Rakesh Kumar Patney

Some studies on nonpolynomial interpolation and error analysis

Abstract In this paper, we propose nonpolynomial and Hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation. The generalized error analysis results for nonpolynomial and Hermite nonpolynomial interpolations are derived. We establish theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series, and propose solution to Runge’s phenomenon through nonpolynomial interpolation. The Hermite nonpolynomial cubic spline, nonpolynomial cubic spline interpolation methods and their error analysis are presented. Numerical simulations are carried out for the analysis of error in cubic spline interpolations. Proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations, and to Empirical Mode Decomposition (EMD) to illustrate practical usefulness of the results.

The Fourier decomposition method for nonlinear and non-stationary time series analysis

for many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory, and demonstrate its efficacy for the analysis of nonlinear and non-stationary time series. The proposed FDM decomposes any data into a small number of ‘Fourier intrinsic band functions’ (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM. The proposed MFDM generates a finite number of band-limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.

Finding nonfaulty subtrees in faulty binary tree architectures

Abstract In this paper we have studied fault tolerance of a full binary tree in terms of availability of non-faulty (full) subtrees. When an unaugmented full binary tree is faulty, then the computation can be carried out on the largest available non-faulty (full) binary subtree. It is shown that the minimum number of faulty nodes required to destroy all subtrees of height h in a full binary tree of height n is given as f bt ( n, h ) = ⌊ (2 n −1) (2 h −1) ⌋ . It follows that the availability of a non-faulty subtree of height h = n−w, in an n level full binary tree containing u faulty nodes, can be ensured, where w is the smallest integer such that u ≤ 2w. An algorithm which evaluates whether a given set of faulty nodes will destroy all subtrees of some specified height, is given. This algorithm can also evaluate the largest available non-faulty subtree in a faulty full binary tree. We also study the availability of a non-faulty subtree in some augmented binary tree architectures.

A novel approach to design of signal matched QMF and DFT filter bank

In this paper, we propose three different filter bank structures matched to a signals or its statistics namely: 2-channel uniform filter bank, M-channel dyadic nonuniform filter bank and M-channel modified DFT filter bank. First, 2-channel QMF analysis filter bank matched to a signal or its statistics is obtained. In order to obtain this filter bank, the given sequence is first divided into even and odd subsequences. For each subsequence predictor is estimated. By combining these predictors in such a way that the resultant signals represent the different frequency bands, analysis low pass and high pass filters are obtained and then by combining the inverses of linear predictors corresponding synthesis filters are easily obtained. In this manner two channel signal matched QMF bank is estimated by using same approach, we present, M-channel NUFB model matched to the signal or its statistics. To estimate the filters of this model, first we estimate two channel QMF analysis bank and then using the signal across the low pass subband, 2-channel analysis bank is obtained again in the same fashion. By cascading these two 2-channel analysis banks, 3-channel non-uniform filter with decimation factors {2 ,4, 4} matched to signal or its statistics is estimated. Further, proposed approach is also extended to find M-channel residual error DFT filter bank (modified DFT filter bank). In this approach, first given sequence is divided into M-subsequences and for each subsequence predictor coefficients are estimated and then these predictors are combine using DFT matrix. Filter banks design using proposed approach are computationally inexpensive and also they give compression results equal to or better than uniform counter part. To validate the theory the results of compression on speech clips are tabulated in the table.

Fault-tolerant analysis and algorithms for a proposed augmented binary tree architecture

An augmented binary (AB) tree architecture is proposed with a view to providing fault tolerance. This architecture is an augmentation of an n-level full binary tree with n redundant nodes and 2/sup n/+3n-6 redundant links. The AB tree can be configured into a full binary tree even when one node is faulty at each level. While functionally equivalent to the RAE-tree, the proposed AB tree has a regular topology, reduced number of maximum input-output channels per processor, and fewer wire crossovers when implemented using very large-scale integration layout. A reconfiguration algorithm, which constructs an n-level full binary tree from an n-level faulty AB tree, is given. A distributed fault diagnosis algorithm is given which runs concurrently on each nonfaulty processor, enabling each nonfaulty processor to identify all faulty processors.<<ETX>>

Generation Mechanism for Cyclostationary and Self-Similar Processes

This paper proposes a generation mechanism for cyclostationary and self-similar processes. The proposed model extracts the information from the immediate coarser scale and adds the innovations to it to obtain the finer scale representation of the stochastic process. Basic block of the proposed model is the subband coder. By cascading the blocks of subband coder and passing white noise as one of input in addition to coarser information at each stage, cyclostationary process is generated. For generation of self-similar processes, white noise input for each stage is given in particular fashion. The mapping from finer scale to the immediate coarser scale is obtained using proposed blurring model. We have also given scheme for estimation of parameters of the proposed generation mechanism. The parameters are estimated for a given statistics case as well as the given data case. The proposed model can be used in variety of applications such as speech and image processing, biomedical signal processing, seismic data processing etc

Perfect Reconstruction in Non-biorthonormal Filter Banks using Vector Space Approach

A formulation is proposed for construction of PRFB from a given non-PRFB and is described using vector space framework for filter banks. To construct PRFB, a transmultiplexer (TMUX) structure is inserted into the subband such that the synthesis and analysis parts of the TMUX are biorthonormal to analysis and synthesis bank of the given filter bank. The TMUX is a represented by transformation matrix. In addition to PR, in this paper, another objective is to study and exploit the properties of transformation matrix corresponding to non-PR TMUX. The transformation matrix is portioned into distinct subblocks. In case of uniform filter bank (UFB) it is shown that each subblock of transformation matrix has convolution matrix structure. Whereas in case of nonuniform filter bank (NUFB) it is shown that each of these subblock has a structure consisting of interspersed convolution matrices. Implementation of these matrices using discrete time FIR or IIR filters are also shown in this paper. It is also shown that implementation of convolution matrices involve linear time invariant filters whereas interspersed convolution matrices involve the time varying filters. During the implementation of transformation matrix it is also found that some of the blocks can be derived by using implemented blocks. By inserting one or more TMUXs in the subbands and merging with subchannels of UFB we can obtained NUFB from given UFB. There are various application of NUFB over UFB such as speech and audio signal processing where nonuniform division of bands is important