# An Overview and Classification of Machine Learning Approaches for Radar Signal Deinterleaving

Louis Lesieur, Jean-Marc Le Caillec, Ali Khenchaf, Vincent Guardia, Abdelmalek Toumi

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Louis Lesieur, Jean-Marc Le Caillec, Ali Khenchaf, Vincent Guardia, Abdelmalek Toumi. An Overview and Classification of Machine Learning Approaches for Radar Signal Deinterleaving. IEEE Access, 2025, 13, pp.28008-28028. 10.1109/ACCESS.2025.3539589. [hal-04950695](https://hal.science/hal-04950695v1)

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Distributed under a Creative Commons Attribution 4.0 International License
Received 10 December 2024, accepted 21 January 2025, date of publication 5 February 2025, date of current version 13 February 2025.

Digital Object Identifier 10.1109/ACCESS.2025.3539589

TOPICAL REVIEW

# An Overview and Classification of Machine Learning Approaches for Radar Signal Deinterleaving

LOUIS LESIEUR<sup>1,2</sup>, JEAN-MARC LE CAILLEC<sup>1,3</sup>, (Senior Member, IEEE),

ALI KHENCHAF<sup>1,1</sup>, (Senior Member, IEEE), VINCENT GUARDIA<sup>2</sup>, AND ABDELMALEK TOUMI<sup>1</sup>

Corresponding author: Louis Lesieur (louis.lesieur@ensta-bretagne.org)

This work was supported in part by the Association Nationale de la Recherche Technologique (ANRT) under Grant 2021-1016. It was carried out within the GIS CORMORANT scientific interest group, in the context of a partnership between Thales and Laboratoire des Sciences et Techniques de l'Information, de la Communication et de la Connaissance (Lab-STICC), unit UMR6285 affiliated to the Centre National de la Recherche Scientifique (CNRS).

**ABSTRACT** Electronic Warfare (EW) receivers are passive systems that are designed to detect and identify active radar emitters in the environment. The radar pulses emitted by multiple sources are received and must be deinterleaved, in other words, sorted according to the waveform to which they belong, i.e. according to their emitter. As radar signals are more complex and observations are denser, new Machine Learning (ML) approaches appear in the literature to enhance traditional Radar Signal Deinterleaving (RSD). In this paper, we propose an overview of the ML approaches to RSD. To this end, we identify some criteria to characterize a method: its used technique, underlying assumptions, exploited parameters, input characterization, and architectural pattern. First, the problem of RSD is detailed with its challenges and operational requirements. We then outline the methods of the literature inside a taxonomy based on the technique criterion: the first category includes PRI estimation methods including histogram-based methods, and ML techniques constitute three other categories: clustering, RNN, CNN. Finally, the other identified criteria are explained and discussed.

**INDEX TERMS** Electronic warfare, radar, deinterleaving, pulse sorting, machine learning, segmentation.

# I. INTRODUCTION

## A. CONTEXT

In the battlefield, it is essential to identify active radar emitters present in the environment. Electronic Warfare (EW) receivers are passive systems that aim to detect and identify these emitters, in particular, for foes early detection. At the beginning of the monitoring chain of an EW receiver, a digital processing extracts and characterizes pulses. The radar pulses are interleaved; it is a matter of sorting them according to the emitter. This is the role of the Radar Signal Deinterleaving (RSD) step: processing pulses over a time window to form groups. The last step of the chain involves

The associate editor coordinating the review of this manuscript and approving it for publication was Hasan S. Mir.

identifying the emitters by comparing them to a library of known waveforms. The radar pulses are characterized by a set of parameters that are measured by the receiver. The most significant are the Time Of Arrival (TOA), Direction Of Arrival (DOA), Pulse Width (PW), Pulse Amplitude (PA) and Radio Frequency (RF). To describe a waveform, other parameters are needed, such as the Pulse Repetition Interval (PRI). All these parameters are fully detailed in Section II-B1.

## B. MOTIVATION

In EW, density of the environment is a key criterion for evaluating the difficulty to obtain good performances. This notion is referred to as the time density and the spectral density. The time density refers to the number of devices

<footnote n="1">¹Lab-STICC, UMR CNRS 6285, ENSTA, Institut Polytechnique de Paris, 29806 Brest, France</footnote>

<footnote n="2">²Thales, 29200 Brest, France</footnote>

<footnote n="3">³Lab-STICC, UMR CNRS 6285, IMT Atlantique, 29285 Brest, France</footnote>
that emit simultaneously. In recent (high intensity) conflicts, a lot of operators either surveying or communicating has drastically increased the time/spectral density, making the deinterleaving step more and more complicated (as already observed three decades ago in [1]). In particular, the steadily lowering technology level (and then price) of emitters explains this overcrowded environment (see [2] for instance). In fact, even if many emitters have a circular scanning, which means that they only light the receiver for a short period of time, the receiver can jointly observe several emitters. The spectral density derives from the number of devices emitting within a specific frequency range of interest, for instance, in the range of navigational radars. In parallel, radar waveforms become more and more diverse and complex thanks to the development of software technologies in radar waveform design. The discrimination power of RSD in usual systems, based on well-established algorithms, can be out of date in some high density and high waveform complexity situations. Obviously, to circumvent the cluttering of the time/spectral density, more complex methods emerged from the machine learning domain, in particular Deep Learning (DL). In fact, DL algorithms have the property of inferring in real time, once the learning step is achieved, with the counterpart of being considered as a doubtful black box by the operators.

## C. RELATED WORKS

Few works aim to review machine learning/AI-based methods applied to EW problems. Sharma et al. [3] present recent trends and applications in the use of AI in EW systems, focusing more on radar jamming. In particular, their overview of AI algorithms is interesting and shows their diversity: Convolutional Neural Network (CNN), Recurrent Neural Network (RNN), clustering, hybrid methods, and several types of transformations. Lang et al. [4] survey machine learning applied to radar signal processing, but focus much on SAR image processing. Then, they suggest promising improvements on real-time applications, interpretability, cognitive radar, and overcoming data scarcity.

Some works published in master theses compile RSD methods. Bildøy [5] compiles traditional methods, focusing on threshold equations for Difference of TOA (DTOA) histogram methods. More recently, Manickchand [6] presents a state of the art TOA-based methods, as well as Aldossary [7], who also presents multiparameter methods.

## D. CONTRIBUTION AND OUTLINE

In the literature, there are numerous methods that try to solve the RSD problem, with a wide variety of underlying assumptions, which can lead to confusion. The purpose of this paper is three-fold.

1. We explain in detail the RSD problem.

2. We describe RSD methods inside a technique-based taxonomy and propose a Euler diagram to provide an overview of it.

3. We discuss other aspects of these methods: in particular, the underlying assumptions required by these algorithms, which are rarely explicitly mentioned in the literature, and the new concept of RSD architectural patterns.

We begin in Section II by explaining the context of EW, and the observed data in passive systems. In Section III, we state the problem of RSD and list the assumptions often made to simplify the problem. In Section IV, we present challenges and requirements of RSD. They can be well-established requirements/challenges or new ones raised by the outbreak of AI methods in the RSD field. Then comes the core of the review in Section V, where RSD methods are described within a taxonomy. The four main categories are PRI estimation based, clustering-based, RNN-based, CNN-based. In Section VI, we identify and discuss other relevant criteria: input parameters and the input data format, as well as the algorithmic strategy for RSD. Finally, we end with a conclusion and some perspectives on possible RSD trends and future work.

## II. BACKGROUND

We begin by presenting the background, that is, EW and observed data.

### A. EW RECEIVER

On the battlefield, electromagnetic (EM) waves are used to communicate between the operator, assess the environment, or guide a weapon. For example, the radar emits an EM pulse signal to detect objects in its environment. EW is the set of all the operations aiming to monitor and control the EM spectrum, this being a key stake.

These actions are split into three activities:

* Electronic Support: taking measures in order to warn in case of a new threat, to map radar emissions in a zone, characterize them, and intercept signals of interest, e.g. a foe radar.

* Electronic attack: preventing the adversary from taking measures, by carrying out countermeasures, e.g. electronic jamming.

* Electronic Defense: designing techniques to protect against countermeasures, e.g. frequency-agile radar.

As part of Electronic Support, the EW receiver is the passive system that “listens” to the surrounding radar emissions. Its objective is to detect all active emitters, identify known emitters, and warn in case of presence of foe or unknown emitters. In its generic architecture, shown in Figure 1, the signal received by the antennas is digitally processed to extract and characterize the radar pulses.

The received pulses are interleaved; it is necessary to sort them to identify the different active emitters. There comes the deinterleaving function: it is a question, inside a given time window, to gather pulses with the same radar waveform, and coming a priori from the same emitter. Then, the tracking function slides over several time windows.
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Passive receiver chain.</figcaption></figure>

Therefore, a track is the acknowledgment of the persistence of the same emitter through time then allowing for the false/spurious pulse detection as seen in the following. Lastly, the tracks are compared to a library of known waveforms for identification. Performances on a battlefield cannot be evaluated, but the system is developed so that some detection criteria are optimized. A dangerous failure would be to miss the detection of a close unknown emitter. A more quantitative approach to RSD performance is detailed in Section III-C.

## B. WAVEFORMS

In EW, a waveform names the characterization of the EM wave emitted by a radar. Each waveform corresponds to a specific operational function: target speed measurement, low-altitude aircraft surveillance, etc. The waveforms considered in this paper are pulsed waveforms, that is, a succession of brief EM emissions. In this section, we detail the parameters that allow for waveform characterization.

### 1) PRIMARY PARAMETERS

The pulse intrinsic parameters are called primary parameters, they are summarized in Table 1 and displayed in Figure 2. TOA is the time of arrival of the pulse. DOA is its direction of arrival, that is, a relative angle. PA is the pulse maximum amplitude, it depends on the energy at emission and the attenuation during the propagation. PW is frequently defined as the duration of the pulse below -3 dB of the PA, going from some tens of ns to a few ms. Finally, RF is the frequency within the pulse, sometimes called the carrier frequency, the RF of most radars lies in the [1, 40] GHz interval [8]. PW and RF are the only primary parameters chosen by the emitter, and then are valuable for EW tasks (identification/countermeasure).

<table><caption>TABLE 1. Primary parameters.</caption><thead><tr><td>Abbreviation</td><td>Parameter</td><td>Notation</td><td>Unit</td></tr></thead><tbody><tr><td>TOA</td><td>time of arrival</td><td>t</td><td>μs</td></tr><tr><td>DOA</td><td>direction of arrival</td><td>θ</td><td>degrees</td></tr><tr><td>PA</td><td>pulse amplitude</td><td>a</td><td>dBm</td></tr><tr><td>PW</td><td>pulse width</td><td>w</td><td>μs</td></tr><tr><td>RF</td><td>radio frequency</td><td>f</td><td>GHz</td></tr><tr><td>MOP</td><td>modulation on pulse</td><td>-</td><td>-</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Display of primary parameters and PRI over two pulses.</figcaption></figure>

The vector of parameters is called Pulse Descriptor Word (PDW) and it fully characterizes a pulse. According to the EW receiver, there can be fewer parameters, e.g. DOA can be missing. Besides, there can be additional informative pulse parameters, such as Modulation On Pulse (MOP), either a Phase modulation (PMOP) or a Frequency modulation (FMOP). A FMOP measurement provides more information than RF, i.e. the frequency is not conveyed by just a scalar but by a time-frequency (2-D) map over the pulse duration.

### 2) SECONDARY PARAMETERS

To entirely describe a waveform, it is necessary to use secondary parameters defining the interpulse properties.

The Pulse Repetition Interval (PRI) is a relevant secondary parameter. It quantifies the period between the emission of two consecutive pulses. In the simplest case, this period is stationary (i.e., constant). PRI values are equal to some μs up to some tens of ms. However, various emitters of interest modulate their PRI. The PRI of a jitter waveform is random. In this case, the waveform parameters are the mean PRI and its standard deviation. Other modulations follow a pattern, such as the stagger modulation. The pattern of a stagger waveform is characterized by a Waveform Pattern Period (WPP), equal to the sum of its (possibly varying) PRI.

Interpulse modulations on primary parameters are possible as well. RF is the most frequent modulated parameter.

Figure 3 represents a waveform taking 3 frequency values. These interpulse modulations are also rooted in a pattern.

<figure><img src="image_4.png" /><figcaption>FIGURE 3. Three consecutive patterns of a 4-moment and frequency-agile stagger waveform.</figcaption></figure>

Moreover in the next decades, RSD will be facing smarter systems of waveform design due to the cognitive radar
raise. They will be able to (pseudo-randomly) change the jitter and the stagger parameters and then to hide their WPP. This will probably require new RSD approaches and algorithms [9].

### C. SCANNING

Most radars scan their environment to detect objects; they often have a circular scan. From the receiver point of view, pulse amplitudes of such emitter throughout time are theoretically seen as the radar radiation pattern, which depends on the angle between the radiation and the receiver system direction. The radiation pattern can be modeled as a sinc function, composed of a main lobe, which is always visible, and side lobes, possibly unseen because of their relatively low power. Lobes can be modeled as backward quadratic functions. Note that the pulse maximum, i.e. PA, is a random parameter, mainly due to the propagation conditions. There can be a large standard variation within a main lobe, and it is not always relevant to fit a parabola on this lobe (see below, Figure 4).

### D. OBSERVED SCENE

From the receiver point of view, one can observe a PDW temporal flow, emitted by several independent devices in the nearby environment. A typical order of magnitude of the number of pulses is $10^6$ per second (to be processed in real time). It is a waveform mixture, the waveform number as well as their characteristics being a *priori* unknown. The scene is the data observed in a time window. Table 2 presents a possible occurrence of such data for a scene simulated over 10 ms. Each line is a PDW, i.e. the vector of the pulse primary parameters, and the last column is the corresponding pulse emitter identifier to be found.

<table><caption>TABLE 2. Example of a simulated scene, mixture of 3 waveforms.</caption><thead><tr><th>Pulse</th><th>TOA (µs)</th><th>PW (µs)</th><th>RF (MHz)</th><th>Emitter</th></tr></thead><tbody><tr><td>1</td><td>97.2</td><td>2.0</td><td>9420.0</td><td>1</td></tr><tr><td>2</td><td>151.7</td><td>50.0</td><td>9388.4</td><td>3</td></tr><tr><td>3</td><td>348.3</td><td>2.0</td><td>9419.4</td><td>1</td></tr><tr><td>4</td><td>394.7</td><td>0.4</td><td>9370.7</td><td>2</td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>n</td><td>9890.1</td><td>2.0</td><td>9420.1</td><td>1</td></tr></tbody></table>

A natural way to visualize the data is to display the time series of the non-time primary parameters, i.e. PA over TOA (see Figure 4), RF over TOA (see Figure 5), DOA over TOA (see Figure 6), and PW over TOA. Each point in these figures is the projection of the PDW onto 2-D planes. These figures are real-world data extracted from the same observed scene, each point represents a pulse, and the colors characterize the emitters. Each representation displays useful information. The TOA-RF plane permits one to identify temporal patterns. The PW-RF plane (see Figure 7) is typically used to form possibly homogeneous groups of pulses that can be easier to process.

<figure><img src="image_5.png" /><figcaption>FIGURE 4. Four waveforms, two of which present a well-formed radar lobe (orange and blue).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIGURE 5. Four waveforms, one of which being frequency-agile (blue).</figcaption></figure>

### E. NOISE

Multiple perturbations can occur between a pulse emission and its measurement by the receiver, as shown in Figure 8. A disturbance different from measurement noise is a pulse loss. It indicates a miss in the pulse detection and may have different causes: atmospheric conditions, hindering by obstacles, or a willed turning off of the listening for electromagnetic compatibility reasons. Moreover, there can be spurious pulses due to reflection phenomena, i.e. one pulse can be first directly received and received again later after a reflection over a cliff or a boat. A typical value of the pulse loss rate is 10%. The spurious/missed pulses disturb the PRI estimation-based methods as detailed below.

## III. PROBLEM STATEMENT

### A. PULSE MAPPING

In what follows, we consider a scene with *K* emitters in an observation window of duration *D*. A pulse *x* is conveyed by
<figure><img src="image_7.png" /><figcaption>FIGURE 6. Four waveforms, DOA is a stable parameter over such a small time window; notice that red is moving. In simple this case, DOA clustering would be sufficient for deinterleaving.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIGURE 7. Four waveforms in the RF-PW plane, blue present multiple clusters, unlike the other waveforms which present one cluster each.</figcaption></figure>

its PDW, $x = (t, w, f, a, \theta)$ (see Table 1 and Section II-B1) The $k^{th}$ waveform, $k \le K$, denoted $X^{(k)}$, is a set of $n^{(k)}$ pulses coming from the $k^{th}$ emitter: $X^{(k)} = \{x_i^{(k)}\}_{i \le n^{(k)}}$. Last, we denote $X$ the final set containing all of the observed pulses:

$$X = \bigcup_k X^{(k)} = \{x_i\}_{i \le N} \quad (1)$$

For such a scene, the goal of RSD is to identify the sets $X^{(k)}$ from the observation set $X$, the number of emitters $K$ being unknown. In other words, RSD has to map the pulses onto an integer vector $Y = (y_i)_{i \le n}$, each integer corresponding to an emitter (see the last column of Table 2). As stated above, the upper bound of these integers also has to be estimated.

## B. DIFFICULTY OF A SCENE

Application of AI methods raises the question of the degree of complexity of a scene. This is not a straightforward concept;

<figure><img src="image_9.png" /><figcaption>FIGURE 8. Examples of perturbations.</figcaption></figure>

we identify three main aspects: agility, overlapping, and density.

Interpulse modulation makes a waveform more difficult to process. It defines the characterization of the waveform and can occur in PRI, RF, and PW. The most complex PRI modulations are the jitter/stagger type, especially in case of high number of moments (see section II-B1). Interpulse modulation in RF and/or PW, can also raise issues of fragmentation, in particular for staggers which have an imbalance in terms of RF or PW values proportions, e.g. $f_1$ occurs more often than $f_2$. An RSD method must take into account interpulse modulation to avoid fragmentation of one waveform into multiple predicted waveforms.

Time overlapping refers to the common case when multiple lobes overlap, i.e. multiple radar emitters point approximately towards the EW system. One has to disentangle, or deinterleave, overlapping waveforms. In real-world cases, overlap in RF is not rare (see Figure 5). Inside the same time observation window, there can be multiple RF-agile emitters, the frequency range of which overlap. Then, the frequencies of two different radars cannot be distinguished.

All those issues of interpulse modulation and overlapping can jointly occur. Overall, the most relevant criterion is the number of present emitters, because it exacerbates all the previously mentioned issues.

## C. PERFORMANCE METRICS

Evaluation of the performance of an RSD is not straightforward. The quality of a partitioning $\hat{Y}$ is estimated using metrics, i.e. functions $m: \mathbb{N}^n \to \mathbb{R}$. We focus on clustering metrics, because labels of $\hat{Y}$ are permutation invariant, since the emitter label (number) is not informative. For example, $\hat{Y} = (0, 1, 1, 2, 2, 0)$ and $\hat{Y}' = (1, 2, 2, 0, 0, 1)$ should have the same score, because they both represent the same pulse clustering $\{(x_1, x_6), \{x_2, x_3\}, \{x_4, x_5\}\}$. Clustering metrics are divided between external metrics based on a ground truth $Y$ (the most relevant), and internal metrics that do not rely on the real solution.

The main metric is the Adjusted Rand Index (ARI), (see [10] for instance). The index evaluates all of the pulse pairs of $\hat{Y}$. More precisely, the Rand Index (RI) is the rate of pairs correctly associated or dissociated by the RSD method, according to $Y$. ARI obtains 0 in the random case and 1 in case of an exact clustering, by normalizing:

$$ARI = \frac{RI - E(RI)}{\max(RI) - E(RI)} \in [-1, 1] \quad (2)$$
Rosenberg et al. [11] propose the V-measure, which stands for “validity”, as a general clustering metric. It is a harmonic mean of homogeneity and completeness, two scores derived from the entropy of information theory:

* Homogeneity $h \in [0, 1]$ measures whether only pulses from the same emitter are within the same cluster.

* Completeness $c \in [0, 1]$ measures whether all of the pulses from the same emitter are within the same cluster.

These two scores are complementary and merged by the V-measure, defined as:

$$V_{\beta} = \frac{(1 + \beta) \cdot h \cdot c}{(\beta \cdot h) + c} \in [0, 1] \quad (3)$$

Parameter $\beta$ controls the respective weights of $h$ and $c$. It is equivalent to Normalized Mutual Information (NMI) when $\beta = 1$.

Gasperini et al. [12] propose three interpretable metrics for RSD. Purity and non-fragmentation are variants of homogeneity and completeness, but not relying on information theory. In particular, the third metric is the detection rate, which can be important in RSD: it is the rate of emitters represented by a cluster. In other words, when this detection score does not equal 1, the error propagates in the processing and the some emitters can be undetected. In their experiments, the authors use ARI and NMI as main metrics, while purity, non-fragmentation and detection are modified to be used as loss functions, and also used as complementary interpretable metrics. Other more generic metrics derived from the classification framework, such as the accuracy, i.e. the rate of well classified pulses, may be found in some papers. Finally, reject rate (rate of not classified inside a cluster) can also be mentioned as a metric of RSD quality.

## D. USUAL ASSUMPTIONS OF SIGNAL DEINTERLEAVING ALGORITHMS

All of the articles make (explicitly or not) weak assumptions about the characterization of targets: such assumptions are hyperparameters that can be modified later on: it can be an expected range of PRI, as is the case for most traditional methods, or a restricted PRI modulation range. Aside from this, some articles make stronger assumptions to develop RSD algorithms: these assumptions cannot always be modified since they are inherent to their core. In order to expose our taxonomy in the next section, we have identified four main assumptions.

* $A_1$ Known number of waveforms. The assumption about $K$, the number of waveforms present, allows the use of classic machine learning classification on pulses. Of course, $K$ is unknown in real-world applications. It can be a hard constraint ($K$ exactly known) or a weaker constraint (upper bound for $K$).

* $A_2$ Known waveforms. Assumption about the targets, including their characteristics. Sometimes, this assumption is made jointly with $A_1$. Obviously, the issue with the algorithms developed under such an assumption

is that they could fail when encountering unknown waveforms.

* $A_3$ No Specific Emitter Identification (SEI). SEI refers to a well-known problem aiming at separating the emission of distinct radars with identical waveforms. In some papers, this assumption is not explicitly made by the authors and can be unrealistic in an operational theater.

* $A_4$ Radar scanning. Relevant assumption that radar emitters have mechanical radar scanning. It yields two aspects: time duration and pulse amplitude. First, the duration of a waveform is limited, i.e. not spread over the whole observation window, this being additional information. Second, it produces lobe patterns on PA (see Figure 4). Although traditional methods discard PA, this information can be leveraged by identifying and/or separating lobe patterns.

Other less current assumptions, such as the multistation assumption, will be described when needed.

## E. RELATED TOPICS

For the sake of clarity, in this section, we briefly mention close topics and associated works that can be confused with RSD at first sight, but cannot be considered as being within the scope of this overview.

In the following cases, the input is a pulse sequence from a single emitter, that is, their use case occurs after RSD:

* identification, or waveform classification: [13], [14]

* PRI modulation classification: with Gated Recurrent Units (GRU) and attention [15], with Temporal Convolutional Network (TCN) [16], with DL and attention [17]

* scanning type classification: [18]

There are also more general cases:

* prediction task: from a single emitter pulse sequence, predict the next pulses with an LSTM [19]

* source separation: from the raw signal, i.e. before the creation of PDW, separate sources [20]

* parameter estimation: from the waveform raw signal, estimate its parameters [21]

* cognitive radar: optimize frequency band switching strategy for surveillance [22]

## IV. CHALLENGES AND REQUIREMENTS

### A. MULTIMODALITY

The first difficulty in processing a PDW is its multi-modality, since it contains physical quantities of different nature: TOA and PW are time measures, DOA is an angle, RF is a frequency. In particular, some of these measures exhibit a large range of values, such as PW, the width of the largest pulse being threefold that of the narrowest ones. It induces issues of normalization to avoid the joint processing of inconsistent measures, e.g., computing a distance between pulses in the RF-PW, mixing GHz and $\mu$s. It is often necessary to perform a data preprocessing with a type of
normalization/standardization, e.g. for a distance definition
in a multimodal space (see below Section VI-B3).

## B. A NON-CONVENTIONAL PROBLEM

RSD is not a conventional problem since it differs from the usual ML problems such as classification, clustering, and segmentation. In this paragraph, we analyze some characteristics of the problem.

### 1) CLASSIFICATION, CLUSTERING OR SEGMENTATION ?

Blind deinterleaving, in particular without $A_1$ assumption, is not a typical classification problem as those solved by a NN with a final softmax activation layer because the classes and their number is unknown. It is neither a usual cluster analysis problem. First, because the solution is objective, i.e. there is only one exact partition solution. Then because it is not only a matter of clustering based on the primary parameter, since agility is possible, and since TOA and PA are not suitable for similarity analysis. That being said, clustering is part of many RSD methods, although not sufficient. Formally, blind deinterleaving is closer to a segmentation problem, because we aim to segment $X$ into waveforms. However, in (possibly multivariate) time-series segmentation, the aim is to retrieve the switching regimes of an observed signal, according to the changes of the time-series statistical properties. In RSD, the regimes (that is, the different emitters) are entangled without changing the time-series statistics. As seen in the following, it also differs from the image segmentation problem.

### 2) SIGNAL (SEQUENTIAL) MODELING APPROACH

Some methods model the scanned scene as a multivariate time series, in order to design NNs to process this kind of data support. However, there are three main differences between the usual tasks and pulse deinterleaving:

* The input and output are the same size (each pulse being linked to a unique emitter), unlike language translation tasks, for example, a sequence-to-sequence (seq2seq) problem.

* The sequence can be read forward and backward, unlike most time-series processing and language processing (there is no causality relationship, unlike usual time series). An exception is for the possibly agile emitters, since in this case one must take into account the order of the pattern moment of the waveform (see Section II-B2).

* The pulses from a predicted emitter are not necessarily connected in the sequence, except when there is no temporal overlap.

### 3) IMAGE MODELING APPROACH

Some authors encode the observation scene in a 2-D frame, with TOA and RF axes, to further segment the image with a CNN-based algorithm (see Figures 4 to 7). There are two main characteristics of RSD in this context:

* classes are unknown in blind deinterleaving

* the image is sparse, so the overwhelming class is for non-pulse pixels

* consequently, the areas characterizing the output classes are not connected, unlike the image segmentation of physical objects

## C. DATA SCARCITY

Unlike some domains such as face recognition, for which there are many ready-to-use datasets, EW suffers from data scarcity, which hampers the application of supervised learning methods. Data scarcity is a two-fold concern, that is, the weak amount of correctly labeled real data and obviously the confidential nature of some information.

### 1) SCARCITY OF LABELED

data First, we refer to the weak availability of labeled observed data, which means that we can only access a few input-output pairs $(x, y)$, in an observed scene $X = \{x_i \le N\}$ with their corresponding labels $Y = \{y_i \le N\}$. Large amounts of data are required for deep learning techniques in order to achieve high performance. It is also essential for validation, since an algorithm is evaluated according to truth values (see Section III-C). It is sometimes difficult to know whether two pulses belong to the same waveform, and more generally to label the data. Ideally, each pulse of the time window should have a label. The obvious solution to make fully labeled scenes available is to simulate the data. Questions of reliability of data from the literature arise. The simulation models cannot account for the real complexity of data, but are effective when they take into account physical phenomena of propagation, reflection, and so on. They must be closely tied to the knowledge of the simulated receiver. Some databases containing civil radar waveforms are available as a background for data augmentation/simulation [23].

### 2) SCARCITY OF WAVEFORMS

Another aspect is that almost the waveforms, possibly faced in war operations, are of military usage, i.e. protected by confidentiality. Thus, the number of effectively known waveforms is low. The risk is to encounter generalization difficulties, one of the RSD objectives being to discriminate unknown waveforms. There are two approaches to overcome this. The first is data augmentation; it consists of artificially increasing the number of instances by creating new ones from existing instances. Generative Adversarial Networks (GANs) are neural network architectures that also provide data augmentation. Nevertheless one can wonder about the relevance of such generated instances; the risk is to violate some physics or radar theory rules by generating inconsistent waveforms, or useless waveforms from the operational point of view. It would create a bias in the learning step. In addition, it is possible to use expert knowledge to make data augmentation. The second approach is to develop algorithms that are suitable for this situation. The authors
of [4] mention few-shot learning, a recent technique, the objective of which is to extract representative features from a few examples. Another idea is semi-supervised learning: labeling is necessary for some pulses only, for instance those with less ambiguity. The data augmentation, and more specifically the consistency of the new data, raises also the question of eXplainable Artificial Intelligence.

## D. EXPLAINABILITY

Data augmentation, but also the ethical aspects of AI algorithms applied in the defense field (such as RSD) such as their explainability/interpretability are an operational requirement.

### 1) DEFINITION AND CONTEXT

Explainability is the ability for a given algorithm to be able to provide elements of explanation of the output. An EW receiver is a critical system for which explainability is necessary to some extent. In a war situation, we want the system to provide operative elements to justify the decisions. Yet, a common characteristic of a number of machine learning architectures is the black-box effect, there is no interpretable explanation. It is the case of segmentation-based RSD methods, including [12], [24].

### 2) TECHNIQUES

Explainability is a complex issue; we generally distinguish two approaches. The first approach is to build an explicable model by design. Leaving DL, using algorithms such as Density-based Spatial Clustering of Applications with Noise (DBSCAN) clustering associated with distance between pulses [25], decision trees, or using hybrid solutions with symbolic AI can help build such models for RSD. The attention mechanism [26], used in natural language processing, can also be an option to increase the interpretability level of deep learning architectures. The second approach is to search a *posteriori* explanations inside the neural network, i.e. after inference using perturbation method or Shapley values (see [27], [28]). A useful approach in the defense domain is the counterfactual explanation methods that provide the reasons of a classification result versus another [29].

## E. REAL-TIME PROCESSING

Most of EW receivers are destined to be embedded for tactic missions, e.g. in naval or airborne equipment. It raises two requirements: computation time and computer memory. The computational requirement is less critical when applied to Electronic Intelligence (ELINT) missions, where the signals can be processed later on.

### 1) COMPUTATION TIME AND MEMORY

Limited computation time is a requirement since the receiver must be able to raise warning in case of threat, with a latency typically less than a second, while continuing processing the continuous pulse flow. The supervised-learning-based

processing can be trained upstream on a devoted server, but should be able to manage the real-time pulse flow in inference. With this in mind, a parameter to take into account is the number of pulses to be processed, linked to the deinterleaving time window. However, some large architectures such as Transformer networks could be prohibitive. The other requirement is about computer memory: the mass and volume of the physical receiver must be minimized.

### 2) POSSIBLE SOLUTIONS

A solution to computational issues is to reduce the initial flow of information, by early clustering pulses based on some primary parameters, such as [25]. The risk is to mix multiple waveforms from the beginning, and then to propagate the error in the next steps. There are also neural network compression techniques, including pruning and tensor decomposition that allow the reduction of the number of neurons and then the reduction in computational time.

## V. PULSE DEINTERLEAVING TAXONOMY

In this section, we analyze the different approaches used in pulse deinterleaving methods. We propose a taxonomy of four main classes (with obviously some subclasses).

1) PRI estimation

2) Clustering-based

3) RNN-based

4) CNN-based

The first class gathers the usual RSD methods. These methods aim to exploit the TOAs and, more specifically, the PRIs. Their principle is to estimate the PRI of only one waveform (with possible modulation) among all of the detected pulses, then extract relevant pulses, and start anew. Classes 2, 3 and 4 are based on the ML technique used. The methods of the second class model the RSD problem as a clustering problem within a multidimensional space. Generally, all PDW parameters can be considered, except for TOA.

In recent years, ML algorithms, including DL, have been greatly developed. They have shown their potential with remarkable progress, notably in image processing and natural language processing. ML approaches include RNN-based methods that rely on pulse sequence modeling, while CNN-based methods assume image encoding. Finally, other approaches are introduced in an additional paragraph. Figure 9 provides an overview of the methods and categories.

The next four subsections detail these four taxa. In these sections, the methods are presented in chronological order.

## A. PRI ESTIMATION METHODS

PRI estimation methods are recursive deinterleaving, i.e. they follow a principle of recursive waveform extraction. They aim to sequentially extract waveforms one by one according to a PRI estimation strategy until there are no or very few unlabeled pulses. At each step, the strategy theoretically yields *I* the set of selected pulses to be extracted, as detailed in Algorithm 1. Historical methods follow this framework,
<figure><img src="image_10.png" /><figcaption>FIGURE 9. Euler diagram of deinterleaving methods. The four main categories are represented by blue rectangles. The recursive methods are written in red. Finally, the location on the diagram also indicates the exploited primary parameters (TOA and/or other primary parameters) and whether the methods are based on supervised learning.</figcaption></figure>

exclusively exploiting the times of arrival of pulses. Lately, some variants of this principle also exploit other parameters. Most of the time, PRI estimation methods make none of the strong assumptions presented in Section III-D.

**Algorithm 1 Recursive Waveform Extraction**

RecursiveDeinterleaving

**Input:** X

**Output:** Y

label ← 0

$y_{i \in 1..N} \leftarrow \text{label}$

while there are still enough pulses

$I \leftarrow \text{Selection}(\mathbf{X})$

label ← label + 1

$y_{i \in I} \leftarrow \text{label}$

$\mathbf{X} \leftarrow \mathbf{X} \setminus \{x\}_{i \in I}$

As stated in Section III-A, the input data **X** is made up of *n* PDW vectors, and $\mathbf{y} \in \{1, \dots, K\}^N$ is to be the output vector, initialized to 0.

At each step, it is a question of selecting some (not yet labeled) pulses. This selection function is a key problem and many algorithms have been proposed (as seen below). The selection of pulses is represented by the set *I* of its pulse indices. Then $\mathbf{y}_i$ is updated for all $i \in I$ with a new label value, representing their waveform, and the pulses under inspection are removed from **X**, before a new iteration. The stopping

condition is based on the number of unlabeled pulses, i.e. below a threshold.

In what follows, we focus on the three main pulse selection functions: Sequence Search, BTDOAH, Transforms.

1) HISTORICAL METHODS

a: *SEQUENCE SEARCH (SS)*

SS is an intuitive and naive algorithm that can be used to
select pulses without prior information (when all functions
are used) or with prior information (with the FindPulses
function) [30]. It is presented in Algorithm 2. It is based on
pulse pairing: two pulses i and j are randomly chosen, under
the assumption that they have been emitted by the same radar
(EvaluationPRI function). This allows one to estimate the PRI
of the waveform: pri is instantiated as pri = t_j - t_i, and the
temporal phase of the waveform as τ = t_i mod pri (then
τ ∈ [0, pri]). For each pulse k, we verify whether it matches
the waveform, that is, t_k mod pri ≃ τ under a tolerance ε
(FindPulses function). If there are enough pulses selected to
fit a waveform, then the PRI is validated, otherwise a new pair
of pulses is chosen (Selection function).

**Algorithm 2 Sequence Search (SS)**

EstimatePRI

**Input:** X

**Outputs:** *i*, *pri*

*i* ← random pulse index

*j* ← random pulse index close to *i*

*pri* ← *t*<sub>*j*</sub> − *t*<sub>*i*</sub>

FindPulses

**Inputs:** (X, i, pri, ε)

**Output:** *I*

$\tau \leftarrow t_i \bmod pri$

$I \leftarrow \{k, |(t_k \bmod pri) - \tau| \le \varepsilon\}$

Selection

**Inputs:** (X, ε, N<sub>min</sub>)

**Output:** *I*

$I \leftarrow \emptyset$

while $|I| \le N_{\min}$ do

*i*, *pri* ← EstimatePRI(*X*)

*I* ← FindPulses(*X*, *i*, *pri*, *ε*)

b: DIFFERENTIAL TIME OF ARRIVAL HISTOGRAM (DTOAH)

This (very conventional) method exploits the time intervals
between pulses by computing a Differential Time of Arrival
(DTOA).DTOA of degree/level (or delay) *l* or *l*th order
DTOA is defined as: *dtoa*<sub>*i*</sub><sup>(*l*)</sup> = *t*<sub>*i*+*l*</sub> − *t*<sub>*i*</sub>. TheDTOA
Histogram method (DTOAH) is based on computation of
DTOA histograms for several degrees (*l* ∈ [1, *L*])DTOA
histograms. It makes it possible to highlight the waveform
PRI. In fact, if two pulses *x*<sub>*i*</sub> and *x*<sub>*i*+*l*</sub> belong to the same
waveform, then theDTOA is equal to the PRI (or a multiple
of it, called harmonics). If they do not belong to the same
waveform, then the DTUA does not have a particular value, as sampled from a uniform-like random distribution. As a consequence, the DTUAH exhibits peaks corresponding to the PRI of the observed waveforms. In fact, estimating DTUA with different degrees circumvent the pulse overlapping problem.

The main process of this method is the selection of waveforms. A threshold function is computed to select peaks in the histogram. When a DTUA peak is selected, the entire set of pulses corresponding to it is extracted to form a predicted waveform, as in Algo 3.

<caption>**Algorithm 3** Differential TOA Histogram (DTUAH)</caption>

Selection
Inputs: X, L
Output: l
    Δ ← 0
    for each pulse xᵢ ∈ X
        for l ∈ [1, L]
            dtoa ← tᵢ₊₁ - tᵢ
            Δ[i, l] ← dtoa
        H ← Histogram(Δ)
        σ ← ThresholdForPeak(H)
        if max(H) ≥ σ
            pri ← arg max dtoa (H)
            I ← set of indices contributing to H(pri) peak.

There are several improvements of this method, differing in the histogram computation and the threshold function. A notable variant, Cumulative Difference Histogram (CDIF) [30], builds a cumulated histogram of several degrees of DTUA. The Sequential Difference Histogram (SDIF) version [31] does not accumulate DTUA degrees, but rather verifies the unique-degree histograms sequentially. DTUAH methods are still studied today. Ge et al. [32] compute boxplots of DTUA distributions for bins above the threshold on a smoothed histogram. It allows one to identify PRI modulation types.

The DTUAH only exploit the time parameter. The use of histograms makes DTUAH well suited to fixed PRI waveforms, but not to jittered PRI, because it flattens PRI peaks on the histogram. There are also issues of harmonics, in particular in case of a high rate of missing pulses: PRI harmonics might be more frequent than the PRI.

### c: TRANSFORMS

Nishiguchi and Kobayashi [33] introduce the PRI Transform, which replaces the histogram by the TOA autocorrelation, to reduce harmonic issues in DTUAH.

Li et al. [34] propose a temporal deinterleaving method based on the Hough transform. The time axis is represented as a large binary vector, with ones placed at the location of TOA. The idea is to divide the vector into constant periods *P* and to form a matrix by stacking segments. Within this representation, pixels from the pulses of a fixed-PRI

waveform of PRI ≤ *P* draw a line, the parameters of which depending on PRI and the phase *τ* with respect to *P* (*P* = *τ* mod PRI). The Hough transform identifies such lines; one can extract the corresponding pulses and rebuild the waveform. With this method, a stagger should be seen as multiple lines of the same slope, and a jitter as a blurred line.

## 2) RECENT ENHANCEMENTS

Recently, Noone and Howard [35] propose to enhance SS. They use a RNN with 4 layers of 2 neurons each, to predict the next DTUA of a jitter waveform, the idea being to improve the central PRI estimate.

Guo-Liang et al. [36] propose an improvement of SS. They add parameters such as PW and RF to select only one waveform when several pulses are encountered within a tolerance window. This selection is based on a pulse similarity metric *d*, which is calculated based on a weighted Euclidean distance between normalized parameters as described in Eq. (4), where $N_w$ and $N_f$ are normalization functions for PW and RF.

$$d_{ij} = \sqrt{\delta_{ij}^T W \delta_{ij}} = \left( \frac{N_w(w_i) - N_w(w_j)}{N_f(f_i) - N_f(f_j)} \right) \quad (4)$$

Ge et al. [32] preprocess the cumulative DTUAH with a smoothing function to analyze the DTUA distribution derived from pulses contributing to the DTUAH spikes. In fact, the PRI distribution of a jitter is more spread out than that of a fixed PRI waveform. In that respect, the DTUA distribution analysis enables one to separate waveforms centered on the same PRI, in particular if there is a jittered PRI waveform and a fixed PRI waveform.

Xie et al. [37] propose a graph-based method for the extraction of a jittered PRI waveform under the assumption of known PRI mean value and jitter rate, that is, under assumptions $A_1$ and $A_2$. While the graph vertices convey pulses, edges exist when the DTUA between two pulses is within the expected range. They solve the longest-path problem, making the method robust to spurious pulses. It is also robust to pulse loss since the expected DTUA range includes multiples of the PRI.

Scholl and Brüggenwirth [10] focus their solution on the incremental RSD. In other words, in order to detect and track the emitters in real time, they adapt some methods (including DBSCAN and Fuzzy Adaptive Resonance Theory network) to immediately classify a pulse when detected (without a long monitoring time). The real-time aspect (see Section IV-E) needs to address the cluster instancing problem (for a new encountered emitter) as well as the discarding of spurious pulses. Their results are close to best non incremental method (DBSCAN) for clustering based on RF and PW or DOA and PW. Moreover, the proposed methods are shown to be robust to false alarms.

### B. CLUSTERING-BASED

In this approach we have to distinguish between the one-step and two-step approaches, The clustering technique does not
require $A_1$ nor $A_2$ assumptions. However, the assumption $A_3$ (no SEI) is made by the vast majority, because it is the main case of failure of such methods: if multiple emitters share their parameters, their pulses will be gathered within the same cluster.

## 1) ONE-STEP

Digne et al. [38] propose an RSD robust to SEI. It is an original pre-processing before cluster analysis. The Instantaneous Frequency (IF) signal inside each pulse is analyzed and described by a Bézier curve with a previously fixed number of Bézier points. The points take place in the time-IF plane (see Figure 5), and are then used as features for a DBSCAN clustering. The only assumptions are the availability of intrapulse raw signal, and the matching between the intrapulse signature and its emitter.

**Algorithm 4** Digne et al. [38]

Deinterleaving

**Inputs:** X, d

**Output:** DBSCAN output

**for** each pulse $x_i$ **do**

Consider its IF signal $f_i \in \mathbb{R}^{2 \times n}$

Estimate the Bézier points $b_i \in \mathbb{R}^{2 \times d}$ best fitting $f_i$

Apply DBSCAN in the space of Bézier points **B**

Dudczyk and Kawalec [39] use clustering on the entire PDW for SEI ($A_3$), in particular the PA. In this paper, they propose to analyze six copies of the same radar with jitter and stagger. They ground the discrimination on possible PRI distortion, as shown in a single example. Unfortunately, they do not provide quantitative results.

Su et al. [40] cluster pulses in the RF/PW/DOA space with K-Means and Self-Organizing Feature Map (SOFM), which are unsupervised ML techniques, under the assumption of variable cluster number. The idea is to determine the number of clusters through the K-mean algorithm. This step has to be performed with care, in particular, to be unsensitive to the center initialization and robust to outliers. Then, a SOFM (Kohonen) is designed with the neuron number given by the K-mean cluster number. Some additional adjustments can be performed during the final clustering performed by the SOFM

Mahmod [41] shows that DBSCAN clustering has better results than F-ART clustering on the same input PDW. F-ART, although not used much today, is interesting because it can work online, i.e. with flows of PDW that is an identified requirement in the RSD case (see Section IV-E).

Martin et al. [42] also propose a DBSCAN-based deinterleaving based on PA, PW, and three MOP frequency parameters. The principle of their method is to separately tune the $\epsilon$ parameter of DBSCAN for each parameter.

Dadgarnia and Sadeghi [43] introduce an unusual RSD method, based on a “temporal” clustering called modified DBSCAN (MDBSCAN), in the TOA-DTOA plane. The

principle is recursive extraction by PRI estimation, as the first class of methods, but it is clustering-based and adapted to jitter, unlike DTOAH. In fact, it comes with a variable jitter tolerance that increases with iteration to extract more and more variable PRIs. It leads to fixed PRI waveform extraction first and jittered PRI waveforms later. For the selection step, the pulses are placed in a TOA-DTOA plane, DTOA being the TOA difference with the adjacent next pulse, as detailed in Algorithm 5. MDBSCAN goes together with a neighbor zone definition trying to link pulses which DTOA is the same, modulo the jitter tolerance. In other words, it allows for the finding of sets of adjacent pulses with the same DTOA, which is why the assumption $A_4$ is made (a part of the waveform must be observed alone for a duration at least equal to a few PRI). This occurs for fixed PRI and jittered PRI waveforms that are isolated on a time window, even if short, because once a PRI is identified, one can find other pulses with SS. The largest group is selected. If only the first-order DTOA is considered, then the method is equivalent to the search of the longest constant DTOA sequence among all of the pulses.

**Algorithm 5** Dadgarnia and Sadeghi [43]

Selection

**Inputs:** X, jit

**Output:** I

**for** each pulse $x_i$

$$ dtoa \leftarrow t_{i+1} - t_i $$

$$ x_i \leftarrow (t_i, dtoa) $$

$$ l \leftarrow 2 \cdot jit \cdot dtoa $$

$$ \text{leftZone}_i \leftarrow (t_i - dtoa_i, dtoa_i)\text{-centered } l\text{-square} $$

$$ \text{rightZone}_i \leftarrow (t_i + dtoa_i, dtoa_i)\text{-centered } l\text{-square} $$

Apply DBSCAN on X with the defined neighborhoods.

$$ I \leftarrow \text{pulses the most populated cluster} $$

$$ I \leftarrow I \cup \{\text{pulses extracted by SS from } I\} $$

Unlike PRI estimation methods, one-step clustering methods are not based on TOA but on RF, PW and/or DOA. Their limitation is that they do not usually account for the possibility of parameter overlap. $A_3$ is a key requirement for this class of RSD. There is the exception to intrapulse clustering, which is theoretically a better method, but much more difficult to implement because intrapulse is not always measured.

## 2) TWO-STEP

Ata'a and Abdullah [44] start by a F-ART clustering in the DOA-RF plane. Then they use temporal RSD techniques including DTOAH to separate waveforms possibly coexisting inside each cluster, unlike Aldossary [7], who used DBSCAN clustering and DTOAH jointly for RSD.

Mu et al. [45] first cluster pulses and second merge some of them. The first step is to cluster pulses that have consistent RF and PRI. The consistency in PRI is based on rules rather than clustering. The second step then involves considering
each pair of clusters to decide whether they are from the same source. This second-step decision is made with an SVM binary classifier, which is a supervised learning algorithm.

Guo et al. [46] propose a segmentation algorithm of a 3-D space (PW, RF, DOA) based on the modeling of similarity (Gaussian transform of the Euclidean distance of each pulse) as a graph, the pulses being the vertices of this graph. Their similarity matrix defines the adjacency matrix from which the degree matrix and the Laplacian matrix are derived. They propose finding subgraphs, i.e. clusters, by detecting the “abrupt” changes of eigenvalues of the normalized Laplacian matrix. Then, the number of clusters, that is, emitters, is adaptive.

Mottier et al. [25], [47], use the same idea of clustering pulses and then possibly merging some clusters. The first step is an Hierarchical DBSCAN clustering in the RF-PW plane. With the assumption that the RF-PW clusters are homogeneous, the next step is an unsupervised merging of some of them to solve agility issues (see Figure 7). The chosen algorithm is TOA-based Agglomerative Hierarchical Clustering (AHC), which is unsupervised. A distance is needed between clusters. Each cluster is described by the set of its pulse TOAs. The distance between two clusters is then defined by the Optimal Transport Distance (OTD) between their respective set of TOAs. The motivation is that the clusters that overlap the most are likely to be from the same emitter (under $A_4$).

**Algorithm 6** Mottier et al. [25]

Step 1: pulse separation

**Input:** $\mathbf{X}$

**Output:** $C_1$

Normalize RF and PW

$C_1 \leftarrow \text{DBSCAN}(\mathbf{X})$ in the RF/PW plane

Step 2: cluster aggregation

**Input:** $C_1$

**Output:** $C_2$

$\mathbf{D} \leftarrow 0$

for clusters $c_i$ and $c_j \in C_1$ do

$\mathbf{D}_{i,j} \leftarrow \text{OTD}(\{t_k, k \in c_i\}, \{t_k, k \in c_j\})$

$C_2 \leftarrow \text{AHC}(C_1)$ based on the distance matrix $\mathbf{D}$

In a previous work [48], we analyzed the latter method and proposed an alternative to the distance definition of the merging step. Optimal transport distance can indeed have long computation time. In order to shorten the computational time, we proposed a simple Jaccard distance between time supports. This intersection over the union distance comes down to measuring the overlay of radar lobes constituted by first-step clusters. Hyperparameters are optimized by means of Bayesian optimization.

In summary, two-step clustering methods are more sophisticated, as parameter agility is often taken into account through the second step. They often exploit both time

and non-time parameters. However, like one-step clustering methods and for the same reason, they often have limitations with handling the overlapping of PWD parameters that leads to connected clusters.

### C. RNN-BASED

Recurrent neural networks (RNN) are architectures devoted to sequence processing. Vanilla RNN being limited in practice regarding the sequence length, Long Short Term Memory (LSTM) and Gated Recursive Unit (GRU) have been then designed to improve the RNN performances. Most of the RNN-based networks introduced for RSD use these two approaches and are bidirectional.

Liu and Philip [49] propose a parallel approach, where each known waveform is represented by a specific denoising GRU network. The GRU models aim to denoise a sequence of their waveform by pointing towards the next correct pulse at each step. When the waveform of interest is alone, it should just point to all of the pulses successively. When there are spurious pulses, it should be able to ignore such pulses. In this way, each GRU can be seen as a denoiser. They are tested in parallel for temporal deinterleaving.

Guo et al. [50] propose a method for RSD with an LSTM, exploiting the shape of the radar lobes (see Figure 4). In fact, they propose to separate radars having exactly the same features ($A_3$) on the basis of TOA and PA (the other primary features being not informative). The main challenge is to identify time segments with overlapping pulses (of different radars) and the segment with only one radar. For segments with overlapping pulse, the distance to cluster barycenters criterion is computed to separate the sources. This two steps segmentation is performed through the PA using time neighborhood (TOA). Once the lobes perfectly separated ($A_4$), the LSTM tracks the emitters. Their simulations show the efficiency for two copies of a same radar.

Zhu et al. [51] propose an algorithm based on a Bidirectional LSTM (BLSTM) architecture withDTOA as input features (no other PDW parameters). Although the number of interleaved radars is weak ($K = 3$ as exact number or upper bond), they propose to solve the RSD problem on advanced and difficult WPPs (see section II-B2) for tracking future cognitive radars. In particular, they assume random parameters for PRIs of the jitter case and briefly address the problem of SEI $A_3$.

In [24] and [52] Chao et al. propose a semantic segmentation based on Dilated Convolutional Network withDTOA as input (with a comparison with bidirectional LSTM and GRU). Semantic segmentation is mainly based on two classification steps. The first is based on the MOP (fixed, staggered, etc.), and the second step on the PRI (emitters) characteristics. They propose two strategies; the first is a parallel strategy involving classifying all of the possible MOPs and second, the PRI for each subclass (of modulation). The other strategy is a serial scheme for which, after each PRI modulation classification, the corresponding PRI is classified
(all possible RSD strategies are detailed in Section VI-C). They also propose a scheme with smaller NNs devoted to specific subtasks. Their method does not directly address RSD, but it solves deinterleaving in some cases. They improved their method in [53] by adding parameters such as PW, RF, and PA to improve the accuracy of classification, particularly when multiple target radars have the same PRI modulation modes and the PRI value ranges overlap.

Wang et al. [54] use a BGRU for pulse classification based onDTOA and PA. They model the scanning as a sinc function ($A_4$) and include pulse loss and noise in the simulated TOA and PA measurements. They tested the BGRU on overlapping radar scanning (emission). The BGRU exhibits loss of accuracy between the optimal case and the pulse loss and noisy case. Similarly, Tang et al. [55] propose a BGRU with TDOA, PW and RF as input features. They face the problem of normalizing and encoding these multi-modal inputs (see VI-B1 for a discussion on input data format).

**Algorithm 7** Segmentation for Deinterleaving

SupervisedLearning

**Input:** $\mathcal{D}$

**Output:** the NN weights are updated

while training do

Take a scene $(X, Y) \in \mathcal{D}$

$E_X, E_Y \leftarrow \text{Encoding}(X, Y)$

$E_{\hat{Y}} \leftarrow \text{NN}(E_X)$

Compute loss between $E_{\hat{Y}}$ and $E_Y$

Update weights of NN

Inference

**Input:** $X$

**Output:** $\hat{Y}$

$E_X, E_Y \leftarrow \text{Encoding}(X, Y)$

$E_{\hat{Y}} \leftarrow \text{NN}(E_X)$

Retrieve $\hat{Y}$ from $E_{\hat{Y}}$

Recently, Zhou et al. [56] design a network for the multiscale extraction of pulses. Multiscale attention allows the network to gather several pulses from the same emitter, without estimating the PRI. Then, this approach is robust to jitter and can simultaneously process different emitters. This first part of the network is made up of five Attention Down-Sampling Blocks (for the multiscale analysis) and the second part of four Feature Fusion Up-Sampling Blocks to retrieve the signal time sampling. The down-sampling/up-sampling blocks have to be linked for the attention mechanism.

RNN-based methods have the classic advantages and drawbacks of neural network methods. They are adapted to complex waveforms because they can learn the underlying patterns from the training data. However, they are not interpretable and should be considered as black-box systems (Section IV-D). More specifically to RSD, the above-mentioned RNN-based segmentation methods require strong

assumptions $A_1$ (known $K$) and $A_2$ (known waveforms), because the network architecture requires a fixed output size.

## D. CNN-BASED

In the last decade, CNN led to significant breakthroughs in the computer vision domain. Then, it is logical to apply them to RSD. The starting point is to reshape the pulse flow in order to have 2-D inputs, either by using several parameters or simply by rearranging the TOA as seen in the next subsection.

### 1) TOA ONLY

Nuhoglu et al. [57] propose to segment time-PRI images with the following two metaclasses: fixed PRI waveform versus jitter or stagger PRI waveform. The preprocessing consists of computing time-PRI images by the means of PRI Transform, which is a traditional RSD method. There are two additional variants of PRI Transform, yielding two images, one fostering fixed PRI detection, and the other jitter/stagger PRI detection. Then, the point is to segment such images with a U-Net-based network. Pixels are segmented into three classes: nothing, fixed PRI, jitter/stagger PRI. A very close approach is proposed by Liu et al. [58] to segment aDTOA image (i.e., all the TOA differences between the detected pulses). They aim to retrieve different PRI modulations (fixed, jitter, nothing and others) with an encoder-decoder network, i.e. an improved U-Net.

Xiang et al. [59] propose a recursive RSD method to extract waveforms one after the other. Their architecture is an encoder-decoder, with convolutive layers and attention, which performs binary segmentation on pulses. At each step, pulses predicted on the positive class are extracted, and form a predicted emitter. The core of the network exploits TOA, but other parameters are used in a post-processing step.

Mao et al. [60] propose to segment aDTOA image. Their encoding is detailed in Algo 8. After initialization to zero, the encoding is filled withDTOA up to the order 10, which makes an upper triangular matrix. Then, the values of the matrix are replaced by their probability of occurrence $P$, and the Min-Max normalized to make pixels of values 0 to 255. They propose Sep-RefineNet, a neural network for segmentation, with convolutive layers and residual connections.

### 2) MULTIPARAMETER

Gasperini et al. [12] try to segment images, under a weak $A_1$ assumption, the number of waveforms is upper-limited to 15, but these waveforms are *a priori* unknown (without $A_2$). Their method being able to deinterleave 2 to 15 waveforms. They introduce a new scene encoding as well as new loss functions derived from the confusion matrix. Their encoding is an image of fixed size, here 512 × 512: the pulse image $E_X$ has two channels, while the label image $E_Y$ has only one. The image axes are time and frequency. The pixel values take the other values of the PDW parameters, such as PW and PA, which make the image a several-channel image. Details are given in Algorithm 9. They use U-Net to segment pulses of
**Algorithm 8 Mao et al. [60]**

Encoding

Inputs: X, Y

Outputs: E<sub>X</sub>, E<sub>Y</sub>

$E_X \leftarrow 0$

$E_Y \leftarrow 0$

for each pulse $x_i \in X$ do

for each pulse $x_j \in X$ s.t. $j \in [i+1, i+10]$ do

$E_X[i, j] \leftarrow t_j - t_i$

if $y_i = y_j$ then

$E_Y[i, j] \leftarrow y_i$

for i, j do

$E_X[i, j] \leftarrow P(E_X[i, j])$

$E_X[i, j] \leftarrow \lceil 255 \times \frac{E_X[i, j] - \min E_X}{\max E_X - \min E_X} \rceil$

the pulse encoding. Of course, U-Net is trained by supervised learning, and the predicted pulse labels $\hat{Y}$ can be directly retrieved from the label encoding $E_{\hat{Y}}$.

**Algorithm 9 Gasperini et al. [12]**

Encoding

Inputs: X, Y

Outputs: E<sub>X</sub>, E<sub>Y</sub>

$E_X(512 \times 512 \times 2) \leftarrow 0$

$E_Y(512 \times 512) \leftarrow 0$

for each pulse $x_i$ s.t. $t_i \le 512 \cdot 5\mu s$ do

$E_X[t_i, N_f(f_i)] \leftarrow (N_w(w_i), N_a(a_i))$

$E_Y[t_i, N_f(f_i)] \leftarrow y_i$

Kang et al. [61] also propose a method for pulse classification, by segmentation with U-Net. Its uniqueness lies in its preprocessing. The input data are not PDW but a time-frequency map with fine granularity, i.e., one pulse without RF modulation, is represented by several connected points on the map. In fact, the pre-processing consists in flattening the map, keeping only one value per column. The resulting frequency vector feeds the U-Net.

Cain et al. [62] propose a method for waveform identification. The encoding is interesting, even if it is not RSD. The problem is a classification problem, where each input instance is a set of pulses from the same emitter. The encoding of an instance is the concatenation of three filters of size 100 × 100, with the x axis being the time axis. The y-axis of each filter corresponds to a parameter: RF, PW,DTOA. Their architecture is a CNN inspired by U-Net.

CNN-based methods share the same advantages and drawbacks as RNN-based methods: they are able to capture complex information, but mostly require strong assumptions ($A_1, A_2$) and are not interpretable.

## E. HYBRID METHODS

In this last section, we have gathered all the methods that are not fully within the scope of the previously exposed taxa.

In particular, other NN architectures that are neither RNN nor CNN and some methods are not fully time domain of clustering approaches.

### 1) NEURAL NETWORKS

Willson et al. [63] compare several algorithms for pulse classification, i.e. with known emitters. They conclude that Multi Layer Perceptron has better results than the KNN and Template Matching.

Li et al. propose an Autoencoder (AE) approach with TOA for RSD with knowledge of the possible present waveforms. First, for each known waveform, they train a specific MLP to be an AE, i.e. to be able to reconstruct a sequence [64]. The sequence is a binary representation of the presence of pulses along the time axis. It is self-supervised learning, as the network learns to output the input as accurately as possible (sequence-to-sequence), there is no need to know the denoised signal. AE acts as an information compressor. In the related article [65], they use the trained AE for RSD, still under the assumption of potential radar emitters, which have their corresponding AE. The principle is to extract pulses each time an AE recognizes a waveform inside the observed scene.

Lin et al. [66] try to adapt voice separation algorithms to radar deinterleaving. Their impractical assumption is that there are exactly two active emitters of the same power. The idea is to train an MLP on time-frequency representations of the raw EM signal to output to indicate the weight of each emitter on each part of the signal.

Can et al. [67] propose one of the scarce Transformer-based method for RSD. Their approach is a bit far-fetched; the initial idea is to come down to an optimization problem from graph theory called a minimum-cost flow problem. If one has access to a cost matrix, then one can deinterleave by solving the minimum-cost flow problem, which can be done by linear programming. After rephrasing the problem and its constraints, the authors come to a problem statement and suggest to use Transformer-type encoders, as a supervised learning architecture. The input of the network is a first-orderDTOA sequence. The output is a matrix representing flows in graphs, from which RSD is trivial.

Lang et al. [68] propose a method for pulse classification based on Graph Neural Network (GNN). The steps are the following:

1) Min-max normalization of features

2) Initialization of the graph: vertices are pulses, binary edges are present for the nearest neighbors

3) Residual Graph Conv Net with a final softmax layer, to predict the label of each vertex

### 2) OTHERS

Derived from the blind source separation problem, Keshavarzi et al. [69] propose a geometric approach for RSD. First, they select instants for which they estimate that there is only one emitter, which gives the column of the mixing matrix
(then PA is the only radar parameter used). A fuzzy C-mean algorithm denoises the observation, the cluster center being the column of the mixing matrix that is inverted to separate over all the observation. Obviously, the emitter number sets the dimension matrix and is assumed to be known.

Yang et al. [70] propose the to deinterleave by PA envelop (i.e. radar lobes, $A_4$) tracking, with interpolations or Kalman filter. The interpolation, based on Hermite polynomials, aims to separate two emitters having identical DOAs (with different PA levels, blue and red in figure 4) while Kalman filter tracks the PA of two crossing radar lobes, that is, having closed DOA (orange and blue in figure 4). Then a pre-screening is necessary and RF is used as an additional parameter to separate the radar signals.

Liu et al. [71] propose an online pulse deinterleaving method with finite automata, under $A_2$ and $A_3$. Each known waveform is associated with a unique finite automaton, the size of which is its pattern length. Vertices are theoretical pulses, corresponding to a position in the waveform pattern, and are characterized with RF and PW. The edges between adjacent pulses are the PRI. To take into account the possible missing pulses, there are also edges between nonadjacent pulses that are sums of corresponding PRI values. At first, all the automata are initialized. Then, for each received pulse, every automaton is checked, and the matching automaton is updated with a new current state. The pulses are processed sequentially. This method performs waveform identification at the same time as deinterleaving but is unable to process unknown waveforms.

The next two methods use only TOA et PA and try to exploit the structure of the radar lobe (the $A_4$). We can see them as parabola separation methods. Dutt et al. [72] propose using the wavelet transform, which they use directly on input data as a preprocessing. Then, they define three steps to iteratively process the transform output. Enderli et al. [73] propose using Expectation Maximization (EM). The EM algorithm requires making a hypothesis about the number of parabola *K*, but it is possible to overcome this limit by trying multiple hypotheses on *K*. With this method, the issue is that each pulse must be associated with a parabola, which leads to weak performances in the case of a high outlier rate. The two methods have the same limitation; they are not robust to the presence of pulses not belonging to a lobe. However, real observations are often composed of multiple noisy lobes and spurious pulses, which can contribute to lack of performance (as depicted in Figure 8).

Ford et al. [74] propose a method based on Markov process estimation. The idea is to model emitters by Markov chains of symbols, each symbol representing an RF-PW cluster. The observed scene, i.e. the sequence of pulses from multiple emitters, is modeled with a Markov switch. Sequentially, at each pulse, a possible emitter switch is performed. The first step is DBSCAN clustering in RF-PW, each cluster is assigned a symbol. Then, each PDW sequence is replaced by its symbol, i.e. the primary parameters are set aside including the TOA. The problem comes down to a symbol partition. It is

done with a local search, evaluating the relevance of multiple partitions, with penalized maximum likelihood entropy as a loss.

Zhao et al. [75] propose the uncommon approach of multi-station RSD, the assumption being to have multiple receivers. The idea is to use TDOA, which is a well-known localization method. A close idea is given in [76], in which pulses are also recorded by several stations (nodes). The interest of this approach is that TOAs are much richer from an information point of view than the methods presented above. In fact, it contains the range information between the emitter and all the stations that can be merged with the DOA to accurately locate the emitters. This mapping, along with pulse features such as RF and PW, highly facilitates pulse clustering (in particular $A_3$, SEI can be released).

Yang et al. [23] use the contour lines of the radar ambiguity function to extract the pulse features, after minimizing a possible Doppler frequency. The similarity of the contour lines is learned through a Contrastive Self-Supervised Deep Attention Network. This network is made up of several dual branches. These dual branches allow contrastive learning by backpropagating the similarity from the same pulse occurrence but also the dissimilarity for two different pulses. Once the pulse features are extracted, an adaptive clustering performs the RSD. This algorithm is adaptive since the number of clusters is estimated by optimizing some performance metrics (ARI, NMI, see Section III-C).

Walenczykowka and Kawalec in [77] propose a pulse classification based on intrapulse modulation (chips, stepped frequencies, phase modulation, etc.). They first perform a time-frequency analysis through the generalized Morse wavelet. Then, statistical features of the wavelet coefficients, such as higher order statistics, are extracted, and a dimension reduction is performed before the classification through a NN. In a second paper [78], they directly apply a CNN to time-frequency analysis, to mainly distinguish between frequency-modulated and phase-modulated waveforms. In this paper, they also test other time-frequency tools, such as the Short Time Fourier Transform or the Wigner-Ville distribution.

## F. SUMMARY

In this last section, we briefly sum up the limitation and robustness of the four classes according to some points relevant in operational cases.

* Missing and Spurious peaks: Obviously SS/DTOA approaches are the most disturbed by missing/spurious pulses since the TOA/DTOA are the background of these approaches. Moreover, when an error occurs, it is propagated afterwards. Unfortunately, robustness measures are barely given in the aforementioned papers. Clustering-based, RNN/CNN-based are more robust since they mainly use other PDW parameters.

* Jitter and stagger waveforms: The same remarks as the previous point hold for the robustness with regards to
jitter. On the other hand, SS/DTOA are insensitive to
the stagger, unlike others methods. RNN-based methods
could be more impacted when including PW or RF,
MOP in their input data as well as the CNN methods
that are not only based on TOA, Clustering methods
are theoretically the less robust to stagger in particular
when parameters of different waveforms overlap, the
clusters being then connected and the emitter possibly
confused. Unfortunately, the agility of future cognitive
radars could weaken the robustness of these methods.

*   **Real time:** SS/DTOA algorithms verify the real time requirement for the simplest methods but can be questionable for the most complicated ones. RNN/CNN-based methods can also achieve real time inferences, once the NN trained. Again, this has to be inspected again for the more complex NN architectures (along with the memory limitations). On the other hand, some Clustering-based methods with their recursive procedures (with two steps in some cases) seem to be the class the less in accordance to this requirement.

* Explainability: As previously stated, in the defense
domain, results provided by an algorithm have to be
understandable for ethic reasons. Obviously SS/DTOA
approaches completely verify this requirement as well
as well clustering-based methods. RNN/CNN-based
methods do not verify (and by far), but the development
of eXplainable Artificial Intelligence (XAI) could bring
key improvements on this requirement.

*   Known number of emitters and known waveforms ($\mathcal{A}_1$ and $\mathcal{A}_2$): SS/DTOA methods do not have theoretical limitations over these two assumptions. Only limits due to their complexity and the memory limitations could lead to fix an upper bound of jointly processed waveforms. They do not require a priori knowledge of the waveforms. Clustering-based methods generally do not assume the number of clusters/emitters. Until recently, RNN/CNN-based methods assumed the number of clusters/emitters (since it was the size of the output layer). More recent papers proposed to try several numbers of classes (always limited by an upper bound), the choice (between the possible numbers) of the final solution is not straightforward. The main difference between the clustering-based and the RNN/CNN-based methods is that the waveforms are generally known in the DL approaches while clustering based are unsupervised methods. In this case also, XAI, in particular counterfactual explanations, could lead to improvements for detecting not learned waveforms.

* SEI and rotating scanning ($\mathcal{A}_3$ and $\mathcal{A}_4$): In order to discriminate two emitters having the same waveform features, the DOA (angle) and/or PA (distance through electromagnetic attenuation) are the necessary parameters. Thus, SS/DTOA methods (that do not use these parameter) and RNN/CCN that cannot learn the operational (attenuation) conditions are not fitted

for this task. Then, clustering is the fitted approach.
Moreover, the fitting to a parabola can remove wrong
classifications. Others possible solutions are to detect
possible PRI defaults of each emitter (when existing)
and to use several receptors and merge the received
signal for distinguishing the similar emitters.

VI. OTHER FRAMES OF CLASSIFICATION

In the previous section, we organized our presentation
according to a methodological criterion (PRI estimation-
based, clustering, RNN-based, and CNN-based methods).
However, each method can also be characterized by a larger
set of criteria, such as the features used for RSD, for instance.
In this section, we discuss the methods according to these
other criteria, their purpose, advantages and limitations, and
challenges. The main identified criteria are the parameters,
input data, type of problem solved, principle, and underlying
technique.

A. PRIMARY PARAMETERS AS INPUT FEATURES

We first discuss the use of primary parameters, focusing
on the less common parameters DOA, PA, and MOP.
In fact, as seen above, choosing the parameters on which
the clustering/mapping is performed is not straightforward in
PD. Algorithms must find a trade-off between a high number
of PDW components, thus increasing the computational time
but also introducing more noisy parameters such as PA and a
too low number of components, making the RSD somewhat
difficult.

1) TOA AND DTOA

TOA has no meaning in itself, the information is in the
differences with other TOA, that is, DTOA. Usual methods
heavily rely on DTOA by trying to estimate a waveform
period (i.e. PRI). However, NN-based methods replace TOA
in the PDW by the first-order DTOA, and use it as a feature
inside the PDW. This is because the standard variation of
the DTOA feature is less than the TOA feature, so it eases
the training step. Note that Ford et al. [74] encode temporal
information of pulses without TOA nor DTOA, they only
keep the chronological order of pulses for Markov process
modeling.

2) DOA

When available, the DOA provides valuable information.
It is an important parameter since it cannot be agile, i.e. the
DOA from an emitter stays around the same value through
time in a small time window. An obvious processing for
DOA is cluster analysis [40], [44]. If there is a sufficiently
wide gap in the DOA values, it can lead to an effortless
separation into multiple subproblems. In this way, one divides
the initial deinterleaving problems into independent problems
with lower density. If remaining DOA values are too close,
then the parameter might not be more useful and DOA
exploitation ends here.
### 3) PA

PA is barely used in RSD problem. Martin et al. [42], [79] do not assume that radar emitters have rotational mechanical scanning but instead that the pulses from one emitter are received at the same amplitude, leading them to use simple clustering rules. Mu et al. [45] use PA to overcome RF agility, using SVM to merge pulse clusters. NN approaches can more easily incorporate all pulse parameters as features, because they rely on the capacity of the network to find the underlying patterns [12], [53], [54].

To our knowledge, only two approaches are highly dependent on PA and are fitted to radar main lobes: using the wavelet transform [72] and using the Expectation-Maximization algorithm [73], which are approaches to separation of the main lobe.

### 4) RF AND PW

RF and PW are used in most methods that are not based only on time. They are useful as they are the only primary depending only on the emitted waveform design. However, the possible RF and PW agility makes a clustering based on these parameters fairly unreliable, so they are not sufficient to perform RSD.

### 5) MOP

Information about intrapulse modulation is unusual but is of interest when available. To obtain MOP, the EM signal must be transformed into the IF signal, then digitized with an adapted sampling rate that requires additional hardware devices [43]. In the literature about deinterleaving, only frequency modulation on pulse (FMOP) is considered. FMOP can be intentional, for example, a chirp pulse, or unintentional (see Blunt et al. [80] for an overview of radar waveform diversity, including MOP).

Martin et al. [79] characterize intentional FMOP with three frequencies instead of one, that is, the instantaneous frequency at the beginning, center, and end of the pulse. Unintentional FMOP refers to an EM signature due to a technical defect in the radar emitters. The signature can be unique for each radar instance [81], making it efficient to circumvent the SEI problem. Digne et al. [38] characterize the pulse IF signal by Bézier curves, that is, with only a few points in the TOA-IF plane, then cluster pulses with Bézier points as features. Theoretically, if unintentional MOP description is accurate enough, then other primary parameters, i.e. TOA, DOA, PA, RF and PW, could be unnecessary.

## B. CONSIDERATIONS ABOUT INPUT DATA

In this paragraph we comment on input formatting, input length, and preprocessing.

### 1) FORMAT

It is interesting to analyze the input formats, i.e. the way pulses are described and passed to the algorithms. Three main

formats are different expressions of the relationship between pulses.

#### a: ORDERED PDW

The natural way of formatting input data is to simply process concatenated PDW vectors of one or several pulses, the only relationship between pulses being their chronological order. This format is used by historical, clustering-based, and RNN-based deinterleaving methods. Note that sometimes, particularly for RNN-based methods, a transformation of the TOA parameter into first-order DTOA is performed to reduce the standard variation of the time parameter. In this case, DTOA is an explicit connection towards the adjacent pulse.

#### b: IMPLICIT CONNECTION

Another format is to encode pulses within a new space represented by a tensor. Each dimension of the tensor is related to a physical quantity, that is, often time and/or frequency. A pulse, instead of filling a row, as in the ordered PDW format, is located inside a tensor entry (i.e. a pixel in the 2-D case). This form is exclusively adapted to image processing techniques, i.e. CNN-based methods. A simple case is [65], which sets pulses as 1 within a zero-initialized time vector. Another interesting case is [12], which encodes pulses inside a tensor of shape 255 × 255 × 2: time and frequency are the first two dimensions; then the third dimension is used to encode PW and PA, as pixel values. The main drawback is the computing time, due to the sparse nature of such encoding.

#### c: EXPLICIT RELATIONSHIP

Another form is to encode the relationship between pulses in a graph-like structure. The aim is to highlight the temporal relations between pulses through a DTOA matrix [58], [60], which encodes the duration between two TOA. Another example is a pulse graph; for example [68] uses a binary graph to encode the pulse K-nearest neighbors. This format is used by CNN-based and graph-based methods. There is no clear statement on the best input format, each format being adapted to a technique. Note that there are other minor forms: TOA-DTOA plane [43], [57], RF vector [61].

### 2) LENGTH

Depending on the type of processing, constraints on the input length can apply. Histogram-based, clustering-based, and RNN-based are not affected by such constraints. It deals mainly with CNN-based methods due to the steady size of the CNN input layer. Depending on the input format, it constraints either the pulse number or a duration. The first case occurs for the ordered PDW format or some explicit connection format such as a DTOA matrix. The second case occurs for the implicit connection form as one dimension of the input size is a duration. That is the reason why the length question has to be combined with the global processing, in particular the processing of successive time windows. For
instance, it can be useful to overlap adjacent time windows
to enable the visibility of a full waveform lobe over a single
window.

3) PREPROCESSING

Preprocessing is a common step among RSD methods; it is
often normalization and/or quantization.

Normalization is motivated by the joint processing of
multiple primary parameters. For example, clustering-based
methods often normalize data to make the values consistent
with the use of a distance within a multidimensional space.
For example, a Euclidean distance has no sense in the RF-PW
place without normalization, the RF values being in MHz and
the PW values in ns. Another use case of normalization is for
neural network learning optimization.

Quantization is used mainly for CNN-based methods. The
main issue is to deal with the trade-off between the image
size (i.e., computing time) and the loss of resolution. Another
issue can occur in the case of pulses represented by a pixel
inside the time axis vector: information about the number of
pulses is lost when two closed pulses are mapped by the same
pixel.

C. ARCHITECTURAL PATTERNS

In this paragraph, we introduce a last classification frame,
which is the Deinterleaving Architectural Pattern (DAP).
We saw in Section V-A that some methods are recursive:
instead of directly partitioning all observed pulses into
multiple classes, like most neural network-based methods
do, they identify and extract waveforms one by one. The
concept of DAP intends to generalize these strategies. All
methods follow a strategy, or DAP, describing the relation
between partitioning modules. We identify two types of
modules:

* bipartition ($P_2$), into 1. a subset containing pulses from a waveform of interest, 2. a subset of other pulses.

* arbitrary partition ($P_K$): into $K$ subsets, where $K$ is either forced as in a segmentation module under the assumption $A_1$, or not as a DBSCAN clustering module.

They are the basic elements of the following patterns
represented in Figure 10.

1) DEINTERLEAVING ARCHITECTURAL PATTERNS

a: RECURSIVE WITH UNIQUE (R-U) PARTITION

Recursive deinterleaving consists in recursive extraction of
one waveform, using a bipartition module. We specify the
“unique partition” to identify modules where only one
partition is proposed, in contrast to the R-W architectural
pattern (see below). The R-U pattern (see Figure 10.a) is
the historical DAP, it is detailed in Algorithm 1. Selection
functions of Algorithms 2, 3, and 5 are examples of such
bipartitioning modules. Most of the time, P₂ is implemented
by a PRI estimation method such asDTOAH [30], but it can
also be based on neural networks [59].

<figure><img src="image_11.png" /><figcaption>FIGURE 10. Types of DAP. Bipartition is figured in green and denoted $P_2$, a special case is denoted $E$ for PRI estimation. Arbitrary partitioning is figured in red and denoted $P_k$, the special case of clustering is denoted $C$.</figcaption></figure>

b: DIRECT WITH UNIQUE (D-U) PARTITION

Direct deinterleaving aims to solve the problem in one
iteration, in contrast to the recursive pattern. The D-U
pattern (see Figure 10.b) is the most straightforward DAP,
and the output classes represent the emitters. Most of the
time, the $P_k$ module is a supervised learning segmentation
algorithm, as detailed in Algorithm 7. It can also be a one-
step clustering [38].

c: RECURSIVE WITH WINNING (R-W) PARTITION

The “winning partition” means that multiple bipartitions are
computed, and the best one is chosen for the actual waveform
extraction. In other words, the parallelization of $P_2$ modules
consists of trying several waveform selections, then picking
the best one for extraction, and iterating (see Figure 10.c).
An example of such a DAP is [65], which uses multiple
denoisers implemented by trained MLP to select specific
waveforms.

d: DIRECT WITH WINNING (D-W) PARTITION

Similarly to the previous pattern, D-W consists of trying
several partitions and then picking the best one (see
Figure 10.d). To our knowledge, this DAP has never been
applied.

e: DIRECT WITH TWO-STEP (D-C2) PARTITION

What happens when $P_K$ modules are in series (see
Figure 10.e) ? This pattern exists only for the two-step
clustering methods presented in Section V-B2. An example is
Algorithm 6: first make homogeneous groups of pulses, then
merge some of them [25].

2) DISCUSSION AND PERSPECTIVES

A combination of $P_2$ and $P_k$ can also make new architectural
patterns, as that presented as in Figure 10.f. It consists of
clustering to separate pulses that are obviously independent,
e.g., with a wide gap in frequency, before applying the R-U pattern on each sub-problem [44]. Then $P_2$ is the DAP of PRI estimation methods, while the D-U is the overwhelming pattern among the exposed algorithms. We think it should be profitable to design new ways to implement $P_2$ and new parallel-based DAP methods. To that extent, Xiang et al. [59] propose an interesting method, with a CNN-based encoder-decoder as the R-W pattern (type c), which combines DL performance and the absence of constraint on $K$.

Finally, it should also be interesting to investigate methods splitting RSD into multiple modules, each solving a simple task, e.g. predicting the homogeneity score of a pulse selection. In addition, such methods should exploit as many primary parameters as possible. In future work, we will propose a method towards this goal.

## VII. CONCLUSION

In this paper, we have presented the problem of radar signal deinterleaving, and proposed several frames of classification for the existing methods.

We proposed to study the RSD literature in terms of the following criteria: the main assumptions, the techniques used, the parameters exploited, the input characterization, and the deinterleaving architectural pattern. In particular, the technical criterion is used as a main classification of methods, leading on the one hand to the PRI estimation category including traditional algorithms, and on the other hand to three machine learning categories: clustering-based, RNN-based, and CNN-based methods. ML techniques have the advantage of capture more complex information than traditional techniques, but they suffer from the lack of explainability and generally need more assumptions such as the knowledge of the number of emitters.

It is difficult to quantitatively evaluate and compare the methods because the domain of electronic warfare is not suited to the open data philosophy, so there are no benchmark data or open source code.

As perspectives, we can refer to the ideas of Section VI-C2 about future works around the concept of deinterleaving architectural pattern. Lastly, one must keep in mind that RSD is not a purpose per se. Beyond RSD, what is important is the ability of the EW receiver as a whole, i.e. the identification and tracking of emitters in the environment, and this is why joint deinterleaving-identification methods should also be considered.

## ACKNOWLEDGMENT

All images are under the copyright THALES ©2023 THALES DMS SAS All rights reserved.

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<figure><img src="image_12.png" /></figure>

**LOUIS LESIEUR** received the Engineering degree in robotics from the École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne), Brest, France, in 2021, and the Ph.D. degree from ENSTA Bretagne in collaboration with Thales and Lab-STICC, in 2024. His current research interests include machine learning, evolutionary algorithms, and signal processing.

<figure><img src="image_13.png" /></figure>

**JEAN-MARC LE CAILLEC** (Senior Member, IEEE) received the Engineering degree in telecommunications from Telecom Bretagne, in 1992, and the Ph.D. degree in mathematics and signal processing from the University of Rennes 1, in 1997. From 1997 to 1999, he worked with Thomson AirSys (now Thales Land Air System). He joined Telecom Bretagne (now IMT Atlantique) as an Associate Professor, in 1999, and became a Full Professor, in 2007. From 2014 to 2020, he was

in charge of the Division CID (Knowledge, Information, Decision) of the Laboratory of Sciences and Technics for Information, Communication and Knowledge (Lab-STICC, UMR 6285. He initiated work on Explainable AI for active sensors (Radar and Sonar), in 2020. He is currently a Professor with the Data Science Department, IMT Atlantique. His main research interests are statistics, nonlinear system modeling, mathematics, and signal processing for applications in remote sensing, and finance.

<figure><img src="image_14.png" /></figure>

**ALI KHENCHAF** (Senior Member, IEEE) received the M.S. degree in statistical data processing from the University of Rennes 1, Rennes, France, in 1989, and the Ph.D. degree in electronic systems and computer networks from the University of Nantes, Nantes, France, in 1992. From 1989 to 1993, he was a Researcher with the IRCCyN Laboratory, where he was an Assistant Professor, from 1993 to 2001. Since 2001, he has been with ENSTA Bretagne, France, where he was the Head of the E3I2 Laboratory (EA 3876), from 2001 to 2011. Since 2012, he has been with Lab-STICC, UMR CNRS 6285, Brest, where he was, from 2012 to 2020, the Head of PIM Team (Propagation and Multiscale Interaction), and since January 2021, he has been the Head of SyPH Pole (Photonic and Microwave Systems) which brings together two teams. He is currently a Full Professor. He has edited or co-authored two books and authored or co-authored over 390 scientific articles. He assumed responsibility for more than 55 scientific projects contracted in partnership with industry and other organizations. He also supervised over 46 students' Ph.D. thesis. His research interests include radar applications, radar wave scattering, wave propagation, microwave remote sensing, scattering in random media, monostatic and bistatic scattering of electromagnetic waves, target radar cross-section, sea clutter, radar imagery, target parameter estimation, and GNSS applications. He has been a member of the editorial board of four scientific journals. He is an expert with several agencies and organizations in France and abroad.

<figure><img src="image_15.png" /></figure>

**VINCENT GUARDIA** received the Engineering degree in telecommunications from Telecom Bretagne, in 1987. After one year of military service as a Scientist with the Centre Electronique de l'ARmement, Bruz, he joined THALES Defense Mission Systems, in 1988. He is currently with the Technical Direction as a Senior Expert. His field of expertise covers digital signal processing, data processing, and information processing for electronic warfare and electronic intelligence on platforms ranging from satellites and aircraft to naval platforms. He is the co-author of several patents on localization algorithms, signal detection algorithms, and wideband digital receiver architectures. He was awarded the THALES Award for a new passive technique for real-time location of weapon systems, in 2004.

<figure><img src="image_16.png" /></figure>

**ABDELMALEK TOUMI** received the Engineering degree in computer science from the École Nationale Supérieure d'Informatique (ESI), Algiers, Algeria, in 2001, the M.Sc. degree in computer science, automatic, and distributed systems from the University of Franche-Comté, France, in 2003, and the Ph.D. degree in radar recognition, information systems, and machine learning from the University of Western Brittany, France, in December 2007. Since 2008, he has been an Associate Professor with ENSTA Bretagne, Brest, France. He is actually interested in pattern recognition, machine learning, and deep learning dedicated to remote sensing applications. He is the Coordinator of the master specialization "Observation Systems and Artificial Intelligence" in ENSTA Bretagne.