# Multi-level Design for Multiple-Symbol Non-Coherent Unitary Constellations
for Massive SIMO Systems ††thanks: The authors are with the Department of
Electrical and Computer Engineering, University of Saskatchewan, Saskatoon,
Canada S7N 5A9. Emails: {wve765, ha.nguyen<EMAIL_ADDRESS>††thanks: The
co-authors dedicate this work to Prof. Ha H. Nguyen, who passed away before
the submission of the paper.

Son T. Duong, Ha H. Nguyen, and Ebrahim Bedeer

###### Abstract

This paper investigates non-coherent detection of single-input multiple-output
(SIMO) systems over block Rayleigh fading channels. Using the Kullback-Leibler
divergence as the design criterion, we formulate a multiple-symbol
constellation optimization problem, which turns out to have high computational
complexity to construct and detect. We exploit the structure of the formulated
problem and decouple it into a unitary constellation design and a multi-level
design. The proposed multi-level design has low complexity in both
construction and detection. Simulation results show that our multi-level
design has better performance than traditional pilot-based schemes and other
existing low-complexity multi-level designs.

###### Index Terms:

Constellation design, energy encoding and decoding, Kullback-Leibler (KL)
divergence, non-coherent detection, ultra-reliable low-latency communications
(URLLC).

##  I Introduction

Ultra-reliable low-latency communication (URLLC) has been introduced as one of
the pillars of the fifth generation (5G) and beyond cellular networks. The
critical requirements of URLLC, i.e., block error rate (BLER) $<10^{-5}$ and
latency $<$ 1 ms, are expected to facilitate emerging applications such as
self-driving cars, and industrial automation [1]. Short packet communications
(SPCs) with packets length of tens of bytes have been widely adopted to
achieve low latency, while massive multiple-antenna systems with coherent
detection have been well-studied as a diversity resource to achieve the
stringent BLER requirement. However, to enable coherent detection, the
transmitter must acquire accurate channel state information by sending a long
sequence of training symbols, which decreases the spectral efficiency (SE) of
SPCs. Thus, it is important to investigate the non-coherent detection of SPCs.

Among different non-coherent detection schemes, energy-based transmission and
detection have been well-studied in massive single-input multiple-output
(SIMO) systems [2, 3]. However, non-coherent energy-based schemes suffer from
loss of the SE since only the amplitude of the signal carries information, and
only one symbol is encoded and decoded at a time. That said, and to improve
the SE, we aim to design non-coherent constellations that carry information in
both amplitude and phase and are encoded and decoded over several symbols
duration. Please note that non-coherent detection of information in the phase
is possible at the receiver by calculating the phase difference between
symbols.

For designing non-coherent constellations over several symbols, we utilize the
Kullback-Leibler (KL) divergence, which is recognized as an effective design
criterion for non-coherent communications [4]. In particular, the authors in
[4] proved that the pairwise error probability performance achieved by the
maximum likelihood (ML) detector of any two symbols is bounded by its KL
distance. This means that by maximizing the KL divergence between any two
symbols, one also maximizes the symbol error rate (SER) performance of the
constellation. By utilizing the KL divergence as a constellation design
criterion, the authors in [4] divided the constellation into different energy
levels, where each level contains constellation points with the same energy.
The optimization of the energy levels is called multi-level design while
designing the constellation points within the same energy level is called
unitary constellation design.

While researchers had focused on designing unitary constellations, e.g., see
[5, 6, 7] and the references therein, the multi-level design over several
symbols has attracted little attention [4, 8]. For instance, the authors in
[4] found the optimal multi-level design using an exhaustive search, and the
ML detection of the optimal multi-level design was also obtained through an
exhaustive search. The prohibitive computational complexity of the exhaustive
search for the construction and the detection of the multi-level design in [4]
limits its application to a small number of symbols.

In [8], the authors provided a low-complexity multi-level design over only two
symbols. However, the design in [8] applies only to a specific unitary
constellation and cannot be generalized to other structures or to other
numbers of symbols.

To the best of our knowledge, a low-complexity multi-level design over any
number of symbols for non-coherent detection of SPCs has not been considered
in the literature. That said, we design a multi-level scheme based on the KL
divergence to improve the SER performance of existing non-coherent unitary
constellations designs. The proposed multi-level constellation has low
complexity in both its construction and detection to meet the latency
requirement of URLLC systems. Simulation results show that our multi-level
design significantly improves the SER performance of existing unitary
constellations and outperforms other low-complexity non-coherent schemes.

_Notation:_ Matrices, column vectors, and scalar variables are denoted by
uppercase bold letters (e.g., $\mathbf{A}$), lowercase bold letters (e.g.,
$\mathbf{a}$) and lowercase letters (e.g., $a$), respectively. We use
$(\cdot)^{*}$, $(\cdot)^{T}$, and $(\cdot)^{H}$ to denote the conjugate,
transpose, and conjugate transpose, respectively. We use $|\cdot|$,
$\|\cdot\|$, $\det(\cdot)$, and $\rm{tr}(\cdot)$ to denote absolute value,
Euclidean norm, determinant, and trace operations, respectively. We use
$\mathbb{R}^{m\times n}$ and $\mathbb{C}^{m\times n}$ to indicate the set of
real and complex matrices with dimension $m\times n$, respectively.
$\mathbf{I}_{K}$ denotes the $K\times K$ identity matrix and
$\rm{card}\\{\Omega\\}$ represents the cardinality of the set $\Omega$. The
circularly symmetric complex Gaussian distribution with mean $\mu$ and
variance $\sigma^{2}$ is represented as $\mathcal{CN}(\mu,\sigma^{2})$, and we
use $\mathbb{E}(\cdot)$ to denote the expectation of a random variable.

The remainder of the paper is organized as follows. The system model is
presented in Section II. We overview the design method based on KL divergence
and propose our multi-level structure in Section III. The optimization
formulation of our proposed multi-level structure is provided in Section IV.
Simulation results and related discussion are given in Section V. Section VI
concludes the paper.

##  II System Model

Consider the uplink transmission of a SIMO system in which a single-antenna
transmitter communicates with an $M$-antenna base station. Let
$\mathbf{h}\sim\mathcal{CN}(0,\mathbf{I}_{M})$ be the vector of independent
Rayleigh fading channel coefficients between the transmitter and the receiver,
and it is unknown to both of them. We assume a block fading channel where the
channel coefficients stay constant over a block of consecutive $K$ symbols and
then change to an independent realization in the coming block. Let
$\mathbf{s}\in\mathbb{C}^{K\times 1}$ be a sequence of $K$ transmit symbols
that represents one constellation point. The received signal
$\mathbf{Y}\in\mathbb{C}^{M\times K}$ at the receiver can be formulated as:

$\mathbf{Y}=\mathbf{h}\mathbf{s}^{T}+\mathbf{N},$ (1)

where $\mathbf{N}\in\mathbb{C}^{M\times K}$ is the noise matrix, whose
elements are independent zero-mean circular Gaussian random variables with
variance $\sigma^{2}$. The transmitted sequence is assumed to satisfy the
power constraint $\mathbb{E}(\|\mathbf{s}\|^{2})=1$. Thus, the average SNR at
each receiving antenna is given as $\text{SNR}=1/(K\sigma^{2})$.

Given the transmit signal $\mathbf{s}$, the probability density function of
the received signal $\mathbf{Y}$ can be written as follows [4], [8]:

$\displaystyle f(\mathbf{Y}|\mathbf{s})$
$\displaystyle=\frac{\exp\left(-\rm{tr}\left(\mathbf{Y}^{H}\mathbf{Y}(\mathbf{s}^{*}\mathbf{s}^{T}+\sigma^{2}\mathbf{I}_{K})^{-1}\right)\right)}{\pi^{KM}\left[\det(\mathbf{s}^{*}\mathbf{s}^{T}+\sigma^{2}\mathbf{I}_{K})\right]^{M}},$
$\displaystyle=\frac{\exp\left(-\frac{\rm{tr}(\mathbf{Y}^{H}\mathbf{Y})}{\sigma^{2}}+\frac{\rm{tr}\left(\mathbf{Y}^{H}\mathbf{Y}\
\mathbf{s}^{*}\mathbf{s}^{T}\right)}{\sigma^{2}(\sigma^{2}+\|\mathbf{s}\|^{2})}\right)}{\pi^{KM}\left[(\sigma^{2}+\|\mathbf{s}\|^{2})\sigma^{2K-2}\right]^{M}}.$
(2)

Given a known constellation of $\mathbf{s}$, i.e., $\Omega_{s}$, the ML
detector to detect the transmit signal $\mathbf{s}$ is [8]:

$\mathbf{s}_{\rm{ML}}=\operatorname*{argmax}_{\mathbf{s}\in\Omega_{s}}\left(\frac{\rm{tr}\left(\mathbf{Y}^{H}\mathbf{Y}\
\mathbf{s}^{*}\mathbf{s}^{T}\right)}{\sigma^{2}(\sigma^{2}+\|\mathbf{s}\|^{2})}-M(\sigma^{2}+\|\mathbf{s}\|^{2})\right).$
(3)

For any two transmit signals $\mathbf{s}_{i}$ and $\mathbf{s}_{k}$, the KL
distance between them is defined as the average KL divergence of their two
conditional probability density functions per antenna, which can be formally
written as[4]:

$\displaystyle
D_{\rm{KL}}\left(f(\mathbf{Y}|\mathbf{s}_{i}),f(\mathbf{Y}|\mathbf{s}_{k})\right)=\frac{1}{M}\mathbb{E}_{f(\mathbf{Y}|\mathbf{s}_{i})}\left(\frac{f(\mathbf{Y}|\mathbf{s}_{i})}{f(\mathbf{Y}|\mathbf{s}_{k})}\right)$
$\displaystyle=\left(\frac{\|\mathbf{s}_{k}\|^{2}\|\mathbf{s}_{i}\|^{2}-\|\mathbf{s}_{k}^{T}\mathbf{s}_{i}^{*}\|^{2}}{\sigma^{2}(\sigma^{2}+\|\mathbf{s}_{k}\|^{2})}\right)$
$\displaystyle\qquad+\left(\frac{\sigma^{2}+\|\mathbf{s}_{i}\|^{2}}{\sigma^{2}+\|\mathbf{s}_{k}\|^{2}}-\rm{ln}(\frac{\sigma^{2}+\|\mathbf{s}_{i}\|^{2}}{\sigma^{2}+\|\mathbf{s}_{k}\|^{2}})-1\right).$
(4)

##  III KL-Based Multi-Level Constellation Design

As pointed out in [4], the KL distance between any two transmit sequences
$\mathbf{s}_{i}$ and $\mathbf{s}_{k}$ determines the pairwise symbol error
probability between them. For this reason, the pair with the smallest KL
distance will produce the highest error probability and contribute the most to
the error rate of the whole constellation set. As a result, we aim to design a
constellation set $\Omega_{s}$ that maximizes the minimum KL distance, which
can be formally expressed as follows:

$\displaystyle\max_{\Omega_{s}}\min_{\mathbf{s}_{i},\mathbf{s}_{k}\in\Omega_{s},\mathbf{s}_{i}\neq\mathbf{s}_{k}}D_{\rm{KL}}\left(\mathbf{s}_{i},\mathbf{s}_{k}\right).$
(5)

Since any sequence $\mathbf{s}$ is a $(K\times 1)$-dimensional complex vector,
it can be characterized by its magnitude $\alpha\in\mathbb{R}$ and its
direction vector $\mathbf{v}\in\mathbf{C}^{K\times 1}$, which are defined as
follows:

$\displaystyle\alpha=\|\mathbf{s}\|,\
\mathbf{v}=\frac{\mathbf{s}}{\|\mathbf{s}\|}.$ (6)

Mathematically, $\mathbf{v}$ is a unitary vector obtained by normalizing the
original vector $\mathbf{s}$ such that any two sequences $\mathbf{s}_{i}$ and
$\mathbf{s}_{k}$ with the same direction will have the same unitary vector
$\mathbf{v}$.

The maximum likelihood detector in (3) can be rewritten in terms of $\alpha$
and $\mathbf{v}$ as follows:

$\\{\alpha_{\rm{ML}},\mathbf{v}_{\rm{ML}}\\}=\operatorname*{argmax}_{(\alpha,\mathbf{v})}\left(\frac{\alpha^{2}\rm{tr}\left(\mathbf{Y}^{H}\mathbf{Y}\mathbf{v}^{*}\mathbf{v}^{T}\right)}{\sigma^{2}(\sigma^{2}+\alpha^{2})}-M(\sigma^{2}+\alpha^{2})\right).$
(7)

Similarly, the KL divergence in (II) is also rewritten in terms of $\alpha$
and $\mathbf{v}$ as follow:

$\displaystyle
D_{\rm{KL}}\left(\alpha_{k},\mathbf{v}_{k},\alpha_{i},\mathbf{v}_{i}\right)=D_{1}\left(\alpha_{k},\mathbf{v}_{k},\alpha_{i},\mathbf{v}_{i}\right)+D_{2}\left(\alpha_{k},\alpha_{i}\right),$
$\displaystyle=\frac{\alpha_{k}^{2}\alpha_{i}^{2}\left(1-|\mathbf{v}_{k}^{T}\mathbf{v}_{i}^{*}|^{2}\right)}{\sigma^{2}(\sigma^{2}+\alpha_{k}^{2})}+\left(\frac{\sigma^{2}+\alpha_{i}^{2}}{\sigma^{2}+\alpha_{k}^{2}}-\rm{ln}\left(\frac{\sigma^{2}+\alpha_{i}^{2}}{\sigma^{2}+\alpha_{k}^{2}}\right)-1\right).$
(8)

As can be seen from (8), the KL distance between two constellation points
$\mathbf{s}_{i}$ and $\mathbf{s}_{k}$ consist of two terms $D_{1}$ and
$D_{2}$. The first term $D_{1}$ is due to the difference of direction vectors
$\mathbf{v}_{i}$ and $\mathbf{v}_{k}$, and it is scaled by the energy levels
$\alpha_{i}^{2}$ and $\alpha_{k}^{2}$ of the two points. On the other hand,
the second term $D_{2}$ is mainly due to the difference between the energy
levels of the two points.

Equation (8) indicates that designing the optimal $\Omega_{s}$ based on the KL
divergence requires a joint optimization over both $\alpha$ and $\mathbf{v}$,
which is an extremely complex process. One can observe that the KL divergence
in (8) reduces to $D_{2}$, i.e., $D_{1}=0$, when the two constellation points
have the same direction $\mathbf{v}$ but different energies; and reduces to
$D_{1}$, i.e., $D_{2}=0$, when the two constellation points have the same
energy levels. That said, to strike a balance between the SER performance and
complexity, we split the constellation set $\Omega_{s}$ into different subsets
$\mathcal{W}_{n}$, $n=0,...,N-1$, where each subset $\mathcal{W}_{n}$
comprises of $w_{n}$ points with the same energy level $\alpha_{n}^{2}$. By
doing so, we transform the constellation design problem in (8) into optimizing
the KL distance between constellation points inside a given subset
$D_{\rm{intra}}(\mathcal{W}_{n})$ (which only contains $D_{1}$) and optimizing
the KL distance between different subsets
$D_{\rm{inter}}(\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}})$. Please note that
both $D_{\rm{intra}}(\mathcal{W}_{n})$ and
$D_{\rm{inter}}(\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}})$ will be formally
defined in Section IV.

For $l_{s}$ bits allocated to this constellation set $\Omega_{s}$, the number
of points of all subsets $\mathcal{W}_{n}$, $n=0,...,N-1$, must be summed up
to $2^{l_{s}}$, i.e., $\sum_{n=0}^{N-1}w_{n}=2^{l_{s}}$. To find the optimal
combination of $\\{w_{0},w_{1},...,w_{N-1}\\}$, an exhaustive search algorithm
can be used [4]; however, its computational complexity prohibits its practical
implementation, especially for a higher number of bits per constellation. To
overcome such a high computational complexity, we propose a constellation
structure with subset $\mathcal{W}_{n}$, $n=0,1,...,N-1$, defined as

$\displaystyle\mathcal{W}_{n}$
$\displaystyle=\\{\alpha_{n}\mathbf{v}_{0},\alpha_{n}\mathbf{v}_{1},...,\alpha_{n}\mathbf{v}_{2^{l_{v}}-1}\\},\quad
n=0,1,...,N-1,$ (9)

where $N=2^{l_{\alpha}}$. We define the set
$\Omega_{\alpha}=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{2^{l_{\alpha}}-1}\\}$
as the set of $\alpha$ that satisfies
$\alpha_{0}<\alpha_{1}<\dots<\alpha_{2^{l_{\alpha}}-1}$, and we define the set
$\Omega_{v}=\\{\mathbf{v}_{0},\dots,\mathbf{v}_{2^{l_{v}}-1}\\}$ as the set
containing all possible unitary vectors $\mathbf{v}$, with $l_{\alpha}$ and
$l_{v}$ be the number of bits allocated to $\Omega_{\alpha}$ and $\Omega_{v}$,
respectively.

One can observe that the set $\Omega_{s}$ can be seen as the Cartesian product
of $\Omega_{\alpha}$ and $\Omega_{v}$, and such an observation serves two
purposes: the low-complexity construction and detection of our proposed multi-
level constellation design as discussed below.

####  III-1 Low-complexity construction of the multi-level constellation

For our proposed structure in (9), the number of possible choices for $w_{n}$,
$n=0,...,N-1$, are only $l_{s}+1$, which corresponds to
$N\in\\{1,2,4,\dots,2^{l_{s}}\\}$. In contrast, the method in [4] requires an
exhaustive search over approximately
$e^{\pi\sqrt{2^{l_{s}+1}/3}}/(4*2^{l_{s}}\sqrt{3})$ choices of partitioning
$w_{n}$ (Hardy–Ramanujan partition formula). Furthermore, [4] requires a large
number of available unitary constellations $\Omega_{v}$ whose number of points
ranges from one to $2^{l_{s}}$, which requires a gigantic computational
complexity. Instead, our method only requires unitary constellations whose
cardinality range is the power of two, i.e., $\\{1,2,4,\dots,2^{l_{s}}\\}$,
which are readily available in the literature.

####  III-2 Low-complexity ML detection of the multi-level constellation

Instead of searching over all possible $\mathbf{s}=\alpha\mathbf{v}$, as shown
in (7), we only need to search over $\alpha$ and $\mathbf{v}$ separately,
which is represented as follows:

$\displaystyle\mathbf{v}_{\rm{ML}}$
$\displaystyle=\operatorname*{argmax}_{\mathbf{v}\in\Omega_{v}}\
\rm{tr}\left(\mathbf{Y}^{H}\mathbf{Y}\mathbf{v}^{*}\mathbf{v}^{T}\right).$
(10) $\displaystyle\alpha_{\rm{ML}}$
$\displaystyle=\operatorname*{argmax}_{\alpha\in\Omega_{\alpha}}\left(\frac{\alpha^{2}\rm{tr}\left(\mathbf{Y}^{H}\mathbf{Y}\mathbf{v}_{\rm{ML}}^{*}\mathbf{v}_{\rm{ML}}^{T}\right)}{\sigma^{2}(\sigma^{2}+\alpha^{2})}-M(\sigma^{2}+\alpha^{2})\right).$
(11)

By doing this, the ML decoding complexity of the whole constellation can be
reduced from $\mathcal{O}(2^{l_{\alpha}+l_{v}})$ to
$\mathcal{O}(2^{l_{\alpha}})+\mathcal{O}(2^{l_{v}})$, where
$\mathcal{O}(2^{l_{\alpha}})$ and $\mathcal{O}(2^{l_{v}})$ are the
complexities of detecting $\alpha$ and $\mathbf{v}$, respectively. The
decoding complexities can be further reduced if we embed special unitary
constellations with low decoding complexity.

##  IV Optimization of Multi-level Constellations Under Fixed Bit Allocation

To find the optimal multi-level constellation $\Omega_{s}$, we find sub-
optimal constellations $\Omega_{\alpha}$ and $\Omega_{v}$ with given bit
allocations $l_{\alpha}$ and $l_{v}$, respectively. Then we choose
$l_{\alpha}$ and $l_{v}$ corresponding to constellation $\Omega_{s}$ with the
highest KL distance. The optimization for given fixed values of $l_{\alpha}$
and $l_{v}$ is discussed in this Section IV.

Given the proposed structure in (9), $D_{\rm{intra}}(\mathcal{W}_{n})$ and
$D_{\rm{inter}}(\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}})$ can be written as
follows:

$\displaystyle
D_{\rm{intra}}(\mathcal{W}_{n})=\frac{\alpha_{n}^{4}}{\sigma^{2}(\sigma^{2}+\alpha_{n}^{2})}\min_{\mathbf{v}_{k}\neq\mathbf{v}_{i}}(1-|\mathbf{v}_{k}^{T}\mathbf{v}_{i}^{*}|^{2}),$
(12) $\displaystyle
D_{\rm{inter}}(\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}})=\frac{\sigma^{2}+\alpha_{n_{1}}^{2}}{\sigma^{2}+\alpha_{n_{2}}^{2}}-\text{ln}\left(\frac{\sigma^{2}+\alpha_{n_{1}}^{2}}{\sigma^{2}+\alpha_{n_{2}}^{2}}\right)-1.$
(13)

For a given $l_{\alpha}$ and $l_{v}$, our problem in (5) can be reformulated
as follows:

$\displaystyle\\{\Omega_{\alpha},\Omega_{v}\\}=\operatorname*{argmax}_{\begin{subarray}{c}\Omega_{\alpha},\Omega_{v}\
\end{subarray}}$
$\displaystyle\left(\min\left[\min_{\mathcal{W}_{n}\subseteq\Omega_{s}}D_{\rm{intra}}(\mathcal{W}_{n}),\min_{\begin{subarray}{c}n_{1}\neq
n_{2}\\\
\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}}\subseteq\Omega_{s}\end{subarray}}D_{\rm{inter}}(\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}})\right]\right),$
(14a) $\displaystyle\text{s.t.}\ \|\mathbf{v}_{i}\|=1,\
\forall\mathbf{v}_{i}\in\Omega_{v},\ \rm{card}\\{\Omega_{v}\\}=2^{l_{v}},$
(14b)
$\displaystyle\qquad\frac{1}{2^{l_{\alpha}}}\sum_{\alpha_{i}\in\Omega_{\alpha}}\alpha_{i}^{2}=1,\
\rm{card}\\{\Omega_{\alpha}\\}=2^{l_{\alpha}},$ (14c)

Since $\alpha_{n}^{4}/(\sigma^{2}(\sigma^{2}+\alpha_{n}^{2}))$ is an
increasing function of $\alpha_{n}$ for $\alpha_{n}\geq 0$,
$D_{\rm{intra}}(\mathcal{W}_{n})$ is minimized when $\alpha_{n}$ is minimized,
i.e., $\alpha_{n}=\alpha_{0}$. Since $1/x-\rm{ln}(1/x)-1<x-\rm{ln}(x)-1$ for
$x>1$,
$D_{\rm{inter}}(\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}})<D_{\rm{inter}}(\mathcal{W}_{n_{2}},\mathcal{W}_{n_{1}})$
for $n_{1}<n_{2}$. Thus, the minimum inter distance must be in the cases of
$D_{\rm{inter}}(\mathcal{W}_{n_{1}},\mathcal{W}_{n_{2}})$ where $n_{1}<n_{2}$.
Also, since $D_{\rm{inter}}$ increases when the energy difference between two
subsets increases, the minimum inter distance must be the cases of two
consecutive subsets. As a result, we simplify our problem in (14) as follows:

$\displaystyle\\{\Omega_{\alpha},\Omega_{v}\\}$
$\displaystyle=\operatorname*{argmax}_{\begin{subarray}{c}\Omega_{\alpha},\Omega_{v}\
\end{subarray}}\left\\{\min\left(D_{\rm{intra}}(\mathcal{W}_{0}),\
D_{\rm{inter}}(\mathcal{W}_{n},\mathcal{W}_{n+1})\right)\right\\},$ (15a) s.t.
$\displaystyle\eqref{eqn:optProb_multilevel_constraint_v}\>\text{and}\>\eqref{eqn:optProb_multilevel_constraint_alpha}.$

We can see that the distance $D_{\rm{intra}}(\mathcal{W}_{0})$ contains the
variable $\mathbf{v}$ in the form of
$\min_{\mathbf{v}_{k}\neq\mathbf{v}_{i}}(1-|\mathbf{v}_{k}^{T}\mathbf{v}_{i}^{*}|^{2})$,
and this optimization problem of $\mathbf{v}$ is not affected by $\alpha$.
Thus, we can decouple our problem into two separate optimization problems,
i.e., unitary set optimization and multi-level optimization. The unitary set
optimization is given by:

$\displaystyle\Omega_{v}$
$\displaystyle=\operatorname*{argmax}_{\Omega_{v}}\left\\{\min_{\mathbf{v}_{k}\neq\mathbf{v}_{i}}(1-|\mathbf{v}_{k}^{T}\mathbf{v}_{i}^{*}|^{2})\right\\}$
(16a) s.t. $\displaystyle\eqref{eqn:optProb_multilevel_constraint_v},$

which is a classic problem called sphere packing on Grassmannian manifolds and
has been well-studied in the literature [5, 6, 7]. Thus, we can use any
available unitary set $\Omega_{v}$ from the literature and focus on the multi-
level optimization. Given $\Omega_{v}$, let
$T_{v}=\min_{\mathbf{v}_{k}\neq\mathbf{v}_{i}}(1-|\mathbf{v}_{k}^{T}\mathbf{v}_{i}^{*}|^{2})$
be the minimum distance of the unitary constellation $\Omega_{v}$. The multi-
level optimization problem for a given $T_{v}$ can be written as follows:

$\displaystyle\Omega_{\alpha}$
$\displaystyle=\operatorname*{argmax}_{\Omega_{\alpha}}\left\\{\min_{\begin{subarray}{c}\alpha_{0}\\\
r_{0},\dots,r_{N-2}\end{subarray}}\left(\frac{\alpha_{0}^{4}T_{v}}{\sigma^{2}(\sigma^{2}+\alpha_{0}^{2})},\frac{1}{r_{n}}-\rm{ln}\left(\frac{1}{r_{n}}\right)-1\right)\right\\},$
(17a) s.t. $\displaystyle\eqref{eqn:optProb_multilevel_constraint_alpha}.$

with $r_{n}=(\sigma^{2}+\alpha_{n+1}^{2})/(\sigma^{2}+\alpha_{n}^{2})$. The
optimal $\\{\bar{\alpha}_{0},\bar{r}_{0},\dots,\bar{r}_{N-2}\\}$ must satisfy
the conditions:

$\displaystyle\bar{r}_{0}=\bar{r}_{1}=\dots=\bar{r}_{N-2},$ (18a)
$\displaystyle\frac{\bar{\alpha}_{0}^{4}\
T_{v}}{\sigma^{2}(\sigma^{2}+\bar{\alpha}_{0}^{2})}=\frac{1}{\bar{r}_{0}}-\rm{ln}\left(\frac{1}{\bar{r}_{0}}\right)-1.$
(18b)

The proof is presented in the Appendix. Now we proceed to construct the
optimal set
$\Omega_{\bar{\alpha}}=\\{\bar{\alpha}_{0},\dots,\bar{\alpha}_{N-1}\\}$ given
the two conditions in (18). From (18a), it is easy to prove that
$\sigma^{2}+\bar{\alpha}_{i}^{2}=(\sigma^{2}+\bar{\alpha}_{0}^{2})(\bar{r}_{0})^{i}$.
Thus,

$\displaystyle
2^{l_{\alpha}}(\sigma^{2}+1)=\sum_{i=0}^{2^{l_{\alpha}}-1}(\sigma^{2}+\bar{\alpha}_{i}^{2})=(\sigma^{2}+\bar{\alpha}_{0}^{2})\sum_{i=0}^{2^{l_{\alpha}}-1}(\bar{r}_{0})^{i}$
$\displaystyle=(\sigma^{2}+\bar{\alpha}_{0}^{2})\left(\frac{r^{2^{l_{\alpha}}}-1}{r-1}\right).$
(19)

The solution of (18b) and (19) can be obtained by conventional methods such as
bisection or Newton method. We will present our bisection method in Algorithm
1.

1 Input: $\sigma^{2}$, $l_{\alpha}$, $T_{v}$.

2 Output: ${\bar{r}_{0}}$, $\bar{\alpha}_{0}$.

3 Parameters: $\epsilon$, $r_{l}=1$, $r_{u}$ satisfies
$\frac{1-r_{u}^{2^{l_{\alpha}}}}{1-r_{u}}\leq
2^{l_{\alpha}}\frac{1+\sigma^{2}}{\sigma^{2}}$.

4 while $r_{u}-r_{l}\geq\epsilon$ do

5 $r=\frac{r_{u}+r_{l}}{2}$,
$\alpha_{0}^{2}=2^{l_{\alpha}}\frac{(1+\sigma^{2})(1-r)}{1-r^{2^{l_{\alpha}}}}-\sigma^{2}$

6 if
$\frac{\alpha_{0}^{4}}{(\alpha_{0}^{2}+\sigma^{2})\sigma^{2}}T_{v}-\left(\frac{1}{r}-\text{ln}(\frac{1}{r})-1\right)\leq
0$ then

7 $r_{u}=r$

8 else

9 $r_{l}=r$

10 end if

11

12 end while

${\bar{r}_{0}}=r$, $\bar{\alpha}_{0}=\alpha_{0}$.

Algorithm 1 Bisection method to solve (18b) and (19)

##  V Simulation Result

In this section, we evaluate the SER performance of our proposed multi-level
constellation and compare it to other constellations from the literature.
Since already-optimized unitary constellations are prerequisites for our
method, we evaluate our proposed multi-level constellations with two types of
unitary constellations: the general unitary constellations obtained by
numerically solving the optimization problem in (16) and the cube-split
constellations in [7]. While the numerically-optimized general unitary
constellations have the best SER performance among all unitary constellations,
the cube-split constellations have low decoding complexity while still
attaining acceptable performance compared to other non-coherent schemes.

###  V-A Optimal Bit Allocation to $\Omega_{\alpha}$

((a)) $K=2$, Multi-level design with cube-split constellations.

((b)) $K=2$, Multi-level design with general unitary constellations.

((c)) $K=3$, Multi-level design with general unitary constellations.

((d)) $K=4$, Multi-level design with cube-split constellations.

Figure 1: Number of bits allocated to $\Omega_{\alpha}$ for optimal multi-
level design.

First, we plot the number of bits allocated to $\Omega_{\alpha}$. Since our
multi-level design is SNR-adaptive, given a collection of unitary
constellations $\Omega_{v}$, the number of bits allocated to
$\Omega_{\alpha}$, and hence, to $\Omega_{v}$, will only depend on the SNR
level. In general, if the number of bits allocated to $\Omega_{\alpha}$ is
greater than zero, it means our multi-level design achieved higher KL distance
than unitary constellations. In contrast, if the number of bits allocated to
$\Omega_{\alpha}$ is equal to zero, our multi-level design simply converges to
unitary constellations, and no performance gain is achieved from our multi-
level design. Figure 1 shows that the number of bits allocated to
$\Omega_{\alpha}$ tends to decrease when the SNR increases and converges to
1-level design (unitary constellations) in the high SNR regime. This is in
line with the statement of optimality of unitary constellations in the high
SNR regime in [4]. It can also be seen from Fig. 1 that for higher $K$, our
multi-level design still has advantages over unitary constellations, though
less frequently than the case of $K=2$.

###  V-B SER Performance Evaluation

Figure 2: SER comparison between the proposed multi-level design, 1-level
design, and other non-coherent schemes in case of $K=2$ and $M=256$. Solid
line: $l_{s}=6$. Dashed line: $l_{s}=8$

In this sub-section, we evaluate the SER performance gain achieved by our
multi-level design (combined with unitary constellations) when compared to
only unitary constellations without multi-level design. We also use two non-
coherent schemes as baselines for comparisons: non-coherent detection of
pilot-based QAM, which includes a pilot symbol and a sequence of QAM symbols,
and the multi-level constellations from [8]. Fig. 2 shows that multi-level
design improves the SER performance in both general unitary constellations and
cube-split constellations for both low and high SNR regimes. By embedding our
multi-level design into cube-split constellations, the cube-split design
achieves a significant improvement in SER performance and outperforms the
conventional pilot-based QAM scheme. Additionally, the proposed multi-level
design improves the performance of general unitary constellations and also
outperforms both of the reference schemes. Please note that the multi-level
design will converge to 1-level design when SNR further increases, as
discussed in Fig. 1.

Figure 3: SER comparison between the proposed multi-level design, 1-level
design, and other non-coherent schemes for different values of $K$ and
$M=256$.

Since our method can be applied to any number of time slots $K$, we also
evaluate the SER performance among different values of $K$ given the same SE
of 4 bits/Hz/s. In the case of $K=1$, energy encoding and detection in [3],
which is also a specific case of our multi-level design for $K=1$, will be
used as a reference. Firstly, Fig. 3 shows that our multi-level design also
improves the performance of the cube-split constellation for $K=4$ and general
unitary constellation for $K=3$. In general, since the overall performance of
non-coherent constellations improves when $K$ increases, it is reasonable to
apply our multi-level design to enhance the performance of unitary
constellations in a higher number of time slots. In contrast, the multi-level
designs in [8] (which is only designed for $K=2$) and [4] (which has high
complexity and therefore cannot be applied for large $K$) cannot achieve
performance gain by increasing the number of time slots.

##  VI Conclusion

In this paper, we proposed a low-complexity design for multi-level
constellations based on KL divergence. Our multi-level design is shown to
improve the SER performance of non-coherent schemes based on unitary
constellations as well as outperforms other non-coherent schemes while having
low complexity in construction and detection.

We prove the set $\Omega_{\bar{\alpha}}$ is the optimal solution of (18) by
contradiction. In this case, the objective value corresponding to
$\Omega_{\bar{\alpha}}$ is $\frac{\bar{\alpha}_{0}^{4}\
T(k_{\alpha})}{\sigma^{2}(\sigma^{2}+\bar{\alpha}_{0}^{2})}$ or
$\frac{1}{\bar{r}_{0}}-\rm{ln}\left(\frac{1}{\bar{r}_{0}}\right)-1$. Now,
assume that $\Omega_{\bar{\alpha}}$ is not the optimal solution, then there
exists at least one other set
$\Omega_{\hat{\alpha}}=\\{\hat{\alpha}_{0},...,\hat{\alpha}_{N-1}\\}$ and its
corresponding $\\{\hat{r}_{0},\dots,\hat{r}_{N-2}\\}$ that has higher
objective value. This is equivalent to:

$\displaystyle\frac{\hat{\alpha}_{0}^{4}\
T_{v}}{\sigma^{2}(\sigma^{2}+\hat{\alpha}_{0}^{2})}>\frac{\bar{\alpha}_{0}^{4}\
T_{v}}{\sigma^{2}(\sigma^{2}+\bar{\alpha}_{0}^{2})}.$ (20)
$\displaystyle\frac{1}{\hat{r}_{n}}-\rm{ln}\left(\frac{1}{\hat{r}_{n}}\right)-1>\frac{1}{\bar{r}_{n}}-\rm{ln}\left(\frac{1}{\bar{r}_{n}}\right)-1,\
\forall n\in\\{0,\dots,N-2\\}.$ (21)

Since $\frac{\alpha_{0}^{4}}{\sigma^{2}(\sigma^{2}+\alpha_{0}^{2})}$ is an
increasing function of $\alpha_{0}>0$ and $1/r-\rm{ln}(1/r)-1$ is an
increasing function of $r>1$, (20) and (21) are equivalent to
$\hat{\alpha}_{0}>\bar{\alpha_{0}}$ and $\hat{r}_{n}>\hat{r}_{n}$,
respectively. Thus,

$\displaystyle\sigma^{2}+\hat{\alpha}_{n}^{2}$
$\displaystyle=(\sigma^{2}+\hat{\alpha}_{0}^{2})\prod_{i=0}^{n-1}\hat{r}_{i}>(\sigma^{2}+\bar{\alpha}_{0}^{2})\prod_{i=0}^{n-1}\bar{r}_{i}=\sigma^{2}+\bar{\alpha}_{n}^{2}$
(22)

$\Omega_{\hat{\alpha}}$ clearly does not satisfy the power constraints
because:

$\frac{1}{2^{k_{\alpha}}}\sum_{i=0}^{2^{k_{\alpha}-1}}\hat{\alpha}_{i}^{2}>\frac{1}{2^{k_{\alpha}}}\sum_{i=0}^{2^{k_{\alpha}-1}}\bar{\alpha}_{i}^{2}=1,$
(23)

which concludes the proof. $\hfill\blacksquare$

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