A Lattice Basis Reduction Approach for the Design of Finite Wordlength FIR Filters

International audienceMany applications of finite impulse response (FIR) digital filters impose strict format constraints for the filter coefficients. Such requirements increase the complexity of determining optimal designs for the problem at hand. We introduce a fast and efficient method, based on the computation of good nodes for polynomial interpolation and Euclidean lattice basis reduction. Experiments show that it returns quasi-optimal finite wordlength FIR filters; compared to previous approaches it also scales remarkably well (length 125 filters are treated in < 9s). It also proves useful for accelerating the determination of optimal finite wordlength FIR filters

NATURAL ALGORITHMS IN DIGITAL FILTER DESIGN

Digital filters are an important part of Digital Signal Processing (DSP), which plays vital roles within the modern world, but their design is a complex task requiring a great deal of specialised knowledge. An analysis of this design process is presented, which identifies opportunities for the application of optimisation. The Genetic Algorithm (GA) and Simulated Annealing are problem-independent and increasingly popular optimisation techniques. They do not require detailed prior knowledge of the nature of a problem, and are unaffected by a discontinuous search space, unlike traditional methods such as calculus and hill-climbing. Potential applications of these techniques to the filter design process are discussed, and presented with practical results. Investigations into the design of Frequency Sampling (FS) Finite Impulse Response (FIR) filters using a hybrid GA/hill-climber proved especially successful, improving on published results. An analysis of the search space for FS filters provided useful information on the performance of the optimisation technique. The ability of the GA to trade off a filter's performance with respect to several design criteria simultaneously, without intervention by the designer, is also investigated. Methods of simplifying the design process by using this technique are presented, together with an analysis of the difficulty of the non-linear FIR filter design problem from a GA perspective. This gave an insight into the fundamental nature of the optimisation problem, and also suggested future improvements. The results gained from these investigations allowed the framework for a potential 'intelligent' filter design system to be proposed, in which embedded expert knowledge, Artificial Intelligence techniques and traditional design methods work together. This could deliver a single tool capable of designing a wide range of filters with minimal human intervention, and of proposing solutions to incomplete problems. It could also provide the basis for the development of tools for other areas of DSP system design

Efficient FPGA implementation and power modelling of image and signal processing IP cores

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Field Programmable Gate Arrays (FPGAs) are the technology of choice in a number ofimage and signal processing application areas such as consumer electronics, instrumentation, medical data processing and avionics due to their reasonable energy consumption, high performance, security, low design-turnaround time and reconfigurability. Low power FPGA devices are also emerging as competitive solutions for mobile and thermally constrained platforms. Most computationally intensive image and signal processing algorithms also consume a lot of power leading to a number of issues including reduced mobility, reliability concerns and increased design cost among others. Power dissipation has become one of the most important challenges, particularly for FPGAs. Addressing this problem requires optimisation and awareness at all levels in the design flow. The key achievements of the work presented in this thesis are summarised here. Behavioural level optimisation strategies have been used for implementing matrix product and inner product through the use of mathematical techniques such as Distributed Arithmetic (DA) and its variations including offset binary coding, sparse factorisation and novel vector level transformations. Applications to test the impact of these algorithmic and arithmetic transformations include the fast Hadamard/Walsh transforms and Gaussian mixture models. Complete design space exploration has been performed on these cores, and where appropriate, they have been shown to clearly outperform comparable existing implementations. At the architectural level, strategies such as parallelism, pipelining and systolisation have been successfully applied for the design and optimisation of a number of cores including colour space conversion, finite Radon transform, finite ridgelet transform and circular convolution. A pioneering study into the influence of supply voltage scaling for FPGA based designs, used in conjunction with performance enhancing strategies such as parallelism and pipelining has been performed. Initial results are very promising and indicated significant potential for future research in this area. A key contribution of this work includes the development of a novel high level power macromodelling technique for design space exploration and characterisation of custom IP cores for FPGAs, called Functional Level Power Analysis and Modelling (FLPAM). FLPAM is scalable, platform independent and compares favourably with existing approaches. A hybrid, top-down design flow paradigm integrating FLPAM with commercially available design tools for systematic optimisation of IP cores has also been developed

Finite state machine representation of digital signal processing systems

A new method for implementing digital filters is discussed. The met11od maximises the output signal to noise ratio of a filter by assigning at each of the filter variables an optimal quantization law. A filter optimised for a gaussian process is considered in detail. An error model is developed and applied to first and second order canonic form filter sections. Comparisons are drawn between the gaussian optimised filter and the equivalent fixed point arithmetic filter. The performance of gaussian optimised filters under sinusoidal input signal conditions is considered ; it is found that the gaussian optimised filter exhibits a lower approximation error than the equivalent fixed point arithmetic filter. It is shown that when high order filters are implemented as a cascade of second order sections - with if necessary one first order section - the section ordering has a very small effect on the overall signal to noise r atio performance. A similar result for the pairing of poles and zeroes is found. Bounds on the maximum limit cycle amplitude for first and second order all-pole sections are presented. It is shown that for a first order all-pole the maximum limit cycle amplitude is lower than would be expected in the equivalent fixed point arithmetic filter, whereas , for the second order all- pole the bound is twice as large. Examples of a low-pass , band-pass and wideband differentiating filter,designed using free quantization law techniques,are presented. This new design method leads to a filter whose arithmetic operations can not be performed using fixed point arithmetic hardware. Instead, the filter must be represented as a finite state machine and then implemented using sequential logic circuit synthesis techniques. The logic complexity is found to depend - amongst other considerations - on the so called state (code) assignment. Some preliminary results on this problem are presented for the case of a next state function computed using the AND/EXCLUSIVE- OR (ring-sum) logic expansion. A review of the state assignment techniques in the literature is included. A part of the state assignment problem - for the case of AND/EX'·/OR logic - requires the numerous and consequently rapid computation of the Reed-Muller Transformation. A hardware processor - designed as an add-on to a minicomputer - is described; speed comparisons are drawn with the equivalent software algorithm.Digitisation of this thesis was sponsored by Arcadia Fund, a charitable fund of Lisbet Rausing and Peter Baldwin

A robust and scalable implementation of the Parks-McClellan algorithm for designing FIR filters

Preliminary version accepted for publicationInternational audienceWith a long history dating back to the beginning of the 1970s, the Parks-McClellan algorithm is probably the most well-known approach for designing finite impulse response filters. Despite being a standard routine in many signal processing packages, it is possible to find practical design specifications where existing codes fail to work. Our goal is twofold. We first examine and present solutions for the practical difficulties related to weighted minimax polynomial approximation problems on multi-interval domains (i.e., the general setting under which the Parks-McClellan algorithm operates). Using these ideas, we then describe a robust implementation of this algorithm. It routinely outperforms existing minimax filter design routines

Dsign of I-D Recursive Digital Filters With Linear Phase Using Two All-Pass Filters With/Without Integer Coefficients

Digital signal processing is becoming increasingly important, and is finding applications in speech processing and telecommunications in the area of 1-D signal processing. One of the important branches 1n digital signal processing 1s digital filtering. Among the numbers of structure of digital filters, the recursive(IIR) filter is known for its computational efficiency compared to the FIR counterparts. In this thesis, an alternative approach to the direct design of 1-D recursive digital filters satisfying prescribed magnitude specifications with or without constant group delay characteristic using two all-pass filters is presented. It is known that, by this approach, the most computationally efficient realization can be obtained among IIR filters for meeting the filter specifications. The method uses unconstrained optimization techniques for the filter design to approximate both the group delay and the magnitude response of the desired filter simultaneously if the constant group delay characteristic is required. Two different approaches are chosen for the stability of the filter. In the first approach, a new stability test is used to generate the stable polynomials. In the second approach, one-variable Hurwitz polynomials(HPs) using properties of positive definite matrices are generated. Bilinear transformations are applied to the HPs to obtain the stable polynomials in z domain. The polynomials generated using the approaches explained above are imposed on the filter\u27s denominator polynomials through the variable subs ti tut ion method, hence ensuring the stability .of the designed filter. The designed filters using this method are stable in nature and neither stability check nor stabilization procedure is required. To illustrate the usefulness of the technique, the results obtained are compared with a well known direct method design using a general 1-D IIR transfer function. Once the infinite precision filter is obtained, through a procedure based on discretization and reoptimization technique we discretize all coefficients to integer values. By this algorithm, the error caused by truncating the filter coefficients is minimized. Examples are given with comparisons in order to demonstrate the usefulness of the algorithm

Design of discrete-time filters for efficient implementation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 325-333).The cost of implementation of discrete-time filters is often strongly dependent on the number of non-zero filter coefficients or the precision with which the coefficients are represented. This thesis addresses the design of sparse and bit-efficient filters under different constraints on filter performance in the context of frequency response approximation, signal estimation, and signal detection. The results have applications in several areas, including the equalization of communication channels, frequency-selective and frequency-shaping filtering, and minimum-variance distortionless-response beamforming. The design problems considered admit efficient and exact solutions in special cases. For the more difficult general case, two approaches are pursued. The first develops low-complexity algorithms that are shown to yield optimal or near-optimal designs in many instances, but without guarantees. The second focuses on optimal algorithms based on the branch-and-bound procedure. The complexity of branch-and-bound is reduced through the use of bounds that are good approximations to the true optimal cost. Several bounding methods are developed, many involving relaxations of the original problem. The approximation quality of the bounds is characterized and efficient computational methods are discussed. Numerical experiments show that the bounds can result in substantial reductions in computational complexity.by Dennis Wei.Ph.D