On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers

We study the transients of matrices in max-plus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by $\max\{T_1,T_2\}$, where $T_1$ is the weak CSR threshold and $T_2$ is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for $T_1$ and $T_2$, naturally leading to the question which matrices, if any, attain these bounds. In the present paper we characterize the matrices attaining two particular bounds on $T_1$, which are generalizations of the bounds of Wielandt and Dulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the non-weighted case.Comment: 42 pages, 9 figure

Weak CSR expansions and transience bounds in max-plus algebra

This paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max-plus matrix powers. To this aim, we introduce the concept of weak CSR expansions: A^t=CS^tR + B^t. We observe that most of the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR expansion to hold, which does not depend on the values of the entries of the matrix but only on its pattern, and (ii) a bound for the CS^tR term to dominate. To improve and analyze (i), we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here. We also show how to make use of various parameters of digraphs. To improve and analyze (ii), we introduce three different kinds of weak CSR expansions (named after Nachtigall, Hartman-Arguelles, and Cycle Threshold). As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.Comment: 32 page

Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure

Numerical characterisation of quasi-orthogonal piecewise linear frequency modulated waveforms

This paper presents an analysis of the Doppler tolerance and isolation properties of five different sets of piecewise linear frequency modulated (PLFM) waveform triplets consisting of a combination of LFM subchirps. Different combinations of PLFM signals are used to produce waveforms with the same time-bandwidth product and optimise them with respect to isolation. The performance of the proposed waveforms are numerically investigated and a comparison between sets is presented. Results confirm that the waveforms have quasi-orthogonal properties and exhibit a degree of Doppler tolerance

Multibeam radar based on linear frequency modulated waveform diversity

Multibeam radar (MBR) systems based on waveform diversity require a set of orthogonal waveforms in order to generate multiple channels in transmission and extract them efficiently at the receiver with digital signal processing. Linear frequency modulated (LFM) signals are extensively used in radar systems due to their pulse compression properties, Doppler tolerance, and ease of generation. Here, the authors investigate the level of isolation between MBR channels based on LFM chirps with rectangular and Gaussian amplitude envelopes. The orthogonal properties and the mathematical expressions of the isolation are derived as a function of the chirp design diversity, and specifically for diverse frequency slopes and frequency offsets. The analytical expressions are validated with a set of simulations as well as with experiments at C-band using a rotating target

Double diffraction in an atomic gravimeter

We demonstrate the realization of a new scheme for cold atom gravimetry based on the use of double diffraction beamsplitters recently demonstrated in \cite{Leveque}, where the use of two retro-reflected Raman beams allows symmetric diffraction in $\pm \hbar k_{eff}$ momenta. Though in principle restricted to the case of zero Doppler shift, for which the two pairs of Raman beams are simultaneously resonant, we demonstrate that such diffraction pulses can remain efficient on atoms with non zero velocity, such as in a gravimeter, when modulating the frequency of one of the two Raman laser sources. We use such pulses to realize an interferometer insensitive to laser phase noise and some of the dominant systematics. This reduces the technical requirements and would allow the realization of a simple atomic gravimeter. We demonstrate a sensitivity of $1.2\times10^{-7}g$ per shot

Hybridizing matter-wave and classical accelerometers

We demonstrate a hybrid accelerometer that benefits from the advantages of both conventional and atomic sensors in terms of bandwidth (DC to 430 Hz) and long term stability. First, the use of a real time correction of the atom interferometer phase by the signal from the classical accelerometer enables to run it at best performances without any isolation platform. Second, a servo-lock of the DC component of the conventional sensor output signal by the atomic one realizes a hybrid sensor. This method paves the way for applications in geophysics and in inertial navigation as it overcomes the main limitation of atomic accelerometers, namely the dead times between consecutive measurements

New bounds on the periodicity transient of the powers of a tropical matrix: using cyclicity and factor rank

Building on the weak CSR approach developed in a previous paper by Merlet, Nowak and Sergeev, we establish new bounds for the periodicity threshold of the powers of a tropical matrix. According to that approach, bounds on the ultimate periodicity threshold take the form of T=max(T_1,T_2), where T_1 is a bound on the time after which the weak CSR expansion starts to hold and T_2 is a bound on the time after which the first CSR term starts to dominate. The new bounds on T_1 and T_2 established in this paper make use of the cyclicity of the associated graph and the (tropical) factor rank of the matrix, which leads to much improved bounds in favorable cases. For T_1, in particular, we obtain new extensions of bounds of Schwarz, Kim and Gregory-Kirkland-Pullman, previously known as bounds on exponents of digraphs. For similar bounds on T_2, we introduce the novel concept of walk reduction threshold and establish bounds on it that use both cyclicity and factor rank.Comment: 37 page

SAR image dataset of military ground targets with multiple poses for ATR

Automatic Target Recognition (ATR) is the task of automatically detecting and classifying targets. Recognition using Synthetic Aperture Radar (SAR) images is interesting because SAR images can be acquired at night and under any weather conditions, whereas optical sensors operating in the visible band do not have this capability.Existing SAR ATR algorithms have mostly been evaluated using the MSTAR dataset.1 The problem with the MSTAR is that some of the proposed ATR methods have shown good classification performance even when targets were hidden,2 suggesting the presence of a bias in the dataset. Evaluations of SAR ATR techniques arecurrently challenging due to the lack of publicly available data in the SAR domain. In this paper, we present a high resolution SAR dataset consisting of images of a set of ground military target models taken at various aspect angles, The dataset can be used for a fair evaluation and comparison of SAR ATR algorithms. We applied the Inverse Synthetic Aperture Radar (ISAR) technique to echoes from targets rotating on a turntable and illuminated with a stepped frequency waveform. The targets in the database consist of four variants of two 1.7m-long models of T-64 and T-72 tanks. The gun, the turret position and the depression angle are varied to form 26 different sequences of images. The emitted signal spanned the frequency range from 13 GHz to 18 GHz to achieve a bandwidth of 5 GHz sampled with 4001 frequency points. The resolution obtained with respect to the size of the model targets is comparable to typical values obtained using SAR airborne systems. Single polarized images (Horizontal-Horizontal) are generated using the backprojection algorithm.3 A total of 1480 images are produced using a 20° integration angle. The images in the dataset are organized in a suggested training and testing set to facilitate a standard evaluation of SAR ATR algorithms

Characterisation of sidelobes for multibeam radar based on quasi-orthogonal LFM waveforms

Multibeam radars (MBRs) enable multiple independent channels by simultaneously exploiting spatial and waveform diversity. Orthogonal waveforms are employed to form multiple independent antenna beams, each one providing a different function and using different dedicated radar resources. This paper investigates sidelobe levels in MBRs and presents a comparison with those of an Electronic Steerable Array (ESA) that employs a single waveform in transmission to generate multiple simultaneous beams. Simulations are carried out for a 3-channel MBR transmitting quasi-orthogonal Linear Frequency Modulated (LFM) waveforms at Ku band. The response of the MBR to an ideal point target as a function of aspect angle as well as that to multiple targets in different locations has been investigated. Results corroborate the analytical findings and show that the sidelobe levels with respect to angle, at the target range, are attenuated by the cross-ambiguity function properties between the waveforms employed. The range response to a target in low channel isolation suffers from cross-channel interference that may alter the noise floor characteristics of the radar, hence stressing the importance of suitable waveform selection