Minimum-energy filtering on the unit circle

Abstract— We apply Mortensen’s deterministic filtering approach to derive a third order minimum-energy filter for a system defined on the unit circle. This yields the exact form of a minimum-energy filter (namely an observer plus a Riccati equation that updates the observer gain). The proposed Riccati equation is perturbed by a term depending on the third order derivative of the value function of the associated optimal control problem. The proposed filter is third order in the sense that it approximates the dynamics of the third order derivate of the value function by neglecting the fourth order derivative of the value function. Additionally, we show that the nearoptimal filter proposed by Coote et al. in prior work can indeed be derived from a second order application of Mortensen’s approach to minimum-energy filtering on the unit circle

Deterministic attitude and pose filtering, an embedded Lie groups approach

Attitude estimation is a core problem in many robotic systems that perform automated or semi automated navigation. The configuration space of the attitude motion is naturally modelled on the Lie group of special orthogonal matrices SO(3). Many current attitude estimation methods are based on non-matrix parameterization of attitude. Non-matrix parameterization schemes sometimes lead to modelling issues such as the singularities in the parameterization space, non-uniqueness of the attitude estimates and the undesired conversion errors such as the projection or normalization errors. Moreover, often attitude filters are designed by linearizing or approximating the nonlinear attitude kinematics followed by applying the Kalman filtering based methods that are primarily only suitable for linear Gaussian systems. In this thesis, the attitude estimation problem is considered directly on SO(3) along with nonlinear vectorial measurement models. Minimum-energy filtering is adapted to respect the geometry of the problem and in order to solve the problem avoiding linearization or Gaussian assumptions. This approach allows for obtaining a geometric approximate minimum-energy (GAME) filter whose performance is tested by means of Monte Carlo simulations. Many of the major attitude filtering methods in the literature are surveyed and included in the simulation study. The GAME filter outperforms all of the state of the art attitude filters studied, including the multiplicative extended Kalman filter (MEKF), the unscented quaternion estimator (USQUE), the right-invariant extended Kalman filter (RIEKF) and the nonlinear constant gain attitude observer, in the asymptotic estimation error. Furthermore, the proposed GAME filter is shown to be near-optimal by deriving a bound on the optimality error of the filter that is proven to be small in simulations. Moreover, similar GAME filters are derived for pose filtering on the special Euclidean group SE(3), attitude and bias filtering on the unit circle and attitude and bias filtering on the special orthogonal group. The approximation order of the proposed method can potentially be extended to arbitrary higher orders. For instance, for the case angle estimation on the unit circle an eighth-order approximate minimum-energy filter is provided

Trellis-Based Equalization for Sparse ISI Channels Revisited

Sparse intersymbol-interference (ISI) channels are encountered in a variety of high-data-rate communication systems. Such channels have a large channel memory length, but only a small number of significant channel coefficients. In this paper, trellis-based equalization of sparse ISI channels is revisited. Due to the large channel memory length, the complexity of maximum-likelihood detection, e.g., by means of the Viterbi algorithm (VA), is normally prohibitive. In the first part of the paper, a unified framework based on factor graphs is presented for complexity reduction without loss of optimality. In this new context, two known reduced-complexity algorithms for sparse ISI channels are recapitulated: The multi-trellis VA (M-VA) and the parallel-trellis VA (P-VA). It is shown that the M-VA, although claimed, does not lead to a reduced computational complexity. The P-VA, on the other hand, leads to a significant complexity reduction, but can only be applied for a certain class of sparse channels. In the second part of the paper, a unified approach is investigated to tackle general sparse channels: It is shown that the use of a linear filter at the receiver renders the application of standard reduced-state trellis-based equalizer algorithms feasible, without significant loss of optimality. Numerical results verify the efficiency of the proposed receiver structure.Comment: To be presented at the 2005 IEEE Int. Symp. Inform. Theory (ISIT 2005), September 4-9, 2005, Adelaide, Australi

On linear H∞ equalization of communication channels

As an alternative to existing techniques and algorithms, we investigate the merit of the H∞ approach to the linear equalization of communication channels. We first give the formulation of all causal H∞ equalizers using the results of and then look at the finite delay case. We compare the risk-sensitive H∞ equalizer with the MMSE equalizer with respect to both the average and the worst-case BER performances and illustrate the improvement due to the use of the H∞ equalizer

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist