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Digital filter banks have found in a wide variety of applications in data compression, digital communications, and adaptive signal processing. The common objectives of the filter bank design consist of frequency selectivity of the individual filters and perfect reconstruction of the filter banks. The design problems of filter banks are intrinsically challenging because their natural formulations are nonconvex constrained optimization problems. Therefore, there is a strong motivation to cast the design problems into convex optimization problems whose globally optimal solutions can be efficiently obtained. The main contributions of this dissertation are to exploit the convex optimization algorithms to design several classes of the filter banks. First, the two-channel orthogonal symmetric complex-valued filter banks are investigated. A key contribution is to derive the necessary and sufficient condition for the existence of complex-valued symmetric spectral factors. Moreover, this condition can be expressed as linear matrix inequalities (LMIs), and hence semi-definite programming (SDP) is applicable. Secondly, for two-channel symmetric real-valued filter banks, a more general and efficient method for designing the optimal triplet halfband filter banks with regularity is developed. By exploiting the LMI characterization of nonnegative cosine polynomials, the semi-infinite constraints can be efficiently handled. Consequently, the filter bank design is cast as an SDP problem. Furthermore, it is demonstrated that the resulting filter banks are applied to image coding with improved performance. It is not straightforward to extend the proposed design methods for two-channel filter banks to M-channel filter banks. However, it is investigated that the design problem of M-channel cosine-modulated filter banks is a nonconvex optimization problem with the low degree of nonconvexity. Therefore, the efficient semidefinite relaxation technique is proposed to design optimal prototype filters. Additionally, a cheap iterative algorithm is developed to further improve the performance of the filter banks. Finally, the application of filter banks to multicarrier systems is considered. The condition on the transmit filter bank and channel for the existence of zero-forcing filter bank equalizers is obtained. A closed-form expression of the optimal equalizer is then derived. The proposed filter bank transceivers are shown to outperform the orthogonal frequency-division multiplexing (OFDM) systems.

Digital filter banks have found in a wide variety of applications in data compression, digital communications, and adaptive signal processing. The common objectives of the filter bank design consist of frequency selectivity of the individual filters and perfect reconstruction of the filter banks. The design problems of filter banks are intrinsically challenging because their natural formulations are nonconvex constrained optimization problems. Therefore, there is a strong motivation to cast the design problems into convex optimization problems whose globally optimal solutions can be efficiently obtained. The main contributions of this dissertation are to exploit the convex optimization algorithms to design several classes of the filter banks. First, the two-channel orthogonal symmetric complex-valued filter banks are investigated. A key contribution is to derive the necessary and sufficient condition for the existence of complex-valued symmetric spectral factors. Moreover, this condition can be expressed as linear matrix inequalities (LMIs), and hence semi-definite programming (SDP) is applicable. Secondly, for two-channel symmetric real-valued filter banks, a more general and efficient method for designing the optimal triplet halfband filter banks with regularity is developed. By exploiting the LMI characterization of nonnegative cosine polynomials, the semi-infinite constraints can be efficiently handled. Consequently, the filter bank design is cast as an SDP problem. Furthermore, it is demonstrated that the resulting filter banks are applied to image coding with improved performance. It is not straightforward to extend the proposed design methods for two-channel filter banks to M-channel filter banks. However, it is investigated that the design problem of M-channel cosine-modulated filter banks is a nonconvex optimization problem with the low degree of nonconvexity. Therefore, the efficient semidefinite relaxation technique is proposed to design optimal prototype filters. Additionally, a cheap iterative algorithm is developed to further improve the performance of the filter banks. Finally, the application of filter banks to multicarrier systems is considered. The condition on the transmit filter bank and channel for the existence of zero-forcing filter bank equalizers is obtained. A closed-form expression of the optimal equalizer is then derived. The proposed filter bank transceivers are shown to outperform the orthogonal frequency-division multiplexing (OFDM) systems.

This thesis is concerned with the issues of design and optimisation of digital filters and multirate filter banks. The main focus and contribution of this thesis is to apply the genetic algorithm (GA) technique and to draw some comparison with the standard gradient and non-gradient based optimisation methods. The finite word length (FWL) constraint affects the accuracy of a real-time digital filter requency response. For the case of digital filters, this study is concerned with the optimisation of FWL coefficients using genetic algorithms. Some comparative study with the simple hill climber algorithms is also included. The outcome of this part of the study demonstrates a substantial improvement of the new results when compared with the simply rounded FWL coefficient frequency response. The FWL coefficient optimisation process developed in the earlier Chapters is extended to the field of multirate filter banks. All multirate filter banks suffer from the problems of amplitude, phase and aliasing errors and, therefore, constraints for perfect reconstruction (PR) of the input signal can be extensive. The problem, in general, is reduced to relaxing constraints at the expense of errors and finding methods for minimising the errors. Optimisation techniques are thus commonly used for the design and implementation of multirate filter banks. In this part of the study, GAs have been used in two distinct stages. Firstly, for the design optimisation so that the overall errors are minimised and secondly for FWL coefficient optimisation of digital filters that form the sub-band filters of the filter bank. This process leads to an optimal realisation of the filter bank that can be applied to specific applications such as telephony speech signal coding and compression. One example of the optimised QMF bank was tested on a real-time DSP target system and the results are reported. The multiple M-channel uniform and non-uniform filter banks have also been considered in this study for design optimisation. For a comparative study of the GA optimised results of the design stage of the filter bank, other standard methods such as the gradient based quasi-Newton and the non-gradient based downhill Simplex methods were also used. In general, the outcome of this part of study demonstrates that a hybrid approach of GA and standard method was the most efficient and effective process in generating the best results.

This thesis investigates several design problems concerning two-channel conjugate quadrature (CQ) filter banks and orthogonal wavelets, as well as orthogonal cosine-modulated (OCM) filter banks. It is well known that optimal design of CQ filters and wavelets and optimal design of prototype filters (PFs) of OCM filter banks in the least squares (LS) or minimax sense are nonconvex problems and to date only local solutions can be claimed. In this thesis, we first make some improvements over several direct design techniques for local design problems in terms of convergence and solution accuracy. By virtue of the recent progress in global polynomial optimization and the improved local design methods mentioned above, we describe an attempt at developing several design strategies that may be viewed as our endeavors towards global solutions for LS CQ filter banks, minimax CQ filter banks, and OCM filter banks. In brief terms, the proposed design strategies are based on several observations made among globally optimal impulse responses of low-order filter banks, and are essentially order-recursive algorithms in terms of filter length combined with some techniques in identifying a desirable initial point in each round of iteration. This main idea is applied to three design scenarios in this thesis, namely, LS design of orthogonal filter banks and wavelets, minimax design of orthogonal filter banks and wavelets, and design of orthogonal cosine-modulated filter banks. Simulation studies are presented to evaluate and compare the performance of the proposed design methods with several well established algorithms in the literature.

<p>Starting from Maxwell’s equations, we derive a sensor model for three-axis magnetometerssuitable for localization and tracking applications. The model dependson the relative position between the sensor and the target, orientation of the targetand its magnetic signature. Both point targets and extended target modelsare provided. The models are validated on data taken from various road vehicles.The suitability of magnetometers for tracking is analyzed in terms of local observabilityand Cramér Rao lower bound as a function of the sensor positions in atwo sensor scenario. Also the signal to noise ratio is computed to determine theeffective range of the magnetometer. Results from field test data indicate excellenttracking of position and velocity of the target, as well as identification of themagnetic target model suitable for target classification.</p><p> </p>

En filterbank består av flera filter som arbetar tillsammans för att dela upp en signal i olika frekvensband. De kan också användas för att slå ihop signaler separerade i frekvensplanet till en enda. Sedan tidigt 70-tal har man lärt sig att designa förlustfria filterbankar som alltså inte introducerar några som helst fel i systemet. Sådana filterbankar kallas PR-filterbankar, där PR står för 'perfekt rekonstruktion'. Exempel på applikationer där filterbankar används är bildkodning, audiokodning, kommunikationssystem och omvandling av analoga signaler till digitala (A/D-omvandling). Under de senaste åren har det framkommit att genom att lätta på kraven gällande perfekt rekonstruktion, går det att markant minska den erforderliga aritmetiska komplexiteten. Eftersom de flesta system i sig inte är förlustfria, kan man utan att egentligen påverka den totala prestandan tillåta små fel i filterbanken, så l¨ange dessa fel är försumbara i jämförelse med andra felkällor som t.ex. kvantisering och avrundning. Avhandlingen behandlar digitala filter och likformiga icke-PR-filterbankar. Merparten av filterbankarna är realiserade med någon slags moduleringsteknik (cosinus-, sinus- eller komplexmodulering). Den röda tråden genom avhandlingen är kombinationen av tämligen smala övergångsband och samtidigt låg aritmetisk komplexitet. Ett sätt att uppnå denna kombination är att använda sig av en teknik som heter frekvenssvarsmaskning och förkortas FRM. Denna metod har på ett framgångsrikt sätt använts i avhandlingen. En potentiell nackdel med FRMmetoden är att den medför en längre fördröjning genom systemet. Därför föreslås också ett sätt att syntetisera FRM-filter med låg fördröjning. Här optimeras filtren både med avseende på komplexitet och fördröjning samtidigt. En annan metod som utnyttjats för att kombinera relativt smala övergångsband med låg aritmetisk komplexitet är att använda IIR filter istället för FIR filter. Ett flertal exempel på filter och filterbankar, optimerade och syntetiserade i Matlab, illustrerar fördelarna med de föreslagna filter- och filterbanks-klasserna. / Filter banks are systems of several filters with a common input or a common output. They are used whenever a signal needs to be split into different frequency bands. Since the early seventies, the theory of digital filter banks has developed to a mature state. Today there exist numerous ways to design filter banks for different applications, such as image and audio coding, transmultiplexing in communication systems, echo cancellation, and analog-to-digital (A/D) conversion systems. However, earlier work has to a large extent been on the transfer function level, whereas in this thesis work, efficient realizations, important in e.g. low-power applications, are in focus. Further, most of the previous work have been focused on the perfect reconstruction (PR) case, which is, for many applications an unnecessarily severe restriction. It has been show that by relaxing the requirements on perfect reconstruction, and allowing the filter banks to have some errors, the arithmetic complexity can be reduced significantly. This thesis treats digital filters and uniform non-PR filter banks. A major part of the filter banks are realized using different modulation schemes (complex, cosine, or sine modulation). The governing idea through the thesis is the combination of frequency selectivity and low arithmetic complexity. One example on how to achieve frequency selective digital filters and filter banks with low arithmetic complexity is to use the frequency-response masking (FRM) approach. This approach together with the idea of using IIR filters instead of FIR filters is successfully used in the thesis. The price to pay for the reduced arithmetic complexity using FRM filters is unfortunately a longer overall delay. Therefore, some work has ben done in the field of low-delay FRM FIR filters as well. These filters are optimized on both low delay and low arithmetic complexity simultaneously. A number of design examples are included in order to demonstrate the benefits of the new classes of filters and filter banks.

Starting from Maxwell’s equations, we derive a sensor model for three-axis magnetometerssuitable for localization and tracking applications. The model dependson the relative position between the sensor and the target, orientation of the targetand its magnetic signature. Both point targets and extended target modelsare provided. The models are validated on data taken from various road vehicles.The suitability of magnetometers for tracking is analyzed in terms of local observabilityand Cramér Rao lower bound as a function of the sensor positions in atwo sensor scenario. Also the signal to noise ratio is computed to determine theeffective range of the magnetometer. Results from field test data indicate excellenttracking of position and velocity of the target, as well as identification of themagnetic target model suitable for target classification.