounding operation.

$\hat{s}_{k}=\exp\left\\{j\left[\frac{\pi}{2}\left\lfloor\left(\angle{p}_{k}-\phi_{0}\right)\frac{2}{\pi}\right\rceil+\phi_{0}\right]\right\\},$
(18)

where $\phi_{0}$ is the phase of the first $M$-PSK symbol associated with all-
zero bits.

Similarly, as shown in [57, 58], the QAM constellation may be decoupled into
two real-valued APSK signal sets as $s={s}_{I}+j{s}_{Q}$. Since ${s}_{I}$ and
${s}_{Q}$ are from orthogonal dimensions, we have

$\hat{l}_{k}=\underset{l}{\arg\min}\left[\left(s_{I}-\Re\left(p_{k}\right)\right)^{2}+\left(s_{Q}-\Im\left(p_{k}\right)\right)^{2}\right],$
(19)

which may be further decoupled into

$\Re\left(\hat{s}_{k}\right)=\min\left[\max\left(2\left\lfloor\frac{\alpha\Re\left(p_{k}\right)+1}{2}\right\rceil-1,-M+1\right),M-1\right],$
(20)

and

$\Im\left(\hat{s}_{k}\right)=\min\left[\max\left(2\left\lfloor\frac{\alpha\Im\left(p_{k}\right)+1}{2}\right\rceil-1,-M+1\right),M-1\right].$
(21)

We note that
$\alpha=\sqrt{\frac{\sum_{m=0}^{\frac{\sqrt{M}}{2}-1}\left(\sqrt{M}-2m-1\right)^{2}}{\frac{\sqrt{M}}{4}}}$
is the normalization factor of the QAM constellation that yields
$\hat{s}_{k}=\frac{1}{\alpha}\left[\Re\left(\hat{s}_{k}\right)+\Im\left(\hat{s}_{k}\right)\right]$
and ensures unity average symbol power [57, 58].

From (18), (20) and (21) we obtain the total number $N$ of demodulated symbols
$S_{min}=\left\\{\hat{s}_{1},\hat{s}_{2},\cdots,\hat{s}_{N}\right\\}$
associated with the $N$ hypotheses, for all $1\leq k\leq N$. Thus, we have

$\begin{split}\left[\hat{k}_{DMLD},\hat{s}_{DMLD}\right]&=\underset{k,s}{\arg\min}\left\|\mathbf{y}-\mathbf{g}_{k}s_{l}\right\|^{2}\\\
&=\underset{k}{\arg\min}\left\|\mathbf{y}-\mathbf{g}_{k}\hat{s}_{k}\right\|^{2}.\end{split}$
(22)

Once ${\hat{k}_{DMLD}}$ is estimated, we have
${\hat{s}_{DMLD}}={\hat{s}_{{{\hat{k}}_{DMLD}}}}$, which is the
${\hat{k}_{DMLD}}$-th element of $S_{min}$.

The DMLD’s complexity order does not grow with the number $M$ of modulation
levels, which constitutes the most appealing advantages of the optimal LCIT-
GSM detection, especially when employing high-order $M$-PSK/QAM schemes. More
importantly, the DMLD of (17)-(22) substantially reduces the detection
complexity without imposing any performance loss on the MLD of (11).

##  IV Performance Analysis

In this section, we present analytical results for characterising the error
performance of the uncoded LCIT-GSM system, where the MLD is assumed to obtain
the optimal reference bound666We note that the theoretical performance bounds
are different for different detectors. Here we derive the ultimate theoretical
bound for the MLD.. A tight UB on the BEP is derived. Additionally, we also
analyze the computational complexity of the proposed LCIT-GSM detectors.

###  IV-A Uncoded Performance

####  IV-A1 Classic Union Bound

First of all, the analytical performance of our LCIT-GSM system is evaluated
using the well-known union bounding technique. The average BEP of LCIT-GSM is
given by:

$P_{b}\leq\frac{1}{MN}\sum_{k=1}^{N}\sum_{l=1}^{M}\sum_{\tilde{k}=1}^{N}\sum_{\tilde{l}=1}^{M}N\left(\mathbf{x}_{k,l}\rightarrow\mathbf{x}_{\tilde{k},\tilde{l}}\right)P\left(\mathbf{x}_{k,l}\rightarrow\mathbf{x}_{\tilde{k},\tilde{l}}\right),$
(23)

where $\mathbf{x}_{k,l}$ indicates the symbol $s_{l}$ transmitted using the
$k$-th TA activation pattern, while
$N\left(\mathbf{x}_{k,l}\rightarrow\mathbf{x}_{\tilde{k},\tilde{l}}\right)$ is
the number of bits in error between $\mathbf{x}_{k,l}$ and
$\mathbf{x}_{\tilde{k},\tilde{l}}$, and
$P\left(\mathbf{x}_{k,l}\rightarrow\mathbf{x}_{\tilde{k},\tilde{l}}\right)$
denotes the probability of deciding on $\mathbf{x}_{\tilde{k},\tilde{l}}$,
given that $\mathbf{x}_{k,l}$ is transmitted.

Following this, the probability
$P\left(\mathbf{x}_{k,l}\rightarrow\mathbf{x}_{\tilde{k},\tilde{l}}\right)$
can be computed by using (12) as follows

$\begin{split}P\left(\mathbf{x}_{k,l}\rightarrow\mathbf{x}_{\tilde{k},\tilde{l}}\right)&=P\left(\left\|\mathbf{y}-{\bf{g}}_{k}s_{l}\right\|^{2}>\left\|\mathbf{y}-{\bf{g}}_{\tilde{k}}s_{\tilde{l}}\right\|^{2}\right)\\\
&=Q\left(\sqrt{\frac{1}{2\sigma_{n}^{2}}\left\|{\bf{g}}_{k}s_{l}-{\bf{g}}_{\tilde{k}}s_{\tilde{l}}\right\|^{2}}\right).\end{split}$
(24)

According to Eq. (14-4-15) in [59], we obtain

$\displaystyle{P_{b}}$
$\displaystyle\leq\frac{1}{{MN}}\sum\limits_{k=1}^{N}{\sum\limits_{l=1}^{M}{\sum\limits_{\tilde{k}=1}^{N}{\sum\limits_{\tilde{l}=1}^{M}{}}}}$
$\displaystyle\quad\times
N\left({{{\bf{x}}_{k,l}}\to{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)\mu_{\alpha}^{{N_{r}}}\sum\limits_{n=0}^{{N_{r}}-1}{C_{{N_{r}}-1+n}^{n}{{\left({1-{\mu_{\alpha}}}\right)}^{n}}},$
(25)

with

${\mu_{\alpha}}=\mu\left(\sigma_{\alpha}^{2}\right)=\frac{1}{2}\left({1-\sqrt{\frac{{\sigma_{\alpha}^{2}}}{{1+\sigma_{\alpha}^{2}}}}}\right),$
(26)

and

$\sigma_{\alpha}^{2}=\frac{1}{4\sigma_{n}^{2}}\left({{{\left|{{s_{l}}}\right|}^{2}}+{{\left|{{s_{\tilde{l}}}}\right|}^{2}}}\right).$
(27)

####  IV-A2 Improved Upper Bound

The theoretical UB above is derived based on the conventional SM study of
[59], which does not take into account the specific effects of the spatial
constellation and symbol constellation. To avoid these limitations, an
improved UB is proposed in [51]. In this subsection, we will derive our
improved performance bound for LCIT-GSM based on the methodology of [51]. We
firstly offer _Proposition 1_ that is conceived for the sake of improving the
UB of our LCIT-GSM system, and then we further derive a simplified UB under
Rayleigh flat fading from _Proposition 1_.

_Proposition 1:_ The average BEP of LCIT-GSM can be tightly upper bounded as
follows:

${P_{b}}\leq{P_{b,signal}}+{P_{b,spatial}}+{P_{b,joint}},$ (28)

where ${P_{b,signal}}$, ${P_{b,spatial}}$, ${P_{b,joint}}$ are defined as

$\left\\{{\begin{array}[]{*{20}{l}}{{P_{b,signal}}=\frac{{{{\log}}M}}{{{N}{{\log}}\left({M{N}}\right)}}\sum\limits_{k=1}^{{N}}{{P_{b,MOD}}\left(k\right)}}\\\
{{P_{b,spatial}}=\frac{{{{\log}}{N}}}{{M{{\log}}\left({M{N}}\right)}}\sum\limits_{l=1}^{M}{{P_{b,SSK}}\left(l\right)}}\\\
{{P_{b,joint}}=\frac{{\sum\limits_{k=1}^{{N}}{\sum\limits_{l=1}^{M}{\sum\limits_{\tilde{k}=1}^{{N}}{\sum\limits_{\tilde{l}=1}^{M}{N\left({\mathbf{x}_{k,l}\to{\mathbf{x}_{\tilde{k},\tilde{l}}},{s_{\tilde{l}}}}\right)\Upsilon\left({\mathbf{x}_{k,l},\mathbf{x}_{\tilde{k},\tilde{l}}}\right)}}}}}}{{M{N}{{\log}}\left({M{N}}\right)}}},\end{array}}\right.$
(29)

with

$\left\\{{\begin{array}[]{*{20}{l}}{{P_{b,MOD}}\left(k\right)=\frac{{\sum\limits_{l=1}^{M}{\sum\limits_{\tilde{l}=1}^{M}{\left[{N\left({{s_{l}}\to{s_{\tilde{l}}}}\right){E_{{\mathbf{g}_{k}}}}\left\\{P\left(s_{l}=s_{\tilde{l}}\right)\right\\}}\right]}}}}{{M{{\log}}\left(M\right)}}}\\\
{{P_{b,SSK}}\left(l\right)=\frac{{\sum\limits_{k=1}^{{N}}{\sum\limits_{\tilde{k}=1}^{{N}}{N\left({\mathbf{b}_{s,k}\to{\mathbf{b}_{s,\tilde{k}}}}\right){\Psi_{l}}\left({\mathbf{b}_{s,k},\mathbf{b}_{s,\tilde{k}}}\right)}}}}{{{N}{{\log}}\left({{N}}\right)}}},\end{array}}\right.$
(30)

where

$\left\\{{\begin{array}[]{*{20}{l}}{{\Psi_{l}}\left({\mathbf{b}_{s,k},\mathbf{b}_{s,\tilde{k}}}\right)=\frac{1}{\pi}\int\limits_{0}^{{\pi\mathord{\left/{\vphantom{\pi
2}}\right.\kern-1.2pt}2}}{{{\mathcal{M}}_{\gamma\left({\mathbf{b}_{s,k},\mathbf{b}_{s,\tilde{k}}}\right)}}\left({\frac{{{\left|{{s_{l}}}\right|^{2}}}}{{4\sigma_{n}^{2}{{\sin}^{2}}\theta}}}\right)d\theta}}\\\
{\Upsilon\left({\mathbf{x}_{k,l},\mathbf{x}_{\tilde{k},\tilde{l}}}\right)=\frac{1}{\pi}\int\limits_{0}^{{\pi\mathord{\left/{\vphantom{\pi
2}}\right.\kern-1.2pt}2}}{{{\mathcal{M}}_{\gamma\left({\mathbf{x}_{k,l},\mathbf{x}_{\tilde{k},\tilde{l}}}\right)}}\left({\frac{1}{{4\sigma_{n}^{2}{{\sin}^{2}}\theta}}}\right)d\theta}},\end{array}}\right.$
(31)

and

$\left\\{{\begin{array}[]{*{20}{l}}\gamma\left({\mathbf{b}_{s,k},{\mathbf{b}_{s,\tilde{k}}}}\right)=\left\|\mathbf{g}_{k}-\mathbf{g}_{\tilde{k}}\right\|^{2}\\\
\gamma\left({\mathbf{x}_{k,l},\mathbf{x}_{\tilde{k},\tilde{l}}}\right)=\left\|\mathbf{g}_{k}s_{l}-\mathbf{g}_{\tilde{k}}s_{\tilde{l}}\right\|^{2}.\end{array}}\right.$
(32)

_Proof:_ Please refer to Appendix A in [51].

As shown in (31), in order to obtain the ABEP upper bound described in (28),
we have to derive the MGFs of
$\gamma\left({\mathbf{b}_{s,k},{\mathbf{b}_{s,\tilde{k}}}}\right)$ and
$\gamma\left({\mathbf{x}_{k,l},\mathbf{x}_{\tilde{k},\tilde{l}}}\right)$.
According to [49, 60], we know that

$\begin{split}\gamma\left({{{\bf{b}}_{s,k}},{{\bf{b}}_{s,\tilde{k}}}}\right)&={\left\|{{{\bf{g}}_{k}}-{{\bf{g}}_{\tilde{k}}}}\right\|^{2}}={\left\|{{\bf{G}}\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right)}\right\|^{2}}\\\
&=\sum\limits_{n=1}^{{N_{r}}}{{{\bf{G}}_{n,:}}\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right){{\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right)}^{H}}{\bf{G}}_{n,:}^{H}}\\\
&={{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}^{T}}\left[{{{\bf{I}}_{N}}\otimes\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right){{\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right)}^{H}}}\right]{{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}^{*}},\end{split}$ (33)

and

$\begin{split}\gamma\left({{{\bf{x}}_{k,l}},{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)&={\left\|{{{\bf{g}}_{k}}{s_{l}}-{{\bf{g}}_{\tilde{k}}}{s_{\tilde{l}}}}\right\|^{2}}={\left\|{{\bf{G}}\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}\right\|^{2}}\\\
&=\sum\limits_{n=1}^{{N_{r}}}{{{\bf{G}}_{n,:}}\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right){{\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}^{H}}{\bf{G}}_{n,:}^{H}}\\\
&={{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}^{T}}\left[{{{\bf{I}}_{N}}\otimes\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right){{\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}^{H}}}\right]{{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}^{*}},\end{split}$ (34)

where ${{{\bf{G}}_{n,:}}}$ represents the $n$-th row of ${\bf{G}}$,
${{\bf{e}}_{k}}$ is the ${N_{t}}\times 1$ unit vector with only the $k$-th
entry is $1$, and ${\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}=vec\left({{{\bf{G}}^{T}}}\right)$
is a ${N_{r}}N\times 1$ vector obtained by the vectorization of
${{\bf{G}}^{T}}$. In this paper, we explicitly investigate the canonical
i.i.d. Rayleigh fading scenarios. Nevertheless, the equivalent LCIT-GSM
channels are correlated. Hence, the MGFs of
$\gamma\left({\mathbf{b}_{s,k},{\mathbf{b}_{s,\tilde{k}}}}\right)$ and
$\gamma\left({\mathbf{x}_{k,l},\mathbf{x}_{\tilde{k},\tilde{l}}}\right)$ can
be derived in the same way as in [49, 60], which can be expressed as

$\displaystyle{M_{\gamma\left({{{\bf{b}}_{s,k}},{{\bf{b}}_{s,\tilde{k}}}}\right)}}\left(t\right)$
$\displaystyle=\det\left[{{{\bf{I}}_{{N_{r}}N}}+t{{\bf{C}}_{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}}\left({{{\bf{I}}_{N}}}\right.}\right.$
$\displaystyle\quad{\left.{\left.{\otimes\left({\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right){{\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right)}^{H}}}\right)}\right)}\right]^{-1}},$
(35)

and

$\displaystyle{M_{\gamma\left({{{\bf{x}}_{k,l}},{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}}\left(t\right)$
$\displaystyle=\det\left[{{{\bf{I}}_{{N_{r}}N}}+t{{\bf{C}}_{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}}\left({{{\bf{I}}_{N}}}\right.}\right.$
$\displaystyle\quad{\left.{\left.{\otimes\left({\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right){{\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}^{H}}}\right)}\right)}\right]^{-1}},$
(36)

where $\det\left({\bf{S}}\right)$ denotes the determinant of matrix
${\bf{S}}$, ${{\bf{C}}_{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}}=\frac{{\rm{1}}}{{{N_{r}}}}{{\bf{I}}_{{N_{r}}}}\otimes
E\left({{{\bf{G}}^{T}}{{\bf{G}}^{*}}}\right)$ is the covariance matrix of
${\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}$, depending on the specific
spatial mapping scheme introduced in Section II-B. More specifically, the
$\left({k,\tilde{k}}\right)$ element of
${{\bf{C}}_{{{\bf{G}}^{T}}}}=\frac{1}{{{N_{r}}}}E\left({{{\bf{G}}^{T}}{{\bf{G}}^{*}}}\right)$
is equal to $1$ when the $k$-th and $\tilde{k}$-th TA activation patterns
share more than one activated TA, and $0$ for the rest.

After substituting (IV-A2) and (IV-A2) into (29) $\sim$ (31), we arrive at

$\displaystyle{P_{b,spatial}}$ $\displaystyle=\frac{1}{{\pi
MN{{\log}}\left({MN}\right)}}\sum\limits_{l=1}^{M}{\sum\limits_{k=1}^{N}{\sum\limits_{\tilde{k}=1}^{N}{N\left({{{\bf{b}}_{s,k}}\to{{\bf{b}}_{s,\tilde{k}}}}\right)}}}$
$\displaystyle\times\int\limits_{0}^{{\pi\mathord{\left/{\vphantom{\pi
2}}\right.\kern-1.2pt}2}}{\det\left[{{{\bf{I}}_{{N_{r}}N}}+\frac{{{{\left|{{s_{l}}}\right|}^{2}}}}{{4\sigma_{n}^{2}{{\sin}^{2}}\theta}}{{\bf{C}}_{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}}}\right.}$
$\displaystyle\quad\times{\left.{\left({{{\bf{I}}_{N}}\otimes\left({\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right){{\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right)}^{H}}}\right)}\right)}\right]^{-1}}d\theta$
$\displaystyle\leq\frac{1}{{2\pi
MN{{\log}}\left({MN}\right)}}\sum\limits_{l=1}^{M}{\sum\limits_{k=1}^{N}{\sum\limits_{\tilde{k}=1}^{N}{N\left({{{\bf{b}}_{s,k}}\to{{\bf{b}}_{s,\tilde{k}}}}\right)}}}$
$\displaystyle\times\det\left[{{{\bf{I}}_{{N_{r}}N}}+\frac{{{{\left|{{s_{l}}}\right|}^{2}}}}{{4\sigma_{n}^{2}}}{{\bf{C}}_{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}}}\right.$
$\displaystyle\quad\times{\left.{\left({{{\bf{I}}_{N}}\otimes\left({\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right){{\left({{{\bf{e}}_{k}}-{{\bf{e}}_{\tilde{k}}}}\right)}^{H}}}\right)}\right)}\right]^{-1}},$
(37)

and

$\displaystyle{P_{b,joint}}$ $\displaystyle=\frac{1}{{\pi
MN{{\log}}\left({MN}\right)}}\sum\limits_{k=1}^{N}{\sum\limits_{l=1}^{M}{\sum\limits_{\tilde{k}=1}^{N}{\sum\limits_{\tilde{l}=1}^{M}{N\left({{{\bf{x}}_{k,l}}\to{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}}}}$
$\displaystyle\times\int\limits_{0}^{{\pi\mathord{\left/{\vphantom{\pi
2}}\right.\kern-1.2pt}2}}{\det\left[{{{\bf{I}}_{{N_{r}}N}}+\frac{1}{{4\sigma_{n}^{2}{{\sin}^{2}}\theta}}{{\bf{C}}_{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}}}\right.}$
$\displaystyle\quad\times{\left.{\left({{{\bf{I}}_{N}}\otimes\left({\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right){{\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}^{H}}}\right)}\right)}\right]^{-1}}d\theta$
$\displaystyle\leq\frac{1}{{2\pi
MN{{\log}}\left({MN}\right)}}\sum\limits_{k=1}^{N}{\sum\limits_{l=1}^{M}{\sum\limits_{\tilde{k}=1}^{N}{\sum\limits_{\tilde{l}=1}^{M}{N\left({{{\bf{x}}_{k,l}}\to{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}}}}$
$\displaystyle\times\det\left[{{{\bf{I}}_{{N_{r}}N}}+\frac{1}{{4\sigma_{n}^{2}}}{{\bf{C}}_{{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{g}}}}}}}\right.$
$\displaystyle\quad\times{\left.{\left({{{\bf{I}}_{N}}\otimes\left({\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right){{\left({{{\bf{x}}_{k,l}}-{{\bf{x}}_{\tilde{k},\tilde{l}}}}\right)}^{H}}}\right)}\right)}\right]^{-1}},$
(38)

where the Chernoff bound is used for simplifying the numerical integrals [49].

On the other hand, ${P_{b,signal}}$ can also be further simplified to

${P_{b,signal}}=\frac{{{{\log}}M}}{{{{\log}}\left({MN}\right)}}P_{b,MOD}^{Rayleigh}.$
(39)

TABLE III:  Contrast between three detectors for LCIT-GSM system. | BER performance | Order of complexity | Computational complexity
---|---|---|---
MLD | optimal | ${\mathcal{O}}\left({MN}\right)$ | ${\delta_{MLD}}=6M{N_{r}}N$
TMLD | sub-optimal |  ${\mathcal{O}}\left({N}\right)$, for a small $c$ and $M$ | ${\delta_{TMLD}}=6M{N_{r}}{N_{t}}+6\beta M{N_{r}}N$
DMLD | optimal | ${\mathcal{O}}\left({N}\right)$ | ${\delta_{DMLD}}=\left\\{{\begin{array}[]{*{20}{l}}{\left({6{N_{r}}+10}\right)N,}&{PSK}\\\ {\left({6{N_{r}}+12}\right)N,}&{QAM}\end{array}}\right.$

For PSK and QAM constellations, $P_{b,MOD}^{Rayleigh}$ can be expressed
similarly to [61] as

$P_{b,PSK}^{Rayleigh}\simeq\frac{{2\sum\limits_{k=1}^{\max\left({{M\mathord{\left/{\vphantom{M4}}\right.\kern-1.2pt}4},1}\right)}{R\left(\frac{{{{\sin}^{2}}{\frac{{\left({2k-1}\right)\pi}}{M}}}}{2\sigma_{n}^{2}}\right)}}}{{\max\left({{{\log}}M,2}\right)}},$
(40)

and

$P_{b,QAM}^{Rayleigh}=\frac{4}{{\sqrt{M}{{\log}}M}}\sum\limits_{l=1}^{\frac{1}{2}{{\log}}M}{\sum\limits_{k=0}^{\left({1-{2^{-l}}}\right)\sqrt{M}-1}{{S_{l,k}}}},$
(41)

where

${S_{l,k}}={\left({-1}\right)^{\left\lfloor{\frac{{{2^{l-1}}k}}{{\sqrt{M}}}}\right\rfloor}}\left({{2^{l-1}}-\left\lfloor{\frac{{{2^{l-1}}k}}{{\sqrt{M}}}+\frac{1}{2}}\right\rfloor}\right)R\left({\frac{{3{{\left({2k+1}\right)}^{2}}}}{4\sigma_{n}^{2}\left(M-1\right)}}\right).$
(42)

and

$R\left(x\right)={\left({\mu\left(x\right)}\right)^{{N_{r}}}}\sum\limits_{{n_{r}}=0}^{{N_{r}}-1}{\left\\{{C_{{N_{r}}-1+{n_{r}}}^{{n_{r}}}{{\left({1-\mu\left(x\right)}\right)}^{{n_{r}}}}}\right\\}},$
(43)

The formulas in (IV-A2), (IV-A2) and (39) provide useful insights into the
performance of LCIT-GSM systems. For example, (IV-A2) shows that the bit
mapping of the PSK/QAM constellation diagram plays an important role in
${P_{b,joint}}$. In particular, while conventional bit mappings based on the
Euclidean distance of the signal constellations turn out to be optimal for
minimising ${P_{b,signal}}$, additional constraints may be imposed onto the
best choice of the signal constellation diagram and onto the related bit
mapping for minimising ${P_{b,signal}}$ and ${P_{b,joint}}$ simultaneously.

###  IV-B Complexity Analysis

In this part, we compare the computational complexity of the detectors
proposed for our LCIT-GSM scheme. The detection complexity is quantified in
terms of the number of real-valued multiplications required. The complexity of
GSM in [39] is given by
${\delta_{GSM}}=6M{N_{r}}{2^{\left\lfloor{{{\log}}C_{{N_{t}}}^{{N_{a}}}}\right\rfloor}}$.
Similarly, for the proposed LCIT-GSM scheme, the detection complexity of the
full-search based MLD of (11) is given by

${\delta_{MLD}}=6M{N_{r}}N.$ (44)

When the near-optimal reduced-scope TMLD of (16) is used, the LCIT-GSM
scheme’s complexity becomes

${\delta_{TMLD}}=6M{N_{r}}{N_{t}}+6\beta M{N_{r}}N,$ (45)

where $\frac{1}{M}\leq\beta\leq 1$ is a parameter that increases with the
constant $c$ of (15). More specifically, we have $\beta=\frac{1}{M}$ for $c=1$
and $\beta\to 1$ for $c\to\infty$, respectively. We note that as $c\to\infty$,
the complexity of TMLD becomes higher than that of MLD, which is due to the
fact that, for a large $c$, TMLD does not reduce the cardinality of the symbol
constellation, but increases the complexity of the first step. However, in
practice, a small $c$ is adequate for achieving near-ML performance at a low
complexity.

When the single-stream-based DMLD of (17)-(22) is invoked, the LCIT-GSM
detection complexity encountered for a given $k$ consists of three parts: 1)
the computation of ${p_{k}}$ in (17) requires $\left({6{N_{r}}+2}\right)$
real-valued multiplications [58]; 2) $2$ and $4$ real-valued multiplications
are required for obtaining ${\hat{s}_{k}}$ in PSK and QAM, respectively; 3)
$6$ real-valued multiplications are required for performing (22). Therefore,
the LCIT-GSM detection complexity is reduced to

${\delta_{DMLD}}=\left\\{{\begin{array}[]{*{20}{l}}{\left({6{N_{r}}+10}\right)N,}&{PSK}\\\
{\left({6{N_{r}}+12}\right)N,}&{QAM}\end{array}}\right..$ (46)

The complexity comparison of the three different detectors is summarized in
Table III, where the order of complexity is defined as the number of
optimization metric evaluations [58]. The numerical complexity comparisons are
offered in Section V.

We note that as previously shown by (3), the throughput of LCIT-GSM is higher
than that of GSM. This implies that for the sake of achieving the same SE,
LCIT-GSM requires less TAs than its conventional GSM counterpart. As a result,
although the signal detection complexity of LCIT-GSM of (44)-(46) is generally
higher than that of GSM, the proposed LCIT-GSM is still advantageous in terms
of the overall energy efficiency, which may be regarded as the SE normalised
by the hardware power consumption [3, 11, 22].

##  V Simulation Results

In this section, the BEP performance of LCIT-GSM systems is investigated for
different numbers of TA/RAs and using different modulation schemes. For the
sake of clarity, the parameters used for generating the results in this
section are detailed in the figures.

Figure 5: ABEP of LCIT-GSM (QPSK, $N_{t}=4$) versus $E_{m}/N_{0}$ under our
three types of spatial mapping rules, i.e., DTAA-R in (6), DTAA-D in (9) and
LUT in (10). The classic UB and improved UB are drawn with solid line and
dash-dot line, respectively. Note that LUT uses the same mapping design as
conventional VGSM of [36, 49]. Figure 6: ABEP of LCIT-GSM (16QAM, $N_{t}=4$)
versus $E_{m}/N_{0}$ under our three types of spatial mapping rules, i.e.,
DTAA-R in (6), DTAA-D in (9) and LUT in (10). The classic UB and improved UB
are drawn with solid line and dash-dot line, respectively. Note that LUT uses
the same mapping design as conventional VGSM of [36, 49].

Figs. 5 $\sim$ 8 contrast our simulations and theoretical derivations for the
LCIT-GSM system in different scenarios, where the circle, square and cross
patterns represent DTAA-R, DTAA-D, and LUT, respectively, while the dash-dot
and solid lines represent the improved performance bound and the classic union
bound, respectively. The MLD is adopted in Figs. 5 $\sim$ 8 for contrasting
its performance to the corresponding theoretical BEP bound. In Figs. 5 and 6,
the number of TAs is set to $N_{t}=4$. Although the three mapping arrangements
only exhibit modest performance differences in Figs. 5 $\sim$ 8, the DTAA-R is
shown to have a worse BEP than DTAA-D and LUT, since it transmits one more bit
using the same number of TAs. Furthermore, as evidenced by Figs. 5 $\sim$ 8,
the LUT performs the best. However, it is worth noting that compared to the
simple DTAA-R/DTAA-D mapping, the LUT has to rely on a pre-defined LUT, which
imposed a higher design complexity.

Figure 7: ABEP of LCIT-GSM (QPSK, $N_{t}=6$) versus $E_{m}/N_{0}$ under our
three types of spatial mapping rules, i.e., DTAA-R in (6), DTAA-D in (9) and
LUT in (10). The classic UB and improved UB are drawn with solid line and
dash-dot line, respectively. Note that LUT uses the same mapping design as
conventional VGSM of [36, 49]. Figure 8: ABEP of LCIT-GSM (16QAM, $N_{t}=6$)
versus $E_{m}/N_{0}$ under our three types of spatial mapping rules, i.e.,
DTAA-R in (6), DTAA-D in (9) and LUT in (10). The classic UB and improved UB
are drawn with solid line and dash-dot line, respectively. Note that LUT uses
the same mapping design as conventional VGSM of [36, 49]. Figure 9: ABEP of
LCIT-GSM (QPSK, $N_{t}=6$) versus $E_{m}/N_{0}$ for our three types of
detectors. The LUT is used to illustrate the compatibility of the proposed
low-complexity detectors with conventional VGSM of [36, 49]. Figure 10: ABEP
of LCIT-GSM (16QAM, $N_{t}=6$) versus $E_{m}/N_{0}$ for our three types of
detectors. The LUT is used to illustrate the compatibility of the proposed
low-complexity detectors with conventional VGSM of [36, 49].

Moreover, it is demonstrated by Figs. 5 $\sim$ 8 that the improved theoretical
bound is about 1.5dB tighter than the conventional bound. Figs. 7 and 8 depict
the theoretical derivation and simulation results of using ${N_{t}}=6$ and
QPSK/16QAM signalling. It can be seen in Figs. 7 and 8 that, once again, the
improved performance bound has a 1.5dB performance improvement over the
conventional performance bound. Furthermore, as the number of RAs increases,
the performance advantage of LUT over DTAA-R/DTAA-D improves, and the UB also
becomes tighter in Figs. 7 and 8, which once again verifies the reliability of
our conclusion.

Figure 11: The performance comparison between LCIT-GSM (QPSK) and GSM for
different ${N_{a}}$. The LUT-LCIT-GSM with $N_{t}=5$ is used to maintain a
overall rate of 6bpcu. Figure 12: The performance comparison between LCIT-GSM
(QPSK) and GSM for different ${N_{a}}$. The LUT-LCIT-GSM with $N_{t}=6$ is
used to maintain a overall rate of 7bpcu.

Figs. 9 and 10 compare the performance of different detectors for our LCIT-GSM
system having ${N_{t}}=6$. As discussed in Section IV-B, the TMLD exhibits a
lower detection complexity, but it also imposes a performance loss compared to
the MLD and DMLD, as confirmed by Figs. 9 and 10. We note that the constant
$c$ of (15) is set to 1.5 in Figs. 9 and 10, which may be increased for the
sake of improving the performance, but this would also result in a higher
detection complexity according to (45). Moreover, Figs. 9 and 10 show that
although TMLD performs similarly to MLD at low SNRs, the TMLD may exhibit an
error floor. The error floor will move down upon increasing ${N_{r}}$ because
the channels between different TAs tend to become more orthogonal for an
increased number of RAs. More specifically, the index of the modulated symbol
found in the first step becomes more accurate and hence supports the global
optimality of the second step of local search. Fortunately, DMLD always
maintains the same performance as MLD, which benefits from the fact that for a
regular constellation, the rounding-based detector and the exhaustive search
return the same result. The simulation results of 16QAM associated with
${N_{t}}=6$ are presented in Fig. 10, which leads to the same observations on
TMLD as Fig. 10. Moreover, by comparing the performance of QPSK and 16QAM in
Figs. 9 and 10, respectively, we can see that the performance of QPSK is
always better than that of 16QAM.

Figure 13: Complexity comparison of different detectors for LCIT-GSM with
${N_{r}}=6$, where DTAA-D and LUT are adopted.

The BEP performance comparisons between GSM and LCIT-GSM are offered in Figs.
11 and 12. In Fig. 11, the number of TAs is set to $N_{t}=5$ and QPSK is used
for LCIT-GSM. In order to maintain the overall throughput of 6bpcu, 16QAM and
8QAM are used for GSM with the fixed number of active TAs ${N_{a}}=1$ and
${N_{a}}=2$, respectively. It can be seen in Fig. 11 that LCIT-GSM outperforms
GSM for ${N_{a}}=2$ by about 2dB for different numbers of RAs of $N_{r}=3$ and
$N_{r}=7$. When GSM associated with ${N_{a}}=1$ is considered, a 4dB
performance gain can be observed in Fig. 11. In Fig. 12, our performance
comparisons between GSM and LCIT-GSM are portrayed for the case of $N_{t}=6$.
The QPSK constellation is also adopted for LCIT-GSM, while 16QAM and 8QAM are
used for GSM at ${N_{a}}=2$ and ${N_{a}}=3$, respectively, hence maintaining a
overall rate of 7bpcu. The 2dB performance gain compared to GSM using
${N_{a}}=3$ can also be observed in Fig. 12 to verify the advantage of the
proposed LCIT-GSM scheme.

Finally, in Fig. 13, the computational complexity of different detectors is
compared in terms of the number of real-valued multiplications, where we set
$c=1$ and when the DTAA-D and LUT are adopted. As seen in Section IV-B, the
complexity of MLD is prohibitive and increases exponentially with ${N_{t}}$,
while our proposed detectors achieve lower complexity for all the scenarios
considered. More specifically, the complexity of the TMLD is lower than that
of the DMLD for a small constellation, and the complexity-difference will
widen as the number of TAs increases. This is because the ratio of the DMLD’s
complexity to $N$ is higher than that of TMLD, bearing in mind that
$N={2^{{N_{t}}-1}}$ for DTAA-D and LUT. By contrast, for a moderate number of
TAs, the DMLD is more competitive in conjunction with high-order modulation,
which is due to the fact that the complexity of the DMLD does not increase
with the constellation size $M$. In summary, the proposed low-complexity
detectors strike an attractive BEP performance vs. detection complexity
tradeoff and can be applied flexibly in various scenarios. Moreover, we
emphasize that the DMLD is always optimal but it is only suitable for regular
constellations, while the TMLD is suboptimal, but it can strike a flexible BEP
performance vs. computational complexity trade-off by adjusting the constant
$c$ in (15).

##  VI Conclusions

A novel LCIT-GSM scheme was proposed, which is capable of maximising the GSM
throughput without imposing IAI while reducing the system complexity of the
conventional VGSM of [36, 49]. In contrast to the conventional GSM that always
invokes a fixed number $N_{a}$ of activated TAs, the proposed LCIT-GSM
utilises all legitimate TA activation patterns, where $N_{a}$ becomes a
variable that has to be recovered by the receiver. Nonetheless, in the absence
of IAI, a low-complexity single-stream based detector may be invoked by the
LCIT-GSM receiver. Both our theoretical derivation and our numerical
simulation have confirmed that LCIT-GSM increases the overall SE
logarithmically with the number of TAs. More specifically, for the LCIT-GSM
mapping, the arrangements of DTAA-R, DTAA-D and LUT are conceived in order to
facilitate a flexible MIMO deployment with special focus on the all-zero
spatial beam stream. Furthermore, as for the LCIT-GSM signal detection, we
have developed the low-complexity TMLD and DMLD schemes, where the former
adopts the sphere decoding philosophy for the sake of striking a compelling
performance vs complexity trade-off, while the latter is capable of achieving
the ML performance at a single-stream detection complexity that does not grow
with the number of modulation levels. Furthermore, an improved theoretical UB
is derived for the proposed LCIT-GSM, which is confirmed by our Monte-Carlo-
based simulation to be about 1.5dB tighter than the existing conventional
solution. Finally, our simulation results demonstrate that the proposed LCIT-
GSM scheme achieves improved performance over the conventional GSM and lower
complexity than the VGSM. In this paper, three spatial mapping schemes are
presented for exploiting the low complexity of the bitwise TA activation. We
believe that the family of mapping schemes designed by formal optimization
methods may achieve a better BEP at the cost of a moderate design complexity,
which is left as a topic for our future research.

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