R-3 Repository :: Browsing by Author "Burrus, C. Sidney"

Browsing by Author "Burrus, C. Sidney"

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A linear time-varying model for nonlinear systems
(1969) Zhang, Dianlin; Burrus, C. Sidney
In this thesis we present an approach to the approximate solution of a class of nonlinear autonomous systems. We use a linear time-varying model, x + a(t)x = 0 to approximate the nonlinear system x + f(x) = 0, i.e., the solution of the former system is a good approximation to the solution of the latter one. The function a(t) will depend on the nature of f(x) and on the initial condition x(0). If we choose a(t) to be a constant, then this method reduces to the conventional iterative method which uses the linear time-invariant model lc + ax = 0 as an approximation to the nonlinear system. We may choose a(t) in several ways. Here we assume a form with an undetermined parameter for a(t) and then we determine the parameter by matching the trajectories of both systems in the phase plane. This method can also be applied to second order system with some modification. Excellent results are obtained when applied to specific examples. Also it is a promising idea to extend this method to driven nonlinear systems.
A method of computing the fast Fourier transform
(1968) Read, Randol Robert; Burrus, C. Sidney
The fast Fourier transform is investigated. It is proved that the number of real (as opposed to complex) multiplications necessary to implement the algorithm M for complex input sequence of length N = 2 is 2N(M -7/2) + 12. Methods which do not avoid the unnecessary multiplications predict 2N(M-1) or 2NM. It is shown experimentally that for at least one implementation of the algorithm, it is faster to take advantage of the multiplication savings mentioned above. Some theorems regarding computational savings when transforming real data are presented. A system of subroutines for calculating finite discrete Fourier transforms by the fast Fourier transform method is given. The results of applying this system to two specific problems is presented.
A numerical method for the synthesis of distributed RC networks
(1969) McKnight, Randy Sherwood; Burrus, C. Sidney
Synthesis of RC distributed networks is considered in this thesis. The behavior of these networks is formulated as a differential equation in terms of impedance. Using this behavioral equation and a boundary condition, an optimization problem is constructed so as to yield a solution that approximates a specified impedance characteristic. The technique is demonstrated for several example network configurations. By assuming an initial structure, a gradient algorithm is employed to solve the optimization problem. Two S-plane representations of the impedance, one along the oj-axis and the other along the a-axis, were used in the solution. Both representations afforded rapid convergence, with the o-axis examples achieving more accurate structures in less time. These examples clearly demonstrate the effectiveness and practicability of this method.
Adaptive Iterative Reqeighted Least Squares Design of LP FIR Filters
(IEEE, 1999-03-01) Vargas, Ricardo; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
This paper presents an efficient adaptive algorithm for designing FIR digital filters that are efficient according to an Lp error criteria. The algorithm is an extension of Burrus' iterative reweighted least-squares (IRLS) method for approximating Lp filters. Such algorithm will converge for most significant cases in a few iterations. In some cases however, the transition bandwidth is such that the number of iterations increases significantly. The proposed algorithm controls such problem and drastically reduces the number of iterations required.
ANALYSIS AND DESIGN OF PERIODICALLY TIME-VARYING DIGITAL FILTERS
(1982) LOEFFLER, CHARLES MONROE; Burrus, C. Sidney
A description of periodically time-varying systems was developed which completely characterizes the relationship between the spectra of the input and output signals of these systems. The description is the bi-frequency map. This description completely separates the time-invariant and time-varying portions of the system. It can be used to analyze each of these portions of the system. When this description was applied to a number of periodically time-varying systems, the analysis often provided new insights and facts about the characteristics of the systems which led to new and improved designs. For example, it was shown that the bi-frequency map of any discrete time-varying system could be approximated with a periodically time-varying system and an optimal design procedure was developed. New criteria for designing the filters with decimator-interpolator structures were derived. With these new design criteria more efficient filters were designed. When these techniques were applied to filters with highly quantized coefficients that dither or vary periodically with time, it was shown that under general circumstances the optimal structure is time-invariant (i.e., non-dithering). In addition to analyzing previously developed filters, a new class of filters, called operator filters, was found using the techniques developed in this thesis.
Application of distributed arithmetic to digital signal processing
(1979) Chu, Shuni; Burrus, C. Sidney; Glantz, Raymon M.; Johnson, Don H.
Distributed arithmetic trades memory for logic circuits and speed, it is suitable for some fixed computations like the DFT computation and the filter calculation with fixed coefficients. A prime length N DFT computation can be converted to two length (Nl)/2 real convolutions and distributed arithmetic can be applied to these convolution computations. Since all the computations of a prime factor FFT reside in a few short length DFT computations, we can do all the prime factor FFT computations by distributed arithmetic. When the input to a DFT is read, we can save half of the computations of a prime factor FFT algorithm by computing only half of the output without computing the other half and get the other half by the symmetric relation. Using an input index table and an output index table in a prime factor FFT algorithm, we avoid any index calculations for any dimension transform. The transpose form of filter structures using distributed arithmetic have a different arrangement of memory and accumulators from that of direct structures. In software implementation, the transpose structure has the advantage of less process with the input or output data to get the address to address the table in the memory but with the disadvantage of more accumulations when compared to the direct structure. Altogether, an IIR filter with transpose structure will have a little higher speed than that with direct structure when implemented on a microprocessor. Distributed arithmetic reduces the DFT and filter computations to simple and repeated addressing and accumulating operations which can be done by simple logic. A general, external logic can be designed to do both the DFT and filter calculations with a microprocessor.
Approximate continuous wavelet transform with an application to noise reduction
(1998-05-20) Lewis, James M.; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
We describe a generalized scale-redundant wavelet transform which approximates a dense sampling of the continuous wavelet transform (CWT) in both time and scale. The dyadic scaling requirement of the usual wavelet transform is relaxed in favor of an approximate scaling relationship which in the case of a Gaussian scaling function is known to be asymptotically exact and irrational. This scheme yields an arbitrarily dense sampling of the scale axis in the limit. Similar behavior is observed for other scaling functions with no explicit analytic form. We investigate characteristics of the family of Lagrange interpolating filters (related to the Daubechies family of compactly-supported orthonormal wavelets), and finally present applications of the transform to denoising and edge detection.
Automatic Generation of Prime Length FFT Programs
(Rice University, 2009-09-16) Burrus, C. Sidney
This collection of modules is from a Rice University, ECE Department Technical Report written around September 1994. It grew out of the doctoral and post doctoral research of Ivan Selesnick working with Prof. C. Sidney Burrus at Rice. Earlier reports on this work were published in the ICASSP and ISCAS conference proceedings in 1992-94 and a fairly complete report was published in the IEEE Transaction on Signal Processing in January 1996
Automatic Generation of Prime Length FFT Programs
(1996-01-01) Selesnick, Ivan W.; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
We describe a set of programs for circular convolution and prime length FFTs that are short, possess great structure, share many computational procedures, and cover a large variety of lengths. The programs make clear the structure of the algorithms and clearly enumerate independent computational branches that can be performed in parallel. Moreover, each of these independent operations is made up of a sequence of sub-operations which can be implemented as vector/parallel operations. This is in contrast with previously existing programs for prime length FFTs: they consist of straight line code, no code is shared between them, and they can not be easily adapted for vector/parallel implementations. We have also developed a program that automatically generates these programs for prime length FFTs. This code generating program requires information only about a set of modules for computing cyclotomic convolutions.
Basic Vector Space Methods in Signal and Systems Theory
(Rice University, 2013-01-10) Burrus, C. Sidney
Linear algebra, vector space methods, and functional analysis are a powerful setting for many topics in engineering, science (including social sciences), and business. This collection starts with the simple idea of a matrix times a vector and develops tools and interpretations for many signal processing and system analysis and design methods.
Brief Notes on Signals and Systems
(Rice University, 2012-11-17) Burrus, C. Sidney
This collection is a brief and concise set on notes on continuous and discrete time signals and systems with some information on up and down sampling and a brief introduction to wavelets.
Complex Wavelet Transforms with Allpass Filters
(2003-08-20) Fernandes, Felix; Selesnick, Ivan W.; van Spaendonck, Rutger; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
Complex discrete wavelet transforms have significant advantages over real wavelet transforms for certain signal processing problems. Two approaches to the implementation of complex wavelet transforms have been proposed earlier. Both approaches require discrete-time allpass systems having approximately linear-phase and (fractional) delay. This paper compares the results when different allpass systems are used. In the earlier work, maximally flat delay allpass systems were used. In this paper, it is shown that an allpass system designed according to the minimax criterion yields improvements for the complex discrete wavelet transforms.
Constrained FIR Filter Design for 2-band Filter Banks and Orthonormal Wavelets
(1994-10-20) Markus, Lang; Selesnick, Ivan W.; Odegard, Jan E.; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
2-band paraunitary FIR filter banks can be used to generate a multiresolution analysis with compactly supported orthonormal (ON) wavelets. The filter design problem is formulated and solved (a) as a constrained L_{Â â ¡} optimization problem and (b) as a constrained L₂ optimization problem which allows arbitrary compromises between an L₂ and an L_{Â â ¡} approach with both of them as special cases. Additional flatness constraints can also be easily included. The L₂ and the L_{Â â ¡} design are based on the Kuhn-Tucker (KT) conditions and the alternation theorem, respectively. Therefore, optimality of the solution is guaranteed. The method (a) is a simpler alternative to a known method. The method (b) solves a more general problem than the approaches known in the literature including all of them as special cases.
Constrained Least Square Design of FIR Filters Without Specified Transition Bands
(1996-01-15) Selesnick, Ivan W.; Lang, Markus; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
We consider the design of digital filters and discuss the inclusion of explicitly specified transition bands in the frequency domain design of FIR filters. We put forth the notion that explicitly specified transition bands have been introduced in the filter design literature as an indirect and sometimes inadequate approach for dealing with discontinuities in the desired frequency response. We also present a rapidly converging, robust, simple algorithm for the design of optimal peak constrained least square lowpass FIR filters that does not require the use of transition bands. This versatile algorithm will design linear and minimum phase FIR filters and gives the best L₂ filter and a continuum of Chebyshev filters as special cases.
Constrained Least Square Design of FIR Filters Without Specified Transition Bands
(1995-01-15) Selesnick, Ivan W.; Lang, Markus; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
We consider the design of digital filters and discuss the inclusion of explicitly specified transition bands in the frequency domain design of FIR filters. We put forth the notion that explicitly specified transition bands have been introduced in the filter design literature as an indirect and sometimes inadequate approach for dealing with discontinuities in the desired frequency response. We also present a rapidly converging, robust, simple algorithm for the design of optimal peak constrained least square lowpass FIR filters that does not require the use of transition bands. This versatile algorithm will design linear and minimum phase FIR filters and gives the best L₂ filter and a continuum of Chebyshev filters as special cases.
Design of Linear Phase Cosine Modulated Filter Banks for Subband Image Compression
(1994-01-15) Odegard, Jan E.; Gopinath, Ramesh A.; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/); CML (http://cml.rice.edu/)
Wavelet methods give a flexible alternative to Fourier methods in non-stationary signal analysis. The concept of band-limitedness plays a fundamental role in Fourier analysis. Since wavelet theory replaces frequency with scale, a natural question is whether there exists a useful concept of scale-limitedness. Obvious definitions of scale-limitedness are too restrictive, in that there would be few or no useful scale-limited signals. This paper introduces a viable definition for scale-limited signals, and shows that the class is rich enough to include bandlimited signals, and impulse trains, among others. Moreover, for a wide choice of criteria, we show how to design the optimal wavelet for representing a given signal, and how to design robust wavelets that optimally represent certain classes of signals.
Design of oversampled multichannel filter banks
(2005) von Borries, Ricardo Freitas; Burrus, C. Sidney
This thesis introduces a general approach for the design of oversampled multichannel filter banks including one-dimensional and nonseparable multidimensional filter banks, as well as filter banks with linear phase and complex-valued coefficients. The approach is carried within the framework of rectangular polyphase matrices, developed to design perfect reconstruction filter banks with uniform sampling in the subbands and finite impulse response filters. This thesis introduces an approach for the design of two-dimensional oversampled filter banks that allows directional frequency selectivity with low redundancy. Directional two-dimensional filter banks find application in texture classification, denoising, segmentation, and enhancement. In several of these applications, the frequency directionality relies on oversampled filter banks with redundancy factors that range from two to four. This thesis uses non uniform frequency division of the unit frequency cell to design multichannel oversampled filter banks that have frequency directionality and redundancy factor less than two. Additionally, this thesis expands the tools available for the design of wavelet systems by providing a formulation of the problem of filter bank design that covers polyphase matrices of one-dimensional perfect reconstruction filter banks, oversampled or critically sampled, with real- or complex-valued coefficients. Although in the past the design of critically sampled filter banks was addressed using a formulation based either on the factorization of polyphase matrices or on the modulation of prototype windows, the design of oversampled filter banks was more restricted to modulation. No formulation based on the polyphase matrix existed in the oversampled case for multichannel filter banks with complex-valued coefficients. Furthermore, no approach in the oversampled case had a formulation for linear phase filter banks with non integer rational oversampling ratios for non uniform bandwidth filters.
Digital Signal Processing Structures: Block and Multidemensional Formulation and Distributed Arithmetic
(1978-01-20) Burrus, C. Sidney
In this report we will consider a special class of digital filter structure; and by structure we mean the particular arrangement and sequence of arithmetic and storage operations to realize a desired signal processing function. The more conventional structures consist of an interconnection of arithmetic operations such as addition and multiplication with storage elements such as memory registers. The filter is implemented in hardware by wireing together electronic modules that perform these functions or in software by a program that directs a general purpose computer to carry out the operations. These structures area analogous to analog or continuous time active and passive filters; indeed, many digital structures have exact analog counterparts as active filters.
Directional Complex-Wavelet Processing
(2000-08-20) Fernandes, Felix; van Spaendonck, Rutger; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)
Poor directional selectivity, a major disadvantage of the separable 2D discrete wavelet transform (DWT), has previously been circumvented either by using highly redundant, nonseparable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees. In this paper, we demonstrate that superior directional selectivity may be obtained with no redundancy in any separable wavelet transform. We achieve this by projecting the wavelet coefficients to separate approximately the positive and negative frequencies. Subsequent decimation maintains non-redundancy. A novel reconstruction step guarantees perfect reconstruction within this critically-sampled framework. Although our transform generates complex-valued coefficients, it may be implemented with a fast algorithm that uses only real arithmetic. We also explain how redundancy may be judiciously introduced into our transform to benefit certain applications. To demonstrate the efficacy of our projection technique, we show that it achieves state-of-the-art performance in a seismic image-processing application.
Directional, shift-insensitive, complex wavelet transforms with controllable redundancy
(2002) Fernandes, Felix Carlos A.; Burrus, C. Sidney
Although the Discrete Wavelet Transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages. First, the DWT is shift sensitive because input-signal shifts generate unpredictable changes in DWT coefficients. Second, the DWT suffers from poor directionality because DWT coefficients reveal only three spatial orientations. Third, DWT analysis lacks the phase information that accurately describes non-stationary signal behavior. To overcome these disadvantages, we introduce the notion of projection-based complex wavelet transforms. These two-stage, projection-based complex wavelet transforms consist of a projection onto a complex function space followed by a DWT of the complex projection. Unlike other popular transforms that also mitigate DWT shortcomings, the decou pled implementation of our transforms has two important advantages. First, the controllable redundancy of the projection stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, non-redundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and non-redundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. We exploit this flexibility to create the Complex Double-density DWT (CDDWT): a shift-insensitive, directional, complex wavelet transform with a low redundancy of 3m-12m-1 in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy. Besides the mitigation of DWT shortcomings, our transforms have unique properties that will potentially benefit a variety of signal processing applications. As an example, we demonstrate that our projection-based complex wavelet transforms achieve state-of-the-art results in a seismic signal-processing application.

Browsing by Author "Burrus, C. Sidney"

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