\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/diagram-T.eps}}}\;\rightsquigarrow\;\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/CT.eps}}}\simeq\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/pulled-out.eps}}}.
\end{equation}

The grey box consists of a string link $\xi$ with a degree $1$ tree clasper $C_T$ on it, satisfying (A). Thus we compute

\begin{align*}
Z&\paren{R\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/diagram-T.eps}}}}}=Z\paren{\sigma\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/xi-out.eps}}}}}\circ Z\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/tangle-box.eps}}}}\\
&\overset{(\textbf{A})}{=}\paren{Z\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/box-straight.eps}}}}+\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/box-chord.eps}}}+O(2)}\circ Z\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/tangle-box.eps}}}}\\
&=Z\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/box-straight.eps}}}}\circ Z\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/tangle-box.eps}}}}+\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/box-chord.eps}}}\circ\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/pulling-claspers-out/tangle-box-empty.eps}}}+O(2)\\
\end{align*}

where the first equality uses Figure <ref> and multiplicativity. Then the left factor can be computed by the above since (A) is satisfied in the small grey box. Then we distribute the product over the sum on the left to obtain the second line, where everything that has degree $\geq 2$ is contained in $O(2)$. Use multiplicativity again to reconstruct the first term as the Kontsevich integral of a string link isotopic to the unlink, which becomes $1$, while the second term gives the diagram $T$ when attached to the outside. This concludes the proof of the case $n=1$.

Now we show the induction step. Let $n\geq 2$ and assume that the statement holds in degree $n-1$.

First, consider a degree $n$ tree diagram $T\in\Ds^T_n(n+1)$ on $n+1$ strands and satisfying (A). It looks like the diagram on the left below, where the black dashed rectangle contains the $n-1$ other legs. Realize it as a tree clasper $C_T$, assuming that the additional half-twist is in the blue blob.

\begin{align*}
\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/tree_T.eps}}}\:\rightsquigarrow\;\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/clasper_C_T.eps}}}.
\end{align*}

The core of the induction step is to apply Habiro's move 2 and 10 <cit.> to the visible tripod:

\begin{align*}
\vcenter{\hbox{\includegraphics[scale=1.3]{Ressources/Kontsevich_integral/induction-TPT/move-2.eps}}}\;\overset{H10}{\sim}\;\vcenter{\hbox{\includegraphics[scale=1.3]{Ressources/Kontsevich_integral/induction-TPT/move-10.eps}}}\;\simeq\:\vcenter{\hbox{\includegraphics[scale=1.3]{Ressources/Kontsevich_integral/induction-TPT/combinatorial-form.eps}}}.
\end{align*}

The right-hand-side is obtained by isotopy. The new tree clasper $C'$ ($C_{T}$ minus the tripod) realizes a degree $n-1$ tree diagram $T'$.

Thus, to compute $R(T)$ we can first apply moves $2$ and $10$ as above, then perform surgery on $C'$. This surgery is entirely contained in the grey rectangle shown in the first line below, and we obtain that first equality by multiplicativity of $Z$:

\begin{align*}
Z&(R(T))=Z\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/clasper-in-box.eps}}}}\circ Z\paren{\vcenter{\hbox{\includegraphics[scale=1]{Ressources/Kontsevich_integral/induction-TPT/tangle-without-box.eps}}}}\\
&=\underbrace{\paren{(R_{n+1})_*(D_n)_*\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/Tprime.eps}}}}}\circ Z_1\paren{\vcenter{\hbox{\includegraphics[scale=1]{Ressources/Kontsevich_integral/induction-TPT/tangle-without-box.eps}}}}}_{(1)}\\ &\quad+\underbrace{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/box-empty.eps}}}\circ Z\paren{\vcenter{\hbox{\includegraphics[scale=1]{Ressources/Kontsevich_integral/induction-TPT/tangle-without-box.eps}}}}}_{(2)}+\underbrace{Z_{\geq 1}\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/clasper-in-box.eps}}}}\circ\vcenter{\hbox{\includegraphics[scale=1]{Ressources/Kontsevich_integral/induction-TPT/empty-without-box.eps}}}}_{(3)}\;+\; O(n+1).
\end{align*}

From the first to the second line, the induction hypothesis $Z(R(T'))=1+T'+O(n)$ is used, with additional strand doubling and reversing. Then the third line is obtained by distributing this product[Observe that the $1$'s are not all the same, they stand for the empty diagram on the adequate tangle skeleton.], and putting all the terms of degree $\geq n+1$ inside $O(n+1)$. The resulting three terms are called (1), (2) and (3). Let us compute them:

* The first term is a product of degree $(n-1)$ terms with degree $1$ terms. First, we compute those two factors individually. Since we are only computing the degree $1$ of $Z$, only the braidings matter. The first factor is obtained by doubling the $n$-th strand and reversing the new $(n+1)$-st strand

\begin{align*}
(R_{n+1})_*(D_n)_*\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/Tprime.eps}}}}=(R_{n+1})_* &\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/lift-Tpr-left.eps}}}+\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/lift-Tpr-right.eps}}}}\\
\end{align*}

and the second factor is a sum of Kontsevich integrals of braidings

\begin{align*}
\end{align*}

Combining them yields

\begin{align*}
\end{align*}

where the last three columns vanish, since the leg attached to the blob is free to slide across the maximum. In the first column, the additional chord prevents that leg from sliding. This last column gives the tree diagram $T$ by $\STU$.

* This part is easier. Indeed, the left factor is $Z$ applied to a trivial string link (but whose rightmost strand is reversed). By mutliplicativity of $Z$, we can combine the two factors into

\begin{equation*}
\end{equation*}

* Since terms (1) and (2) equal $T$ and $1$, respectively, we expect (3) to vanish. Although the first factor might be very complicated, all we know is that it is obtained from a sum of diagrams by doubling the last strand, then reversing the new last strand. This fact alone actually suffices to show that (3) vanishes.

Indeed, suppose that $D$ is any nonempty diagram on $n$ strands, with $k\geq 1$ legs attached to the last strand. Doubling the last strand turns that diagram into a sum of all its $2^k$ lifts. Such a lift is indexed by a word $e_1\dots e_k\in\{0,1\}^{k}$ where a $e_i=0$, resp. $e_i=1$, means that the $i$-th leg is attached to the $n$-th, resp. $n+1$-st strand. Reversing the last strand has the effect of mutliplying by $\pm 1$, depending on the number of legs attached to the last strand. The resulting sum

\begin{align*}
(R_{n+1})_*(D_n)_*\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/diagram-D.eps}}}}=\sum_{e_2,\dots,e_k\in\{0,1\}}(-1)^{\sum_{j\geq 2}e_j}\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/D-lift-left.eps}}}-\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/D-lift-right.eps}}}}
\end{align*}

is a sum of differences of the form "$e_1=0$" minus "$e_1=1$". Multiplying each such difference with the second factor of (3) gives $0$ since the first leg is now free to go from one position to the other:

\begin{align*}
\paren{\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/D-lift-left.eps}}}-\vcenter{\hbox{\includegraphics[scale=1.2]{Ressources/Kontsevich_integral/induction-TPT/D-lift-right.eps}}}}\circ\vcenter{\hbox{\includegraphics{Ressources/Kontsevich_integral/induction-TPT/empty-without-box.eps}}}=0.
\end{align*}

The above shows that all terms in $Z(R(T'))$ that have at least one leg on the last strand vanish once we apply $(R_{n+1})_{*}(D_n)_*$ and multiply the result with the second factor.

In fact, all the terms in $Z(R(T'))$ must have at least 1 leg on the last strand. Indeed, by definition of the Kontsevich integral, the sum of all terms with no leg on the last strand is equal to the Kontsevich integral of $R(T')$ minus the last strand. If the last strand is removed, then $T'$ gets a trivial leaf and the surgery is trivial. In other words, this comes from the fact that a string link obtained by surgery along a tree clasper with exactly one leg on each strand is a Brunnian string link <cit.>.

This concludes the proof that (3) vanishes.

By the above computations of (1,2,3), we obtain

as desired. This concludes the case (A) in degree $n$.

The general case is shown exactly as in degree $1$, by pulling the clasper out to extract a box containing the clasper (as in (<ref>)), in which assumption (A) is satisfied. This concludes the induction step, and the proof of (<ref>).

It remains to show that $Z$ induces an inverse to $R$.

By (<ref>), the degree $n$ part of $Z$ induces a map $Z_n\colon L_n(m)\rightarrow\As^{\FI}_n(m)_{\Q}$, which is constant on $C_{n+1}$-equivalence classes. Thus it factors through

$$Z^{GH}_n\colon\Ls_n L(m)\rightarrow\As^{\FI}_n(m)_{\Q},$$

whose image lies in $\Prim(\As^{\FI}(m)_{\Q})_n$. Identifying the latter space with $\Ls_n^{\FI}(m)_{\Q}$ by Theorem <ref>, direct summing all the degrees at once and tensoring with $\Q$, we finally obtain

$$Z^{GH}\colon\Ls L(m)_{\Q}\rightarrow\Ls^{\FI}_n(m)_{\Q},$$
which is an inverse to $R$ by (<ref>).

§ APPLICATION TO THE RATIONAL GOUSSAROV–HABIRO CONJECTURE

§.§ The rational Goussarov–Habiro conjecture

§.§.§ Vassiliev filtration and associated graded algebra.

The monoid ring[If $M$ is a monoid and $R$ a ring, the monoid ring $RM$ is the abelian group of formal sums $\sum_{i=1}^n r_i x_i$, where $r_i\in R$, $x_i\in M$, wih multiplication given by $rx\cdot r'x'=rr' xx'$ and extended linearly.] $\Z L(m)$ of string links is filtered by the Vassiliev filtration (<ref>), with associated graded algebra $\As L(m)$ (<ref>):

\begin{align}
\Z L(m)&\supset \del_1(\Z S_{1} L(m))\supset\del_2(\Z S_{2} L(m))\supset\dots,\label{eq:Vassiliev_filtration}\\
\As L(m)&\coloneqq\bigoplus_{n\geq 0}\As_nL(m)\coloneqq\bigoplus_{n\geq 0}\frac{\del_n(\Z S_{n} L(m))}{\del_{n+1}(\Z S_{n+1} L(m))},\label{eq:graded_algebra_associated_with_Vassiliev_filtration}
\end{align}

where $S_{\bullet} L(m)$ denotes the graded monoid of (isotopy classes of) singular[Singularities are only allowed to be transversal double points, and isotopies must preserve the double points.] string links, graded by the number of double points. The $\Z$-linear maps $\del_n\colon\Z S_n L(m)\rightarrow \Z L(m)$ are defined by applying the following Skein relation <cit.> to each double point

\begin{equation}\label{eq:skein_relation}
\includegraphics[width=25pt,valign=c]{Ressources/double_point_skein_relation}\longmapsto\includegraphics[width=25pt,valign=c]{Ressources/double_point_skein_relation_+}-\includegraphics[width=25pt,valign=c]{Ressources/double_point_skein_relation_-},
\end{equation}

where it is understood that the terms are identical outside of the part shown. Thus, for $\gamma\in S_nL(m)$ a singular string link with $n$ double points, its image $\del_n(\gamma)\in\Z L(m)$ is an alternating sum of all possible resolutions of the $n$ double points.

A string link invariant $V\colon L(m)\rightarrow A$ valued in an abelian group $A$ naturally extends to a $Z$-linear map $V\colon\Z L(m)\rightarrow A$. We say that $V$ is a Vassiliev invariant or finite type invariant of degree $n$ if it vanishes on $\del_{n+1}(\Z S_{n+1} L(m))$. In particular, a Vassiliev invariant of degree $n$ induces a functional on $\As_nL(m)$. Two string links $\gamma,\gamma'\in L(m)$ are $V_n$-equivalent if they cannot be distinguished by Vassiliev invariants of degree $<n$ or, equivalently, if $\gamma-\gamma'$ belongs to $\del_{n}(\Z S_{n} L(m))$.

There is a realization map $R^v\colon\As^{\FI}(m)\epic\As L(m)$ (the $v$ refers to Vassiliev), which is a morphism of $\Z$-algebras. Originally, $R^v$ was defined on the presentation of $\As^{\FI}(m)$ by chord diagrams, by sending a chord diagram to a singular string link realizing it <cit.>. However, one can define this realization map on Feynman (forest) diagrams using claspers[This defines the same realization map since degree $1$ clasper surgeries are crossing changes] <cit.>. This was made precise by Conant and Teichner in the proof of <cit.>. Following their argument, we can construct a surjective[Surjectivity follows from Section 6 in <cit.>, where Habiro defines the Vassiliev filtration using forest claspers instead of singular string links.] morphism of graded $\Z$-algebras

\begin{equation}\label{eq:Vassiliev_realization_map}
R^v\colon\As^{\FI}(m)\epic\As L(m)
\end{equation}

which in degree $n$ sends a forest diagram $F\in\Ds_n(m)$ to
$$[\gamma_0;T_1,\dots,T_s]\coloneqq \sum_{\Ss'\subset\{T_1,\dots,T_s\}}(-1)^{s-\abs{\Ss'}}(\gamma_0^{\Ss'})\in\As_n L(m)$$
where $T_1\cup\dots\cup T_s$ is a forest clasper realizing $F$, $\gamma_0$ denotes the trivial string link on $m$ strands and $\gamma^{\Ss'}_0$ denotes the result of clasper surgery on $\gamma_0$ along the trees contained in $\Ss'$. That $R^v$ is well-defined follows from <cit.> and <cit.>.

A forest clasper realizing a forest diagram $F$ is constructed as in the proof of Proposition <ref>, but with multiple trees at the same time and with an additional positive half-twist near a chosen leaf of each tree. Those half-twits are important in order for $\STU$ relations to be realized with correct signs. This explains the footnote on page 68 in <cit.>.

Over $\Q$, the realization map $R^v\colon\As^{\FI}(m)_{\Q}\xrightarrow{\cong}\As L(m)_{\Q}$ is an isomorphism, with inverse induced by the Kontsevich integral <cit.>.

§.§.§ The Goussarov–Habiro conjecture

From Habiro's definition of the Vassiliev filtration using claspers[And from Goussarov's work as well.], it follows that $C_n$-equivalence implies $V_n$-equivalence for string links in $L(m)$. Goussarov and Habiro independently conjectured the converse:
[Goussarov–Habiro conjecture for string links in the cylinder]
For all $m\geq 1$ and $n\geq 1$, any $V_n$-equivalent string links on $m$ strands are $C_n$-equivalent.

Since $C_n$-equivalence implies $V_n$-equivalence, any $C_n$-trivial string link $\gamma\in L_n(m)$ is automatically $V_n$-equivalent to the trivial string link $1$, hence $\gamma-1$ belongs to $\del_n(\Z S_nL(m))$. This yields the comparison map <cit.>

\begin{equation*}
\chi\colon\Ls L(m)\rightarrow\As L(m)\colon [\gamma]\mapsto [\gamma-1],
\end{equation*}

which is injective if and only if the Goussarov–Habiro conjecture is true. In particular, it is injective when $m=1$ by the Goussarov–Habiro theorem for knots <cit.>.

Exploiting the dimension subgroup property in characteristic zero, Massuyeau showed the following rational version of the Goussarov–Habiro conjecture:

Over $\Q$, the map $\chi\colon\Ls L(m)_{\Q}\rightarrow\As L(m)_{\Q}$  is injective.

The realization maps $R,R^v$ and the comparison map $\chi$ combine into the following commutative square of graded (Lie or Hopf) $\Z$-algebras:

\begin{equation}\label{eq:integral_diagram}
\begin{tikzcd}
{\Ls^{\FI}(m)} & {\Prim(\As^{\FI}(m))} & {\As^{\FI}(m)} \\
{\Ls L(m)} && {\As L(m)}
\arrow["\chi", from=2-1, to=2-3]
\arrow["{R}"', two heads, from=1-1, to=2-1]
\arrow["{R^v}",two heads, from=1-3, to=2-3]
\arrow[from=1-1, to=1-2]
\arrow[hook, from=1-2, to=1-3]
\end{tikzcd}
\end{equation}

When tensoring everything with $\Q$, we obtain:

\[\begin{tikzcd}
{\Ls^{\FI}(m)_{\Q}} & {\Prim(\As^{\FI}(m)_{\Q})} & {\As^{\FI}(m)_{\Q}} \\
{\Ls L(m)_{\Q}} && {\As L(m)_{\Q}}
\arrow["\chi", hook, from=2-1, to=2-3]
\arrow["{R}"',"\cong", two heads, from=1-1, to=2-1]
\arrow["{R^v}"',"\cong", bend right = 25,two heads, from=1-3, to=2-3]
\arrow["\cong"',"\iota", from=1-1, to=1-2]
\arrow[hook, from=1-2, to=1-3]
\arrow["{Z^{GH}}"', from=2-1, to=1-2]
\arrow["{Z^{v}}"', bend right=25, from=2-3, to=1-3]
\end{tikzcd}\]

where $Z^v$ is the inverse to $R^v$ by Kontsevich's theorem, $\iota$ is an isomorphism by Theorem <ref>, the left triangle commutes and $R$ becomes an isomorphism by Theorem <ref>. Therefore, $\chi$ is injective, which gives an alternative proof the rational Goussarov–Habiro conjecture.

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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2023.0322000

Corresponding author: D. Turlykozhayeva (e-mail:
<EMAIL_ADDRESS>
# Automatic modulation classification for MIMO system based on the mutual
information feature extraction

N. USSIPOV1  S. AKHTANOV 1  Z. ZHANABAEV1  D. TURLYKOZHAYEVA 1  B. KARIBAYEV 2
T. NAMAZBAYEV 1  D. ALMEN 1  A. AKHMETALI 1  X. TANG 3  (Member  IEEE)
Faculty of Physics and Technology, Al-Farabi Kazakh National University, 71
al-Farabi Ave., Almaty, Kazakhstan  Department of Telecommunication
Engineering, Almaty University of Power Engineering and Telecommunications
named after Gumarbek Daukeyev, Baytursynuli 126/1, Almaty, Kazakhstan  School
of Electronics and Information, Northwestern Polytechinical University, 127
West Youyi Road, Xi’an, China

###### Abstract

Automatic Modulation Classification (AMC) is an essential technology that is
widely applied into various communications scenarios. In recent years, many
Machine Learning and Deep-Learning methods have been introduced into AMC, and
a lot of them apply different approaches to eliminate interference in complex
Multiple-Input and Multiple-Output (MIMO) signals and improve classification
performance. However, in practical communication systems, the perfect
elimination of MIMO signal interference is impossible, and therefore
classification performance suffers. In this paper, we propose a new AMC
algorithm for MIMO system based on mutual information (MI) features
extraction, which does not require a large amount of training data and the
elimination of MIMO signal interference. In this approach, features based on
mutual information are extracted using In-Phase and Quadrature (IQ)
constellation diagrams of MIMO signals, which have not been explored
previously. Our method can be effective since mutual information considers the
interdependencies among variables and measures how much information about one
variable reduces uncertainty about another, providing a valuable perspective
for extracting higher-level and interesting features from the data. The
effectiveness of our method is evaluated on several model and real-world
datasets, and its applicability is proven.

###### Index Terms:

Automatic modulation classification, classifier, feature extraction, mutual
information, entropy, complex MIMO signals

=-21pt

##  I Introduction

Automatic Modulation Classification (AMC) is an important technology that
provides modulation information of the incoming radio signals and plays a
significant role in practical civilian and military applications such as
cognitive radio, interference recognition, spectrum management, data
encryption, etc. AMC has gained considerable research interest in recent years
since it aims to automatically identify the modulation types of wireless
communications signals without prior information [1, 2, 3, 4, 5]. Among the
various modulation types, quadrature amplitude modulation (QAM) and phase-
shift keying (PSK) find widespread application in communication systems.
Additionally, when compared to other modulation types, classifying higher-
order QAM poses a greater challenge [6]. Hence, this study specifically
concentrates on high order 16QAM and 64QAM modulations.

Many modulation classification algorithms have been proposed for both single-
input and single-output (SISO) systems and MIMO systems and these methods are
usually classified into two major groups: likelihood-based (LB) and feature-
based (FB). LB methods are based on the likelihood function and are considered
optimal but generally complex to implement [7, 8]. FB methods depend on
feature extraction and classifier design and compared to LB methods are more
applicable in practical implementations and more popular in AMC due to the
ease of implementation [9, 10]. Various types of features, such as
instantaneous time-domain features, transform-domain features, and statistical
features have been investigated and used in AMC algorithms [11, 12]. After
extracting the features, the classifier is used to determine the class of
modulated signals. Until now, various types of classifiers have been used in
the classification stage of AMC, including artificial neural networks (ANN),
k-nearest neighbors (KNN), and support vector machines (SVM) [13].

Recently, numerous machine learning and deep learning-based AMC methods [14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] have been
developed for the classification of SISO and MIMO signals. These methods, as
demonstrated in recent surveys [31, 32, 33, 34, 35, 36], show promise for
achieving near-optimal performance with acceptable computational complexity.
Some of these methods use various technologies to eliminate interference in
complex MIMO signals and enhance classification performance. For instance, in
[27, 28] authors propose a convolutional neural network-based zero-forcing
equalization AMC method for MIMO systems. However, it is worth noting that the
perfect elimination of MIMO signal interference cannot be achieved in real
communication systems, and therefore the classification performance is
decreased.

In addition, authors in [37, 38, 39, 40] have proposed AMC algorithms for the
classification of communication signals based on information entropy. Entropy,
which measures the uncertainty of signal distribution and represents the
complexity degree of the signal, plays a key role in these methods. Although
these methods are competitive among other approaches, feature extraction using
mutual information (MI) can be preferable over Shannon entropy, since MI
considers the dependencies between two or more variables. Unlike entropy,
which primarily focuses on the uncertainty of individual variables, MI
measures how much information about one variable reduces uncertainty about
another [41, 42, 43]. This nuanced perspective is particularly valuable for
extracting higher-level and interesting features from the data.

In this work, we propose a new AMC algorithm designed for MIMO system based on
the mutual information feature extraction. Our algorithm stands out by
specifically extracting features using IQ constellation diagrams of MIMO
signals, a unique approach that has not been explored before. Furthermore,
this method does not require the elimination of MIMO signal interference and a
large amount of training data.

The rest of the paper is organized as follows. System model, data set
generation and experimental setup are introduced in section II. In Section
III, MI background, the proposed AMC method, and SVM classifier are proposed.
Section IV contains simulation results that demonstrate the correctness and
validity of our method. Finally, conclusions are given in Section V.

##  II MIMO SYSTEM MODEL, DATA SET GENERATION AND EXPERIMENTAL SETUP

###  II-A MIMO SYSTEM MODEL

In this paper, we consider a MIMO system that consists of $N_{s}$ transmitting
antennas and $N_{r}$ receiving antennas ($N_{r}$$\geq$$N_{s}$). Assuming that
the MIMO channel is an uncorrelated Rayleigh flat-fading and time-invariant
channel [28], the received signal at the $n$-th sampling moment can be
expressed as:

$r(n)=Hs(n)+g(n),$ (1)

where $H$ denotes $N_{r}$×$N_{s}$ dimensional MIMO channel matrix, and it
follows circular symmetric complex normal distribution; $r(n)$ =
$[r_{1}(n),r_{N_{r}}(n)]^{T}$ is the received signal vector with a size
$N_{r}$×1; $s(n)$ = $[s_{1}(n),s_{N_{t}}(n)]^{T}$ is the transmitted signal
vector with a size $N_{t}$×1; and $g(n)$ = $[g_{1}(n),g_{N_{r}}(n)]^{T}$ is
the additive white Gaussian noise vector with a size $N_{s}$ × 1 [2, 3, 24].

###  II-B Data set generation

Here we consider the system model of the complex multi- antenna, and the data
set generating process is presented in Fig. 1. Particularly, random data are
modulated using various modulation types, such as binary phase shift keying
(BPSK), quadrature phase shift keying (QPSK), 8-phase shift keying (8PSK),
16-quadrature amplitude modulation (16QAM), 64-quadrature amplitude modulation
(64QAM). The modulation signal vector, represented as $S$, with size 1 × $N$
(where $N$ is the number of symbols, and $N$ = 256, 512, 1024). To ensure a
fair comparison, $S$ is normalized with unit power, i. e., $\parallel
S\parallel_{2}^{2}$ = 1. After normalization, $S$ is reshaped into a $N_{s}$ ×
$N/N_{s}$ matrix, represented as $[s_{1},s_{2},\ldots,s_{N_{s}}]$, here
$s_{i}$ = $[s_{i}(1),s_{i}(2),\ldots,s_{i}(N/N_{s})$], $i\in[1,N_{s}];$ \- is
the transmitted signal vector at the $i$-th antenna.

Figure 1: The process of dataset generation

The received signal vector at the $j$-th receive antenna passing through the
MIMO channel, is denoted as $[r_{1},r_{2},\ldots,r_{N_{s}}]$, with size
$N_{r}$ × $N/N_{s}$, $r_{j}$ =
$[r_{j}(1),r_{j}(2),\ldots,r_{j}(N/N_{s})],j\in[1,N_{r}]$. Next, the received
signal matrix can be vectorized into a 1 × $N$ vector $r$. Test samples and
training data are retrieved from $r$. Specifically, the real and imaginary
parts of $r:\mathcal{R}(r)$ and $\mathcal{I}(r)$ are separated and combined
into a 2 × $N$ matrix $[\mathcal{R}(r)$ and $\mathcal{I}(r)]$, which is a
sample for training and testing. It should be noted that 6,000 samples are
prepared for training and 4,000 samples for testing for each SNR value [25,
27].

###  II-C Experimental setup

To evaluate the effectiveness and reliability of the proposed AMC method in
real communication scenarios, we conducted a series of experiments using an
integrated software and hardware complex for data collection of IQ
constellation diagrams. The software component of this complex is implemented
in the Matlab+Simulink program, while the hardware component comprises the
Zedboard+AD9361 (Fig. 2). The Zedboard, connected to the PC via an Ethernet
cable, provides communication between the software component and the RF module
AD9361. The AD9361 RF module is compulsory as it provides wireless
communication between receiver and transmitter.

Figure 2: Transmitter and receiver block diagram

The transmitting unit of software part consists of a random bit generator, a
modulator, a reshape demultiplexer device, while the hardware part comprises
AD9361 with 2 transmitting antennas.

The receiving unit of the software part accomplishes multiplexing for
vectorization and signal processing for IQ constellation diagrams. Signal
processing, in turn, includes tasks such as automatic gain control (AGC),
frequency compensation, and timing recovery. The hardware part of this
receiving unit consists of AD9361 with 2 antennas (Fig 2). IQ constellation
diagram values were saved and used for testing the proposed AMC algorithm.
Fig. 3 below illustrates the hardware part of the experimental setup designed
for receiving and transmitting signals in MIMO channels.

Figure 3: The hardware part of the experimental setup

The configuration of the AD9361 RF module was performed using Simulink, and
the following parameters were selected: a frequency of 2.4 GHz, a baseband
sample rate of 520.841 kHz. The transmitter and receiver gain parameters were
adjusted to control the Signal-to-Noise Ratio (SNR) value. The experimental
results confirm our theoretical data and serve as a qualitative addition to
this research.

##  III The proposed AMC method based on mutual information feature extraction

In this section, we describe the structure of the proposed AMC method, which
is shown in Fig 4. The method consists of two main parts: feature extraction
based on mutual information $I(X;Y)$ and SVM classifier. To make it easier to
understand, we introduce the section from three aspects: MI background,
detailed explanation of the proposed method and SVM classifier.

Figure 4: The structure of the proposed AMC method

###  III-A MI background

In probability theory and information theory, mutual information is a basic
concept that measures the statistical dependence between two random variables.
Specifically, it represents the amount of information learned about one random
variable by observing another. Mutual information is closely related to the
entropy because knowing information about one random variable can decrease the
level of uncertainty of another variable. Therefore, a high value of $I(X;Y)$
shows a substantial reduction of uncertainty, while a low value signifies a
minor reduction. If the mutual information between two random variables is
zero, it indicates that they are independent of each other [41, 42].

The mutual information between two discrete random variables $X$ and $Y$ is
defined as a double sum:

$I(X;Y)=\sum_{j\in Y}^{N}\sum_{i\in
X}^{M}P_{(i,j)}(x,y)log_{2}\left(\frac{P_{(i,j)}(x,y)}{P_{i}(x)P_{j}(y)}\right),$
(2)

where $N$ is the number of possible values for $X$ and $M$ is the number of
possible values for $Y,P_{(i,j)}(x,y)$ is the joint probability density
function, $P_{i}(x)$ and $P_{j}(y)$ denote the marginal probability density
function of $X$ and $Y$ [41, 42, 43, 44, 45].

As explained before, there is a strong relationship between mutual information
and entropy. The following formula illustrates this relation:

$I(X;Y)=H(X)-H(X/Y),$ (3)

where $H(X)$, $H(X/Y)$ are the marginal and conditional entropies. To
comprehend the meaning of $I(X;Y)$, it is necessary to explain both entropy
and conditional entropy. Entropy ($H$) quantifies the degree of expected
uncertainty of a random variable, and it can be expressed as follows [44]:

$H(X)=-\sum_{i}P_{i}(x)log_{2}P_{i}(x),$ (4)

where $P_{i}$ is the probability of event $x_{i}$. The conditional entropy
quantifies the amount of uncertainty that a random variable $X$ has when the
value of $Y$ is given, and it can be defined using the following formula [45]:

$H(X/Y)=-\sum_{i=1}^{N}\sum_{j=1}^{M}P_{(i,j)}(x,y)log_{2}(P_{(i,j)}(x/y)),$
(5)

where $N$ is the number of possible values for $X$ and $M$ is the number of
possible values for $Y,P_{(i,j)}(x,y)$ denotes the joint probability,and
$P_{(i,j)}(x/y)$ denotes the conditional probability.

###  III-B The proposed method

The proposed method involves the extraction of features based on mutual
information (MI) using a histogram of IQ diagram. The process begins with
constructing an IQ diagram that depicts the complex values of modulated
signals. The in-phase component of the received signal is chosen as variable
$X$, while the quadrature component is denoted as variable $Y$. Then, to
calculate MI we built histogram (Fig. 5) of $XY$ plane divided into bins of
equal size ($\Delta X\times\Delta Y$) with coordinates $i,j$. By counting the
number $k_{(i,j)}$ of samples in bin $(i,j)$, MI can be calculated using the
following formula:

$I(X;Y)=\sum_{j=1}^{k_{Y}}\sum_{i=1}^{k_{X}}\left(\frac{k_{ij}}{N}\right)\log\left(\frac{k_{ij}N}{k_{i}k_{j}}\right)$
(6)

To implement the proposed algorithm, the following steps must be performed:

1\. Construct IQ diagram.

2\. Build histogram of two-dimensional IQ diagram.

3\. MI calculation according to equation (6).

4\. In the final step, apply an SVM classifier to identify the type of
modulation.

In Fig. 5, the histogram of the IQ diagrams shows significant differences
corresponding to the type of modulation. The point density decreases as the
number of symbols increases.

Figure 5: Histogram of IQ diagrams of four modulation schemes at SNR=25

In the literature, several rules have been proposed to select the number of
bins ($b$), such as the choice of the square root (where $b=\sqrt{N}$), the
Sturges formula (where $b=1+\log_{2}{N}$), the Rice rule (where $b=2N^{1/3}$)
[46]. The comparison of these rules is given in the results.

###  III-C SVM classifier

Various types of classifiers have been used in the classification stage of AMC
until those days. In this work, we evaluated the performance of selected
classifiers based on their Probability Correct Classification (Table 1) and
chose the SVM classifier.

Support Vector Machine (SVM) is a supervised machine learning algorithm
developed by Vapnik and Chervonenkis based on statistical learning theory
[47]. It is useful for solving small-sample learning problems and has been
successfully applied in various fields, including pattern recognition,
regression analysis, and density estimation. SVM using kernel functions maps
the input data into a high-dimensional feature space, where a nonlinear
classification problem can be treated as a linear one.

By applying the rule of structural risk minimization, SVM aims to not only
reduce classification errors but also enhance the generalization ability of
classifiers. Compared to traditional Neural Network algorithms, SVM is
advantageous due to its simple structure, high success rates, better
generalization, and small-sample-problem-solving ability. Detailed
mathematical theories of SVM are available in references [48] and [49].

##  IV Results and discussion

This section presents simulation results to evaluate the performance of the
suggested modulation classification scheme. We analyze a MIMO communication
system with $N_{s}=2$ transmitting antennas and $N_{r}=2$ receiving antennas.
A channel with relay fading and AWGN noise is assumed. Six modulation schemes
are considered: BPSK, QPSK, 8PSK, 16QAM, 64QAM. The extraction of signal
features using mutual information is calculated by averaging 100 simulations
using the Monte Carlo method (Fig. 6).

Figure 6: Feature extraction based on mutual information under different SNR

According to Fig. 6, as the number of symbols on the constellation increases,
the distance between the points increases, which leads to a decrease in the
density of points on the constellation and, consequently, to a decrease in
mutual information. To calculate the probability of correct classification,
the following formula is used:

$P_{cc}=N_{cc}/N_{rs},$ (7)

where $N_{cc}$ is the number of correctly classified signals, $N_{rs}$ is the
total number of received signals. Fig. 7 below shows a comparison of the
existing rules for selecting bins.

Figure 7: Comparison of empirical rules for selecting bins (N=1024).

According to Fig. 7, the results of the square root and Rice rule demonstrate
better performance compared to the Sturges rule. This is because, from the
perspective of the integral root-mean-square error, a larger number of bins
enhances the accuracy of the density estimate.

Also, for the proposed AMC algorithm, we assessed the average probability of
correct classification across different classifiers at SNR from -10 to 15, and
we also examined the simulation time of these classifiers (Table 1).
Simulations were conducted using a computer equipped with an 11th Gen Intel(R)
Core (TM) i5-11400F processor operating at 2.60 GHz and 16.0 GB of RAM. The
probability of correct classification of the classifier is generally
considered the most crucial indicator for evaluating a feature extraction
algorithm. We use five classifiers, including KNN, SVM, Adaboost, Random
Forest (RF) and MLP to classify the original feature set extracted by our
proposed AMC method. These classifiers are evaluated using different methods
for determining the number of bins in a histogram, including Sturges, Rice,
and Square Root. The specific parameter for the KNN classifier was set as
follows: the nearest neighbor number ($k$) was set to 5. For the SVM
classifier, the kernel function was set as RBF kernel function. The Adaboost
classifier’s parameters were set with a depth of 10, a learning rate of 0.1,
and 10 iterations. The Random Forest classifier had a depth of 9, a learning
rate of 0.1, and 10 iterations. The MLP classifier was configured with a
learning rate of 0.1, and 10 iterations. According to the data set, the
simulation time and average probability of correct classification of different
classifiers are calculated.

TABLE I: Performance Comparison of Classifiers for the Proposed AMC Algorithm | Sturges | Rice | Square-root
---|---|---|---
| $P_{cc}$ | t (s) | $P_{cc}$ | t (s) | $P_{cc}$ | t (s)
SVM | 79.21% | 0.56 | 84.79% | 0.48 | 85.24% | 0.41
Adaboost | 62.60% | 0.25 | 66.29% | 0.11 | 68.72% | 0.25
MLP | 78.92% | 0.37 | 83.89% | 0.34 | 84.37% | 0.62
RF | 78.84% | 0.09 | 81.41% | 0.19 | 82.31% | 0.19
KNN | 80.25% | 0.10 | 83.85% | 0.09 | 84.06% | 0.09

Table 1 reveals that when applying the square root rule, SVM, MLP, and KNN
classifiers achieve the highest average probability of correct classification
compared to the Sturges and Rice methods. Specifically, SVM classifier leads
with the highest average probability of correct classification, closely
followed by MLP and KNN. On the other hand, Adaboost and RF classifiers
exhibit lower average probabilities of correct classification across all
methods. In terms of simulation times, KNN stands out as the most efficient
among the three top-performing classifiers, demonstrating shorter times
compared to SVM and MLP. Based on these results, in this paper, we favor the
use of the SVM classifier because of its highest average probability of
correct classification, prioritizing the sensitivity of feature extraction
using MI over the simulation time of the classifiers.

Fig. 8 presents the simulation results of our proposed AMC method with a MIMO
configuration of two transmit and two receive antennas (2x2). Fig. 8 shows the
$P_{cc}$ curves against different SNR values for various classifiers,
including SVM and KNN classifiers. The proposed KNN and SVM classifiers based
on mutual information is superior to the KNN and SVM classifier based on HOC,
given in [50].

Figure 8: Performance Comparison between MI-based and HOC-based Classification
Methods

Our proposed future based classifier outperforms classifier based on HOC and
achieves perfect classification at 5 dB SNR, whereas the compared classifier
attains 100% classification at SNR of 13 dB. We conducted a comparison of the
simulation times between our proposed method and the HOC-based approach. The
algorithm execution time for our method was 0.1 seconds for KNN and 0.56
seconds for SVM, slightly longer than the HOC-based method, which recorded
times of 0.53 seconds for SVM and 0.07 seconds for KNN.

Figure 9 shows the effect of N on the performance of our suggested method
under different SNR values, where N = 256, 512, 1024. Figure 9 shows that the
probability of correct classification increases with the growth of SNR, and
then stabilizes. The five modulation types BPSK, QPSK, 8PSK, 16QAM, 64QAM are
classified correctly with 100% probability when the SNR is about 5 dB.
According to Figure 9, the probability of correct $P_{cc}$ classification from
the number of points varies with a small error, since mutual information is
insensitive to the size of the data. Figure 10 below illustrates a comparison
of the probability of correct classification for each modulation scheme
relative to the other considered modulations.

Figure 9: Average probability of correct classification for different N Figure
10: Probability of correct classification ($P_{cc}$) among various modulation
schemes (N=1024)

In general, the classification accuracy of QAM signals is determined better in
comparison with PSK signals. Since with an increase in the modulation order of
QAM signals, the points on the constellation are distributed more evenly. It
is revealed that BPSK signals can be well identified using the classifier
proposed by us even at low SNR = -5 dB. The QPSK and 8PSK signals can be
correctly recognized at SNR above 3 dB. Perfect 16QAM and 64 QAM detection is
achieved at 5 dB SNR. Figure 11 shows the probabilities of correct
classification of BPSK and QPSK signals obtained using the experimental setup.

Figure 11: Average probability of correct classification of experimental data

As shown in Figure 11, the QPSK signal can be correctly recognized at SNR
above 3 dB, as in the theoretical results.

##  V Conclusion

In this article, we proposed a new classifier based on the extraction of
mutual information features for recognizing the type of modulation schemes
with low SNR for MIMO systems. This method does not require the elimination of
MIMO signal interference and a large amount of training data, since our method
is more resistant to noise uncertainty and spatial correlation in MIMO
systems.

In addition, it is worth noting that feature extraction methods based on
mutual information were not applied in AMC algorithms for MIMO systems. Our
approach proves its effectiveness by using mutual information that considers
the interdependencies between variables. The modeling and experimental results
showed that our algorithm can work properly even at low SNR with a small
number of observation samples, thereby representing a promising advancement in
the recognition of modulation schemes within MIMO systems.

##  VI Acknowledgement

We thank the anonymous referee for constructive comments that helped to
improve this article. This research was funded by the Committee of Science of
the Ministry of Science and Higher Education of the Republic of Kazakhstan
(Grant No. AP14872061).

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|  N. Ussipov Author received the master’s degree in engineering sciences from
Taraz State University named after M. Kh. Dulaty, Taraz, Kazakhstan, in 2016.
He is currently a senior researcher and chief programmer at the Department of
Solid State Physics and Nonlinear Physics of the Al-Farabi Kazakh National
University, where he is pursuing the Ph.D. degree. His current research
interests include signal modulation classification, signal processing, network
implementation, network information theory, routing algorithms.
---|---
|  S. Akhtanov Author received the Ph.D. degree in physics from Al-Farabi
Kazakh National University, Almaty, Kazakhstan, in 2019. He is currently a
senior researcher at the Department of Solid State Physics and Nonlinear
Physics of the Al-Farabi Kazakh National University. His current research
interests include signal modulation classification, signal processing, network
implementation, and optimization algorithms in wireless networks.
---|---
|  Z. Zhanabayev Author received the degree of Doctor of Physical and
Mathematical Sciences from Al-Farabi Kazakh National University, Almaty,
Kazakhstan, in 1996. In 1997, he received the title of professor from Al-
Farabi Kazakh National University. Under his leadership, 9 candidate
dissertations, 3 doctoral dissertations, 5 doctoral dissertations of
international level (PhD) were defended. Professor Zhanabaev Z.Zh. is the
initiator of the development of a new scientific direction in Kazakhstan
‘nonlinear physics and synergetics’. Professor Z.Zh. Zhanabayev has written 2
monographs, 12 textbooks, more than 300 scientific articles on modern areas of
science.
---|---
|  D. Turlykozhayeva Author received the master’s degree in physics from Tomsk
Polytechnical University, Tomsk, Russia, in 2017. She is currently a senior
researcher and chief editor at the Department of Solid State Physics and
Nonlinear Physics of the Al-Farabi Kazakh National University, where she is
pursuing the Ph.D. degree. Her current research interests include signal
modulation classification, signal processing, network implementation, network
theory, routing of wireless networks.
---|---
|  B. Karibayev Author received the Ph.D. degree in radio engineering,
electronics and telecommunication from Al-Farabi Kazakh National University,
Almaty, Kazakhstan, in 2019. He is currently a senior researcher at the
Department of Telecommunication Engineering of the Almaty University of Power
Engineering and Telecommunications named after Gumarbek Daukeyev. His current
research interests include signal modulation classification, signal
processing, computer modeling, MIMO antennas.
---|---
|  T. Namazbayev Author received the master’s degree in radio engineering,
electronics, and telecommunication from Al-Farabi Kazakh National University,
Almaty, Kazakhstan, in 2019. He is currently a senior researcher at the
Department of Solid State Physics and Nonlinear Physics of the Al-Farabi
Kazakh National University. His current research interests include signal
modulation classification, signal processing, computer modeling, Mimo
antennas.
---|---
|  D. Almen Author received the master’s degree in radio engineering,
electronics and telecommunication from Al-Farabi Kazakh National University,
Almaty, Kazakhstan, in 2021. She is currently a researcher at the Department
of Solid State Physics and Nonlinear Physics of the Al-Farabi Kazakh National
University. Her current research interests include signal modulation
classification, signal processing, wireless networks theory, wireless
communication.
---|---
|  A. Akhmetali Author student of Al-Farabi Kazakh National University,
Almaty, Kazakhstan. He is currently a junior researcher at the Department of
Solid State Physics and Nonlinear Physics of the Al-Farabi Kazakh National
University. His current research interests include signal processing, network
information theory, network implementation, routing algorithms.
---|---
|  Xiao Tang Author (Member, IEEE) received the B.S. degree in information
engineering (Elite Class Named After Tsien Hsue-shen) and Ph.D. degree in
information and communication engineering from Xi’an Jiaotong University,
Xi’an, China, in 2011 and 2018, respectively. He is currently with the
Department of Communication Engineering, Northwestern Polytechnical
University, Xi’an, China. His research interests include wireless
communications and networking, game theory, and physical layer security.
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