ShivDutt Joshi

Some Studies on the Structure of Covariance Matrix of Discrete-Time fBm

This paper presents some results on the structure of autocovariance matrix of discrete-time fractional Brownian motion. Since these processes are nonstationary, the autocovariance matrix is a function of time. The eigenvalues associated with the autocovariance matrix are dependent on Hurst exponent characterizing the discrete-time fractional Brownian motion. However, it is seen that only one eigenvalue of this autocovariance matrix depends on time index n in the asymptotic sense, and it increases as the time index increases. All other eigenvalues are observed to be invariant with time index in the asymptotic sense. It is observed that the eigenvectors associated with these eigenvalues have an interesting structure and these eigenvectors can be used as filters of an M-band orthogonal wavelet system. The eigenvector associated with the time-varying eigenvalue is a lowpass filter. The subband signal from this lowpass filtered branch imbibes the nonstationary attribute of the input process and is a nonstationary signal whereas all other subband signals are stationary in nature.

Characterization of Discrete-time Fractional Brownian motion

In this paper, we present the characterization of the discrete-time fractional Brownian motion (dfBm). Since, these processes are non-stationary; the auto-covariance matrix is a function of time. It is observed that the eigenvalues of the auto-covariance matrix of a dfBm are dependent on the Hurst exponent characterizing this process. Only one eigenvalue of this auto-covariance matrix depends on time index n and it increases as the time index of the auto-covariance matrix increases. All other eigenvalues are observed to be invariant with time index n in an asymptotic sense. The eigenvectors associated with these eigenvalues also have a fixed structure and represent different frequency channels. The eigenvector associated with the time-varying eigenvalue is a low pass filter

On the concept of Intrinsic Wavelet Functions

This paper presents the concept of Intrinsic Wavelet Functions (IWFs) for efficient analysis of signals with transitory behavior. An approach is proposed for the decomposition of a given signal, consisting of localized waves, into IWFs that can best capture these small waves. To this end, first, we present a method to design a 2-channel signal-matched wavelet system based on least squares criterion. Next, IWFs are designed using this matched wavelet system. The proposed work differs from empirical mode decomposition (EMD) in one fundamental way. While EMD decomposes small segments of a given signal into oscillatory intrinsic mode functions (IMFs) similar to the concept of Short-Time Fourier transform, we extract localized small waves, namely, IWFs based on the concept of wavelets from a given signal. Simulation results demonstrate that this theory is aptly suited to analyze signals consisting of localized waves.

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.

A Novel Least Mean Squares Algorithm for tracking a Discrete-time fBm Process

This paper presents a novel variable step-size LMS (VSLMS) algorithm for tracking a discrete-time fractional Brownian motion that is inherently non-stationary. In the proposed work, one of the step-size values requires time-varying constraints for the algorithm to converge to the optimal weights whereas the constraints on the remaining step-size values are time-invariant in the decoupled weight vector space. It computes the step-size matrix by estimating the Hurst exponent required to characterize the statistical properties of the signal at the input of the adaptive filter. The experimental set-up of an adaptive channel equalizer is considered for equalization of these signals transmitted over stationary AWGN channel. The performance of the proposed variable step-size LMS algorithm is compared with the unsigned VSLMS algorithm and is observed to be better for the class of non-stationary signals considered

Performance analysis of space time coded fractal modulation in the presence of frequency selective fading

The use of orthogonal space time block codes (OSTBC) in conjunction with fractal modulation in the presence of frequency selective fading channels is considered. An L scale fractal modulation and coherent demodulation effectively converts the frequency selective channel into L flat fading channels per information symbol with random scale gains. The bit error performance analysis of the proposed scheme has been done. It has been shown that a diversity order of mTmRL can be achieved. This increase in diversity order in comparison with that achieved for the OFDM case is at a cost of lower spectral efficiency. The transceiver complexity is comparable with the IFFT/FFT operations used in case of OFDM

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

An Application of Multi Wavelets in Wireless Communication In The Presence Of Frequency Selective Fading

Multi wavelet multiplexing is a signal transmission technique in which the message symbols are coded onto the multiplicity `R wavelet basis functions for transmission. The proposed transmission scheme comes under the class of orthogonally multiplexed communication. The use of multiplicity `R wavelet basis functions enhances the data transmission rate. On the other hand, the property of orthogonality enables the design of a simple correlation receiver. The use of `L scale multiplicity `R multi-wavelet multiplexing along with matched filtering at the receiver effectively converts a frequency selective fading channel into `RL flat fading channels. We have examined the performance of this multiplexing scheme from a rate-diversity trade-off point of view. Further, a simple channel estimation technique for lower data rates has been proposed

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