Rates of convergence for the approximation of dual shift-invariant systems in l2(Z)l_2(Z)

A shift-invariant system is a collection of functions $\{g_{m,n}\}$ of the form $g_{m,n}(k) = g_m(k-an)$. Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system $\gamma_{m,n}(k) = \gamma_m(k-an)$ such that each function $f$ can be written as $f = \sum g_{m,n}$. The mathematical theory usually addresses this problem in infinite dimensions (typically in $L_2(R)$ or $l_2(Z)$), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in $l_2(Z)$. For compactly supported $g_{m,n}$ (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper

Weyl-Heisenberg Spaces for Robust Orthogonal Frequency Division Multiplexing

Design of Weyl-Heisenberg sets of waveforms for robust orthogonal frequency division multiplex- ing (OFDM) has been the subject of a considerable volume of work. In this paper, a complete parameterization of orthogonal Weyl-Heisenberg sets and their corresponding biorthogonal sets is given. Several examples of Weyl-Heisenberg sets designed using this parameterization are pre- sented, which in simulations show a high potential for enabling OFDM robust to frequency offset, timing mismatch, and narrow-band interference

A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured "noises". As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data

Efficient Multiband Algorithms for Blind Source Separation

The problem of blind separation refers to recovering original signals, called source signals, from the mixed signals, called observation signals, in a reverberant environment. The mixture is a function of a sequence of original speech signals mixed in a reverberant room. The objective is to separate mixed signals to obtain the original signals without degradation and without prior information of the features of the sources. The strategy used to achieve this objective is to use multiple bands that work at a lower rate, have less computational cost and a quicker convergence than the conventional scheme. Our motivation is the competitive results of unequal-passbands scheme applications, in terms of the convergence speed. The objective of this research is to improve unequal-passbands schemes by improving the speed of convergence and reducing the computational cost. The first proposed work is a novel maximally decimated unequal-passbands scheme.This scheme uses multiple bands that make it work at a reduced sampling rate, and low computational cost. An adaptation approach is derived with an adaptation step that improved the convergence speed. The performance of the proposed scheme was measured in different ways. First, the mean square errors of various bands are measured and the results are compared to a maximally decimated equal-passbands scheme, which is currently the best performing method. The results show that the proposed scheme has a faster convergence rate than the maximally decimated equal-passbands scheme. Second, when the scheme is tested for white and coloured inputs using a low number of bands, it does not yield good results; but when the number of bands is increased, the speed of convergence is enhanced. Third, the scheme is tested for quick changes. It is shown that the performance of the proposed scheme is similar to that of the equal-passbands scheme. Fourth, the scheme is also tested in a stationary state. The experimental results confirm the theoretical work. For more challenging scenarios, an unequal-passbands scheme with over-sampled decimation is proposed; the greater number of bands, the more efficient the separation. The results are compared to the currently best performing method. Second, an experimental comparison is made between the proposed multiband scheme and the conventional scheme. The results show that the convergence speed and the signal-to-interference ratio of the proposed scheme are higher than that of the conventional scheme, and the computation cost is lower than that of the conventional scheme

Noise Covariance Properties in Dual-Tree Wavelet Decompositions

Dual-tree wavelet decompositions have recently gained much popularity, mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy. When the decomposition of a random process is performed -- which occurs in particular when an additive noise is corrupting the signal to be analyzed -- it is useful to characterize the statistical properties of the dual-tree wavelet coefficients of this process. As dual-tree decompositions constitute overcomplete frame expansions, correlation structures are introduced among the coefficients, even when a white noise is analyzed. In this paper, we show that it is possible to provide an accurate description of the covariance properties of the dual-tree coefficients of a wide-sense stationary process. The expressions of the (cross-)covariance sequences of the coefficients are derived in the one and two-dimensional cases. Asymptotic results are also provided, allowing to predict the behaviour of the second-order moments for large lag values or at coarse resolution. In addition, the cross-correlations between the primal and dual wavelets, which play a primary role in our theoretical analysis, are calculated for a number of classical wavelet families. Simulation results are finally provided to validate these results

Multicarrier Faster-than-Nyquist Signaling Transceivers: From Theory to Practice

The demand for spectrum resources in cellular systems worldwide has seen a tremendous escalation in the recent past. The mobile phones of today are capable of being cameras taking pictures and videos, able to browse the Internet, do video calling and much more than an yesteryear computer. Due to the variety and the amount of information that is being transmitted the demand for spectrum resources is continuously increasing. Efficient use of bandwidth resources has hence become a key parameter in the design and realization of wireless communication systems. Faster-than-Nyquist (FTN) signaling is one such technique that achieves bandwidth efficiency by making better use of the available spectrum resources at the expense of higher processing complexity in the transceiver. This thesis addresses the challenges and design trade offs arising during the hardware realization of Faster-than-Nyquist signaling transceivers. The FTN system has been evaluated for its achievable performance compared to the processing overhead in the transmitter and the receiver. Coexistence with OFDM systems, a more popular multicarrier scheme in existing and upcoming wireless standards, has been considered by designing FTN specific processing blocks as add-ons to the conventional transceiver chain. A multicarrier system capable of operating under both orthogonal and FTN signaling has been developed. The performance of the receiver was evaluated for AWGN and fading channels. The FTN system was able to achieve 2x improvement in bandwidth usage with similar performance as that of an OFDM system. The extra processing in the receiver was in terms of an iterative decoder for the decoding of FTN modulated signals. An efficient hardware architecture for the iterative decoder reusing the FTN specific processing blocks and realize different functionality has been designed. An ASIC implementation of this decoder was implemented in a 65nm CMOS technology and the implemented chip has been successfully verified for its functionality

Intelligent detectors

Die vorliegende Arbeit stellt eine Basis zur Entwicklung von On-Board Software für astronomische Satelliten dar. Sie dient als Anleitung und Nachschlagewerk und zeigt anhand der Projekte Herschel/PACS und SPICA/SAFARI, wie aus den Grundlagen weltraumtaugliche Flugsoftware entsteht. Dazu gehören das Verstehen des wissenschaftlichen Zwecks, also was soll wie gemessen werden und wofür ist das gut, sowie die Kenntnis der physikalischen Eigenschaften des Detektors, das Beherrschen der mathematischen Operationen zur Verarbeitung der Daten und natürlich auch die Berücksichtigung der Umstände, unter welchen der Detektor zum Einsatz kommt.This thesis contains the knowledge and a good deal of experience that are necessary for the development of such astronomical on-board software for satellites. The key elements in the development are the understanding of the scientific purpose, knowledge of the physical properties of the detector, the comprehension of the mathematical operations involved in data processing and the consideration of the technical and observational circumstances