Benchmarking Frequency Hopping Algorithms for Resilient Wireless Communications under Multiple Jamming Environments

1. Introduction

At present, there is a dynamic development and growth in the number of wireless communication systems. These systems form the information and communications infrastructure for a significant number of technical devices, such as autonomous robotic platforms, unmanned aerial vehicles, Internet of things systems and others. This, in turn, significantly increases the demands placed on the resilience of information transmission channels in the face of deliberate electronic jamming. One of the most widespread and dangerous threats to wireless information and communication networks is active jamming. Indeed, such radio-electronic jamming is capable of significantly reducing throughput, increasing the probability of packet loss and disrupting the functioning of systems as a whole [1,2,3]. Consequently, the development and implementation of effective methods to counter deliberate jamming are among the main areas of current research in protecting information and communication systems and networks [4].

A method for ensuring interference-resistant communication that has become widely used in modern practice is FHSS [4,5]. This method is based on periodically varying the operating frequency according to a defined hopping sequence. However, despite extensive global experience in the application of FHSS, modern electronic warfare systems are increasingly utilizing adaptive and intelligent algorithms capable of analyzing hopping patterns and predicting future transmission frequencies [4,6]. Recent scientific and applied research shows that a significant number of modifications to the FHSS have been proposed to date. In particular, relevant scientific studies have focused on the use of multi-sequential frequency hopping [7], the optimization of FHSS index modulation [8], the increase in throughput of frequency-hopping systems [9], and the application of reinforcement [10,11] and deep [12] learning algorithms. A separate area of research involves the use of quantum random number generators (QRNGs). This approach allows the generation of practically unpredictable frequency-hopping sequences, thereby enhancing the cryptographic security of FHSS systems [13].

Along with the development of FHSS, models of intentional jamming are also being actively modernized. In addition to traditional (narrowband and broadband) jamming, reactive jamming is attracting considerable attention from researchers. Devices that generate reactive interference begin to emit it almost instantly upon detection of the useful signal. Such models are considerably more dangerous because they consume virtually no energy in standby mode and can precisely focus their interference on active transmission channels [14,15]. It is also worth noting that intelligent jamming systems have been the subject of intensive research in recent times. Such systems predominantly use deep and reinforcement learning methods to adapt their own strategy for suppressing wireless networks [16,17].

Much current scientific research and practical development also focuses on using artificial intelligence algorithms to improve the interference resilience of wireless systems. The application of reinforcement learning enables the dynamic adaptation of transmission parameters based on the current state of the radio channel, the history of interference sources, and accumulated statistics on frequency usage [6,11,12]. Similar approaches to optimizing technical and functional modes are actively used in satellite-ground information and communication systems [18] and reconfigurable intelligent surfaces (RIS)-assisted telecommunication systems [19].

A particular area of research that warrants attention in this article is the comparison of FHSS algorithms and the evaluation of their effectiveness in various scenarios involving intentional interference, using computer modeling techniques. Such research allows for an objective analysis of the advantages of different FHSS strategies at the early stages of designing wireless information and communication systems, with the possibility of repeatedly replicating the computerized experiment. For example, in [9], the authors propose a method for increasing the throughput of FHSS systems using statistical analysis of the bit error rate (BER) and interference resilience. Such an approach has confirmed the validity of using modeling as the primary tool for evaluating the effectiveness of FHSS algorithms. At the same time, recent research increasingly uses adaptive methods for selecting frequency channels. In particular, the scientific work [20] proposes a frequency-hopping algorithm utilizing reinforcement learning. This research is based on the application of modeling methods that enable the formulation of a channel selection strategy based on the current state of the interference environment. A similar line of research is presented in [21], where adaptive frequency hopping is implemented using an algorithm that estimates the interference environment, thereby enhancing the effectiveness of FHSS under variable operating conditions. Furthermore, a significant number of contemporary scientific studies consider FHSS not only as a means of countering radio-electronic interference, but also as a subject of adaptive analysis and classification [22]. This indicates that a promising direction for the development of FHSS is the combination of classical frequency-hopping algorithms with adaptive decision-making mechanisms and software-oriented modeling [5,23]. This allows their effectiveness to be assessed across a wide range of scenarios involving intentional interference.

Considering the significant number of studies devoted to FHSS systems, the analysis of relevant scientific publications confirms the need for further research on developing approaches for the comprehensive evaluation of various FHSS algorithms across different operational scenarios. This conclusion is based on the fact that most authors analyze a limited number of algorithms for a specific type of interference within a restricted range of signal-to-noise ratios (SNR). This makes it difficult to objectively compare the effectiveness of different frequency-hopping methods under reproducible operating scenarios.

Models of electronic warfare jamming require further attention. This is because most published works use additive white Gaussian noise (AWGN) or a single type of deliberate jamming. Most real-world electronic warfare systems combine broadband, narrowband and reactive jamming mechanisms. Consequently, the known results require further investigation when implemented across a wide range of practical application scenarios.

Particular attention should be paid to adaptive FHSS algorithms. The vast majority of modern approaches utilize deep learning or reinforcement learning methods, which are characterized by high computational complexity and the need for lengthy training. For real-world wireless devices with limited computational resources, such solutions can be difficult to integrate into software. Therefore, the search for algorithms that combine ease of implementation, low computational complexity and the ability to adapt to changes in the interference environment remains a priority.

An analysis of existing literature also shows that FHSS algorithms are most often evaluated solely on the basis of traditional metrics such as BER or throughput. At the same time, error vector magnitude (EVM) characteristics and frequency channel utilization statistics are considered far less frequently. It is precisely the comprehensive use of such criteria that allows for a more objective assessment of the effectiveness of FHSS algorithms, not only in terms of noise immunity but also in terms of the predictability of their operation.

A distinctive feature of the approach proposed and implemented in this article is the formalization and software implementation of six different FHSS algorithms: conventional fixed-frequency transmission (NoFHSS), random frequency-hopping spread spectrum (RandomFHSS), linear feedback shift register-based frequency-hopping spread spectrum (LFSRFHSS), chaotic frequency-hopping spread spectrum based on the tent map (ChaoticFHSS), quantum random number generator-based frequency hopping spread spectrum (QRNGFHSS) and adaptive frequency hopping spread spectrum with statistical channel learning (AdaptiveFHSS).

Another important contribution of this work is the formalization and software implementation of a comprehensive set of jamming models, which includes AWGN, narrowband, broadband, reactive and combined jammers. The combined interference model integrates broadband, narrowband and reactive interference mechanisms, enabling the simulation of more realistic operational scenarios for electronic warfare systems.

Unlike most well-known studies, the performance of the algorithms is evaluated through a comprehensive statistical experiment that covers various types of interference, several SNR levels, and multiple repetitions of each scenario via Monte Carlo simulation. In addition to the traditional metrics of BER, throughput and EVM, the statistical characteristics of frequency-hopping algorithms are also analyzed, in particular, channel reuse and the ranking of algorithms according to integral performance metrics.

So, the main aim of this article is to conduct a comprehensive comparative analysis of FHSS algorithms to ensure interference-resistant wireless communication across various intentional interference models, using a unified Monte Carlo-based computer simulation environment. The article compares the NoFHSS, RandomFHSS, LFSRFHSS, ChaoticFHSS, QRNGFHSS and AdaptiveFHSS algorithms under identical operating conditions, and their performance is evaluated in terms of BER, throughput and EVM for AWGN, narrowband, broadband, reactive and combined jammer models.

2. Materials and Methods

2.1. Simulation Framework

The computer experiments conducted as part of the research were performed within a single modeling software environment written in Python 3.12.13. The following libraries were used in the software’s implementation: math, random, numpy==2.0.2, pandas==2.2.2, matplotlib==3.10.0. The model architecture is designed to ensure uniform evaluation conditions across all investigated FHSS algorithms, regardless of interference type. Each experiment simulates the complete transmission cycle of a single data packet from the transmitter to the receiver, passing through all stages of the wireless communication system sequentially.

The structure of a single computer experiment is shown in Figure 1. At the beginning of each iteration, a new data packet consisting of random bits of a fixed length (4096 bits by default) is generated. The data packet is then modulated using quadrature phase-shift keying (QPSK). Next, a signal propagation environment is established, in which information is accumulated regarding the utilization of frequency channels, transmission results and the impact of intentional interference. Once the frequency-hopping sequence has been generated by the FHSS algorithm, the transmitted signal is formed and then the corresponding jamming model is applied. At the receiving side, the signal is demodulated and transmission quality metrics are calculated. The results are then used to update the statistics of the adaptive algorithm and the channel model.

To ensure the statistical reliability of the results obtained, the Monte Carlo method was used [24]. The space of computer experiments was defined taking into account the possibility of generating all possible combinations of the FHSS algorithm, the type of jammer, the SNR and the number of experiment repetitions. The study utilized six frequency-hopping algorithms (NoFHSS, RandomFHSS, LFSRFHSS, ChaoticFHSS, QRNGFHSS, and AdaptiveFHSS), five jammer models (AWGN, narrowband, broadband, reactive and combined jammer), and six SNR levels ranging from −5 dB to 20 dB in 5 dB steps. For each parameter combination, 30 independent replicates were performed. Thus, the total number of experiments amounted to 5400.

Unlike the traditional Monte Carlo scheme, where each iteration is performed independently, the model used in this article employs a persistent learning mechanism. For each series of experiments, a single instance of the FHSS algorithm and a single instance of the environment model are created, which retain the accumulated statistics between individual packets [25]. As a result, the adaptive algorithm gradually forms estimates of the quality of the frequency channels and utilizes them when generating subsequent frequency-hopping sequences, as illustrated in the detailed block diagram in Figure 2.

Thus, once the transmission of each packet is complete, the reception results are used to update the medium statistics and the adaptive algorithm in accordance with (1):

$$S_{k+1}=f\left(S_k,R_k\right),$$

(1)

where S_k+1 is the updated state of the modeled environment; S_k is the current state of the modeled environment; R_k is the result of transmitting the kth data packet; $f\left(·\right)$ is the statistics update function

So, the approach employed, as set out in (1), enables the accumulation of experience by an adaptive algorithm to be modeled during the system’s long-term operation.

2.2. Infocommunication System Model

A model of a digital wireless communication system using FHSS was proposed and implemented in this article. The overall system architecture comprises a data sequence generator, a digital modulator, an FHSS transmitter, a transmission channel model, a noise generator, an FHSS receiver, a demodulator and a transmission quality assessment unit. A generalized view of the software-implemented system model is shown in Figure 3.

When performing computer experiments, data packets of equal length were used, which ensured the accuracy of the comparative evaluation of FHSS algorithms. In accordance with the model’s basic configuration, each packet contains 4096 data bits. Following QPSK modulation, each complex symbol corresponds to two bits of information (the order of quadrature phase shift keying is 4). Thus, the number of complex symbols in a packet was 2048. Information is transmitted using frequency hopping. Based on the adopted model parameters, each frequency hop corresponds to 16 QPSK symbols. Therefore, the total number of frequency hops per packet is 128. These parameters remained constant for all investigated algorithms and all simulation scenarios. However, they can be adapted to the conditions of specific practical applications by adjusting the numerical values of relevant variables.

QPSK was used for digital data transmission, which enables two information bits to be transmitted using a single complex symbol. To verify the correct implementation of the physical layer, timing diagrams of the quadrature components of the basic QPSK signal and its spectral characteristics were plotted, as shown in Figure 4 and Figure 5, respectively.

A preliminary spectral analysis confirms that the composite reference signal has been correctly formed and that there are no unwanted spectral components that could distort the results of the jamming simulator modeling.

Following digital modulation, the resulting sequence of symbols is transmitted using a frequency-hopping approach. In this case, the software-implemented structure of a single packet is described by the following logical sequence:

$$s_i=\left\{s_1,s_2,\mathrm{...},s_{2048}\right\}\to H_j=\left\{h_1,h_2,\mathrm{...},h_{128}\right\}\to Segment{s}_k=\left\{Seg{m}_1,Seg{m}_2,\mathrm{...},Seg{m}_{128}\right\},$$

(2)

where s_i is a formed sequence of complex symbols; H_j is a generated hopping sequence; Segments_k are individual signal segments transmitted at the corresponding frequencies.

A specialized software class has been created to model the transmission process. This class implements the interface between the FHSS algorithm and the jamming models. The main functions are: accumulating usage statistics for each frequency channel; storing information about channels affected by interference; generating a quality estimate for each channel; and providing a learning mechanism for the adaptive FHSS algorithm. A quality estimate (Q_i) is maintained for each frequency channel, which is updated upon completion of the transmission of each data packet. If transmission via the channel took place without the influence of intentional interference, the channel estimate is increased by a factor of α:

$$Q_i\leftarrow \alpha Q_i,\alpha >1.$$

(3)

If an interference source is detected, the channel estimate is reduced by a factor of β. This reduces the probability of it being reused in subsequent hopping sequences:

$$Q_i\leftarrow \beta Q_i,\beta <1.$$

(4)

Once each packet has been transmitted, the estimates are normalized:

$$Q_i\leftarrow {\displaystyle \frac{Q_i}{\max (Q)}},0<Q_i\le 1.$$

(5)

The estimates obtained using (3)–(5) are used by the proposed AdaptiveFHSS algorithm to generate the next sequence of frequency hops.

Consequently, the block diagram shown in Figure 3 was translated into a software model based on the following formalized description. The general form of the signal transmission model is as follows:

$$RS=ISM\left(H\left(M\left(b\right)\right)\right),$$

(6)

where RS is the received symbol; ISM is the formalized model of the jammer; H is the formalized frequency-hopping procedure in accordance with the selected FHSS algorithm; M is the formalized QPSK modulation procedure; b is the information bit sequence.

After reception, the inverse transformation is carried out:

$$\widehat{b}=D\left(RS\right),$$

(7)

where $\widehat{b}$ is the result of the inverse transformation of the bit sequence; D is the formalized procedure for demodulation and reconstruction; RS is – the received symbol.

Thus, the results of executing the formalized procedures (6) and (7) provide the mathematical basis for the software implementation of the information and communication system’s functional principles under investigation. The resulting bit sequences are used to compute metrics that comprehensively characterize the quality of the FHSS algorithms.

2.3. Implemented FHHS Algorithms

This article is devoted to a comparative analysis of six frequency-hopping algorithms (NoFHSS, RandomFHSS, LFSRFHSS, ChaoticFHSS, QRNGFHSS and AdaptiveFHSS), which differ in the way they generate the hopping sequence. All algorithms use the same digital communication system model, the same QPSK modulation, the same packet size and operate under the same conditions of intentional interference (see Figure 3). This ensures an objective comparison of their performance. Figure 6 shows examples of frequency-hopping sequences generated by all investigated FHSS algorithms.

It can be seen that RandomFHSS, QRNG HSS and AdaptiveFHSS ensure an almost uniform utilization of the available channels. LFSRFHSS generates a deterministic pseudo-random sequence, whilst ChaoticFHSS uses the tent map chaotic mapping.

The NoFHSS algorithm is used as a baseline model for comparison. The entire data packet is transmitted over a single fixed-frequency channel without any hopping. In this case, the formalized description of the NoFHSS algorithm is as follows:

$$h_i=h_0,\forall \mathrm{i},$$

(8)

where h_i is the frequency-hopping sequence; h₀ is the constant data transmission channel.

This NoFHSS scheme, described by (8), is most vulnerable to narrowband and reactive jammers. This is because the entire data packet is transmitted on a single fixed frequency.

The implemented RandomFHSS algorithm generates a hopping sequence by independently selecting each subsequent frequency channel at random. In this case, the formal description of the RandomFHSS algorithm is as follows:

$$h_i\sim U\left(0,N_c-1\right),$$

(9)

where h_i is the frequency hopping sequence; U is the uniform discrete distribution function; N_c is the number of available frequency channels.

This implementation of the RandomFHSS algorithm, as described in (9), does not use information from previous transmissions and does not accumulate statistics on channel usage.

The LFSRFHSS algorithm, which was implemented in software in this study, is formalized by the following equations:

$$\begin{array}{l}{x}_{k+n}=\underset{i\in T}{\oplus }{x}_{k+i,};\\ h_i=x_i\mathrm{mod}{N}_c\end{array}$$

(10)

where $x_{k+n}$ is the new register state; T is the feedback taps; $\oplus$ is the XOR logical operation; h_i is the frequency-hopping sequence; N_c is the number of available frequency channels.

This approach, based on (10), ensures a high rate of pseudo-random sequence generation. However, because they are deterministic, they can be partially predicted.

The implemented ChaoticFHSS algorithm uses a chaotic tent map. This article employs an approach based on iterative comparison:

$$x_{k+1}=\left\{\begin{array}{l}2x_k,{\mathrm{x}}_k<0.5\\ 2\left(1-x_k\right),{\mathrm{x}}_k\ge 0.5\end{array}\right.,$$

(11)

where x_k is the current value of the system state; x_k+1 is the next value of the system state; k is the iteration number.

Consequently, the frequency channel number is defined as:

$$h_i=x_i{N}_c,$$

(12)

where h_i is the sequence of frequency hopping; x_i is the sequence of system states; N_c is the number of available frequency channels.

Thus, the chaotic mapping using (11) and (12) ensures sufficient sensitivity to the initial conditions and a complex structure of the hopping sequence.

In implementing the QRNGFHSS algorithm, a quantum random number generator simulation was used to generate independent, equally probable channel numbers. This approach made it possible to simulate the statistical properties of quantum random sequences without the need to use external quantum services. In this case, the formalized description of the channel number determination process is as follows:

$$h_i=q_i\mathrm{mod}{N}_c,$$

(13)

where h_i is the frequency-hopping sequence; q_i is an independent random number generated by the quantum generator emulator; N_c is the number of available frequency channels.

Unlike the LFSRFHSS implementation, sequence (13) lacks a deterministic structure and is difficult to predict.

The AdaptiveFHSS algorithm implemented in this article utilizes statistical learning of frequency channel quality. A similar principle of adaptive channel selection, based on accumulated information about the spectrum state, is widely used in modern intelligent anti-jamming methods [26,27]. In contrast to these, the proposed implementation employs a simplified statistical mechanism for channel estimation that does not require model training or the construction of a reward function. This approach has been chosen based on the potential for applying the research results to devices with limited computational resources. In the proposed implementation, upon completion of the transmission of each data packet, estimates (Q_i) are generated that characterize the reliability of each frequency channel. When generating the next hopping sequence, priority is given to channels with higher indicator values (Q_i). Thus, the generalized formalized description is as follows:

$$h_i=\underset{c}{\arg \max Q_c},$$

(14)

where h_i is the frequency hopping sequence; c is the channel index, determined by scanning through all available channels; Q_c is the quality estimate for channel c.

After each packet is transmitted, the channel estimates are updated in accordance with the transmission success statistics, thereby implementing a mechanism of continuous statistical learning.

2.4. Implemented Jamming Models

To evaluate the effectiveness of the investigated FHSS algorithms, five jammer models were implemented: AWGN, narrowband, broadband, reactive, and combined jammers. These models differ in how they affect the communication system and in how they reflect the most common scenarios for deliberate radio-electronic jamming of wireless networks [5,15,28,29,30,31]. All models are implemented as separate classes within the software model and share a common interface to interact with the environment, ensuring consistent and reproducible test conditions across all frequency-hopping algorithms.

The implemented AWGN model is a basic model of random interference and is used as a reference when analyzing the stability of digital communication systems. This jammer does not selectively affect individual frequency channels, but adds AWGN to the received signal. The software implementation of this model in this case is as follows:

$$\begin{array}{l}{r}_k=s_k+n_k;\\ n_k\sim {\rm N}\left(0,{\sigma }^2\right),\end{array}$$

(15)

where r_k is the received complex symbol; s_k is the kth complex QPSK symbol; n_k is complex Gaussian noise; N is the normal distribution; 0 is the mean of AWNG; σ² is the variance of AWNG, which is determined on the basis of a given SNR in the range from −5 dB to 20 dB in 5 dB steps.

The implemented narrowband jammer model affects only a specific set of frequency channels in accordance with the following formalized description:

$$J\left(c\right)=\left\{\begin{array}{l}1,\mathrm{c}\in {\mathrm{C}}_J\wedge Rand\left(0,1\right)<p_{jam};\\ 0,\mathit{otherwise},\end{array}\right.$$

(16)

where J(c) is the jammer function; с is the index of the available frequency channel; C_J is the set of channels subject to deliberate jamming; Rand is a random variable with a uniform distribution; p_jam is the probability of narrowband jamming being activated.

The implemented broadband jammer model generates interference simultaneously across the entire operating frequency range. In the model, each hop can be suppressed with a given probability in accordance with the following formalized description:

$$P_{jam}\left(J=\left(0,1\right)\right)=p,$$

(17)

where P_jam is the probability of interference from the jammer; J is a random variable describing the state of the jammer (J=1 – jammer active, J=0 – jammer inactive); p is the probability of the jammer being activated.

The software implementation of the reactive jammer uses a mechanism to store the history of frequency channel usage. To this end, after each hopping event, the number of the channel used is recorded in the history buffer. To limit the amount of information stored, a parameter specifies the maximum number of recent hopping events to be considered during analysis. In this way, the jammer analyses only the most recent time interval of the FHSS system’s operation, enabling it to respond quickly to changes in the hopping sequence. Once the history has been accumulated, the usage frequency for each channel is determined. Based on the frequencies obtained, a set of the most frequently used channels is formed. This set is used by the jammer to suppress the most likely future frequency hops. Another feature of the implementation is that it takes into account the reaction time of the jammer. Consequently, the jammer utilizes information about previous frequency hopping events rather than the current state of the system, which more realistically reflects the operation of modern reactive electronic warfare systems. A generalized formalized description of the software implementation of the reactive jammer model is as follows:

$$\begin{array}{l}T=TopN\left(f\left(c\right)\right);\\ f\left(c\right)={\displaystyle \sum _{h_i\in H}1\left(h_i=c\right)},\end{array}$$

(18)

where T is the set of channels to be suppressed; TopN is the set of channels with the highest number of uses; f(c) is the number of times channel c, for which the usage history is being tracked, has been used out of the total set of channels; H is the history of the set of used channels; h_i the number of the channel used during the ith hopping; $1\left(h_i=c\right)$ is an indicator function equal to 1 if the condition that the channel under analysis matches a channel in the usage history is met, and equal to 0 if this condition is not met.

To simulate the most challenging operating conditions, a combined jammer has been implemented, which integrates reactive, narrowband and broadband signal suppression mechanisms. Unlike the independent use of individual models, the combined jammer employs a hierarchical decision-making mechanism in which different types of interference are assigned varying priorities. When processing each frequency hop, the possibility of reactive jamming is checked first. If the current channel belongs to the set of predicted channels generated by the reactive jammer, AWNG is added to the signal. If reactive jamming is not applied, a check is performed to determine whether the channel belongs to the set of narrowband interference channels. For such channels, jamming is performed with probability p₁. Only in the absence of the effects of the first two models is wideband jamming performed with probability p₂. Consequently, the generalized formalized description of the combined jammer is as follows:

$$J_{combined}\left(c_i\right)=\left\{\begin{array}{l}{J}_{reactive}\left(c_i\right),{\mathrm{c}}_i\in T,\\ J_{narrowband}\left(c_i\right),{\mathrm{c}}_i\in C_J\wedge u_1<p_1,\\ J_{broadband}\left(c_i\right),u_2<p_2,\\ 0,\mathit{otherwise},\end{array}\right.$$

(19)

where J_combined, J_reactive, J_narrowband, J_broadband are the functions of the combined, reactive, narrowband and broadband jammers, respectively; c_i is the frequency channel number used during the ith hopping; T is the set of channels defined by the reactive jammer model; C_J is the set of narrowband jammer channels; u₁, u₂ are independent random variables with a uniform distribution; p₁, p₂ are the probabilities of activating the narrowband and broadband jammers, taken in this study to be 0.7 and 0.45, respectively.

It is also worth noting that, once the processing of each frequency hop has been completed, the current channel number is added to the reactive jammer’s history. This ensures that the jammer adapts to the behavior of the FHSS system and makes it possible to predict the most probable subsequent frequency hops.

A graphical representation of the effect of the specified models on the investigated signal, in accordance with (15)–(19), is shown in Figure 7.

2.5. Performance Evaluation Metrics

One of the key quality characteristics of digital transmission is the BER, which quantifies the fraction of bits received in error [32,33]. In a software implementation, the BER is calculated by directly comparing the transmitted and received bit sequences:

$$BER={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^{N}1\left(b_i^{tx}\ne b_i^{rx}\right)},$$

(20)

where BER is the bit error rate metric; N is the number of bits in the data packet; I is the ordinal number of the bit; $b_i^{tx}$, $b_i^{rx}$ are the transmitted and received bits, respectively; $1\left(\cdot \right)$ is the indicator function.

In this article, the quality of demodulation is assessed using the EVM metric. This metric characterizes the average deviation of the received complex symbols from their ideal positions within the signal constellation [34]. In the software implementation, the EVM is calculated based on the Euclidean distance between the corresponding complex symbols using the following formula:

$$EVM=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left|s_i^{rx}-s_i^{tx}\right|}^2}},$$

(21)

where EVM is the error vector magnitude metric; N is the number of complex QPSK symbols; I is the symbol index; $s_i^{tx}$, $s_i^{rx}$ are the transmitted and received complex symbols, respectively.

The throughput is used to assess the efficiency of a communication channel. In the proposed software implementation, it is defined as:

$$Throughput={\displaystyle \frac{N_{total}-N_{error}}{N_{total}}},$$

(22)

where Throughput is a metric for throughput; N_total is the total number of bits transmitted; N_error is the total number of bits received in error.

It is worth noting that, in addition to analyzing individual data transmission quality metrics, this study employs a comprehensive methodology for comparing FHSS algorithms. This approach enables a comprehensive assessment of their effectiveness across all the investigated scenarios. Once the Monte Carlo simulation has been completed, the mean values of metrics (20)–(22) are calculated for each algorithm. A summary table of the mean results is compiled based on these statistical characteristics.

To ensure a fair comparison of metrics with different units of measurement and ranges, all criteria are first normalized to the interval from 0 to 1. For metrics where a decrease indicates an improvement in system quality (BER and EVM), normalization is performed using (23). For the Throughput, where an increase is desirable, normalization is carried out using (24):

$$Metri{c}_i^{norm}={\displaystyle \frac{Metri{c}_{\max }-Metri{c}_i}{Metri{c}_{\max }-Metri{c}_{\min }}},$$

(23)

$$Metri{c}_i^{norm}={\displaystyle \frac{Metri{c}_i-Metri{c}_{\min }}{Metri{c}_{\max }-Metri{c}_{\min }}},$$

(24)

where $Metri{c}_i^{norm}$ is the normalized value of the relevant metric for the ith algorithm; $Metri{c}_i$ is the current value of the relevant metric for the ith algorithm; $Metri{c}_{\max }$, $Metri{c}_{\min }$ are the maximum and minimum values of the relevant metric across all algorithms.

Once normalization has been performed, an integral performance index (IS) is calculated for each analyzed algorithm using the following formula:

$$I{S}_i=a_{1}BE{R}_i^{norm}+a_{2}Throughpu{t}_i^{norm}+a_{3}EV{M}_i^{norm},$$

(25)

where$I{S}_i$ is the integral quality assessment of the ith algorithm; $BE{R}_i^{norm},Throughpu{t}_i^{norm},EV{M}_i^{norm}$ are the normalized metrics for the ith algorithm; a₁, a₂, a₃ are the weighting coefficients for the metrics.

As can be seen from the analysis of (25), the software implementation allows for the adjustment of the weighting of metrics on the overall quality of the assessment, depending on the specific objectives of the analysis. In the current implementation, the highest weight (a₁=0.4) is assigned to BER as the primary indicator of a digital communication system’s noise immunity. The Throughput and EVM metrics have identical weighting coefficients (a₂=a₃=0.3). This has made it possible to take into account both the efficiency of channel utilization and the quality of signal demodulation.

3. Results

3.1. Performance Analysis of FHSS Algorithms

This subsection presents the results of a comprehensive comparative analysis of implemented FHSS algorithms under identical computer simulation conditions. The aim of the study was to evaluate the performance of FHSS algorithms using a single set of quality metrics (BER, Throughput and EVM). This work utilizes integrated dashboards that represent the behavior of each algorithm simultaneously across different interference scenarios. This approach provides a comprehensive view of interference resilience, communication channel utilization efficiency and received signal quality. This enhances the objectivity of the comparison between the investigated algorithms and enables identification of their strengths and weaknesses before a detailed analysis of individual jamming models is conducted. A graphical interpretation of the performance analysis results for FHSS algorithms is shown in Figure 8.

A comparison of the obtained results, shown in Figure 8, reveals that RandomFHSS, QRNGFHSS, and ChaoticFHSS exhibit similar behavior across all the investigated metrics. For these algorithms, with integrated consideration of jamming models, as the SNR increases from −5 dB to 10 dB, the BER decreases from approximately 0.2 to 0.0008, the Throughput increases from 0.90 to almost 1.0, whilst the EVM decreases from 3.1 to 0.3. The almost complete overlap of the corresponding curves indicates the following. Under the modeling conditions used, the type of hopping sequence generator is not a decisive factor in improving the system’s resilience to interference. The results obtained suggest that the main potential for improving FHSS efficiency lies in the use of adaptive control mechanisms. Consequently, investigating potential synergies between different methods for generating frequency-hopping sequences and adaptive strategies is a promising area for further scientific research.

The LFSRFHSS algorithm also demonstrates high transmission-quality metrics across most of the scenarios studied. However, it is characterized by the highest EVM values, excluding NoFHSS. These values reach 5.5, which may be due to the deterministic nature of the frequency hopping pattern.

In contrast, the AdaptiveFHSS implemented in this study delivers the best results across virtually all the metrics examined. Even at an SNR of 10 dB, the BER falls to less than 0.0001, the Throughput reaches a value close to 1.0, and the EVM does not exceed 0.2, which demonstrates the high efficiency of the adaptive frequency channel selection mechanism.

As expected, the NoFHSS model, which does not use frequency hopping, demonstrates the lowest efficiency. It is characterized by the highest BER and EVM, with maximum EVM exceeding 6, approximately twice that of AdaptiveFHSS. This confirms that frequency hopping is a prerequisite for the interference-resistant operation of modern wireless communication systems.

Thus, when all investigated metrics are taken together, the AdaptiveFHSS algorithm demonstrates the best overall performance. It is worth noting that the RandomFHSS, QRNGFHSS and ChaoticFHSS algorithms form a group of algorithms with similar characteristics. The LFSRFHSS algorithm occupies an intermediate position, whilst NoFHSS provides the lowest transmission quality.

The research findings have confirmed the need for a more detailed analysis of how the specific characteristics of individual jamming models affect the performance of each algorithm. These findings are presented in the following subsection.

3.2. Performance Analysis Under Different Jamming Environments

To assess the resilience of the investigated FHSS algorithms to deliberate electronic jamming, simulations were conducted across five different jamming scenarios. Figure 9 shows the dependence of the analyzed quality metrics on the SNR value for each jamming scenario.

An analysis of the results shown in Figure 9 demonstrates that the effectiveness of FHSS algorithms depends significantly on the type of jammer. For the AWGN and combined jammer models, all algorithms demonstrate the expected improvement as the SNR increases. The BER metric decreases to around 0.001, the Throughput approaches 1 and the EVM is reduced by approximately 11 times.

The most significant negative impact is observed for the broadband and reactive jammer models. In these scenarios, the BER is almost independent of the SNR, the Throughput remains virtually constant, and the EVM exceeds 2 even at high SNR values. The NoFHSS algorithm is particularly vulnerable, whilst AdaptiveFHSS delivers the lowest BER and EVM values of all the algorithms studied.

The narrowband jammer model proved to be the least critical. Most algorithms maintain the Throughput close to 1, whilst the BER does not exceed a few percentage points. AdaptiveFHSS once again demonstrates the best interference resilience, effectively avoiding channels with high levels of interference.

At the same time, the RandomFHSS, QRNGFHSS and ChaoticFHSS algorithms exhibit very similar performance across all scenarios. This suggests that, with the model used, the method for generating the hopping sequence offers no significant advantages without adaptive mechanisms. This effect, in turn, is of interest for further research. The main advantage of AdaptiveFHSS is its use of statistical learning of channel quality, which enables it to deliver the best results across virtually all the metrics considered.

3.3. Comparative Performance Analysis Under Different SNR Levels

To provide a more detailed comparison of the algorithms, taking into account the corresponding jamming models at different SNR levels, Figure 10 shows bar charts of the metrics under analysis.

As shown in Figure 10, AdaptiveFHSS consistently achieves the lowest BER values for narrowband and reactive jammers, whilst NoFHSS exhibits the highest error rates across virtually all SNR levels. For AWGN, broadband and combined jammers, the differences between the algorithms are negligible, which confirms the results of the previous analysis.

An analysis of the Throughput shows that, in the absence of active interference (AWGN model) and under combined jammer conditions, all algorithms deliver a near-maximum value of this metric. The biggest differences are observed for narrowband and reactive jammers, where AdaptiveFHSS maintains the highest throughput values, whilst NoFHSS exhibits the greatest degradation.

A comparison based on the EVM confirms similar patterns. The lowest EVM values are observed for AdaptiveFHSS, particularly under narrowband and reactive jammer conditions. Whereas NoFHSS has the highest EVM values, indicating the highest level of distortion in the received signal. For a broadband jammer, all algorithms exhibit similar EVM values, indicating that the effect of broadband interference is practically identical regardless of the method used to generate the hopping sequence.

Overall, the results obtained confirm the outcomes of the preliminary analysis. The advantages of AdaptiveFHSS are particularly evident in scenarios involving targeted jammers (narrowband and reactive), whilst in the case of AWGN, broadband and combined jammers, the performance of most algorithms is similar. This demonstrates the effectiveness of adaptive frequency channel selection, particularly in the face of adaptive and narrowband attacks.

3.4. Comprehensive Performance Assessment

To summarize the results, a comprehensive comparison of the investigated FHSS algorithms was conducted across all experiments. The comparison was based on the average normalized values of metrics (20)–(24). Subsequently, an integral performance index (25) was calculated for each algorithm, taking all three criteria into account simultaneously. The results of this analysis are presented in Table 1.

As shown in Table 1, the implemented AdaptiveFHSS algorithm delivered the best results among all the algorithms studied, based on a combination of criteria including transmission reliability, channel utilization efficiency, and received signal quality.

The RandomFHSS and ChaoticFHSS algorithms demonstrated fairly high performance, with an integral performance index (IS) of 0.961 and 0.954, respectively. Both algorithms demonstrated very similar normalized values for BER and Throughput (0.978 and 0.973, respectively), but were outperformed by AdaptiveFHSS in terms of EVM. The QRNGFHSS algorithm came fourth with an IS of 0.929, which also indicates its high efficiency, but lower performance compared with AdaptiveFHSS.

The LFSRFHSS received a significantly lower integral performance index (IS = 0.627). Despite satisfactory results in individual scenarios, its normalized values for BER, Throughput, and EVM (0.653, 0.653, and 0.567, respectively) were significantly lower than those of the best-performing group of algorithms. As expected, the NoFHSS system achieved the lowest results, reflecting its lack of interference resilience in the absence of a frequency-hopping mechanism.

The practical effectiveness of the implemented AdaptiveFHSS is further confirmed by the results of comparisons with other algorithms, as shown in Table 2. It is worth noting that, in order to ensure greater methodological objectivity, the average absolute values of the BER, Throughput and EVM metrics were used when calculating the relative improvement, rather than their normalized values.

Compared with the baseline NoFHSS algorithm, the BER was reduced by 64.23%, the Throughput was increased by 22.76% and the EVM was reduced by 59.32%. Compared with LFSRFHSS, the implemented AdaptiveFHSS algorithm delivered a 38.40% improvement in BER, a 6.88% increase in Throughput and a 38.70% reduction in EVM. Even when compared with the best competing algorithms, AdaptiveFHSS demonstrates a consistent advantage. In particular, compared with QRNGFHSS, the improvements are 7.66% in BER, 0.86% in Throughput and 15.80% in EVM; compared with ChaoticFHSS — 4.57%, 0.50% and 11.68%, respectively; compared with RandomFHSS — 3.80%, 0.41% and 10.42%, respectively.

The results demonstrate that using a persistent learning mechanism, which accumulates statistics on the quality of frequency channels and accounts for them when generating subsequent hopping sequences, ensures sustained improvements in the efficiency of wireless communication systems. Unlike FHSS algorithms, which use exclusively random or deterministic frequency selection, AdaptiveFHSS combines the properties of random hopping with adaptive channel state estimation, enabling the best results to be achieved across all the metrics studied.

4. Discussion

4.1. Scientific Novelty and Practical Significance

The research results obtained have demonstrated that the software-implemented AdaptiveFHSS algorithm consistently delivers better performance in terms of BER, Throughput and EVM metrics compared with other FHSS implementations across a wide range of jammer models. Unlike most recent studies, which focus on improving the hopping sequence generator or on using machine learning methods to predict the spectral environment, this article employs an approach based on adaptive statistical learning of frequency channel quality, without the need for prior model training. This reduces the computational load and makes it possible to utilize the proposed software implementation in information and communication systems with limited computational resources.

Studies on the use of chaotic frequency hopping have shown that chaotic generators enhance the cryptographic strength and unpredictability of hopping sequences. However, they have virtually no effect on noise immunity in the presence of active jammers. Similar conclusions were reached in this study, in which ChaoticFHSS exhibited performance characteristics comparable to those of RandomFHSS and QRNGFHSS. This outcome is that increasing the generator’s entropy does not, in itself, guarantee an improvement in link quality under active electronic jamming.

Some recent studies propose using deep learning or reinforcement learning methods to select operating frequencies. Despite the high efficiency of such approaches, they generally require significant computational resources, large training datasets and a lengthy training process. The implemented AdaptiveFHSS uses a significantly simpler mechanism for accumulating channel quality statistics, can operate without a preliminary training stage, and adds virtually no computational complexity to the system.

The practical value of the proposed approach also lies in the fact that the entire algorithm has been implemented as a software simulator, which allows the modeling of various FHSS algorithms and several jamming scenarios, as well as the performance of statistical evaluation using the Monte Carlo method. Unlike most published works, which limit their analysis to individual scenarios or specific types of interference, the software developed provides a unified environment for the comprehensive comparison of FHSS algorithms.

4.2. Study Limitations

In the research carried out for this article, several assumptions were made, enabling a comparative analysis of FHSS algorithms under identical simulation conditions.

In particular, the simulations were carried out for a communication channel with AWGN and for realized jammer models. This approach allows for a fair comparison of algorithms under identical conditions, although it does not account for certain characteristics of a real radio channel, such as multipath propagation, fading or Doppler effects.

The implemented AdaptiveFHSS model utilizes statistical accumulation of information on the quality of individual frequency channels, which enables adaptation without complex pre-training procedures. At the same time, this article did not consider more complex approaches to predicting the spectral situation, which take into account temporal or spatial patterns of spectrum usage.

To ensure the reproducibility of the experiments, all interference generator models used a fixed set of parameters. The selected values made it possible to generate representative simulation scenarios and carry out a valid comparison of the algorithms. Investigating the impact of individual parameters on the system’s performance may, however, be the subject of future research.

The algorithms were evaluated using a Monte Carlo simulation, which enabled the statistical analysis of a large number of transmission scenarios. The practical implementation of the proposed algorithm on physical computing platforms or other radio equipment is regarded as the logical next stage of the research.

4.3. Future Research Directions

The results obtained open up several promising directions for further research. The first area is the extension of the communication channel model to take into account multipath propagation and the spatial and temporal characteristics of the wireless medium. This will enable an assessment of the effectiveness of the implemented AdaptiveFHSS algorithm under conditions as close as possible to those of real-world communication networks.

Another promising area is the integration of more sophisticated artificial intelligence methods, in particular reinforcement learning or transformer models, for predicting the state of the spectrum and generating hopping sequences that take into account long-term patterns in the use of frequency channels.

Further development is also required to model more complex jamming systems that use machine learning algorithms for adaptive prediction of subsequent frequency hopping. This will enable the creation of more realistic scenarios for the confrontation between FHSS systems and intelligent electronic warfare systems.

Of particular interest is the investigation into the reasons for the virtually identical performance of RandomFHSS, QRNGFHSS and ChaoticFHSS, as revealed during the modeling. The results indicate that, for the noise models used, it is not the method of generating the random hopping sequence that decisively influences noise immunity, but rather the presence of a mechanism to adapt it to the current state of the spectrum. Further research could focus on determining the conditions under which quantum or chaotic random number generators would provide statistically significant advantages.

Finally, an important step in further research is the practical implementation of AdaptiveFHSS on resource-constrained computing platforms and the experimental verification of the results in a real-world wireless environment. This will enable an assessment of the proposed algorithm’s performance in the presence of hardware constraints, time delays and real-world radio frequency interference.

5. Conclusions

A software simulator for FHSS systems has been developed and studied in this article. It implements six frequency-hopping algorithms, five interference source models, and statistical evaluation using the Monte Carlo method based on the BER, Throughput and EVM metrics. The proposed simulator provides a unified environment for comprehensive research on the interference resilience of FHSS algorithms.

In the research, the AdaptiveFHSS algorithm was implemented in software. This algorithm utilizes the statistical accumulation of information on the quality of frequency channels during the generation of hopping sequences. Unlike the implemented RandomFHSS, LFSRFHSS, ChaoticFHSS and QRNGFHSS algorithms, the AdaptiveFHSS-based approach enables adaptation to the characteristics of the radio channel without the use of pre-training or complex machine learning models.

According to comprehensive modeling results, AdaptiveFHSS demonstrated the best overall performance among the algorithms studied, achieving a maximum integral performance index of 1.0, confirming its consistent superiority across the criteria examined.

Compared with the baseline implementation, the proposed AdaptiveFHSS achieved 64.23% reduction in BER, 22.76% improvement in Throughput and 59.32% reduction in EVM relative to a system without frequency hopping.

The results also showed that RandomFHSS, QRNGFHSS and ChaoticFHSS exhibit very similar characteristics, whilst the main improvement in noise immunity is achieved precisely through the use of an adaptive frequency channel selection mechanism. This confirms the potential of using statistical channel quality learning to design modern adaptive FHSS systems in the face of active electronic countermeasures.