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We present the site quality of the Zadko Observatory, focusing on the brightness of the sky and the effects of the weather on the efficiency of the Observatory. We discuss these effects and their consequences for the scientific goals of the Observatory. Without surprise, the overall sky quality had decreased during the last decade. However, this decrease is mostly due to the weather pattern at the Observatory rather than the light pollution. We put an emphasis on how important the preservation of the sky darkness is for small observatories, in order to stabilize the global degradation of the site quality, as this directly impacts the scientific return of the observations., Comment: 5 pages, 4 figures; accepted for publication into the Proceedings of the AstroRob2023 Workshop

Probing the neutral gas distribution at the edge of magnetic confinement fusion devices is critical for divertor physics studies. In the TCV tokamak, a set of ASDEX-type hot ionization pressure gauges (APGs) has been installed for fast, in-situ measurements of the neutral pressure distribution in the TCV chamber. The APGs have been calibrated against baratron pressure gauges (BGs) for pressures ranging from less than 1 mPa to several hundred mPa. A correction to account for the residual pressure in the pumped torus is proposed to improve the measurement accuracy. APG measurements in a series of plasma discharges with varied density ramp rates are analyzed, and are compared with the pressure measurements of the existing BGs. APG measurements feature significantly higher time resolution and faster time response than the BGs, and extend the measurement range to lower pressures. APGs systematically measure higher neutral pressures than the BGs connected to the same TCV port, likely due to the BG time delay and slow time response and nonuniform neutral distribution in gauge ports during the discharge. The initial APG operations in TCV have been proven successful, which validates the APG as an adequate pressure measurement technique for the upcoming TCV divertor upgrades.

A coherent concatenation of multiple Townes solitons may lead to a stable infrared and visible broadband filament in ceramic YAG polycrystal. This self-trapped soliton train is leveraged to implement self-referenced multiplex coherent anti-Stokes Raman scattering imaging. Simulations and experiments illustrating the filamentation process and the concatenation of focusing-defocusing cycles in ceramic and crystal YAG are presented. In addition, our simulations and experiments further examine the dependence of the filamentation onset location and supercontinuum generation upon peak input power. Understanding this dependence is key for implementation of viable CARS imaging techniques, due to the comparatively exceptional ability of YAG to generate supercontinuum which can enable higher-sensitivity imaging without delay lines., Comment: To appear in Optics Letters Vol. 50 No. 1

Learning how to safely handle Plasma-Material Interaction (PMI) is a key challenge towards the commercialisation of energy from nuclear fusion. In this respect, linear plasma devices are ideal experimental testbeds, and numerical codes play a crucial complementary role. In this paper, a numerical investigation of PMI-relevant helium plasma experimental discharges in GyM linear device is presented, in which SOLPS-ITER and ERO2.0 codes are coupled for plasma background generation and material erosion investigation respectively, with the aim to support the interpretation and complement the available experimental dataset. On the plasma side, simulated profiles are validated against experimental data to provide a realistic plasma background, and the role of He metastable states is assessed for the first time in SOLPS simulations. On the material side, the erosion and deposition effects due to the introduction of the sample-holder in the simulation volume are investigated, now considering also the real stainless steel composition as wall material.

The Physics of the Accelerating Universe (PAU) camera is an optical narrow band and broad band imaging instrument mounted at the prime focus of the William Herschel Telescope. We describe the image calibration procedure of the PAU Survey data. We rely on an external photometric catalogue to calibrate our narrow band data using stars that have been observed by both datasets. We fit stellar templates to the stellar broad band photometry of the Sloan Digital Sky Survey and synthesise narrow band photometry that we compare to the PAUS narrow band data to determine their calibration. Consequently, the PAUS data are in the AB system as inherited from its reference calibrator. We do several tests to check the performance of the calibration. We find it self-consistent when comparing repeated observations of the same objects, with a good overall accuracy to the AB system which we estimate to be at the 2\% precision level and no significant trends as a function of narrow band filter or wavelength. Repeated observations allow us to build a spatial map of the illumination pattern of the system. We also check the wavelength dependence of the calibration comparing to stellar spectra. We find that using only blue stars reduces the effects of variations in the stellar template fitting to broad-band colours, improving the overall precision of the calibration to around 1\% and its wavelength uniformity. The photometric redshift performance obtained with the PAUS data attests to the validity of our calibration to reach the PAUS science goals., Comment: 18 pages, 20 figures, accepted for publication in MNRAS

We investigate how elimination of variables can affect the asymptotic dynamics and phenotype control of Boolean networks. In particular, we look at the impact on minimal trap spaces, and identify a structural condition that guarantees their preservation. We examine the possible effects of variable elimination under three of the most popular approaches to control (attractor-based control, value propagation and control of minimal trap spaces), and under different update schemes (synchronous, asynchronous, generalized asynchronous). We provide some insights on the application of reduction, and an ample inventory of examples and counterexamples., Comment: Peer-reviewed and recommended by Peer Community In "Mathematical and Computational Biology": https://doi.org/10.24072/pci.mcb.100318

We study the diffraction of a particular class of beams, composed only by a combination of azimuthally invariant guided modes of an optical fiber. We demonstrate that such beams can be obtained by injecting a Gaussian beam in a small piece of silica graded-index multimode fiber. This minimalistic low-cost method is applied for improving the axial resolution of a two-photon microscope., Comment: 9 pages, 5 figures

Gravitational waves from black-hole merging events have revealed a population of extra-galactic BHs residing in short-period binaries with masses that are higher than expected based on most stellar evolution models - and also higher than known stellar-origin black holes in our Galaxy. It has been proposed that those high-mass BHs are the remnants of massive metal-poor stars. Gaia astrometry is expected to uncover many Galactic wide-binary systems containing dormant BHs, which may not have been detected before. The study of this population will provide new information on the BH-mass distribution in binaries and shed light on their formation mechanisms and progenitors. As part of the validation efforts in preparation for the fourth Gaia data release (DR4), we analysed the preliminary astrometric binary solutions, obtained by the Gaia Non-Single Star pipeline, to verify their significance and to minimise false-detection rates in high-mass-function orbital solutions. The astrometric binary solution of one source, Gaia BH3, implies the presence of a 32.70 \pm 0.82 M\odot BH in a binary system with a period of 11.6 yr. Gaia radial velocities independently validate the astrometric orbit. Broad-band photometric and spectroscopic data show that the visible component is an old, very metal-poor giant of the Galactic halo, at a distance of 590 pc. The BH in the Gaia BH3 system is more massive than any other Galactic stellar-origin BH known thus far. The low metallicity of the star companion supports the scenario that metal-poor massive stars are progenitors of the high-mass BHs detected by gravitational-wave telescopes. The Galactic orbit of the system and its metallicity indicate that it might belong to the Sequoia halo substructure. Alternatively, and more plausibly, it could belong to the ED-2 stream, which likely originated from a globular cluster that had been disrupted by the Milky Way., Comment: 23 pages, accepted fro publication in A&A Letters. New version with small fixes

This paper focuses on the parameterization of the multipath propagation model (MPM) for indoor broadband power line communications (PLC), which up to now has been established in an heuristic way. The MPM model was originally proposed for outdoor channels in the band up to 20 MHz, but its number of parameters becomes extremely large when used to model indoor channel frequency responses (CFR), which are much more frequency-selective than outdoor ones, and the band is extended to 80 MHz. This work proposes a fitting procedure that addresses this problem. It allows determining the model parameters that yield the best fit to each channel of a large database of single-input single-output (SISO) experimental measurements acquired in typical home premises of different European countries. Then, the properties of the MPM parameters are analyzed. The study unveils the relation between the model parameters and the main characteristics of the actual CFR like the frequency selectivity and the average attenuation. It also estimates the probability density function (PDF) of each parameter and proposes a fitting distribution for each of them. Moreover, the relationship among the main parameters of the model is also explored. Provided results can be helpful for the development of MPM-based models for indoor broadband PLC., Comment: 9 pages, 7 figures

We propose a novel approach using Multiplex-Coherent Anti-Stokes Raman Scattering (M-CARS) for la-bel-free discriminations in biomedical tissues. The strategy is based on the evaluation of the contrast be-tween resonant and nonresonant contributions in a M-CARS hyperspectral dataset, and tested to identify and differentiate thin actin filaments from thick myosin filaments in muscle tissue without any labeling. First step consists in ensuring knowledge of the spatial regions containing thick myosin filaments thanks to its endogenous second harmonic signal, deducing expected location for thin actin filaments between myosin filaments. The ratio of resonant and nonresonant contributions for each pixel of the hyperspectral image allows then to discriminate actin from myosin filaments, whose localization is in accordance with the SHG probing. This qualitative imaging represents a proof of principle for highlighting and discriminat-ing purposes in biological microscopy, thanks to the difference in the nonlinear properties of the related proteins. This paves the way for considering label-free imaging through a competition between two third-order nonlinear signatures., Comment: 12 pages, 5 figures

L-mode negative triangularity (NT) operation is a promising alternative to the positive triangularity (PT) H-mode as a high-confinement ELM-free operational regime. In this work, two TCV L-mode lower single null Ohmic discharges with opposite triangularity $\delta \simeq \pm 0.3$ are investigated using SOLPS-ITER modelling. The main focus is the exploration of the reasons behind the experimentally observed feature of NT plasmas being more difficult to detach than similar PT experiments. SOLPS-ITER simulations are performed assuming the same anomalous diffusivity for particles $D_n^{AN}$ and energy $\kappa_{e/i}^{AN}$ in PT and NT. Nonetheless, the results clearly show dissimilar transport and accumulation of neutral particles in the scrape-off layer (SOL) of the two configurations, which consequently gives rise to different ionization sources for the plasma and produces different poloidal and cross-field fluxes. Simulations also recover the experimental feature of the outer target being hotter in the NT scenario (with $T_{e, NT} \gtrsim 5 \, \mathrm{eV}$) than in the PT counterpart., Comment: 18 pages, 8 figures

The European Space Agency's Euclid mission is one of the upcoming generation of large-scale cosmology surveys, which will map the large-scale structure in the Universe with unprecedented precision. The development and validation of the SGS pipeline requires state-of-the-art simulations with a high level of complexity and accuracy that include subtle instrumental features not accounted for previously as well as faster algorithms for the large-scale production of the expected Euclid data products. In this paper, we present the Euclid SGS simulation framework as applied in a large-scale end-to-end simulation exercise named Science Challenge 8. Our simulation pipeline enables the swift production of detailed image simulations for the construction and validation of the Euclid mission during its qualification phase and will serve as a reference throughout operations. Our end-to-end simulation framework starts with the production of a large cosmological N-body & mock galaxy catalogue simulation. We perform a selection of galaxies down to I_E=26 and 28 mag, respectively, for a Euclid Wide Survey spanning 165 deg^2 and a 1 deg^2 Euclid Deep Survey. We build realistic stellar density catalogues containing Milky Way-like stars down to H<26. Using the latest instrumental models for both the Euclid instruments and spacecraft as well as Euclid-like observing sequences, we emulate with high fidelity Euclid satellite imaging throughout the mission's lifetime. We present the SC8 data set consisting of overlapping visible and near-infrared Euclid Wide Survey and Euclid Deep Survey imaging and low-resolution spectroscopy along with ground-based. This extensive data set enables end-to-end testing of the entire ground segment data reduction and science analysis pipeline as well as the Euclid mission infrastructure, paving the way to future scientific and technical developments and enhancements., Comment: 39 pages, 25 figures, A&A submitted

In deep learning, classification tasks are formalized as optimization problems often solved via the minimization of the cross-entropy. However, recent advancements in the design of objective functions allow the usage of the $f$-divergence to generalize the formulation of the optimization problem for classification. We adopt a Bayesian perspective and formulate the classification task as a maximum a posteriori probability problem. We propose a class of objective functions based on the variational representation of the $f$-divergence. Furthermore, driven by the challenge of improving the state-of-the-art approach, we propose a bottom-up method that leads us to the formulation of an objective function corresponding to a novel $f$-divergence referred to as shifted log (SL). We theoretically analyze the objective functions proposed and numerically test them in three application scenarios: toy examples, image datasets, and signal detection/decoding problems. The analyzed scenarios demonstrate the effectiveness of the proposed approach and that the SL divergence achieves the highest classification accuracy in almost all the considered cases., Comment: Accepted at ICML 2024

The discovery of gravitational waves, first observed in September 2015 following the merger of a binary black hole system, has already revolutionised our understanding of the Universe. This was further enhanced in August 2017, when the coalescence of a binary neutron star system was observed both with gravitational waves and a variety of electromagnetic counterparts; this joint observation marked the beginning of gravitational multimessenger astronomy. The Einstein Telescope, a proposed next-generation ground-based gravitational-wave observatory, will dramatically increase the sensitivity to sources: the number of observations of gravitational waves is expected to increase from roughly 100 per year to roughly 100'000 per year, and signals may be visible for hours at a time, given the low frequency cutoff of the planned instrument. This increase in the number of observed events, and the duration with which they are observed, is hugely beneficial to the scientific goals of the community but poses a number of significant computing challenges. Moreover, the currently used computing algorithms do not scale to this new environment, both in terms of the amount of resources required and the speed with which each signal must be characterised. This contribution will discuss the Einstein Telescope's computing challenges, and the activities that are underway to prepare for them. Available computing resources and technologies will greatly evolve in the years ahead, and those working to develop the Einstein Telescope data analysis algorithms will need to take this into account. It will also be important to factor into the initial development of the experiment's computing model the availability of huge parallel HPC systems and ubiquitous Cloud computing; the design of the model will also, for the first time, include the environmental impact as one of the optimisation metrics., Comment: Submitted for the proceedings of CHEP2023

Plasma edge simulations with the SOLPS-ITER code are performed to study the influence of divertor baffles on nitrogen-seeded detachment in TCV single-null, L-mode discharges. Scans of nitrogen seeding rate are conducted in both baffled and unbaffled TCV divertors, where the nitrogen seeding with baffles is found to yield lower target temperatures and heat fluxes than with baffles-only and with seeding-only. The cumulative effects of baffles and seeding on target parameters are explained by the two-point model. The divertor neutral density and neutral compression increase with baffles, due to lower divertor to main chamber neutral conductance, as explained by a schematic neutral transport model with baffles. The nitrogen retention, defined as the ratio of average nitrogen nuclei density in divertor and main chamber, increases with the seeding rate if baffled, and remains constant if unbaffled. At the same outboard mid-plane separatrix plasma density, the nitrogen retention with baffles is lower than the unbaffled retention at low seeding levels and is higher at high seeding levels, which is explained by the changes of nitrogen ion and neutral transport with baffles and seeding. The baffled carbon retention is higher than the unbaffled retention due to lower divertor to main chamber carbon neutral conductance. Baffles increase the divertor radiation. The predicted trends of target parameters, the distribution of neutrals and radiations are well supported by TCV experiments, though discrepancies in the absolute values remain. The simulations yield an overall colder and denser divertor, consistent with previous SOLPS-ITER simulations of Ohmically heated L-modes in TCV. The successful comparison of simulation and experiment, together with the understanding gained from the neutral transport model, increases the confidence in the SOLPS simulations for the next TCV divertor upgrade.

In recent years, negative triangularity (NT) has emerged as a potential high-confinement L-mode reactor solution. In this work, detachment is investigated using core density ramps in lower single null Ohmic L-mode plasmas across a wide range of upper, lower, and average triangularity (the mean of upper and lower triangularity: $\delta$) in the TCV tokamak. It is universally found that detachment is more difficult to access for NT shaping. The outer divertor leg of discharges with $\delta\approx -0.3$ could not be cooled to below $5~\mathrm{eV}$ through core density ramps alone. The behavior of the upstream plasma and geometrical divertor effects (e.g. a reduced connection length with negative lower triangularity) do not fully explain the challenges in detaching NT plasmas. Langmuir probe measurements of the target heat flux widths ($\lambda_q$) were constant to within 30% across an upper triangularity scan, while the spreading factor $S$ was lower by up to 50% for NT, indicating a generally lower integral Scrape-Off Layer width, $\lambda_{int}$. The line-averaged core density was typically higher for NT discharges for a given fuelling rate, possibly linked to higher particle confinement in NT. Conversely, the divertor neutral pressure and integrated particle fluxes to the targets were typically lower for the same line-averaged density, indicating that NT configurations may be closer to the sheath-limited regime than their PT counterparts, which may explain why NT is more challenging to detach.

The Super-X Divertor (SXD) is an alternative divertor configuration leveraging total flux expansion at the Outer Strike Point (OSP). While the extended 2-Point Model (2PM) predicts facilitated detachment access and control in the SXD configuration, these attractive features are not always retrieved experimentally. These discrepancies are at least partially explained by the effect of parallel flows which, when self-consistently included in the 2PM, reveal the role of total flux expansion on the pressure balance and weaken the total flux expansion effect on detachment access and control, compared to the original predictions. This new model can partially explain the discrepancies between the 2PM and experiments performed on tokamak \`a configuration variable (TCV), in ohmic L-mode scenarios, which are particularly apparent when scanning the OSP major radius Rt. In core density ramps in lower Single-Null (SN) configuration, the impact of Rt on the CIII emission front movement in the divertor outer leg - used as a proxy for the plasma temperature in the divertor - is substantially weaker than 2PM predictions. Furthermore, in OSP radial sweeps in lower and upper SN configurations, in ohmic L-mode scenarios with a constant core density, the peak parallel particle flux density at the OSP is almost independent of Rt, while the 2PM predicts a linear dependence. Finally, analytical and numerical modeling of parallel flows in the divertor is presented. It is shown that an increase in total flux expansion can favour supersonic flows at the OSP. Parallel flows are also shown to be relevant by analysing SOLPS-ITER simulations of TCV., Comment: This version of the paper is outdated. The most recent, and published, version of the paper can be found at the corresponding Journal / DOI reference

We propose an optimal destination scheduling scheme to improve the physical layer security (PLS) of a power-line communication (PLC) based Internet-of-Things system in the presence of an eavesdropper. We consider a pinhole (PH) architecture for a multi-node PLC network to capture the keyhole effect in PLC. The transmitter-to-PH link is shared between the destinations and an eavesdropper which correlates all end-to-end links. The individual channel gains are assumed to follow independent log-normal statistics. Furthermore, the additive impulsive noise at each node is modeled by an independent Bernoulli-Gaussian process. Exact computable expressions for the average secrecy capacity (ASC) and the probability of intercept (POI) performance over many different networks are derived. Approximate closed-form expressions for the asymptotic ASC and POI are also provided. We find that the asymptotic ASC saturates to a constant level as transmit power increases. We observe that the PH has an adverse effect on the ASC. Although the shared link affects the ASC, it has no effect on the POI. We show that by artificially controlling the impulsive to background noise power ratio and its arrival rate at the receivers, the secrecy performance can be improved.

Estimating mutual information accurately is pivotal across diverse applications, from machine learning to communications and biology, enabling us to gain insights into the inner mechanisms of complex systems. Yet, dealing with high-dimensional data presents a formidable challenge, due to its size and the presence of intricate relationships. Recently proposed neural methods employing variational lower bounds on the mutual information have gained prominence. However, these approaches suffer from either high bias or high variance, as the sample size and the structure of the loss function directly influence the training process. In this paper, we propose a novel class of discriminative mutual information estimators based on the variational representation of the $f$-divergence. We investigate the impact of the permutation function used to obtain the marginal training samples and present a novel architectural solution based on derangements. The proposed estimator is flexible since it exhibits an excellent bias/variance trade-off. The comparison with state-of-the-art neural estimators, through extensive experimentation within established reference scenarios, shows that our approach offers higher accuracy and lower complexity., Comment: Accepted at NeurIPS 2024. Code available at https://github.com/tonellolab/fDIME

In this paper, the problem of determining the capacity of a communication channel is formulated as a cooperative game, between a generator and a discriminator, that is solved via deep learning techniques. The task of the generator is to produce channel input samples for which the discriminator ideally distinguishes conditional from unconditional channel output samples. The learning approach, referred to as cooperative channel capacity learning (CORTICAL), provides both the optimal input signal distribution and the channel capacity estimate. Numerical results demonstrate that the proposed framework learns the capacity-achieving input distribution under challenging non-Shannon settings., Comment: 5 pages, 6 figures, submitted to IEEE

Identification of attractors, that is, stable states and sustained oscillations, is an important step in the analysis of Boolean models and exploration of potential variants. We describe an approach to the search for asynchronous cyclic attractors of Boolean networks that exploits, in a novel way, the established technique of elimination of components. Computation of attractors of simplified networks allows the identification of a limited number of candidate attractor states, which are then screened with techniques of reachability analysis combined with trap space computation. An implementation that brings together recently developed Boolean network analysis tools, tested on biological models and random benchmark networks, shows the potential to significantly reduce running times., Comment: 13 pages

In this article, we consider the problem of distributed detection of a localized radio source emitting a signal. We consider that geographically distributed sensor nodes obtain energy measurements and compute cooperatively a statistic to decide if the source is present or absent. We model the radio source as a stochastic signal and deal with spatially statistically dependent measurements, whose probability density function (PDF) has unknown positive parameters when the radio source is active. Under the framework of the Generalized Likelihood Ratio Test (GLRT) theory, the positive constraint on the unknown multidimensional parameter makes the computation of the GLRT asymptotic performance (when the amount of sensor measurements tends to infinity) more involved. Nevertheless, we analytically characterize the asymptotic distribution of the statistic. Moreover, as the GLRT is not amenable for distributed settings because of the spatial statistical dependence of the measurements, we study a GLRT-like test where the joint PDF of the measurements is substituted by the product of its marginal PDFs, and therefore, the statistical dependence is completely discarded for building this test. Nevertheless, its asymptotic performance is proved to be identical to the original GLRT, showing that the statistically dependence of the measurements has no impact on the detection performance in the asymptotic scenario. Furthermore, the GLRT-like algorithm has a low computational complexity and demands low communication resources, as compared to the GLRT.

We demonstrate how a narrowband pump and a broadband spectrum can be spatially multiplexed by selective coupling them in two distinct modes of a few-mode microstructure fiber. The first mode carries most of the input pump energy, and experiences spectral broadening. Whereas the second mode preserves the narrow bandwidth of the remaining part of the pump. Bimodal propagation, with a power unbalance strongly in favor of the fundamental mode, is naturally obtained by maximizing coupling into the fundamental mode of the fiber. At the fiber output, the nearly monochromatic beam and the supercontinuum carried by the two different modes are combined by a microscope objective, and used as a pump and a Stokes wave for self-referenced multiplex coherent anti-Stokes Raman scattering micro-spectroscopy. The spectral resolution, the signal-to-noise-ratio, and the possible amplification of the remaining pump beam are discussed., Comment: 10 pages, 9 figures

To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph., Comment: 16 pages

Probability density estimation from observed data constitutes a central task in statistics. Recent advancements in machine learning offer new tools but also pose new challenges. The big data era demands analysis of long-range spatial and long-term temporal dependencies in large collections of raw data, rendering neural networks an attractive solution for density estimation. In this paper, we exploit the concept of copula to explicitly build an estimate of the probability density function associated to any observed data. In particular, we separate univariate marginal distributions from the joint dependence structure in the data, the copula itself, and we model the latter with a neural network-based method referred to as copula density neural estimation (CODINE). Results show that the novel learning approach is capable of modeling complex distributions and it can be applied for mutual information estimation and data generation., Comment: 6 pages

We consider the distributed detection problem of a temporally correlated random radio source signal using a wireless sensor network capable of measuring the energy of the received signals. It is well-known that optimal tests in the Neyman-Pearson setting are based on likelihood ratio tests (LRT), which, in this set-up, evaluate the quotient between the probability density functions (PDF) of the measurements when the source signal is present and absent. When the source is present, the computation of the joint PDF of the energy measurements at the nodes is a challenging problem. This is due to the statistical dependence introduced to the received signals by the radio source propagated through fading channels. We deal with this problem using the characteristic function of the (intractable) joint PDF, and proposing an approximation to it. We derive bounds for the approximation error in two wireless propagation scenarios, slow and fast fading, and show that the proposed approximation is exponentially tight with the number of nodes when the time-bandwidth product is sufficiently high. The approximation is used as a substitute of the exact joint PDF for building an approximate LRT, which performs better than other well-known detectors, as verified by Monte Carlo simulations.