Noels et al [1],[2] X derived a maximum likelihood (ML)
algorithm for the problem of constant **phase** **estimation**, and then applied a first-order and a second order **phase**-locked loop (PLL) based algorithm for the coded BPSK and QPSK **dynamical** **phase** **estimation**. The corresponding performances are limited both by the on-line bound and by a non-zero **phase** MSE floor. On the contrary, this paper deals with the non data aided (NDA) **estimation** of a time-varying **phase** and proposes an off-line Smoothing PLL (S-PLL) algorithm. To assess the performance of such algorithms, Bayesian and hybrid Cramér- Rao Bounds (BCRB and HCRB) associated to this **dynamical** **phase** synchronization problem have already been considered in some recent contributions X [3] XX -[6] X and clearly show the

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proposed in [5],[6] are based on the variational Bayesian methods. [7] composed a partical filtering with a **phase**-locked loop technique and found some interest to partical filtering when there are abrupt changes in the **phase** trajectory. A BCJR based Gaussian sum smoother was proposed in [21] for convolutional turbo code aided **dynamical** **phase** **estimation**, and was derived by extending the original BCJR algorithm [26] to the case of joint **phase** **estimation** and decoding. However, in the general case, Bayesian methods are far more complex to implement than deterministic methods and thus deterministic **phase** algorithms have been employed so far in real systems. The most commonly used deterministic methods are based on the EM algorithm [8],[9] or on gradient-like **phase** locked loops (PLL) techniques. PLL are traditionally known as low-cost algorithms that lead to a good asymptote performance for on-line **estimation** [10]. Their performance can even be improved at low SNRs within the turbo-receiver concept [11]-[13]. Rather than considering actual on-line **estimation** for which estimated values only depend on past observations, one may also consider off-line estimators involving both past and future observations; such a data block approach is a very natural way to proceed, as modern systems use error correcting codes. Bayesian and hybrid Cramer-Rao bounds (BCRBs and HCRBs), associated to this **dynamical** **phase** synchronization problem, have been considered in some recent contributions [14]-[17] and they allow to measure the performance of such algorithms; they clearly show the superiority of the off-line approach, compared to the on-line approach. Contrarily to the performance of [11]-[13] limited by the on-line bound, [17] analyzed a smoothing PLL (S-PLL) algorithm based on two on-line PLLs that was proposed without any performance evaluation in [18]. [17] derived the S-PLL algorithm for a non-coded BPSK system and it is well known that it is not difficult to achieve synchronization in such a context. [19] and [20] extended the study of the S-PLL algorithm to other telecommunication formats.

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J. Yang, B. Geller, and A. Wei 1
Bayesian and Hybrid Cramer-Rao Bounds for QAM **Dynamical** **Phase** **Estimation**
Abstract—In this paper, we study Bayesian and hybrid Cramer-Rao bounds for the **dynamical** **phase** **estimation** of QAM modulated signals. We present the analytical expressions for the various CRBs. This avoids the calculation of any matrix inversion and thus greatly reduces the computation complexity. Through simulations, we also illustrate the behaviors of the BCRB and of the HCRB with the signal-to-noise ratio.

III. HCRB FOR A **DYNAMICAL** **PHASE** **ESTIMATION** PROBLEM
In [6], we have proposed a closed-form expression of the Bayesian CRB for the **estimation** of the **phase** offset for a BPSK transmission in a non data-aided context. In this section, we extend these previous results by providing a closed-form expression of the HCRB for the **estimation** of the **phase** offset and also of the linear drift. In this more realistic scenario, we show that we have to take into account the statistical dependence between the parameters and, consequently, the HCRB given by Eqn. (3) is not adapted to this problem.

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Turbo code aided [12] X and the convolution code aided [13] X
scenarios; the CA CRBs in X [11] X - X [13] X were derived from the
first derivative of the joint probability function between the observations and parameters. All these papers refer to an idealized situation in which the **phase** offset is constant. However, in modern communications, it is common to take into account a time-varying **phase** noise mostly due to oscillator

noise . This scenario is standard in radio-localization from a satellite signal. In [7], we have shown the potential gain for **phase** **estimation** provided by the use of the fractionaly-spaced signal after matched filter, instead of the symbol time-spaced signal. This was done by deriving a closed-form expression of the on-line Bayesian Cram´er-Rao Bound (BCRB) for the oversampled **dynamical** **phase** **estimation**. Now, our goal is to propose an EKF based algorithm which can approach this bound. We have thus to jointly estimate the coloured noise and the **phase** offset because the EKF doesn’t take it into

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The evaluation protocol is as following: we compute the acoustic pressure at several focal points within the volume. On these points we compare the acoustic pressure obtained with our optimized **phase** shifts **estimation** method (8) (we will call it Optimized **phase** in the rest of the paper) with the naive method (2) (Classical **phase** in the rest of the paper)). The results can be seen on Table I. Our method gives better results than the classical **phase** shift **estimation** especially for focal points located less than 30mm from the probe. In fact, for focal points close to the probe surface, we are in a near field case and therefore the waves originating from the same single element face generate their own interference. Taking the element size into account for the **phase** shifts **estimation** allows optimizing the interferences at the focal point.

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are collected are particularly appealing when affordable in terms of computational complexity [ 17 , 46 , 4 , 2 , 28 ].
Unfortunately in many situations, the **estimation** procedure – independently of the chosen method – can lack of robustness when the measurement are too sparse or too noisy, without relying on strong model priors. A way to compensate the underlying observability weakness can be to use the fact that multiple independent measurements have been collected on configuration where the model variables describe a statistical unit of interest belonging to a larger population where the variability is constrained. This is typically the assumption made in nonlinear

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The DIC **phase** **estimation** problem
The DIC image is formed by the interference of two orthogonally polarized beams that have a lateral displacement (called shear ) and are **phase** shifted relatively one to each other. The resulting image has a 3D high contrast appearance, which can be enhanced by introducing a uniform **phase** diﬀerence between the beams (called bias ).

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A method of formulating equations of motion known as Kane's Dynamical Equations is compared to traditional methods of dynamics, specifically Lagrange's equations. A d[r]

64 CHAPTER 4. ISOCHRONY AND EVOLUTION OF SELF-GRAVITATING SYSTEMS
is originally justified by spatial limitation corresponding to the tidal cut-off imposed by the surrounding galaxy in which the globular cluster evolves. When the cut-off is included as a parameter of the statistical problem, it produces an isothermal sphere in a box (ISB) which is a trademark problem of the gravitational statistical physics. This fundamental problem has a centennial history and it finally appears to be closely related with the King model (see [28] and reference therein). Studying the stability properties of this ISB in terms of the conditions imposed on the box, theoretical astrophysicists were gradually able to understand the **dynamical** history of globular clusters through the more and more refined analysis of the so-called gravothermal catastrophe ([100], [82], [120], [27]). The conclusion of this history is now well established concerning the mass density (see for example a very nice synthesis in §7.5 of [78]): after the initial violent relaxation process an isolated N −body system settles down in a spherical core-halo and quasi-equilibrium state. This state is well described by a lowered isothermal sphere (King Model) and evolves adiabatically to a more concentrated lowered isothermal state under the influence of slow relaxation. When the stability of this lowered isothermal state — governed by its density contrast — is no more possible, the core of the system shrinks. For example in our galaxy, about 20 per cent of the globular clusters possess the steep cusp in their surface-brightness profile (e.g. [43]) that is predicted by models of post-collapse evolution. The complementary set is not yet collapsed and possess a core-halo structure with a slope in the halo fixed by the level of relaxation of the system. When the system is assumed to have a non constant mass function this mechanism is reinforced by mass segregation: the most massive objects become concentrated in the high-density central core.

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Different approaches have been followed in previous work, such as the **Phase** Shifting algorithm used by Lin et al. in [4] and the method based on the transport-of-intensity equation exploited by Bostan et al. in [5]. In this paper we have fol- lowed the approach of rotational-diversity proposed by Preza in [6], which provides information about the specimen-**phase** function derivative in multiple directions and, by exploiting a least-squares approach, gives the **phase** **estimation** as a reg- ularized solution of a smooth nonconvex optimization prob- lem. We propose to regularize the least-squares optimization problem by using a smooth total variation term and we intro- duce an efficient gradient-descent method for solving the reg- ularized problem. We have added the polychromatic acqui- sition in order to further increase **phase** diversity. Our main contributions consist in: i) the extension of the model from grayscale to color images; ii) a novel formulation of the gra- dient which allows faster computations at each iteration; iii) a clever choice of the step size parameter for the gradient- descent method to improve the speed of convergence [7].

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In case of TV channels, [8] develops a time-domain MMSE equalizer based on the well-known Basis Expansion Model (BEM) [9]. It also supposes that the channel is perfectly known at the receiver and it seems that the study of TV channel **estimation** is not widely addressed for Continuous **Phase** Mod- ulation. However, for linear modulation and for OFDM, BEM- based channel **estimation** has been well studied using different BEMs [10]–[15] and so we propose in this paper to adapt some of those methods for block-based CPM transmissions over TV channels and to evaluate their performance.

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and perfect channel knowledge. To our knowledge, only a few papers deal with TIV channel **estimation** for CPM. [2] presents simulation results with channel **estimation** errors but no channel estimator. [5] presents a Least Squares (LS) channel **estimation** in the time-domain, based on the polyphase representation of the received signal. It also exploits the a priori positioning in order to develop a parametric model on the delays of the paths and so to enhance the performance of the channel **estimation**. In [4] and [6], the authors perform a FD channel **estimation** with interpolation (using B-spline functions). [7] performs frequency-domain channel **estimation** with superimposed pilots. Those Frequency-Domain channel estimators exploit the diagonal structure of the channel matrix in the Frequency-Domain to perform Channel **Estimation**, which is not the case anymore for TV channels.

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Despite significant algorithmic advances, 3D motion **estimation** remains challenging due to limited field of view, low image quality, and large computation load.
To increase the definition of the motion **estimation** in the lateral and elevation directions (i.e. perpendicular to the beam axis), the transverse oscillations (TO) method has been recently extended in 3D by the group of J.A. Jensen [9]. This method enables one to obtain 2 separate ultrasound fields featuring each one oscillations along the axial dimension plus one perpendicular (lateral or elevation) direction. This approach consists more in a twice 2D tactic than 3D. In this approach, two different apodization functions are used for generating elevation and lateral oscillations, respectively.

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In the second part, the LPV/ H ∞ control strategy based on the
algebraic **estimation** for the suspension systems is developed to enhance the overall vertical dynamics of the vehicle and improve the car safety and stability in critical driving situations (roll-over, obstacles avoidance,...). Indeed, the authors have already provided some promising strategies and first results that adapt the vehicle performance to the driving situations as in Fer- gani et al. (2013b) and Fergani et al. (2013a). In the sequel, the new strategy uses the **estimation** of the effective vehicle roll, the effective road bank angles and the effective lateral acceleration in order to adapt the vertical performances objectives to the real **dynamical** behaviour generated by the lateral load transfers. This paper is structured as follows: Section 2 presents the al- gebraic **estimation** approach for the vehicle dynamic states and road bank angles. Section 3 is devoted to the main contribution of the paper, which is to give a new LPV strategy for suspension control adaption to **dynamical** lateral load transfers based on the non linear algebraic **estimation** strategy. Then, in Section 4, simulations performed on non linear vehicle models with data collected on real cars prove the efficiency of the presented strategy for improving the vehicle **dynamical** performances. Conclusions are given in the last Section

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Lastly, we will need the classification of Fatou components, which is a corollary of Sullivan’s no wandering domain theorem. Note that McMullen (see [McM14] ) has given a direct and purely infinitesimal proof of Sullivan’s theorem, which does notably not rely on the theory of **dynamical** Teichmüller spaces. His proof is based on quasiconformal vector fields and is in the same spirit as the methods used here.

DOI: 10.1103/PhysRevA.86.062320 PACS number(s): 03.67. −a, 07.55.Ge, 76.30.Mi, 76.60.Lz
I. INTRODUCTION
Solid-state qubits have emerged as promising quantum sensors, as they can be fabricated in small volumes and brought close to the field to be detected. Notably, nitrogen-vacancy (NV) centers in nanocrystals of diamond [ 1 ] have been applied for high-sensitivity detection of magnetic [ 2 – 4 ] and electric fields [ 5 ] and could be used either as nanoscale scanning tips [ 6 ] or even in vivo due to their small dimensions and low cytotoxicity [ 7 ]. Unfortunately, solid-state qubits are also sensitive probes of their environmental noise [ 8 – 10 ] and this leads to rapid signal decay, which limits the sensor interroga- tion time and thus its sensitivity. **Dynamical** decoupling (DD) methods [ 11 – 16 ] have been adopted to prolong the coherence time of the sensor qubits [ 2 , 9 , 17 – 19 ]. Although DD techniques prevent measuring constant dc fields, they provide superior sensitivity to oscillating ac fields, as they can increase the sensor coherence time by orders of magnitude. The sensitivity is maximized by carefully matching the decoupling period to the ac field; conversely, one can study the response of a decoupling scheme to fields of various frequencies, thus mapping out their bandwidth. Still, the refocusing power of pulsed DD techniques is ultimately limited by pulse errors and bounds in the driving power. Here we investigate an alternative strategy, based on continuous **dynamical** decoupling (CoDD), that has the potential to overcome these limitations.

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