Numerical implementation of the Hilbert transform

Many people have abnormal heartbeats from time to time. A Holter monitor is a device used to record the electrical impulses of the heart when people do ordinary activities. Holter monitoring systems that can record heart rate and rhythm when you feel chest pain or symptoms of an irregular heartbeat (called an arrhythmia) and automatically perform electrocardiogram (ECG) signal analysis are desirable.The use of the Hilbert transform (HT) in the area of electrocardiogram analysis is investigated. A property of the Hilbert transform, i.e., to form the analytic signal, was used in this thesis. Subsequently pattern recognition can be used to analyse the ECG data and lossless compression techniques can be used to reduce the ECG data for storage.The thesis discusses one part of the Holter Monitoring System, Input processing.Four different approaches, including the Time-Domain approach, the Frequency-Domain approach, the Boche approach and the Remez filter approach for calculating the Hilbert transform of an ECG wave are discussed in this thesis. By comparing them from the running time and the ease of software and hardware implementations, an efficient approach (the Remez approach) for use in calculating the Hilbert transform to build a Holter Monitoring System is proposed. Using the Parks-McClellan algorithm, the Remez approach was present, and a digital filter was developed to filter the data sequence. Accurate determination of the QRS complex, in particular, accurate detection of the wave peak, is important in ECG analysis and is another task in this thesis. A program was developed to detect the wave peak in an ECG wave.The whole algorithm is implemented using Altera’s Nios SOPC (system on a program chip) Builder system development tool. The performance of the algorithm was tested using the standard ECG waveform records from the MIT-BIH Arrhythmia database. The results will be used in pattern recognition to judge whether the ECG wave is normal or abnormal

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

Design, stability and applications of two dimensional recursive digital filters

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Design &implementation of complex-valued FIR digital filters with application to migration of seismic data

One-dimensional (I-D) and two-dimensional (2-D) frequency-space seismic migration FIR digital filter coefficients are of complex values when such filters require special space domain as well as wavenumber domain characteristics. In this thesis, such FIR digital filters are designed using Vector Space Projection Methods (VSPMs), which can satisfy the desired predefined filters' properties, for 2-D and three-dimensional (3-D) seismic data sets, respectively. More precisely, the pure and the relaxed projection algorithms, which are part of the VSPM theory, are derived. Simulation results show that the relaxed version of the pure algorithm can introduce significant savings in terms of the number of iterations required. Also, due to some undesirable background artifacts on migrated sections, a modified version of the pure algorithm was used to eliminate such effects. This modification has also led to a significant reduction in the number of computations when compared to both the pure and relaxed algorithms. We further propose a generalization of the l-D (real/complex-valued) pure algorithm to multi-dimensional (m-D) complex-valued FIR digital filters, where the resulting frequency responses possess an approximate equiripple nature. Superior designs are obtained when compared with other previously reported methods. In addition, we also propose a new scheme for implementing the predesigned 2-D migration FIR filters. This realization is based on Singular Value Decomposition (SVD). Unlike the existing realization methods which are used for this geophysical application, this cheap realization via SVD, compared with the true 2-D convolution, results in satisfactory wavenumber responses. Finally, an application to seismic migration of 2-D and 3-D synthetic sections is shown to confirm our theoretical conclusions. The proposed resulting migration FIR filters are applied also to the challenging SEGIEAGE Salt model data. The migrated section (image) outperformed images obtained using other FIR filters and with other standard migration techniques where difficult structures contained in such a challenging model are imaged clearly

Design of multichannel nonrecursive digital filters with applications to seismic reflection data

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Image filtering using the NTT convolver.

Dept. of Electrical and Computer Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1984 .V377. Source: Masters Abstracts International, Volume: 40-07, page: . Thesis (M.A.Sc.)--University of Windsor (Canada), 1984

Nonlinear Approximations in Filter Design and Wave Propagation

This thesis has two parts. In both parts we use nonlinear approximations to obtain accurate solutions to problems where traditional numerical approaches rapidly become computationally infeasible. The first part describes a systematic method for designing highly accurate and efficient infinite impulse response (IIR) and finite impulse response (FIR) filters given their specifications. In our approach, we first meet the specifications by constructing an IIR filter, without requiring the filter to be causal, and possibly with a large number of poles. We then construct, for any given accuracy, an optimal IIR version of such filter. Finally, also for any given accuracy, we convert the IIR filter to an efficient FIR filter cascade. In this FIR approximation, the non-causal part of the IIR filter only introduces an additional delay. Because our IIR construction does not have to enforce causality, the filters we design are more efficient than filters designed by existing methods. The second part describes a fast algorithm to propagate, for any desired accuracy, a time-harmonic electromagnetic field between two planes separated by free space. The analytic formulation of this problem (circa 1897) requires the evaluation of the Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is large, it can be accurately approximated by asymptotic methods. The computational difficulties arise in the intermediate region where, in order to obtain an accurate solution, it is necessary to apply the oscillatory Rayleigh-Sommerfeld kernel as is. In our approach, we accurately approximate the kernel by a short sum of Gaussians with complex exponents and then efficiently apply the result to input data using the unequally spaced fast Fourier transform. The resulting algorithm has the same computational complexity as methods based on the Fresnel approximation. We demonstrate that while the Fresnel approximation may provide adequate accuracy near the optical axis, the accuracy deteriorates significantly away from the optical axis. In contrast, our method maintains controlled accuracy throughout the entire computational domain