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\begin{document}

\title{A Dual-HMM Design for Detection of LPI Radar Signals\\
{\footnotesize }
\thanks{We acknowledge Atos Bull for their valuable contribution to this research.}
}

\author{
\IEEEauthorblockN{
Reihan Mazouz\IEEEauthorrefmark{1}\IEEEauthorrefmark{2},
Abigael Taylor\IEEEauthorrefmark{2},
José Picheral\IEEEauthorrefmark{1},
Jonathan Bosse\IEEEauthorrefmark{2},
Sylvie Marcos\IEEEauthorrefmark{1}
}

\IEEEauthorblockA{
\IEEEauthorrefmark{1}L2S, CentraleSupélec, Université Paris-Saclay, France\\
Email: \{reihan.mazouz, jose.picheral, sylvie.marcos\}@l2s.centralesupelec.fr
}

\IEEEauthorblockA{
\IEEEauthorrefmark{2}DEMR, ONERA, Université de Paris-Saclay, 91120 Palaiseau, France\\
Email: \{abigael.taylor, jonathan.bosse\}@onera.fr
}
}


\maketitle

\begin{abstract}
Detecting Low Probability of Intercept radar signals in a passive and fully blind setting remains a major challenge in electronic warfare. In this paper, we introduce a Hidden Markov Models architecture, where the transition dynamics are explicitly designed from the physical properties of the signals rather than learned from data. This physically grounded modeling enables the system to handle different modulation types, including both continuous and abruptly shifting frequencies, while preserving interpretability and eliminating the need for labeled data. The architecture can integrate multiple specialized HMMs, whose outputs are combined through a simple yet effective fusion mechanism. Extensive experiments over a wide SNR range show that our method matches the performance of modern deep learning detectors, while offering enhanced transparency and efficient frequency trajectory estimation.
\end{abstract}


\begin{IEEEkeywords}
LPI radar detection, Hidden Markov Models, YOLO, time-frequency analysis, Track Before Detect
\end{IEEEkeywords}


\section{Introduction}

In modern electronic warfare, the ability to passively detect and characterize radar signals plays a crucial role in threat identification and countermeasure deployment. Signal interception is challenging, as the interceptor has no prior knowledge of the signal to be detected, neither in terms of waveform characteristics nor transmission parameters. The absence of such information drastically increases the complexity of detection, as the system must distinguish radar signals from noise and interference in a fully blind setting. A particularly difficult case arises with Low Probability of Intercept (LPI) radars, which are specifically designed to avoid detection. These radars achieve stealth through low-power emissions, spread spectrum techniques, and adaptive waveforms, making conventional energy-based detection methods ineffective \cite{b1}. 

In this context, deep learning approaches have recently gained significant attention due to their ability to learn discriminative features from time-frequency representations. Object detection models such as YOLO have demonstrated strong performance in LPI signal detection tasks \cite{b4, ma2024low}.

In parallel, more traditional methods like track-before-detect (TBD) techniques, aim to associate low-SNR signal components across time before making a detection decision \cite{b3}. Within this class, Hidden Markov Models (HMMs) provide a principled and interpretable framework for sequential data analysis and have historically achieved strong results in domains such as speech recognition and biomedical signal processing \cite{b5, 18626, 67000}. By representing signals as sequences of hidden states with well-defined transition dynamics, HMMs can effectively distinguish between noise and meaningful emissions, enabling joint detection and parameter estimation \cite{b3,b7,b8}. While recent advances in deep learning particularly with recurrent neural networks (RNNs), have demonstrated strong performance in data-rich applications such as speech synthesis and recognition \cite{deshmukh2020comparison, wang2018comparative}, these approaches often require large annotated datasets and lack interpretability. In contrast, HMMs remain particularly appealing for scenarios such as LPI radar interception, where prior knowledge is limited, labeled data are scarce, and model transparency is critical.

In this paper, we revisit the use of HMMs for LPI radar signal detection, acknowledging earlier efforts that applied them to FMCW and CW signals \cite{alp2013hmm, bayindir2020lpi}. While these studies demonstrated the potential of HMMs in interception scenarios, they were limited to continuous waveform types and did not provide a comprehensive comparison with modern detection techniques such as deep learning.
To address these limitations, we propose a HMM-based framework capable of handling a wider class of LPI waveforms for both pulse and continuous signals. A key contribution lies in the design of physically informed transition matrices tailored to the time frequency structure of each waveform type, enabling interpretability without requiring labeled training data. The architecture is based on a dual-HMM structure, with each model specialized to capture a distinct modulation pattern, one targeting frequency continuity, the other abrupt changes. A simple fusion strategy is then used to combine their outputs.
Furthermore, we benchmark our approach against both a conventional energy detector and a modern deep learning based method over a wide SNR range.

The paper is structured as follows. Section~\ref{sec:Def} introduces LPI radar interception and the fundamentals of HMMs. Our proposed HMM-based framework is detailed in Section~\ref{sec:HMM_LPI}. Section~\ref{sec:Perf} presents implementation and experimental results.
% , comparing our approach with conventional and deep learning-based methods. 
Finally, Section~\ref{sec:Conclusion} summarizes the key findings.



\section{Definitions and Problem Statement}\label{sec:Def}

% \subsection{Problem Statement}
The objective of this work is to detect one or multiple unknown signals embedded in an additive Gaussian white noise environment. The received sampled signal can be expressed as:
\begin{equation}
    y(n) = \sum_{i=1}^{M} x_i(n) + w(n), \quad n = 0, \dots, T-1,
\end{equation}
where $x_i$ is one of $M$ deterministic signals emitted sequentially and $w$ is an additive white Gaussian noise with known variance $\sigma^2$. 
% \footnote{Thermic noise from the receptive field is assumed known.}

The set of signals under consideration includes various modulation schemes commonly encountered in radar systems: Linear and Non-Linear Frequency Modulation (LFM/NLFM), Phase Shift Keying (PSK), Frequency Shift Keying (FSK). The challenge lies in detecting the presence of such signals and extracting their defining characteristics from noisy observations.

% \subsection{Preprocessing}
To enhance detectability, the received signal is first mapped onto a time-frequency domain representation using the Short-Time Fourier Transform (STFT) defined as:
% The discrete STFT is defined as:
\begin{equation}
    Y(m, k) = \sum_{n=0}^{N_t-1} g(n) \, y(n + mN_t) \, e^{-j 2 \pi \frac{k n}{N_f}}, \quad 1 \le k \le N_f
\end{equation}
where \(N_t\) is the window length in time, \(N_f\) is the number of frequency bins, and \(g(n)\) is the temporal window applied.\footnote{In our experiments, we used a rectangular window but this formulation supports any conventional window function.}

% This transformation provides the input of our HMMs.
% , facilitating signal detection and characterization.
 
% \section{Background of Hidden Markov Models} \label{sec:HMM}

% \subsection{HMM Analytical Formulation}
% In what follows, we assume familiarity with hidden Markov models. A good reference is an article by Rabiner \cite{18626} from which we borrow some notation:

% \begin{itemize}
%     \item \( \mathcal{S} = \{ s_0, s_1, \dots, s_{N_f} \} \) : a finite set of \( N_f+1 \) hidden states, where: \( s_0 \) represents the absence of any signal across all frequency bins and \( s_i \) (for \( i > 0 \)) indicates the presence of a signal in frequency bin \( i \).
    
%     \item \( \boldsymbol{Y_t} = [Y(t,f_1), Y(t,f_2), \dots, Y(t,f_{N_f})] \) : the observation at \( t \), which is the STFT column at that instant.

%     \item \( \boldsymbol{\pi} = [\pi_i] \) with \( \pi_i = P(q_1 = s_i) \) : the initial state distribution where \(q_t \in \mathcal{S}\) represents the hidden state at \(t\). In our case, we take $\pi_0=1$ and $\pi_i=0$ for $i>0$.

%     \item \( \boldsymbol{A} = [a_{ij}] \) : the state transition matrix, where:
%     \(a_{ij} = P(q_{t+1} = s_j \mid q_t = s_i)\),
%     denotes the probability of transitioning from state \( s_i \) at \( t \) to state \( s_j \) at next time sample.

%     \item \( \boldsymbol{B} = [b_j(\boldsymbol{Y_t})] \) : the emission probability matrix, where: \(b_j(\boldsymbol{Y_t}) = P(\boldsymbol{Y_t} \mid q_t = s_j),\)
%     represents the likelihood of observing the spectral column \( \boldsymbol{Y_t} \) given that the hidden state at time \( t \) is \( s_j \).
% \end{itemize}

% Thus, the complete HMM is denoted:
% \(
%     \lambda = (\boldsymbol{\pi}, \boldsymbol{A}, \boldsymbol{B}).
% \)

The resulting spectrogram \(Y(m, k)\) provides a structured input in which signal patterns manifest as localized energy concentrations. To capture the temporal evolution of these patterns, we model each spectral column \( \boldsymbol{Y}_t = [Y(t,f_1), \dots, Y(t,f_{N_f})] \) as an observation emitted by a Hidden Markov Model. In the following, we assume the familiarity with HMM. A good reference is an article by Rabiner \cite{18626} from which we borrow some notation. The hidden state \( q_t \in \mathcal{S} = \{ s_0, \dots, s_{N_f} \} \) encodes the presence of a signal in frequency bin \(i\), with \( s_0 \) denoting the absence of signal. The initial state distribution is fixed to \( \pi_0 = 1 \) and \( \pi_i = 0 \) for \(1<i<N_f\). The model is defined by the tuple \( \lambda = (\boldsymbol{\pi}, \boldsymbol{A}, \boldsymbol{B}) \), where \( \boldsymbol{A} \) and \( \boldsymbol{B} \) denote the transition and emission probabilities, with transition probabilities \( a_{ij} = P(q_{t+1} = s_j \mid q_t = s_i) \) and emission probabilities \( b_j(\boldsymbol{Y}_t) = P(\boldsymbol{Y}_t \mid q_t = s_j) \).

To estimate the underlying state sequence from observations, we employ the Viterbi algorithm \cite{18626}, which efficiently recovers the most likely state path \( Q^* = \arg\max_{Q} P(Q \mid \boldsymbol{Y}) \) through dynamic programming, where \( Q = \{ q_1, q_2, \dots, q_T \} \) denotes a sequence of hidden states over time. All computations are carried out in the log domain for numerical stability, with GPU parallelization using PyTorch for large-scale spectrograms.


% \textbf{Remarks:}
% The standard HMM framework relies on two key assumptions:

% \begin{itemize}
%     \item \textbf{Markov Property:} The assumption that the state at time \( t+1 \) depends only on the state at time \( t \) is reasonable for many types of signals, depending on the choice of the hidden states and transition matrix.

%     \item \textbf{Observation Independence:} The assumption that \( \boldsymbol{Y_t} \) depends only on \( q_t \), and not on past or future observations, is relevant with a non overlapped window in the STFT. 
% \end{itemize}

% \subsection{State Path Estimation Algorithm}

% State path estimation aims to determine the most probable sequence of hidden states given an observed spectrogram. Several algorithms can be used for this purpose, such as the Forward-Backward algorithm, which computes marginal posterior probabilities for each state at every time step, and the Viterbi algorithm, which identifies the single most probable state sequence over time.

% In this work, we focus exclusively on the Viterbi algorithm, as it provides an efficient and reliable estimation of the most likely state trajectory.
% % , ensuring consistency in the detection and tracking of LPI radar signals.
% The Viterbi algorithm finds the optimal sequence of hidden states \( Q = \{ q_1, \dots, q_T \} \) by maximizing the joint probability of the state sequence given the observations:
% \begin{equation}
%     Q^* = \arg\max_{Q} P(Q \mid \boldsymbol{Y}).
% \end{equation}

% It employs a dynamic programming approach to efficiently compute the most likely sequence. For numerical stability, computations are performed in the log domain with normalization, as described in \cite{b3}. GPU-based parallelization is leveraged for large-scale spectrogram processing using PyTorch.

    \section{Hidden Markov Models to Detect and Track LPI Signals} \label{sec:HMM_LPI}
    
    % \subsection{Initial State Distribution}
    
    % The initial state distribution is defined as $\pi_0 = 1$, with $\pi_i = 0$ for $i \neq 0$ ensuring that the algorithm starts in the noise state.
    
    \subsection{Emission Probabilities}
    
    The likelihood of an observation given a hidden state follows a probabilistic density model based on the statistical properties of the spectrogram coefficients.
    
    When no signal is present at bin \( i \), the spectrogram coefficients \( Y_t(j) \), follow a circularly symmetric complex Gaussian distribution. The corresponding probability density function is:
    \begin{equation}
        p(Y_t(j) \mid q_t = s_i) = \frac{1}{\pi \sigma^2} \exp \left( -\frac{|Y_t(j)|^2}{\sigma^2} \right), \quad \text{for } j \neq i.
    \end{equation}
    
    When a signal is present at bin \( i \), the observed spectrogram coefficient follows a complex Gaussian distribution centered around the signal component \footnote{Fourier transform of the signal without noise at bin $i$ : $\alpha_t(i) = \sum_{n=0}^{N_t-1} g(n) \, x(n + mN_t) \, e^{-j 2 \pi \frac{k n}{N_f}}$} \( \alpha_t(i) \). The probability density function is:
    \begin{equation}
        p(Y_t(i) \mid q_t = s_i) = \frac{1}{\pi \sigma^2} \exp \left( -\frac{|Y_t(i) - \alpha_t(i)|^2}{\sigma^2} \right).
    \end{equation}
    
    % \textcolor{orange}{
    % In practice, however, the signal contribution \( \alpha_t(i) \) is unknown, which complicates direct computation of this density. To circumvent this issue, as in \cite{b7} we formulate a hypothesis test between:
    % \begin{itemize}
    %     \item \( \mathcal{H}_0 \): no signal is present,
    %     \item \( \mathcal{H}_1(i) \): a signal is present at bin \( i \), with unknown deterministic \( \alpha_t(i) \).
    % \end{itemize}
    % We then construct the likelihood ratio:
    % \[
    %     \Lambda_t(i) = \frac{p(Y_t(i) \mid \mathcal{H}_1(i))}{p(Y_t(i) \mid \mathcal{H}_0)}.
    % \]
    % Taking the Maximum Likelihood estimate under \( \mathcal{H}_1(i) \), i.e., setting \( \hat{\alpha}_t(i) = Y_t(i) \), we obtain:
    % \[
    %     p(Y_t(i) \mid \mathcal{H}_1(i)) = \frac{1}{\pi \sigma^2} \exp \left( -\frac{|Y_t(i) - Y_t(i)|^2}{\sigma^2} \right)
    %     = \frac{1}{\pi \sigma^2},
    % \]
    % and
    % \[
    %     p(Y_t(i) \mid \mathcal{H}_0) = \frac{1}{\pi \sigma^2} \exp \left( -\frac{|Y_t(i)|^2}{\sigma^2} \right).
    % \]
    % Hence, the likelihood ratio becomes:
    % \begin{equation}
    %     \Lambda_t(i) = \exp\left( \frac{|Y_t(i)|^2}{\sigma^2} \right),
    % \end{equation}
    % }
    % As described in \cite{b7}, we can use this likelihood for the emission probability. 
    % We can then normalize this vector, obtaining:
    % \begin{align}
    %     P(\boldsymbol{Y_t} \mid s_i) &=
    %     \frac{\exp \left( \frac{|Y_t(i)|^2}{\sigma^2} \right)}
    %     {1+\sum\limits_{j=1}^{N_f} \exp \left( \frac{|Y_t(j)|^2}{\sigma^2} \right)},
    %     \quad \text{if } i > 0, \nonumber \\
    %     P(\boldsymbol{Y_t} \mid s_0) &=
    %     \frac{1}
    %     {1+\sum\limits_{j=1}^{N_f} \exp \left( \frac{|Y_t(j)|^2}{\sigma^2} \right)}.
    % \end{align}


    Because the complex amplitude $\alpha_t(i)$ is unknown, we substitute its ML estimate $\hat{\alpha}_t(i)=Y_t(i)$.  
    Assuming independence across the $N_f$ frequency bins, the joint likelihood factorises,
    $P(\mathbf Y_t\!\mid\!s_i)=\prod_{j=1}^{N_f}p\!\bigl(Y_t(j)\!\mid\!s_i\bigr)$, and normalising over all
    $N_f\!+\!1$ states gives
    \[
    b_i(\mathbf Y_t)=
    \frac{P(\mathbf{Y}_t|q_t=s_i)}{\displaystyle\sum_{k=0}^{N_f}P(\mathbf{Y}_t|q_t=s_k)} =
    \frac{\exp\!\Bigl[-\frac{1}{\sigma^{2}}\sum_{j\neq i}|Y_t(j)|^{2}\Bigr]}
         {\displaystyle\sum_{k=0}^{N_f}\exp\!\Bigl[-\frac{1}{\sigma^{2}}\sum_{j\neq k}|Y_t(j)|^{2}\Bigr]} .
    \]
    Dividing both numerator and denominator by
    $\exp\!\bigl[-\tfrac{1}{\sigma^{2}}\sum_{j=1}^{N_f}|Y_t(j)|^{2}\bigr]$
    yields the closed-form 
    \begin{equation}
    b_i(\mathbf Y_t)=
    \begin{cases}
    \displaystyle
    \dfrac{\exp\!\bigl(\tfrac{1}{\sigma^{2}}|Y_t(i)|^{2}\bigr)}
          {1+\sum_{j=1}^{N_f}\exp\!\bigl(\tfrac{1}{\sigma^{2}}|Y_t(j)|^{2}\bigr)}, & 1\le i\le N_f,\\[10pt]
    \displaystyle
    \dfrac{1}
          {1+\sum_{j=1}^{N_f}\exp\!\bigl(\tfrac{1}{\sigma^{2}}|Y_t(j)|^{2}\bigr)}, & i=0,
    \end{cases}
    \end{equation}
    which supplies the entries of the emission matrix $\mathbf B$.


    % \textcolor{orange}{
    % Following the Neyman-Pearson paradigm, we fix the threshold \( \tau \) such that the probability of false alarm is controlled at level \( \alpha \). Specifically, since under the null hypothesis \( \mathcal{H}_0(i) \), the squared magnitude \( |Y_t(i)|^2 \) follows an exponential distribution with mean \( \sigma^2 \), the likelihood ratio \( \Lambda_t(i) \) has the distribution:
    % \[
    % \Lambda_t(i) = \exp\left( \frac{|Y_t(i)|^2}{\sigma^2} \right) \sim \text{Exp}(1) \quad \text{under } \mathcal{H}_0(i).
    % \]
    % Therefore, the false alarm probability is:
    % \[
    % \mathbb{P}_{\mathcal{H}_0(i)} \left[ \Lambda_t(i) > \tau \right]
    % = \mathbb{P} \left[ |Y_t(i)|^2 > \sigma^2 \log(\tau) \right]
    % = \exp\left( -\log(\tau) \right) = \frac{1}{\tau}.
    % \]
    % To enforce a false alarm rate of \( \alpha \), we thus set:
    % \begin{equation}
    %     \boxed{\;
    %     \tau = \frac{1}{\alpha}
    %     \;}
    % \end{equation}
    % This simple form allows the threshold to be directly tuned based on the desired sensitivity of the detector.}
    
      \subsection{State Transition Model}
    
    The design of the HMM transition matrix plays a crucial role in capturing the temporal dynamics of different waveform classes. In this paper, we focus on a two-HMM architecture: one dedicated to signals exhibiting continuity in the time-frequency plane, and another tailored to signals with abrupt frequency transitions, such as FSK modulations. This choice is made for clarity and conciseness; however, the proposed framework is readily extensible to any number of HMMs. Also, rather than learning the transition matrix from data (e.g., via Baum–Welch \cite{18626}), we define it based on physical signal properties. This provides interpretability and eliminates the need for a labeled training set.
    % , each specialized for detecting a specific waveform type or class.

    
    \paragraph{Transition to or from the noise state}
    Let \(p_{n \to n}\) be the probability of staying in the noise state, and 
    \(p_{s \to n}\) the probability of transitioning from a signal state back to noise. 
    These yield the following transition coefficients:
    \[
    a_{0,0}\!=\!p_{n \to n}, \quad
    a_{0,i}\!=\!\frac{1 - p_{n \to n}}{N_f}, \quad
    a_{i,0}\!=\!p_{s \to n} \quad \text{for } 1 \le i \le N_f
    \]
   These parameters can be tuned to control the false alarm probability.
    
    
    
    \paragraph{Transitioning between signal states with continuity constraint}
    for signals that exhibit continuity in the time-frequency plane, we adopt a Gaussian-based transition model. 
    % This model encourages transitions between neighboring frequency bins.
    % , reflecting the natural spectral smoothness of most signals.
    
    The transition probability from state \( s_i \) to \( s_j \) is defined as:
    \begin{equation}
        a_{ij} =
        \frac{\exp \left( -\frac{(j - i)^2}{2\sigma_{C}^2} \right)}
        {\sum_{k=1}^{N_f} \exp \left( -\frac{(k - i)^2}{2\sigma_{C}^2} \right)},
        \quad \forall i > 0, \quad \forall j > 0,
    \end{equation}
    where \( \sigma_{C} \) controls the expected frequency variation between consecutive time steps.
    
    This formulation ensures smooth transitions and models signals whose frequency evolves continuously over time. This model will be noted \(\lambda_1=(\boldsymbol{\pi},\boldsymbol{A_1},\boldsymbol{B})\).

\paragraph{Transitioning between signal states allowing frequency discontinuity}
For signals such as FSK, where the frequency exhibits abrupt changes, we define a transition model that explicitly allows for discontinuities. The transition probabilities favor persistence within the current frequency bin, but also permit instantaneous transitions.
% , reflecting the hopping behavior of FSK signals.

The transition probabilities are defined as:
\begin{equation}
    a_{ij} =
    \begin{cases}
        p_{\text{stay in bin}}, & \text{if } j = i, \\
        (1 - p_{s \to n}) \frac{1 - p_{\text{stay in bin}}}{N_f - 1}, & \text{if } j \neq i,
    \end{cases}
    \quad \forall i > 0, \quad \forall j > 0,
\end{equation}
where \( p_{\text{stay in bin}} \) is the probability of staying in the same frequency bin. For FSK signals, it is set based on the expected number \( K \) of hops over a signal duration \( T_s\) (in time steps), as:
\(
p_{\text{stay in bin}} = 1 - \frac{K}{T_s},
\)
ensuring consistency with the modulation dynamics.

This model captures the discontinuous nature of FSK transitions and will be noted \(\lambda_2=(\boldsymbol{\pi},\boldsymbol{A_2},\boldsymbol{B})\).

\section{Numerical Results and Performances Comparison} \label{sec:Perf}


\subsection{Simulated Dataset Description}

To evaluate the performance of the HMM-based detection system, we generate a spectrum dataset. Every data is synthetic due to the sensitivity of such electronic warfare signals. The dataset consists of 1000 spectrums for each SNR level, ranging from \(-30\,\text{dB}\) to \(-10\,\text{dB}\). Each spectrum contains between one and three radar emitters transmitting sequentially to avoid overlapping signals. The STFT parameters are configured to produce spectrum of size \(512 \times 512\), using a rectangular window without overlap. The waveform types include chirp (LFM), S-law (NLFM), frank (PSK), P1 to P4 (PSK), and costas (FSK) signals, with the characteristics of each emitter randomly selected. 
% This dataset allows for a thorough evaluation of detection and tracking performance across various SNR conditions.

% \textbf{Remark:}  
% The reported SNR refers to the power ratio between the signal and noise in the time domain. However, after applying the STFT, the effective SNR is higher due to the dispersion of the white noise power across the frequency bins.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.9\linewidth]{assets/time_non_overlapping_spectrum_2.png}
    \caption{Example of a simulated spectrogram with three radar signals at \(-20\,\text{dB}\) SNR containing: an LFM, an FSK and an NLFM signal.}
    \label{fig:example_spectrogram}
\end{figure}

\subsection{Parameter Calibration}

To ensure a fair comparison of algorithm performance, all evaluations are conducted at a constant false alarm rate of \(10^{-3}\) detections per noise-only spectrogram. The false alarm rate is estimated using a dataset composed of \(10\,000\) noise spectrogramswhile the parameter calibration is performed through a grid search. Any detected tracks in these noise-only spectrograms are considered false alarms.

% The calibration of the HMM transition probabilities relies on a grid search over 
% \( p_{\text{s} \to \text{n}} \) and 
% \( p_{\text{n} \to \text{n}} \). 
% The optimal pair is chosen to satisfy \textcolor{orange}{the} predefined false alarm rate constraint.

For the continuity-constrained model \(\lambda_1\), the standard deviation \(\sigma_C\) of the Gaussian transition kernel is set to 1.16. The following transition probabilities are selected to achieve the desired false alarm rate:
\(
p_{n \to n} = 0.99, \quad p_{s \to n} = 0.85.
\)
For the discontinuity-tolerant model \(\lambda_2\), the probability of remaining in the same frequency bin is fixed at \(p_{\text{stay in bin}} = 0.7\). The corresponding transition probabilities, chosen are:
\(
p_{n \to n} = 0.75, \quad p_{s \to n} = 0.80.
\)


\subsection{Detection Performance}

The detection performance of the proposed HMM-based approach is evaluated across multiple SNR levels. Detection probability is computed for each waveform class providing a detailed analysis of the model's behavior across different modulation types. As a post-processing step, a length-based filtering has been applied to the detected state paths: trajectories shorter than a predefined duration threshold of 5\% of the spectrum duration are discarded. This step mitigates spurious detections caused by noise fluctuations.


\paragraph{Per-Class Detection Analysis}

Figure~\ref{fig:hmm_performance_classes} illustrates the detection probability as a function of SNR for each signal class for the models \(\lambda_1\), \(\lambda_2\) and the combination of both. 

As shown in the top-left graph, the \(\lambda_1\) model achieves highly effective detection performance for continuous signals, such as LFM, NLFM and PSK. However, a noticeable degradation occurs with FSK signals, particularly between \(-22 dB\) and \(-12dB\). This limitation stems from \(\lambda_1\)'s assumption of temporal continuity, which leads to erroneous transitions to the noise state when abrupt frequency hops are detected.

Conversely, the \(\lambda_2\) model (top-right graph) demonstrates improved detection performance on FSK signals by explicitly modeling discontinuous frequency transitions. Nonetheless, it exhibits highly reduced performance on continuous signals compared to \(\lambda_1\), as it lacks constraints favoring smooth frequency evolution.

By combining the strengths of both models, the aggregated approach \(\lambda_1 + \lambda_2\) (bottom graph) provides robust detection capabilities across all waveform types. The combination strategy consists in selecting the path with the highest energy when both models detect overlapping paths in time, ensuring reliable trajectory estimation.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.95\linewidth]{assets/results_lambda1_lambda2_2.png}
    \caption{Detection probability as a function of SNR for each waveform class using \(\lambda_1\) (top-left), \(\lambda_2\) (top-right), and their combination \(\lambda_1 + \lambda_2\) (bottom).}
    \label{fig:hmm_performance_classes}
\end{figure}

Figure~\ref{fig:hmm_detection_example} presents an example of our method applied to the same spectrogram as shown in Figure~\ref{fig:example_spectrogram}. The estimated paths closely follow the true signal trajectories, demonstrating the ability of the combined model to accurately track both continuous waveforms and signals with abrupt frequency shifts.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.95\linewidth]{assets/time_non_overlapping_results_2.png}
    \caption{Example of our combined HMM-based method applied to the same spectrogram as in Figure~\ref{fig:example_spectrogram}. The estimated paths closely match the true signal trajectories.}
    \label{fig:hmm_detection_example}
\end{figure}


\paragraph{Comparison with Alternative Detection Methods}

To provide a comprehensive evaluation, the detection performance of the proposed HMM-based approach is compared to two alternative detection methods:
\begin{itemize}
    \item A cell-by-cell Generalized Likelihood Ratio Test (GLRT), which performs independent hypothesis testing on each cell of the time-frequency representation. Equivalent to the Grid-GLRT described in \cite{b4} with a grid \(1 \times 1\).
    \item A YOLOv8-based deep learning detector, which operates on the spectrogram image and outputs bounding boxes around detected signals proposed in \cite{b4} by the authors.
\end{itemize}

Figure~\ref{fig:comparison_detection_methods} presents the detection probability versus SNR for the three methods. The results show that YOLOv8 achieves slightly better detection probabilities on LFM and PSK signals, particularly in very low SNR conditions. For NLFM and FSK signals, the HMM-based approach and YOLOv8 demonstrate comparable detection performance across the tested SNR range. In contrast, the GLRT method consistently yields lower detection probabilities across all waveform classes, especially in high noise scenarios, due to its lack of temporal modeling and reliance on local cell-wise decisions.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.95\linewidth]{assets/hmm_yolo_glrt.png}
    \caption{Detection probability as a function of SNR for each waveform and each detection method: HMM-based (our method), GLRT cell-by-cell, and YOLOv8.}
    \label{fig:comparison_detection_methods}
\end{figure}

Despite YOLOv8's strengths in certain low SNR regimes, the proposed HMM-based method offers several key advantages:

\begin{itemize}
    \item \textbf{Trajectory-Based Detection}: The HMM estimates a detailed state sequence. This instantaneous frequency and time estimation provides richer information for subsequent processing stages (classification / further parameter estimation) than the bounding boxes predicted by Yolo.
    % For example, post-processing algorithms can leverage the extracted paths to enhance waveform classification or further parameter estimation.

    \item \textbf{No Need for Labeled Data}: Our method operates in an unsupervised manner which is particularly advantageous in contexts involving sensitive or classified signals. 

    \item \textbf{Robustness to Unknown Waveforms}: Since the proposed method does not rely on a supervised learning process, it demonstrates inherent robustness to unknown signal waveforms. This is especially relevant for LPI radar detection, where adversaries may employ highly diverse or intentionally unconventional modulation schemes.

    \item \textbf{Interpretability}: The HMM framework provides a transparent and explainable probabilistic model.
    % This level of interpretability facilitates a better understanding of the decision-making process and allows domain experts to integrate specific signal knowledge into the detection framework.
\end{itemize}
  
Overall, the proposed method offers a favorable trade-off between detection performance, interpretability, and practical deployment in scenarios where data availability is a critical factor.


\section{Conclusion}
\label{sec:Conclusion}

This paper introduced a dual Hidden Markov Model based architecture for the detection of Low Probability of Intercept radar signals. By combining models for both continuous and discontinuous time-frequency behaviors, the method achieves robust detection across various signal types without requiring labeled data. Experimental results demonstrate competitive performance compared to deep learning methods, while offering precise trajectory estimation and enhanced interpretability.

\bibliography{refs}


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