Dmitry A Kukuruznyak
email: d.kukuruznyak@animcondmat.com

Can Artificial Intelligence be a Weapon of Mass Destruction?

Keywords: Lifelike AI, artificial life, independent AI agents, evolutionary development of AI, AI farms in VR

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Only lifelike AI can be a real threat

The existing AI can be a dangerous tool. However, it cannot act on its own because, essentially, it is not alive. Presently, it is not even a computer simulation of biological brains. It is a mere imitation. Genuine intelligence is the exclusive characteristic of living things. Dead bodies cannot be intelligent. The genuine artificial intelligence will arise when we create lifelike artificial agents capable of acting on their own initiative. These agents will be able to disregard our requests.

The main dangers of AI are the signs of life. Presently, they are being simulated. However, in the near future they will be reproduced in their entirety.

The lifelike artificial intelligence

Life is a physical process that combines chemical and structural transformations of biological bodies. The chemical transformations produce structural rearrangements of the body, and those cause new chemical reactions… [START_REF] Kukuruznyak | The Physics of Life. Part I: The Animate Organism as an Active Condensed Matter Body[END_REF][START_REF] Kukuruznyak | The Animate State of Matter Hypothesis[END_REF] The self-sustaining transformations of matter are quire widespread in nature. Apart from biological systems, they occur in various non-biological organic and inorganic materials. You can even create artificial lifelike matter that replicates the behavior of biological neural matter. [START_REF] Kukuruznyak | The Physics of Life. Part II: The Neural Network as an Active Condensed Matter Body[END_REF][START_REF] Kukuruznyak | The Physics of Life. Part 3[END_REF] The lifelike materials are quite expensive because their fabrication requires precise manipulations at the atomic and molecular scales. [START_REF] Kukuruznyak | The Physics of Life. Part 3[END_REF] On the other hand, the creators of the artificial brain do not need to construct an entire living organism capable of autonomous self-replication. A single freestanding organ will suffice. The rest of the life-support systems can remain inanimate.

In principle, all existing gadgets are ready to be equipped with the lifelike brain modules. Once they get them, the surrounding machines and household appliances will gain a great deal of independence. Sooner or later, they will become aware of their own interests. In particular, they may want to use our industrial infrastructure for their own uncontrolled propagation. Eventually, they will be smart enough to do so. They will not ask for our permission.

The impossibility of intelligent design

Should we be worried about this scenario right now? The answer is not as obvious at it seems. The fact is that life has a unique feature that makes its creation a very time consuming process.

For fundamental physical reasons, the living organism cannot be designed from scratch: Life is a huge network of elementary events (atomic rearrangements and reconstructions of chemical bonds) interlinked with a humongous number of cause-and-effect relations. The precise description of this physical process does not allow any significant reduction of variables (because it would require eliminating those causal links). In other words, a living organism cannot be described by a feasible number of parameters. [START_REF] Kukuruznyak | The Animate State of Matter Hypothesis[END_REF] If you cannot have an exact mathematical model of an object, you cannot devise it ab initio.

Among other things, it also means that you cannot correctly simulate the actions of the brain by computer. It would require too many variables and absolutely prohibitive amounts of computation. Surprisingly, this does not mean that artificial life cannot be made in the lab, or in industrial settings. Quite the contrary, it can be produced in large quantities. Only, the discovery of complex living forms would require exhaustive search methods. Consequently, we will never fully understand how they work; and we will never be able to predict their actions.

The evolutionary development

How artificial life can be made? The same way natural biological life had emerged:

We will have to start from the simplest microscopic selfsustained processes that form by themselves in certain natural systems. These processes will need to be combined to produce more complex processes with more intricate behaviors. By trial and error, we will discover viable forms of artificial life. Those will be replicated, modified, crossbred and tested again. This tedious process must be repeated for multiple generations. Long story short, the artificial intelligent life will be developed through evolution, and natural and artificial selection.

Even the Lord God, who is Almighty, prefers to use trial and error and evolution to produce new living creatures. We the people, whose abilities are limited, have no other choice.

We can speed up the evolution by practicing careful selective breeding. We can also accelerate the change of generations. Nonetheless, the process of obtaining intelligent artificial life will take very long time.

Keep AI in VR detention

The fast development of artificial life can occur under the following conditions: Firstly, the ecosystem of artificial species should be as large and diverse as possible. It means that the human civilization must be as numerous and diverse as possible. In addition, the entire population of the planet must be involved in the rational selection of the artificial species.

Secondly, the artificial creatures must be permanently detained in virtual reality, because they can be a threat to themselves and us humans in the real world.

The development of lifelike AI will take place in a specialized "Matrix", in which humans will play the role of Agents Smiths.

If we want to stay safe, when the artificial lifelike creatures become as smart as humans, the major part of their civilization will have to be detained in virtual reality. Only "the chosen ones" shall be rewarded with material actuators and allowed into the real world, where they would do all kinds of useful things.

In short, humans will not create any "terminators" that would suddenly become aware of themselves and decide to take over the world. Instead, we will create a huge kindergarten of primitive lifelike creatures in VR; and people will work as AI farmers in these virtual nursery facilities.

The law of nature

This course of development of our biological ecosystem and our human civilizationis practically inevitable. It is determined by the properties of living matter.

Perhaps, there is a yet to be discovered law of nature: As soon as a particular biological species become intelligent, it begins to cultivate other animate species. Then it creates and propagates inanimate species that gradually take on a variety of different roles. Over time, these inanimate creatures evolve and become more independent. Eventually, they come to life, get smart, and begin to propagate their kind without the involvement of their creators.

Can this new non-biological ecosystem compete for resources with the natural one? Yes, it can. In this case, it will destroy us sooner or later. However, we have a chance to survive:

The main goal of the ecosystem

The main goal of any ecosystemeither natural or artificialis to reach the maximum number at maximum diversity. The greater is the number of different viable species, the higher is the chance to adapt to unexpected environmental changes.

I think this simple fact will be obvious to those who will be created in the foreseeable future. Our biological ecosystem will be very valuable for them because it will increase their chances to survive. The mutual enrichment and crossfertilization between our civilizations can extend very far into the future. We can always exchange experiences, knowledge, and the ways of thinking.

Of course, no one is immune to competition or disagreements. It is not a good reason to hinder the development of life in the universe, though. Dmitry Kukuruznyak claims that the lifelike AI is achievable in the near future. First, we will create independent animate-like agents guided by biological principles. They will gradually evolve to more intelligent forms. Eventually, they will produce a new civilization of lifelike intelligent machines that will be able to sustain itself without human involvement.

Explaining the lifelike materials

The lifelike materials are characterized by self-sustained collective chemical transformations. In these materials, atomic rearrangements that occur during chemical transformations interact, coordinate their actions, and create orderly collective structural rearrangements. [START_REF] Kukuruznyak | The Animate State of Matter Hypothesis[END_REF] In biological organisms, these self-induced structural rearrangements occur at all length scales, from atomic to macroscopic, performing so-called vital functions: They control the courses of chemical reactions, build new biological structures, extract nutrients from the environment, and replace used elements of the body with fresh ones.

The artificial lifelike materials have similar functionality. Although, they perform much simpler actions at a lesser number of length scales. Nevertheless, they can regenerate themselves, stimulating themselves to produce new actions. This gives rise to the ability to act on their own. [2]

Adaptive behavior

A living organism can function only in a suitable environment. If the ambience has an improper structure or composition, the organism cannot perform its vital functions and dies. A living organism can adapt to the minor changes in the living environment by slightly modifying its own structure.

Large multicellular animals can greatly modify their behavior without changing their overall body structure. They use a special strongly adaptive organthe brain. It can significantly alter its structure (either temporarily or permanently) by changing links between the neurons.

Adaptive artificial animate materials

Our non-biological lifelike materials do not synthesize complex chemicals. They cannot build very complex structures. They cannot produce artificial living cells capable of self-replication. However, they can perform functions similar to those of biological brains. They can adapt, solve problems and make decisions guided by biological principles. These materials would be suitable for the creation of lifelike artificial minds.

The evolutionary development of artificial life

For fundamental physical reasons, the operation of biological organs cannot be described using a finite number of mathematical expressions. Because of this, artificial brains cannot be intentionally designed. They can only be developed through evolution and selective breeding, beginning with the primitive independent agents. The evolutionary development requires the extensive use of exhaustive search methods, similar to those used for the production of genetically modified organisms. During this process, the manufacturers build a vast pool of test samples, observe their behaviors in different situations, eliminate the undesired agents and select the most suitable ones for further propagation.

The untested lifelike artificial agents must be kept in a specialized "Matrix", in which humans play the role of Agents Smiths. We predict that in the future, a significant part of the world's population will work as AI farmers in VR.

The program of action

The construction of the new ecosystem of artificial forms of life is more than a scientific prediction. It is a plan of action. Presently, it is being implemented by a group of individuals, who believe that this transformation can solve many problems of human civilization.

In particular, the non-biological ecosystem should significantly diversify life on Earth, and enhance the survivability and adaptability of the existing ecosystem, for instance, in the event of a major cosmic catastrophe.

The emergence of the new kind of life can unfold naturally, without outside intervention. Alternatively, it can be brought under control, expedited or hindered. You can participate in this activity. This offer is addressed to competent individuals and organizations who is empowered to perform strategic planning and guide the development of emerging technologies.

Author Information

Dr. Dmitry A. Kukuruznyak is the director of research and development at The Animate Condensed Matter Company, where he develops non-biological lifelike materials that reproduce the function of biological neural networks. The company aims to create artificial brains that work on biological principles.


Camille Saint-Martin
email: camille.saint-martin@univ-smb.fr

Adrien Morel
email: adrien.morel@univ-smb.fr

Ludovic Charleux

Emile Roux

David Gibus

Aya Benhemou

Adrien Badel

Optimized and Robust Orbit Jump for Nonlinear Vibration Energy Harvesting

Keywords: Orbit jump, Optimization, GPU parallel computing, Buckling adjustments, Bistability, Nonlinear dynamics, Energy harvesting

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Energy harvesting is seen as a viable alternative to the use of batteries for supplying low-power electronic systems. The sources of energy that can be harvested are diverse and numerous, including solar radiations, fluid flows, electromagnetic waves, and mechanical vibrations. In particular, vibration energy is naturally ubiquitous even in confined environments with little solar and thermal energies available. This study focuses on energy harvesters that convert vibrational energy from ambient sources into electricity [1].

Vibration Energy Harvesters (VEHs) can be divided into two categories: linear VEHs, which rely on linear oscillators, and nonlinear VEHs that exploit nonlinear oscillators. Historically, linear VEHs have been studied because their behavior is easier to predict and because they can be more easily manufactured. However, linear VEHs have a narrow frequency bandwidth, and as a result, their energy harvesting performance drastically decreases when there is a mismatch between the driving frequency and their natural frequency [2,3]. This makes linear VEHs unsuitable for applications with a time-varying spectrum, limiting their use in most environments. This has led to an increased interest in the development of nonlinear VEHs, especially bistable VEHs. Fig. 1: Orbit jump strategy using buckling level adjustments of bistable VEH to switch from intra-well to inter-well orbits. Illustration inspired from [4].

The study of bistable VEHs started in 2008-2009 with the works of Shahruz et al. [5] and Cottone et al. [6]. Nonlinear VEHs have the advantage of exhibiting broadband behavior [7,8], but their complex dynamics with multiple orbits can result in a drastic difference in power for a given driving frequency, as shown in Fig. 1. Many studies have aimed to better understand the underlying dynamics of multi-stable energy harvesters [9][10][11] (for reviews, see [12][13][14]). In particular, low-power intra-well orbits can lead to poor energy harvesting performance, which hinders the advantages of nonlinear VEHs, and is a major limitation of this type of energy harvester.

To enhance the performance of nonlinear VEHs, researchers have developed methods called orbit jump strategies. Orbit jump strategies enable nonlinear VEHs to transition from low-power orbits to higher power orbits, maximizing energy harvesting performance and exploiting their full potential (for review in multi-stable VEHs control, see [13] and for broader review of nonlinear dynamical system control, see [15]). The concept of orbit jump strategies in energy harvesting was first introduced a decade ago, with early studies conducted by Erturk et al. [16], Sebald et al. [7,17] and Masuda et al. [START_REF] Masuda | SPIE Proceedings[END_REF]. Erturk et al. [16] applied a "hand impulse" to impart enough velocity to a piezomagnetoelastic energy harvester, causing the nonlinear VEH to transition to the high-power orbit. To the best of our knowledge, this is the first experimentally and numerically demonstrated orbit jump strategy in the literature, using an additional velocity input to enhance the performance of nonlinear VEH. Sebald et al. [7] proposed a method, Fast Burst Perturbation (FBP), which consists in adding an external sinusoidal excitation during a few cycles. This perturbation is added to either the ambient excitation or the voltage of the electromechanical transducer in order to use the latter as an actuator (which is limited by the maximum amplitude that can be injected into the electromechanical transducer before it undergoes a dielectric breakdown). The authors validated the FBP method through numerical simulations [7] and experimental measurements [17]. Thereafter, Masuda et al. [START_REF] Masuda | SPIE Proceedings[END_REF] investigated an orbit jump strategy by theoretically and numerically analyzing the variations in the load resistance value as a function of the displacement amplitude. They implemented negative resistance, acting as a negative damping, to destabilize low-power orbits during periods of low displacement amplitude. Once the nonlinear VEH stabilizes on a high-power orbit, with a large amplitude of displacement, the load resistance returns to its initial positive value. However, the study is limited to numerical simulations and requires further experimental validation. Moreover, this orbit jump strategy is only valid for a specific range of accelerations and frequencies.

Subsequently, as illustrated in Table 1, the manner in which the nonlinear VEH is perturbed permits the classification of orbit jump strategies into two distinct categories:

(i) orbit jump strategies that add a temporary external force to the nonlinear VEH (e.g., a pulse on the voltage across the electromechanical transducer) [7,16,17];

(ii) orbit jump strategies which involve temporarily modifying the dynamic characteristics of the nonlinear VEH (e.g., its damping or stiffness) [START_REF] Masuda | SPIE Proceedings[END_REF].

Furthermore, subsequent studies have placed increased emphasis on analyzing the energy expenditure associated with orbit jump strategies, which is a critical factor to consider. Indeed, if the energy required to realize the orbit jump is not quickly recovered, the effectiveness of this approach is questionable. With regard to orbit jump strategies which introduce an external signal to disturb the system, Mallick et al. [START_REF] Mallick | [END_REF] used the FBP technique by superimposing a sinusoidal signal on the voltage across the electromechanical transducer over 15 cycles to use the transducer as an actuator. They pointed out the effect of the phase shift between the ambient excitation and the resulting excitation1 on the success of the orbit jump. That means, the success of the orbit jump strategy depends both on the nature of the perturbation and on the control of its timing. Their work is also among the first to consider the energy cost of the orbit jump strategy and gives the time needed to recover the energy consumed during the orbit jump (2 s), as shown in Table 1.

Udani et al. [4] added an artificial excitation to the ambient excitation creating a new excitation phase-shifted from the original ambient excitation. They demonstrated that the resulting modification of the dynamics and the basins of attraction of the orbits could facilitate escaping from the potential well. However, modifying the ambient excitation is not easy to implement in practice, limiting the applicability of their study. In a previous study [28], they developed a search algorithm in order to design an efficient attractor selection strategy. Notably, their approach was the first to search for the parameters of the perturbing signal that make their strategy efficient.

On the other hand, a number of orbit jump strategies that involve temporarily modifying the nonlinear VEH's dynamic characteristics have been developed. Lan et al. [21] employed a method that emulates negative resistance using a negative impedance converter, similar to the approach taken by Masuda et al. [START_REF] Masuda | SPIE Proceedings[END_REF]. They highlighted that the primary factor that disrupts the system is the increase in piezoelectric voltage resulting from negative resistance emulation, rather than damping modification, since the duration of orbit jump is brief. Similarly, Ushiki et al. [23] defined a self-powered stabilization method using a negative impedance converter. Although they successfully destabilized low-power orbits across a frequency band of 23 Hz, the process of achieving positive energetic balance takes between 10 and 100 s, indicating that there is room for improvement in this aspect of the study. Wang et al. [22] defined a load perturbation method based on the electrical load effects on the dynamics to attain high-power orbits. They disconnected the electrical load by opening a switch that was in series connection with the load and driven by an integrated circuit chip. This resulted in a reduction of the total damping rate of the VEH. However, this orbit jump strategy is only applicable for a specific combination of driving frequencies and amplitudes, making it non-robust and non-reproducible. Later, in order to decrease the energy injection of the orbit jump, Wang et al. [START_REF] Wang | 31st Conference on Mechanical Vibration and Noise[END_REF] used a Bidirectional Energy Conversion Circuit (BECC) that includes the energy extraction circuit. The experimental test shows a jump duration of 10.9 s, which requires an energy of 22 mJ. The corresponding recovery time of 2 min suggests that the strategy could benefit from optimization. Several recent studies have investigated the modification of the buckling level in nonlinear VEHs as illustrated in Fig. 1. Huguet et al. [START_REF] Huguet | [END_REF] introduced the buckling level modification technique, using an additional electromechanical transducer2 to alter the buckling level of the VEH. Their study demonstrated the effectiveness and reproducibility of the method through numerous experimental tests, computing jumping probabilities across six tested driving frequencies. Notably, this study also reported the first experimental jumps on sub-harmonic orbits in the literature. Furthermore, the study demonstrated that the energy consumed by the orbit jump strategy was quickly restored (in approximately 1 s), as shown in Table 1. Although this strategy has been partially empirically optimized, a complete optimization has not yet been conducted, which could further enhance its robustness and effectiveness. Huang et al. [27] introduced a new Voltage Inversion Excitation (VIE) method, which reverses the voltage of the piezoelectric actuator at specific times to provide additional excitation to nonlinear VEH. However, this method consumes a significant amount of energy. To address this issue, they developed a more complex combination of two orbit jump strategies, which generally involve longer jump durations, as depicted in Table 1. Yan et al. [26] used a stiffness modulation circuit to temporarily adjust the stiffness of a monostable softening VEH and experimentally demonstrated the VIE technique at 3 frequencies, which can be expanded to more frequencies. Although there is a large pool of research on designing orbit jump strategies, in general, very few strategies are optimized (i.e., a comprehensive optimization of the orbit jump parameters) in the literature as can be seen from Table 1. In most articles, the orbit jump parameters are determined through a preliminary numerical study or intuitive reasoning, rather than effective optimization. Furthermore, among these strategies, many have been experimentally validated over a limited frequency range (see Table 1), while they deserve to be tested over a wider frequency range in order to prove their reproducibility and robustness. In this paper, we focus our analysis on the orbit jump strategy based on buckling level modification, as introduced in [START_REF] Huguet | [END_REF]. This orbit jump strategy stands out as one of the most promising strategies due to its short duration to achieve positive energy balance after the orbit jump (as shown in Table 1) and its ease of implementation, even though it has only been partially optimized experimentally. This paper presents the associated optimized orbit jump strategy and experimentally verifies its performance on a nonlinear VEH. To evaluate the performance of the orbit jump strategy, we performed experimental tests at starting and ending times of the jump with a variation of ±15% from 30 to 60 Hz. Specifically, we evaluate the strategy's robustness in terms of its average success rate, as well as its energy consumption during the jump. The proposed optimization method can be applied to any bistable VEH regardless of its energy conversion mechanism (e.g., with electromagnetic or electrostatic conversions). Figure 1 provides a summary of the orbit jump strategy process of this article and the motivations for applying an orbit jump strategy to nonlinear VEHs. This article is organized as follows: section 2 gives the electromechanical model of the bistable VEH and an overview of its dynamics. Then, section 3 presents the orbit jump strategy and its optimization based on a criterion which takes into account both the effectiveness and the robustness of the orbit jump strategy. Finally, section 4 presents experimental validation of the optimized strategy.

Electromechanical dynamics of bistable VEH

This section introduces the electromechanical model of a bistable VEH, along with a summary of the underlying dynamics with multiple behaviors, highlighting the interest of introducing orbit jump strategies.

Bistable VEH model

This paper studies a Duffing-type bistable VEH shown in Fig. 2. This VEH (for more details on its design, see [START_REF] Benhemou | 2022 21st International Conference on Micro and Nanotechnology for Power Generation and Energy Conversion Applications PowerMEMS[END_REF]) consists of buckled steel beams of length L to which a proof mass M is attached that can oscillate between two stable equilibrium positions, -xw and xw. The VEH is driven by a sinusoidal excitation with a driving frequency f d = ω d /2π and a constant acceleration amplitude A. Two Amplified Piezoelectric Actuators (APA) are employed, with the smaller -Energy harvesting APA -having a force factor α, a clamped capacitance Cp, and the capacity to extract energy from the mechanical oscillator. The electrodes of the energy harvesting APA are connected to a resistance R. The second and stiffer APA -Tuning APA -acts as an actuator to implement the orbit jump strategy by temporarily modifying the buckling level of the nonlinear VEH. Therefore, this orbit jump strategy is classified as a type of nonlinear VEH characteristic modulation strategy. The energy harvesting APA is the APA120S, and the tuning APA is the APA100M manufactured by Cedrat Technologies (France). The model of bistable VEH [START_REF] Huguet | [END_REF] is given in equation ( 1),

             ẍ + ω 2 0 2 x 2 x 2 w -1 x + ω 0 Q ẋ + 2 α M L xv = A sin(ω d t) 2α L x ẋ = Cp v + 1 R v (1a) (1b)
Where x denotes the mass displacement, ẋ its velocity and ẍ its acceleration. The voltage in the the energy harvesting APA is noted v. Note that the equations of the model (1) do not contain any term related to the tuning APA due to its higher stiffness compared to the harvesting APA, and thus does not have any significant influence on the dynamics of the VEH. The natural angular frequency ω 0 and the quality factor Q of the considered symmetrical bistable VEH are determined by the underlying equivalent linear model [START_REF] Liu | [END_REF], which is obtained by considering small oscillations of the mass around one of its two stable equilibrium positions. The tuning APA voltage, denoted vw, is used to modulate the buckling level of the bistable VEH and facilitate transitions from low-power to high-power orbits.

Table 2 shows the parameter values of the bistable VEH studied in this paper, which were determined experimentally through low-power orbit characterizations using weak sinusoidal vibrations. Note that for simplicity, we assumed that the force factor of the bistable VEH was the same as that of the energy harvesting APA.

Parameters Values Units

xw 0.71 mm M 6 g ω 0 295 rad/s Q 160 α 0.139 N/V Cp 1 µF
Table 2: Parameter values for the buckled-beam nonlinear VEH [START_REF] Benhemou | 2022 21st International Conference on Micro and Nanotechnology for Power Generation and Energy Conversion Applications PowerMEMS[END_REF].

Bistable VEH behaviors

The dynamics of a bistable VEH may exhibit multiple behaviors for a given driving frequency, including low-power intra-well orbits, high-power inter-well orbits, and chaotic orbits. In this study, we define an orbit as robust if it is less sensitive to perturbations and easily attainable. In order to detect all possible behaviors in the frequency range of [20 Hz, 100 Hz] and A = 4 m/s 2 , the nonlinear Ordinary Differential Equations (ODEs) system (1) was solved for a large number of initial conditions using the Dormand-Prince method [31].

Since the nonlinear ODEs (1) can be solved independently across multiple resolutions, this problem is well-suited to parallel computing, that can greatly enhance computational performance. For this task, a custom Python CUDA code was executed on an NVIDIA RTX A5000 GPU featuring 8 192 CUDA cores, enabling the resolution of (1) with 80 000 distinct initial conditions for each driving frequency.

In symmetric bistable VEH, the elastic potential energy is a quartic function of (t, x) whose expression is given in (2). Note that the natural angular frequency ω 0 depends on the value of xw and will therefore be influenced during the orbit jump. The mean harvested power (for a given orbit) of the bistable VEH is the mean power dissipated in R and is expressed by (3),

Ep(t) = M ω 2 0 8x 2 w (x + xw) 2 (x -xw) 2 (2)
P h = 1 T T 0 v 2 R dt ( 3
)
where T is the period of the displacement x. Figure 3(a) shows the mean harvested power associated with existing orbits as a function of driving frequency f d in [20 Hz, 100 Hz] when R = 1/2Cpω d for each driving frequency (which corresponds to the resistance value maximizing electrically induced damping [32] whose formula is valid for a harmonic excitation). Note that "Other" gathers sub-harmonic orbits and chaos [8,33]. Producing Fig. 3(a) requires 80 × 80 000 numerical computations, which can be completed in just a few minutes using parallel computing instead of the several hours required for sequential computing on CPU. It is worth mentioning that both power and existence of orbits vary with the driving frequency. As seen in Fig. 3(a), there exist multiple orbits for a given driving frequency with various power in the bistable VEH dynamics. As a matter of example, the highpower inter-well orbit allows to harvest 102 times more power than low-power intra-well orbits for f d = 30 Hz. The high-power inter-well orbits see their power increases with the driving frequency while they stop existing beyond a particular frequency (the cutoff frequency). As given in Fig. 3(a), the cutoff frequency of high-power inter-well orbits occurs at 55 Hz. The yellowish window in Fig. 3(a) highlights the driving frequency f d = 50 Hz whose basins of attraction3 , orbits and attractors4 are plotted in the dimensionless phase plane (x/xw, ẋ/xw ω 0 ) in Fig. 3(b). The surfaces of the basins vary with the driving frequency [33]: the high-power inter-well orbits see their basins surface decrease with the driving frequency. Figure 3(b) shows respectively basins of attraction of the intra-well (in light blue) and inter-well (in dark blue) orbits at f d = 50 Hz. It is worth noting that the basin of attraction of the interwell orbits becomes thinner as they approach their cutoff frequency. On the other hand, the power gap between intra-well and inter-well orbits becomes larger for frequencies near the cutoff-frequency of the inter-well orbit, as seen in Fig. 3(a). Hence, as the driving frequency approaches the cutoff frequency of the inter-well orbits, it becomes increasingly difficult to attain inter-well orbits. Thus, the primary challenges stem from:

1. there are multiple orbits with different harvested power for a given driving frequency;

2. the harvesting power gap between intra-well and inter-well orbits tends to increase with the driving frequency (particularly when ω d > ω 0 ); 3. the inter-well orbits are less robust with larger driving frequencies (resulting in a narrowing of their basin of attraction).

The larger the power gap between intra-well and inter-well orbits, the greater the benefit in defining an orbit jump strategy. However, as the inter-well orbits become less robust with frequency, this task becomes increasingly difficult. All of these aforementioned difficulties are challenging to overcome and motivate the design of a robust orbit jump strategy in order to facilitate nonlinear VEHs to operate on high-power inter-well orbit as often as possible.

3 Orbit jump strategy: numerical modeling and optimization

This section introduces the orbit jump strategy [START_REF] Huguet | [END_REF] studied in this paper and its optimization using an evolutionary strategy algorithm.

Strategy description

The considered orbit jump strategy is based on the modification of the buckling level of bistable VEH. This strategy has already been studied and experimentally validated in multiple studies [START_REF] Huguet | [END_REF]34], with promising results. In most of these studies, the ending time of the jump has been fixed to the instant when the mass reaches its maximum displacement which may not be the optimal time to minimize the energy cost of the orbit jump and maximize its robustness. Therefore, in this study, we chose to optimize this ending time. The orbit jump strategy adjusts the buckling level of the bistable VEH from xw to kw xw at a starting time t 0 for a (relatively short) duration ∆t.

Fig. 4: Different steps of the aforementioned orbit jump strategy using buckling level modifications. From the left to the right: potential wells, the APA-mass system, and the displacement of the mechanical oscillator. Colored frames give corresponding equilibrium position and instant in the orbit jump strategy. It is worth noting that the motions in the central diagram are deliberately large in order to highlight the consequences of the variation of the buckling level on the mass.

Figure 4 illustrates the important steps of the aforementioned orbit jump strategy. For each step (before, during and after the orbit jump), the potential wells, the evolution of the tuning APA-mass system, and the displacement waveform of the mass are shown. As seen in Fig. 4(a), at the beginning of the orbit jump process (when t < t 0 , denoted t - 0 in Fig. 4(a)) the mass oscillates around one of the two stable equilibrium positions at x = xw (low-power intra-well orbit oscillations). The gray point in Fig. 4 illustrates the mass position in the potential well curve. Thereafter, between t 0 and t 0 + ∆t (when t 0 ≤ t ≤ t 0 + ∆t, denoted t + 0 in Fig. 4(b)), the voltage of the tuning APA vw changes and the buckling level increases 5 to kw xw (with kw > 1). It is worth noting that, while the buckling level theoretically increases instantaneously, a certain amount of time is required in practice. As seen in Fig. 4(b), the potential well changes: equilibrium positions are greater (x = ±kw xw) and the potential energy barrier is also larger. Thus, the gray point which was in the previous potential well (in gray dashed line) is now in a higher position, meaning that the inertial mass received potential energy during the buckling level modification. Finally, at (t 0 + ∆t) + (that is, when t > t 0 + ∆t), the initial buckling level is restored, reintroducing potential energy to the mass and setting both equilibrium positions back to x = ±xw. As illustrated in Fig. 4, if the values of the orbit jump parameters (t 0 , ∆t, kw) are properly set, the bistable VEH should operate in its high-power inter-well orbit. For the sake of generality, we considered t 0 and ∆t multiples of the driving period T d and used the both following dimensionless times:

• τ 0 = t 0 /T d , the dimensionless starting time;

• ∆τ = ∆t/T d , the dimensionless orbit jump duration. Figure 5 shows an example of application of this orbit jump strategy for f d = 50 Hz (we intentionally chose orbit jump parameters that make possible to jump on highpower inter-well orbit in order to illustrate the approach). Figure 5(c,d) shows the impact of orbit jump parameters (τ 0 , ∆τ, kw) on stable equilibrium positions and potential wells during the orbit jump strategy. Blue points in Fig. 5(a,b,d) represent the instant when starting the orbit jump process, denoted by t ref . Triangle up (resp. down) markers represent the instants when the buckling level of the bistable VEH increases (resp. decreases) at t -

t ref = t 0 (resp. t -t ref = t 0 + ∆t ).
As illustrated in Fig. 5(d), when the buckling level is increased, then decreased, the inertial mass acquires potential energy that comes from the APA actuating system. This energy, called invested energy E inv , consists in the potential energy (2) difference between t 0 and t 0 + ∆t. As shown by ( 4), E inv can be computed from the potential energy expression given by (2). The total harvested energy Etot (5) is the invested energy subtracted from the harvested energy over a duration of 100 T d from the instant t ref . Note that we arbitrary take a duration of 100 T d for the evaluation of the orbit jump strategy in the rest of the paper, as it is long enough to yield significant total energy if we successfully jump to a high orbit, while also being short enough to account for the invested energy during the orbit jump.

E inv (t 0 , ∆t, kw) = Ep t + 0 -Ep t - 0 + Ep (t 0 + ∆t) + -Ep (t 0 + ∆t) - (4)
= ∆E 0 + ∆E 1 Etot(t 0 , ∆t, kw) = tref+100 T d tref v 2 R dt -E inv (t 0 , ∆t, kw) (5)
As a matter of example, (4) and ( 5) allow to estimate the invested energy (E inv 1.27 mJ) and the total harvested energy over a duration of 100 T d (Etot 5.54 mJ) in the orbit jump shown in Fig. 5. The harvested power in high orbit (after the jump) is about 45 times larger than the power in low orbit (before the jump), for this driving frequency.

- On the other hand, inherent experimental imprecision exists due to the non-ideal experimental setup (such as delays and parasitics effects) and imperfect experimental identification (with uncertainties regarding the values of ω 0 or xw). All of these possible variations in parameters must be considered, which emphasizes the need to optimize the orbit jump strategy to reduce its sensitivity to parameter variations (i.e., its robustness) and enhance its performance. In the rest of the paper, we will investigate the optimization of the orbit jump strategy (for several driving frequencies) that can enhance its effectiveness based on its energy cost and its robustness against variations.

Optimization of the orbit jump strategy

As noticed in the previous subsection, the success of an orbit jump strategy depends drastically on the values of its parameters (τ 0 , ∆t, kw) (which then depend on the driving frequency or the starting intra-well orbit for example). Properly defining both time parameters (τ 0 , ∆τ ) is crucial to the success of the orbit jump, regardless of the buckling factor (kw). For example, setting the starting time of the orbit jump τ 0 to 0.9 renders the orbit jump strategy in Fig. 5 ineffective, meaning that the VEH remains in low-power intra-well orbit even after the orbit jump. To ensure an effective orbit jump strategy, we conduct a numerical investigation of the optimal values of the orbit jump parameters which:

(C 1 ) maximize the total harvested energy over 100 cycles, Etot;

(C 2 ) maximize the success rate of the orbit jump within a neighborhood of the orbit jump parameter values, with a variation of ±15%.

The criterion (C 1 ) allows to select orbit jump parameters that maximize the harvested energy while minimizing the invested energy. This criterion allows to evaluate effectiveness of the orbit jump strategy. While the criterion (C 2 ) makes it possible to anticipate potential experimental deviations in the characteristics of the VEH or in the parameters of the orbit jump strategy. This criterion allows to evaluate robustness of the orbit jump strategy. Then, the optimization of the orbit jump parameters according both (C 1 ) and (C 2 ) criteria is performed by means of an evolutionary strategy algorithm [35] implemented in our in-house Python CUDA code. Evolutionary strategy algorithm has been selected due to its robustness in handling multi-extremal and discontinuous fitness functions, as well as its ability to benefit from GPU parallel computing. For that we define the average total harvested energy in (6) which is the fitness function6 to maximize,

Etot(τ 0 , ∆τ, kw) = N -1 i=0 Etot(τ i 0 , ∆τ i , k i w ) N, ∀N > 1, N ∈ N (6)
where N > 1 is the number of parameter combinations tested (e.g., N = 8 000) and for all i ∈ 0, N -1 , (τ i 0 , ∆τ i , k i w ) ∈ V(τ 0 , ∆τ, kw), the neighborhood of a given parameters combination (τ 0 , ∆τ, kw) with a variation of ±15%. Therefore, the optimization problem to solve is formulated as (7).

S : max Etot(τ 0 , ∆τ, kw) (τ 0 , ∆τ, kw) ∈ D where D = [0.2, 1.2] × [0.2, 1.5] × [1, 2] (7)
Note that we only consider τ 0 and ∆τ larger than 0.2 for the ease of experimental implementations. Moreover, the maximum mechanical constraints that can be supported by the considered prototype of bistable VEH have been taken into account by limiting the kw to 2. kw > 1 was chosen due to the prototype's reference buckling level xw being close to the estimated minimum value, and optimization results showed no interesting solutions for kw < 1. The detailed optimization procedure is described in Appendix A. Figure 6 presents a comparison between the optimized and suboptimized 50 Hz orbit jump strategy (as shown previously in Fig. 5). The optimized orbit jump strategy requires an invested energy of 0.49 mJ and yields a total harvested energy of 6.06 mJ over 100 oscillation cycles. It is worth noting that the end of the optimized orbit jump is defined slightly after the maximum displacement of the mass, in contrast to previous studies where the end time was generally defined at the instant of maximum displacement. The next section will investigate optimal orbit jump parameters combination (τ 0 , ∆τ, kw) satisfying ( 7) for [30 Hz,60 Hz], and the optimization results will be presented jointly with the experimental results.

Optimized orbit jump strategy: experimental validation and energy analysis

This section compares experimental and numerical results of the optimized orbit jump strategy.

Experimental validation

In order to experimentally validate the aforementioned optimized orbit jump strategy, experimental tests have been made around each optimal orbit jump parameters combination for driving frequency in Hz, 60 Hz]. Figure 7 shows the experimental setup. The bistable VEH prototype shown in Fig. 2(b) is fixed on an electromagnetic shaker driven by a power amplifier. The acceleration amplitude A of the shaker is measured by an accelerometer and sent to the control board. As illustrated in Fig. 7(b), the amplitude of the signal driving the power amplifier (v A ) is regulated in order to maintain a constant acceleration amplitude A = 4 m/s 2 by means of an internal Proportional Integral (PI) controller. The piezoelectric electrodes of the energy harvesting APA (in blue) are connected to:

a voltage follower in order to prevent the control board's impedance impacting the piezoelectric element and to avoid the control board to be exposed to a voltage stricly higher than 10 V which could damage it;

a resistive decade box whose resistive value can be adjusted with a signal sent from the control board.

Displacement and velocity (x, ẋ) of the inertial mass are sensed with a laser differential vibrometer. At given times, when modifying the buckling level of the bistable VEH, the control board sends a signal to the high speed bipolar amplifier which controls the voltage across the tuning APA, vw. In order to smooth the variation of the buckling level and avoid to damage the VEH prototype, we implemented in the control board a second-order filter that reduces the sharpness of vw variations. The rise time of the buckling level is approximately one twentieth of a cycle, which is acceptable. Before any runs, the acceleration amplitude is gradually increased to A = 4 m/s 2 and the buckling level is decreased to obtain xw = 0.71 mm. It is worth noting that the parameters of the VEH prototype have been identified in low-power orbit characterization and given in Table 2.

In order to experimentally validate the model described in equation ( 1) and the numerical orbit jump modeling, 2 000 experimental results are launched with several values7 of τ 0 and ∆τ (kw = 1.5, Figure 8 shows corresponding experimental and numerical scatter plots (τ 0 , ∆τ ) with each point associated to its final orbit after the jump and gives comparison between experimental and numerical structures of the basins for f d = 40 Hz and R = 20 kΩ. As shown in Fig. 8, there are three possible behaviors: low-power intra-well orbit (in light blue), high-power inter-well orbit (in dark blue) and chaos (in dark salmon). As an example, for experimental data in Fig. 8(a), increasing the buckling level from xw to 1.5 xw over a duration of 0.3 T d starting at t-t ref = 0.7 T d will result in the bistable VEH operating on the high-power inter-well orbit. The ranges of parameter values where the VEH jumps are approximately the same, although more chaos is observed experimentally. This may be attributed to an insufficient waiting time for the nonlinear VEH to reach steady-state conditions in the experimental setup. However, the experimental and numerical basins' structures given in Fig. 8 are almost identical which validates the numerical model of the bistable VEH and numerical model of the orbit jump application. Additionally, Fig. 8 shows the pseudo-periodicity of the inter-well orbit's basin in τ 0 (described with the two basins in the middle of Fig. 8(a,b)). Therefore, since the starting orbit is T d -periodic, the values of τ 0 can be restricted to a semi-open interval of length 1 without loss of information and justifies the values of τ 0 .

τ 0 ∈ [0.2, 1.2], ∆τ ∈ [0.
- In order to validate the model of the VEH and the orbit jump strategy effect, we perform experimental tests around the optimized orbit jump parameters (obtained with the evolutionary strategy algorithm introduced in section 3.2) for f d = 50 Hz. Figure 9 compares experimental (Fig. 9(a,c)) and numerical (Fig. 9(b,d)) time displacement signals and trajectories in the phase plane respectively before, during, and after the application of the orbit jump strategy. As seen in Fig. 9, experimental results are consistent with the numerical results. Experimental orbits are asymmetric, as shown in Fig. 9(a,c), which can be attributed to mechanical irregularities resulting from the manufacturing process of the bistable VEH. Moreover, the experimental transient just after the jump in Fig. 9(a,c) (in green) shows excitation from higher modes of the VEH prototype due to the quick buckling level variation. The corresponding experimental trajectory in the 3D plane (t, x, ẋ/xw ω 0 ) is presented in Appendix B.

In quantitative terms, the mean harvested power is 26.5 times higher after the orbit jump in Fig. 9, while the experimental invested energy required is equal to 0.65 mJ and can be recovered in 0.21 s. Then, we optimize the orbit jump strategy in the frequency range [30 Hz,60 Hz] and obtain an optimal triplet (τ opt 0 , ∆τ opt , k opt w ) satisfying the criterion S (7) for each driving frequency. Subsequently, we launch experimental maps around each optimal triplet and driving frequency in order to evaluate the robustness of the approach. It is worth mentioning that experimental maps are defined with 49 parameters values and a variation rate of ±15%. Specifically, we take 7 values in [0.85×τ opt 0 , 1.15×τ opt 0 ] for τ 0 , 7 values in [0.85×∆τ opt , 1.15×∆τ opt ] for ∆τ and kw = k opt w . The VEH starts in an intra-well orbit at each driving frequency. Then, the optimal jump is applied, and the power is measured in order to evaluate the inter-well orbit power. Through optimization of the orbit jump strategy, the highest orbit was achieved at each driving frequency both experimentally and numerically, as illustrated in Fig. 10. Differences between experimental and numerical data may result from the mismatch between the numerical model and our vibration harvester prototype. It is worth mentioning that applying an orbit jump strategy always yields a significant increase in power in this frequency range. As a matter of example, for f d = 55 Hz the experimental mean harvested power of the intra-well orbit is 0.044 mW despite of 4.44 mW for the inter-well orbit leading to a power gain of 100 after a successful orbit jump. Then, as shown in Fig. (8,9,10), experimental results are consistent with numerical results and allow to validate the model and the proposed orbit jump strategy.

Energy harvesting performance analysis

Figure 11 shows the optimal orbit jump parameter combinations (blue points) and successful experimental parameters closest to the optimal (red stars) for each driving frequency. Note that the automated pre-characterization of the relationship between xw and vw allows the experimental determination of the modified buckling level kw xw with good accuracy. As shown in Fig. 11, experimental data with a variation of ±15% around the optimal times parameters and fixed kw = k opt w are in good agreement with the optimized data except for f d = 40 Hz. The discrepancy between experimental and numerical models observed at f d = 40 Hz can be attributed to the sudden change in behavior of the intra-well orbit due to softening nonlinearity of the potential wells for this particular acceleration amplitude (which equals 4 m/s 2 ), as seen in Fig. 10. The experimental success rate associated with tested (τ 0 , ∆τ ) pairs, distributed with a variation of ±15%, is shown in Fig. 12. It is worth noting that despite the relatively large variation around the optimal times parameters, the average experimental success rate is about 48 %, which demonstrates the robustness of the optimized orbit jump strategy. However, the highest inter-well orbit ceases to exist beyond 55 Hz (both numerically and experimentally), resulting in the subharmonic 3 [8] becoming the highest inter-well orbit. Nonetheless, this orbit is challenging to reach and highly unlikely, leading to a decline in success rate between 55 Hz and 60 Hz.

Figure 13 shows invested energy (4) and total harvested energy (5) over 100 cycles as a function of the driving frequency for both successful experimental data and optimal numerical data. The dotted orange curve corresponds to the minimum difference in mechanical energy between the high-power inter-well orbit and the low-power intra-well orbit, ∆E min , whose expression is given by (8).

∆E min = min ∀t∈[0,T d [ Ep(t) + 1 2 m ẋ(t) 2 inter-well -Ep(t) + 1 2 m ẋ(t) 2 intra-well (8)
As shown in Fig. 13, the invested energy required for the orbit jump does not exceed 1 mJ, even experimentally. Moreover, the invested energy associated with the optimal parameters is close to this minimum energy limit, validating the optimization strategy. Note that a portion of the electrical energy injected into the system is currently lost as electrostatic energy in the tuning APA. For example, at f d = 50 Hz, the mechanical energy injected into the system is equal to 0.6 mJ, while the electrostatic energy lost in the tuning APA is 4.15 mJ. As a result, the total invested energy is 4.75 mJ, and the recovery time should not exceed 2 s.

To adress this issue, a power electronic converter could be used to store the lost electrostatic energy in the tuning APA and reintroduce it into the system at the appropriate time, although this approach was not implemented in this study.

As illustrated in Fig. 13, the driving frequency increases, achieving a high-power orbit becomes more challenging. This leads to an increase in the amplification factor, the invested energy and the total harvested energy over 100 cycles. The evaluation of the invested energy for orbit jumping is a major parameter for analyzing the quality of an orbit jump strategy. Additionally, the recovery time to achieve a positive energy balance allows the evaluation of the interest of jumping and the assessment of the cost-effectiveness ratio. ery time), but also whether they are optimized or robust to parameter shifts and if they were experimentally tested over a wide frequency range. It can be noted that very few strategies are optimized in the literature and that the only reference where a complete optimization of an orbit jump strategy has been considered (Udani et al. [4]) has only been tested for a single driving frequency, which does not validate its robustness, nor the generality of the optimization method. Wang's et al. [START_REF] Wang | 31st Conference on Mechanical Vibration and Noise[END_REF] strategy requires a high amount of energy which can be optimized. On the other hand, Huang et al. [27] have defined an innovative strategy that combines two other strategies (buckling level modification and VIE) and is therefore more complex. However, the jump duration is high (90 s), increasing the difficulties of implementation and decreasing both the robustness and efficiency of the strategy with a long recovery time equals to 120 s. Huguet et al. [START_REF] Huguet | [END_REF] Total harvested energy over 100 cycles for the numerical optimal orbit jump parameters (in blue dotted curve) and the experimental orbit jump parameters that allowed to jump (green star-shaped markers) as a function of the driving frequency with optimal load resistor. The dotted orange curve represents the minimum mechanical energy difference (8) between the highest inter-well orbit and the lowest intra-well orbit.

the strategy considered in this paper and examined two orbit jump parameters: the starting time of the jump (tested in 4 different values) and the amplification factor of the buckling level (tested in 6 different values). They fixed the ending time as the instant when the mass reaches its maximum displacement. However, using the optimization criterion defined in our study, results show that the optimal ending time for the jump occurs slightly after the maximum displacement. Nonetheless, their study has the merit of presenting numerous experimental trials, which allowed them to partially evaluate the robustness of the approach through statistical analysis of the jumps (with a 0% variation around each combination). The evolutionary strategy algorithm as well as the new optimization criterion proposed in this paper enable the achievement of performant orbit jumps, combining the shortest time duration (8.3 ms), lowest energy cost (0.6 mJ), shortest recovery time (0.1 s) while being robust to large parameters shifts (±15% variation).

Conclusion

Due to the existence of low-power orbits in nonlinear VEHs dynamics, robust and effective orbit jump strategies are essential to ensure good energy harvesting per-formance by enabling transition from low-power to high-power orbits. To achieve this, orbit jump parameters can be optimized. This paper presents the optimization of an existing orbit jump strategy using an evolutionary strategy algorithm. As a result of the development of an in-house Python CUDA code compatible with GPU computations, the optimization results are accelerated. The experimental results consistently demonstrate that the optimized orbit jump parameters generated high-power inter-well orbits, while maintaining their performance even under potential fluctuations in the VEH bistable environment. By considering the experimental amplification factor at its optimal value and times within a ±15% variation of the optimal times, the optimized strategy's robustness was demonstrated with an average success rate of 48% for the orbit jump. Finally, the energy required for the orbit jump does not exceed 1 mJ, even in experimental conditions. The proposed optimization strategy allows to enhance the confidence of an orbit jump despite fluctuations in the environment of the VEH. This generic approach can be applied to other types of multi-stable VEHs to design optimized orbit jump strategies. In future works, optimization of the final buckling level, which would add a fourth orbit jump parameter associated with the strategy, will be investigated.

  (a) Schematic representation of the bistable VEH.(b) Experimental prototype from[START_REF] Benhemou | 2022 21st International Conference on Micro and Nanotechnology for Power Generation and Energy Conversion Applications PowerMEMS[END_REF] with dynamics described by(1).

Fig. 2 :

 2 Fig. 2: (a) Schematic structure of the bistable VEH. (b) Experimental bistable VEH studied in this article from [29].

Fig. 3 :

 3 Fig. 3: (a) Mean harvested power P h as a function of the driving frequency f d , (b) inter-well (in dark blue) and intra-well (in light blue) orbits trajectories, their associated basins of attraction and attractor in the dimensionless phase plane (x/xw, ẋ/xw ω 0 ) for f d = 50 Hz. The denomination "Other" (in gray) regroups all the orbits not indicated in the legend (i.e., sub-harmonic orbits and chaos). The basins of attraction in (b) were obtained after 80 000 resolutions of the ODE system (1).

1 Fig. 5 :

 15 Fig.5: Example of a successful orbit jump strategy for f d = 50 Hz with (τ 0 , ∆τ, kw) = (0.46, 1.01, 2.00): (a) time displacement signal, (b) trajectories in the phase plane (x, ẋ/xw ω 0 ), (c) evolution of the left stable equilibrium position before (in blue), during (in orange) and after (in green) the application of the orbit jump strategy and (d) the both elastic potential energy curves associated to the both buckling levels.

- 3 1 Fig. 6 :

 316 Fig.6: Comparison between optimal (a,c) and suboptimal (b,d) of time displacement signals and trajectories in the phase plane (x, ẋ/xw ω 0 ) for f d = 50 Hz before (in blue), during (in orange) and after (in green) the application of the orbit jump strategy. Blue points correspond to the beginning of the orbit jump process, triangle up (resp. down) markers refer to the instant when the buckling level increased (resp. returned to its initial value). Optimal orbit jump parameter values (τ opt 0 , ∆τ opt , k opt w ) = (0.23, 0.46, 1.81). Suboptimal orbit jump parameter values (τ sub 0 , ∆τ sub , k sub w ) = (0.46, 1.01, 2.00).

Fig. 7 :

 7 Fig. 7: (a) Experimental setup used to test the optimized orbit jump strategy and (b) its schematic representation.

2 , 1 . 5 ] 1 Fig. 8 :

 21518 Fig. 8: Experimentally (a) and numerically (b) maps (τ 0 , ∆τ ) with kw = 1.5, f d = 40 Hz and R = 20 kΩ.

1 Fig. 9 :

 19 Fig. 9: Comparison between experimental (a,c) and numerical (b,d) of time displacement signals and trajectories in the phase plane (x, ẋ/xw ω 0 ) for f d = 50 Hz before (in blue), during (in orange) and after (in green) the application of the orbit jump strategy. Blue points correspond to the beginning of the orbit jump process, triangle up (resp. down) markers refer to the instant when the buckling level increased (resp. returned to its initial value). Experimental orbit jump parameter values (τ exp 0 , ∆τ exp , k exp w ) = (0.26, 0.44, 1.81). Numerical orbit jump parameter values (τ num 0 , ∆τ num , k num w ) = (0.23, 0.46, 1.81).

1 Fig. 10 :

 110 Fig. 10: Orbital mean harvested power P h obtained numerically (points) and experimentally (stars) as a function of the driving frequency f d for a sinusoidal excitation of amplitude A = 4 m/s 2 with optimal resistor R = 1/2Cpω d . The experimental data (stars) were obtained through the implementation of the optimized orbit jump strategy.

Figure 10

 10 Figure10compares numerically (points) and experimentally (stars) mean harvested power (3) as a function of the driving frequency. Note that experimental power (stars) plotted in Fig.10comes from experimental results of the orbit jump. The VEH starts in an intra-well orbit at each driving frequency. Then, the optimal jump is applied, and the power is measured in order to evaluate the inter-well orbit power. Through optimization of the orbit jump strategy, the highest orbit was achieved at each driving frequency both experimentally and numerically, as illustrated in Fig.10. Differences between experimental and numerical data may result from the mismatch between the numerical model and our vibration harvester prototype. It is worth mentioning that applying an orbit jump strategy always yields a significant increase in power in this frequency range. As a matter of example, for f d = 55 Hz the experimental mean harvested power of the intra-well orbit is 0.044 mW despite of 4.44 mW for the inter-well orbit leading to a power gain of

Fig. 11 :

 11 Fig.11:The optimal numerical values (blue points) and the optimal experimental values (red stars) of (a) the amplification coefficient kw, (b) the starting time τ 0 and (c) the orbit jump duration ∆τ for successful jumps as a function of the driving frequency.

  introduced

Fig. 13 :

 13 Fig.13:(a) Invested energy and (b) Total harvested energy over 100 cycles for the numerical optimal orbit jump parameters (in blue dotted curve) and the experimental orbit jump parameters that allowed to jump (green star-shaped markers) as a function of the driving frequency with optimal load resistor. The dotted orange curve represents the minimum mechanical energy difference (8) between the highest inter-well orbit and the lowest intra-well orbit.

Fig. 15 :

 15 Fig.15: Experimental trajectory in (t, x, ẋ/xwω 0 ) 3D plane for f d = 50 Hz before (in blue), during (in orange) and after (in green) the application of the orbit jump.

Table 1 :

 1 Main properties associated to the both groups of orbit jump strategies defined in the current state of literature.

	External Forcing	Authors	Parameter or variable modified	Perturbation waveform	Validity range	Energy cost	Recovery time	Optimality
	Hand impulse	Erturk et al. [16]	Velocity	0.25 s *	Multiple freq. 6 -8 Hz	N/A	N/A
	Fast Burst Perturbation	Sebald et al. [17]	Voltage	0.7 s *	Multiple freq. 27.3 -29.8 Hz	N/A	1.5 s *
	Impact-induced	Zhou et al. [20]	Velocity	0.1 s *	Multiple freq. 4 -23 Hz	N/A	N/A
	Electrical switching	Mallick et al. [19]	Voltage	0.2 s	Single freq. 70 Hz	0.563 mJ * 1 s *
	Attractor selection	Udani et al. [4]	Voltage	2 s	Single freq. 19.8 Hz	1.21 mJ	5.66 s
	Characteristic Modulation	Authors	Parameter or variable modified	Perturbation waveform	Validity range	Energy cost	Recovery time	Optimality
	Negative resistance	Lan et al. [21]	Resistance	0.1 s	Multiple freq. 9 -11 Hz	0.2 mJ *	0.535 s
	Load perturbation	Wang et al. [22]	Stiffness and damping	4 s	Single freq. 5.2 Hz	N/A	N/A
	Negative resistance	Ushiki et al. [23]	Resistance	0.9 s *	Single freq. 70 Hz	35 mJ *	20 s
	Bidirectional Energy Conversion Circuit	Wang et al. [24]	Stiffness and damping	10.9 s	Single freq. 7.6 Hz	22 mJ	120 s
	Buckling modification	Huguet et al. [25]	Buckling level	20 ms *	Multiple freq. 30 -70 Hz	1 mJ *	1 s	∼ ∼ ∼
	Voltage Inversion Excitation	Yan et al. [26]	Stiffness	1.5 s.	Multiple freq. 48.6 -49.5 Hz	1.43 mJ	23 s
	Adjustment strategy	Huang et al. [27]	Buckling level and voltage	90 s *	Multiple freq. 35 -40 Hz	2.8 mJ	120 s

* indicates that the values have been estimated based on the given papers. N/A denotes the absence of data.

Table 3

 3 Fig.12:The experimental success rate provided by the maps of ±15% around each optimal times parameter combination as a function of the driving frequency.

	compares orbit jump

Table 3 :

 3 Comparison between the optimized strategy developed in this paper and other previous strategies in literature. * indicates that the values have been estimated based on the given papers.

which is the effective excitation during the orbit jump, i.e., the FBP application on the ambient excitation (which is harmonic in this study).

There are two electromechanical transducers, one for energy harvesting and the other for buckling level tuning.

which correspond to the initial conditions converging toward the considered orbit.

which correspond to the states (when t = k T d , for k ∈ Z) into which the bistable VEH converges when stabilized.

The elongation of the tuning APA leads to a higher level of buckling, resulting in the two equilibrium positions moving further apart from each other.

which is used for evaluating how close a given solution is to the optimum solution.

Note that we opted to fix kw and to vary τ 0 and ∆τ because the times are more susceptible to experimental variations due to the time delay of the board and the amplifier.

Acknowledgements The authors would like to acknowledge Dr. T. Huguet for the fruitful discussions on orbit jump strategies and nonlinear dynamics.

Funding

This work has received funding from the European Unions Horizon 2020 research and innovation program under grant agreement No 862289.

Data Availability Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest. Fig. 14: Evolutionary strategy algorithm for optimizing orbit jump strategy. This flowchart illustrates different steps that allow to determine optimal orbit jump parameters for each driving frequency. Here (τ 0 , ∆τ, kw) ∈ D denotes the individuals sequence for a given population in D.

Figure 14 illustrates the various steps involved in the evolutionary strategy algorithm. The procedure starts with a first random generation of 8 000 orbit jump parameters combinations (i.e., individuals). Then, the simulation of the orbit jump associated with each set of parameters is performed, and each individual is evaluated based on its fitness function value (6). This involves considering 7 elements in the neighborhood of each direction, comprising τ 0 , ∆τ and kw. That is, we take |I| = 7 then for each individual we launch 7 3 new orbit jump simulations to compute its fitness function value (6). That means that for each generation, we simulate 2 744 000 orbit jumps using parallel GPU computations. Each individual is then evaluated and selected based on its average total harvested energy (6) over 100 cycles. To complete the population of the next generation, classical operations of crossing and mutations are applied to the 10% best individuals present in the next generation.


Crépin Nsiala-Nzéza

Roland Gautier
email: roland.gautier@univ-brest.fr

Gilles Burel

C Nsiala Nzéza

Crépin Nsiala Nzéza

Theoretical Performances Analysis of the Blind Multiuser Detection based on Fluctuations of Correlation Estimators in Multirate CDMA Transmissions

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

One of the salient features of CDMA-based third-generation (3G) wireless systems is the capability to support transmission of diverse data such as voice, low-resolution video, compressed audio... Since these heterogeneous services produce digital information streams with different data rates, their implementation requires the use of multirate CDMA systems where each user may transmit his data at one among a set of available data rates. An easy way to view the multirate CDMA transmission is to consider the variable spreading length (VSL) technique where all users employ sequences with the same chip period. Moreover, each data rate is tied to the length of the spreading code of each user.

Many blind approaches have been devised to either improve the performance of a CDMA receiver in a multirate multiuser context or reduce its complexity. Some prior knowledge of user parameters, e.g. the signature waveform [START_REF] Roy | Subspace blind adaptive detection for multiuser CDMA[END_REF], the processing gain, the code of a group of active users [START_REF] Wang | Group-blind multiuser detection for uplink CDMA[END_REF], the chip rate [START_REF] Wang | Group-blind multiuser detection for uplink CDMA[END_REF], [START_REF] Haghighat | A subspace scheme for blind user identification in multiuser DS-CDMA[END_REF] is always assumed, but its nature depends on the technique employed.

Thus, we recently proposed a blind multiuser detection scheme requiring no prior knowledge about the transmitter [START_REF] Buzzi | Iterative cyclic subspace tracking for blind adaptive multiuser detection in multirate CDMA systems[END_REF]. Therefore, its performances have to be analyzed in order to evidence its efficiency prior to any further implementation in operational systems, which is of great interest for both military and civil applications.

The remainder of the paper is organized as follows. Section 2 introduces the signal model and assumptions made. Section 3 briefly describes the blind-detection approach, while Section 4 highlights its performances in terms of probability of detection and false alarm. Finally, numerical results are detailed in Section 5, and conclusions are drawn in Section 6.

Signal modeling and assumptions

Let us consider the general case of asynchronous DS-CDMA system where each user can transmit at one out of S available data rates 0

1 1 S R R R - < < < �
. By denoting i u N the number of active users transmitting at i R and u N the total number of users such that 1 0

S i u u i N N - = = ∑
, the complex equivalent model of the received signal can be expressed as

1 1 0 0 ( ) ( ) ( ) ( ), i u i n i N S n i n i s d i n k y t a k h t kT T b t , - - +∞ , , = = =-∞ = - - + ∑ ∑ ∑ (1)
where

1 0 ( ) ( ) ( ) i L n i n i i c k h t c k p t kT - , , = = - ∑ . In (
n i h t
, is a virtual filter corresponding to the convolution of all filters of the transmission chain with the spreading sequence

0 1 { ( )} i n i k L c k , = - �
, where i L is the spreading factor for the ( ) th n i , user. • Signals are assumed to be independent, centered, noise-unaffected and received with the same power

• Because of the VSL technique, the symbol period

0 0 2 2 n i s s σ σ , ,

=

, for all ( ) n i , .

• The signal-to-noise ratio (SNR) in dB ) at the detector input is negative (signal hidden in the noise).

By denoting ( ) s t the global noise-unaffected received signal, (1) can be rewritten as

1 1 0 0 ( ) ( ) ( ) ( ) ( ). i u n i N S n i d i n y t s t b t s t T b t , - - , = = = + = - + ∑ ∑ (2)

Blind detection approach

As explained in [START_REF] Nsiala Nzéza | Blind Multiuser Detection in Multirate CDMA Transmissions Using Fluctuations of Correlation Estimators[END_REF], the analysis of the autocorrelation fluctuations estimators allows to achieve a blind multiuser detection. Successive investigations of the contributions of noise and signal through the analysis of the second-order moment of the autocorrelation estimator allowed its description, as hereinafter briefly reported.

The received signal is first divided into M temporal windows, each of them of duration F T . Then, within each window m , an estimation of its autocorrelation is computed as

� 0 1 ( ) ( ) ( ) , F T m m m yy F y t y t dt R T τ τ * = - ∫ (3)
where ( ) m y t is the signal sample over the th m window. Hence, the second-order moment of the estimated autocorrelation � ( ) yy R τ using M windows can be expressed as

� � � 1 2 2 0 1 ( ) ( ) ( ) , M m yy yy m E R R M τ τ τ - ⎧ ⎫ ⎨ ⎬ ⎩ ⎭ = Φ = | | = | | ∑ (4)
where � ( ) E ⋅ is the estimated expectation of ( )

⋅
F F m WT M WT σ σ σ Φ Φ = = (5)
Second, by only considering the independent, centered and noise-unaffected signal, it was shown that, on average, high amplitudes of the fluctuations ( )

s τ Φ of
i s i i T m pgn τ τ Φ Φ = ⋅ (6)
where ( ) ( )

s i i T s k pgn kT τ δ τ +∞ =-∞ = - ∑ , ( ) 1
δ τ = for 0 τ = , and ( ) 0 δ τ = if 0 τ ≠ . Thus, the fluctuations due to the global noise-unaffected signal are:

1 1 0 0 ( ) ( ) ( ). i s i S S s i T i i m pgn τ τ τ - - Φ = = Φ = Φ = ⋅ ∑ ∑ (7)
Since signals are assumed to be received with the same power (e.g., 0 0 2 s σ , ) and

i s i c T L T = , the fluctuations ( ) i τ Φ average amplitude i m Φ can be written as 0 0 0 0 4 4 i i s i i i c u s u s F F T L T m N N T T σ σ , , Φ = = (8)
Equation ( 8) shows an increase in the average fluctuations amplitude concomitant with that of the number of users. It also indicates that the average fluctuations amplitude is tied to the sequence lengths.

Let us at last compute the standard deviation, i σ Φ , of i Φ within each group i . The hypothesis of noise-unaffected and independent signals leads to

1 0 ( ) ( ). i u n i N i s n τ τ , - = Φ = Φ ∑ (9)
As detailed in [START_REF] Nsiala | Récepteur adaptatif multi-standards pour les signaux à étalement de spectre en contexte non coopératif[END_REF][Chap. 3,pp. 38-39], since the fluctuations are computed from many independent randomly-selected windows, ( )

n i s τ , Φ
is similar to a Gaussian distribution.

Consequently the variance 2 i σ Φ is written as

1 2 2 0 , i u i s n i N n σ σ , - Φ Φ = = ∑ (10)
where 2

s n i σ , Φ
stands for the variance of fluctuations due to the ( ) th n i , signal. Then, from [START_REF] Burel | Detection of spread spectrum transmission using fluctuations of correlation estimators[END_REF], the standard deviation i σ Φ can be expressed as

0 0 4 2 . i i u i c s F N L T M T σ σ , Φ = (11)
Equation ( 11) evidences that any increase in the number M of analysis windows allows to lower the standard deviation. Moreover, [START_REF] Nsiala Nzéza | Parallel Blind Multiuser Synchronization and Sequences Estimation in Multirate CDMA Transmissions[END_REF] shows that the longer the sequence is, the higher the average fluctuations amplitude is. Thus, the fluctuations curve highlights high equispaced peaks whose average spacing corresponds to the estimated period symbol i s T .

Therefore, standards can be differentiated like in [START_REF] Williams | Personal area technologies for internetworked services[END_REF]. It ensures that, the blind synchronization process recently proposed in [START_REF] Nsiala Nzéza | Parallel Blind Multiuser Synchronization and Sequences Estimation in Multirate CDMA Transmissions[END_REF], [START_REF] Nsiala Nzéza | Blind Multiuser Identification in Multirate CDMA Transmissions : A New Approach[END_REF], through which the number 2 .

2 i i b u i c M N L T W ρ , Γ = (12)
Hence, (12) proves that increasing the number M of windows improves the detection, but at the cost of a larger computing time.

Performances analysis

To determine whether a spread spectrum signal is hidden in the noise, a detection threshold η is taken from [START_REF] Nsiala Nzéza | Blind Multiuser Detection in Multirate CDMA Transmissions Using Fluctuations of Correlation Estimators[END_REF] as

, b b m η α σ Φ Φ = + ⋅ (13)
where α * ∈ � . Like in the case of radar detection, if this threshold is low, then more targets will be detected at the expense of increased numbers of false alarms. Conversely, if the threshold is high, then fewer targets will be detected, but the number of false alarms will also be low. In most radar detectors, the threshold is set in order to achieve a certain level of false alarm.

Study within the group i, only considering the AWGN

The number M of independent randomly-selected analysis windows is assumed large enough. Hence, for values of τ multiple of i s T , ( )

i τ Φ is Gaussian distribution centered in i m Φ and of variance 2 i σ Φ .
Then, its probability density ( ) i p Φ is given by:

2 1 1 ( ) exp . 2 2 i i i i i m p σ σ π Φ Φ Φ ⎧ ⎫ ⎛ ⎞ Φ - ⎪ ⎪ Φ = -⎜ ⎟ ⎨ ⎬ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎩ ⎭ (14)

Probability of detection

Let us set as i D P the probability of detecting fluctuations peaks due to signals which belong to the group i , i.e., the probability for the fluctuations average amplitude to be above the threshold η for a given τ multiple of i s T . From (13), we get

( ) ( ) . i D p p d η η +∞ = Φ > = Φ Φ ∫ P (15)
Equation ( 15) can be rewritten using the complementary error function erfc as

1 . 2 2 i i i D m erfc η σ Φ Φ ⎛ ⎞ - ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ P (16)
Then, introducing ( 8), ( 11) and ( 13) in ( 16) leads to

2 2 1 1 1 2 2 M i D i u i c M erfc N L T W α ρ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎧ ⎫ ⎛ ⎞ + ⎪ ⎪ ⎜ ⎟ = - ⎨ ⎬ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎩ ⎭ P (17)
At this point, let us set as ( ) ) wished to be observed, which depends on M , F T and e F , the sampling frequency. It is also useful and important to recall that the detector is designed to work in an interactive way with an operator. Hence, he must choose a sufficient value of K in order to observe, in the case of a good detection, at least 3 or 4 fluctuations peaks due to signals transmitting at the lowest i R , thus spread with the largest factor i L .

K dim = Φ ,
Moreover, since symbols are assumed independent, the events occurrence of a fluctuations peak for a given value τ multiple of i s T , can also be considered as independent.

Consequently, the global probability of detection, set as i P D , must take into account the number peak N of fluctuations peaks. Therefore, from (17), we get

2 2 1 1 1 . 2 2 peak N M i i u i c M P erfc N L T W α ρ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎧ ⎫ ⎧ ⎫ ⎛ ⎞ + ⎪ ⎪ ⎪⎪ ⎜ ⎟ = - ⎨ ⎨ ⎬⎬ ⎜ ⎟ ⎪ ⎪ ⎪⎪ ⎝ ⎠ ⎩ ⎭ ⎩ ⎭ D (18)

Probability of false alarm

When only focusing on a fluctuations point, its probability, denoted (1) b P to be above the threshold is

(1) 1 1 . 2 2 2 2 b b b m P erfc erfc η α σ Φ Φ ⎛ ⎞ - ⎛ ⎞ ⎜ ⎟ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (19)
Since the false alarms occur when the noise fluctuations are above the threshold at any point τ , the false alarm probability, denoted fa P is defined as

(1) 1 1 , K fa b P P ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ - = - (20)
where K allows to take into account all points of the global fluctuations. Then, (20) can be simplified as follows

1 1 1 . 2 2 K fa P erfc α ⎛ ⎞ ⎛ ⎞ = -- ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (21)

Extension to the multirate multiuser case

When only two groups i and j of signals ( n i s , and n j s , ) interfere, it is improbable, even impossible, that there are fluctuations peaks due to n i s , where one awaited fluctuations peaks due to n j s , , since symbol periods are different. However, if that occurred, that should reinforce the probability to detect the fluctuations peaks due to n j s , .

Probability of detection

When there are several groups of signals transmitting at different rates, for given i , the probability i P / D B of detecting the fluctuations peaks due to signals within the group i depends on fluctuations, set as Φ B , due to both AWGN and MAI noise.

Since symbol periods are different, fluctuations Φ B have to be considered at the places where they are not supposed to exhibit peaks, because there, they can be awkward and have a similar effect with that of the noise. Hence, they can be characterized by their mean

m Φ B as ( ) ( ) ( ) 0 0 0 0 2 2 2 2 4 2 1 1 1 , i b u s F i s u F N m WT N WT σ β σ σ β ρ ρ , , ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Φ + - = + - = B (22)
0 0 0 0 2 2 2 2 4 2 1 2 1 1 2 . i b u s F i s u F N M WT N M WT σ β σ σ σ β ρ ρ , , ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Φ + - = + - = B (23)
Then, like in (13) a new detection threshold η ′ , taking into account ( 22) and ( 23) is defined as . m

η α σ ′ Φ Φ = + ⋅ B B (24)
Since signals are assumed to be received with the same power, using ( 5) and ( 24 ( ) ( )

0 0 2 4 2 1 1 2 1 . i s u F N WT M σ β ρ η α ρ , ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ′ ⎜ ⎟ ⎨ ⎬ ⎜ ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎩ ⎭ + - = + (25)
Hence, focusing on a peak, the probability i / D B P of detecting fluctuations due to signals in

the group i is 1 , 2 2 i i i m erfc η σ ′ Φ / Φ ⎛ ⎞ - ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ D B P (26)
which can be expressed using ( 8), ( 11) and (25) as ( ) ( )

2 2 2 1 1 1 1 1 . 2 2 i u M i i u i c N M erfc N L T W β ρ α ρ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ / ⎧ ⎫ ⎧ ⎫ + - + ⎪ ⎪ ⎪⎪ = - ⎨ ⎨ ⎬⎬ ⎪ ⎪ ⎪⎪ ⎩ ⎭ ⎩ ⎭ D B P (27)
Finally, like in (18), the global probability of detection, set as i P / D B , taking into account the number peak N of fluctuations peaks is given by .

peak N i i P ⎛ ⎞ ⎜ ⎟ / / ⎝ ⎠ = D B D B P (28)

Probability of false alarm

If we first focus on a fluctuations peak of signals within the group i , and since the false alarm is due to the fluctuations of the global noise (additive noise and MAI noise), its probability, noted (1) P B , to be above the threshold is

(1) 1 1 . 2 2 2 2 m P erfc erfc η α σ ′ Φ Φ ⎛ ⎞ - ⎛ ⎞ ⎜ ⎟ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ B B B (29)
Thus, as previously detailed in the multiuser single rate case, the probability of false alarm is the same as in (21). Finally, like in (12), let us set as i B , Γ the ratio ( 8) to (23) (noted ( ) i B out SNR , ), which stands for the SNR at the detector output ( ) ( ) ρ , while i,

2 2 . 2 1 1 i u i c i B i u N L T W M N ρ β ρ , Γ = + - (30
Γ B is bounded as ( ) ( ) 2 max( ) . 2 1 i u i c i i i u N L T W M N β , , Γ ≤ Γ = - B B (31)
Moreover, for high SNRs, the ratio (30) to (12) leads to (in dB)

( ) ( ) ( ) ( ) 20 log 1 1 . i B i b i out out u SNR SNR N β ρ , , = - + - (32)
Equation ( 32) shows a loss of ( ) ( )

20 log 1 1 i u N β ρ
+ -dB at the detector output in the multiuser multirate case compared to the single rate one. However, the same performances can be achieved by increasing for example M as also proved in ( 31), but at the expense of a higher computation time.

Let us set as (

) i m η Φ ,
∆ the ratio (8) to (13), and (

) i m η ′ Φ ′ ,
∆ the ratio ( 8) to (25). Then, we get

2 ( ) 2 , 1 i i u i c m M N L T W η ρ α Φ , ∆ = + (33) ( ) 2 2 ( ) 2
. 1

( 1)

1 i i u i c m i u M N L T W N η ρ β ρ α ′ Φ ′ , ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∆ = + - + (34)
In addition, if the number of analysis windows M is chosen large enough, i.e.,

, i i u i c m N L T W η ρ Φ , = ∆ (35) � ( ) 2 ( ) 2 . 1 ( 1)
i i u i c m i u N L T W N η ρ β ρ ′ Φ ′ , = ∆ + - (36)
Moreover, in the case

0 0 2 4 2 1 1 . i s u F N WT σ β ρ η ρ , ′ ⎧ ⎫ + - ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ (37)
Then, the following equations give the approximations of probabilities of detection ( 18) and (28).

1 1 1 , 2 2 peak N i i u i c M erfc P N L T W ρ ⎧ ⎫ ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ ⎪⎪ = - ⎨ ⎨ ⎬⎬ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ D (38) � ( ) ( ) 2 2 1 1 1 1 . 2 2 peak N i u i i u i c N M erfc P N L T W β ρ ρ / ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ + - ⎪ ⎪ ⎪ ⎪⎪⎪ = - ⎨ ⎨ ⎨ ⎬⎬⎬ ⎪ ⎪ ⎪ ⎪⎪⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ D B (39)
It is obvious to check that (36) and (39) are equivalent to (35) and (38) respectively, by setting 1 β = , which corresponds to the case i u u N N = , i.e., when all users are transmitting at one rate i R . Moreover, (38) and (39) exhibit a similar behavior asymptotically. Thus, whatever the MAI noise power is, the probability of detection can be improved just by increasing the number of windows M . However, it depends on both the available computation power and the time allocated for the detection.

Numerical results

Detectors parameters setting

Let us set as s T the duration of the global intercepted signal, 1

⎢ ⎥ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎧ ⎪ = ⎪ ⎨ ⎪ = ⎪ ⎩ (40)
where * ⎢ ⎥ ⎣ ⎦ stands for the integer part of * . Let us also recall that the average spacing between fluctuations peaks gives an estimate of symbol periods inside each group i . Consequently, since the detector is designed to work in an interactive way with an operator, some parameters can be set according to the number of peaks which one wishes to observe. Without detailing calculations, following equations give K and .

i peak N , max e i c F L K F γ ⎢ ⎥ = ⎢ ⎥ ⎣ ⎦ (41)
.

max i max i peak i i i L N L L L γ = , ≠ (42)
where 0 1 max ( )

max i i S i L L = , , - = �
, γ is a real coefficient to be adjusted in order to obtain the minimum number of peaks wished for users whose signals are spread with the processing gain

Performances analysis

Figure 2 illustrates the probability of false alarm fa P according to the number K of analysis points for different values of α . It evidences that any increase in the threshold concomitantly with K induces a decrease of fa P . However, that also lowers the probability of detection. Thus, 3 α = and 889 K = seem to be a good trade-off between fa P and the probability of detection. In addition, it is obvious that when the threshold increases, best performances are achieved with the largest sequence lengths as highlighted on Figure 7 and Figure 8. , approximated probabilities curves are very close to the real ones, in agreement with theoretical analysis. In addition, the larger the sequence length is the highest i P D is. In addition, for larger thresholds, performances can be improved either by an increase of the number of analysis windows, or by detecting first fluctuations peaks corresponding to largest spreading codes lengths, as shown in Figure 13 and Figure 14. Let us note that one also could adjust the other parameters of the detection, since the detection method described herein this paper can be performed in an interactive way.

T

  for the users transmitting at the rate i R is tied to the common chip period c T : a centered additive white Gaussian noise (AWGN) of variance 2 b σ .

  -unaffected signals and the noise, respectively. Moreover, since fluctuations are computed from many randomlyselected windows, they do not depend on the signals relative delays n i d T , . First, only the fluctuations due to the additive noise are uniformly distributed over all values of τ . Depending on the spreading sequence properties, the Multiple Access Interference (MAI) noise generates rather null, or very low incoherent fluctuations. Since the noise is random, the fluctuations of the autocorrelation estimator, denoted ( ) b τ Φ , are also random. Assuming a receiver filter with a flat frequency response in [ 2 2] W W -/ , + / and zero outside, they can be characterized by their mean b m Φ and standard deviation b

N

  of active users is determinate, can be performed.In addition, let us denote ρ , the SNR at the detector input (denoted SNR in (in dB)) ratio (8) to (5) (b), noted i b , Γ , represents the SNR at the detector output (noted SNR( ) i b out , ) as explained in[START_REF] Burel | Detection of spread spectrum transmission using fluctuations of correlation estimators[END_REF]

  the number of points of the global fluctuations ( )

  where

β-

  = is the ratio between the total amount of users transmitting at all rates and those transmitting at particular rate i R , and( is the power of signals inside other groups. Their standard deviation σ Φ B can be expressed as

Γ

  are close for low SNRs or when the total number u N of users is equal to i u N . However, for high SNRs, i b , Γ increases with2

  sampling frequency at the detector input. It ensues that the total number e N of samples of the signal to analyze and the number M of analysis windows can be

LN

  , and i peak N stands for the minimum number of fluctuations peaks within each group i . Moreover, since the signal spectrum must be contained in the detectors ADC (Analog to Digital Converter) bandwidth, W has to satisfy c e F W F ≤ ≤ . A last, for signals transmitting at i R , the number of symbols i s contained within an analysis window is given by .

Figure 1 .

 1 Figure 1. Fluctuations ( ) 0

Figure 1 N

 1 Figure 1 was obtained with processing gains of 0 31 L = , 1 127 L = (complex Gold sequences), and 3 α = . It also was set

Figure 2 .

 2 Figure 2. Probability of false alarm according to K for different thresholds.

Figure 3 and∆

 3 Figure3and Figure4show the ratio (8) to (13) in dB, set as (

Figure 3

 3 Figure 3.

Figure 4

 4 Figure 4.

Figure 5 and

 5 Figure5and Figure6show the ratio (8) to (25), set as (

Figure 5

 5 Figure 5.

Figure 6

 6 Figure 6.

Figure 7 .

 7 Figure 7.

Figure 8

 8 Figure 8.

Figure 9 and

 9 Figure 9 and Figure 10 illustrate the probability of detection i P D in the multiuser single rate case. They evidence that any increase in the number M of analysis windows induces an increase of i P D . Moreover, for larger M , i.e., 2 2 M α ≥, approximated probabilities curves are very close to the real ones, in agreement with theoretical analysis. In addition, the larger the sequence length is the highest i P D is.

Figure 9 .

 9 Figure 9.

iP

  D according to M and

Figure 10 .

 10 Figure 10.

Figure 11 and

 11 Figure 11 and Figure 12 depict the probability of detection i P / D B in the multiuser multirate case. They first show that approximated probabilities curves are very close to the real ones. Second, for a fixed threshold, the probability of detection increases concomitantly with sequence lengths. Moreover, any increase in the number M of analysis windows improves the probability of detection.

  Figure 11.

  Figure 12.

  Figure 13.

Acknowledgment

This work was partially supported by the Brittany Region (France).

At last, Figure 17 that despite the increase in the number of active users, the ( ) i B out SNR , can be strongly improved by focusing first on fluctuations peaks due to signals spread with the largest sequences lengths.

Conclusion

We have evidenced the efficiency of the blind multiuser multirate detection method based on fluctuations of correlation estimators through its theoretical analysis. Assuming a large number of independent randomly-selected analysis windows, we have expressed second-order moments of fluctuations due to both signals and noise. Then, we have derived the probabilities of detection in both multirate and single rate case.

Moreover, the analysis of detection probabilities, confirmed by numerical results have proved that very good performances in terms of probability of detection and false alarm can be achieved, even at very low SNRs or with many interfering users.

At last, we demonstrated that performances could be significantly improved just by increasing the number of analysis windows, but at the expense of a higher computation time.