1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

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Wavelet Methods for Time Series Analysis

In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of $n$-by-$n$ Toeplitz systems is reduced to $O(n \log n)$ operations as compared to $O(n \log ^2 n)$ operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.

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Conjugate Gradient Methods for Toeplitz Systems

We develop a new multiscale modeling framework for characterizing positive-valued data with long-range-dependent correlations (1/f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing N-point data sets. We study both the second-order and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variance-time plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.

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A multifractal wavelet model with application to network traffic

This chapter contains sections titled: Introduction Overview of Multicarrier CDMA Systems Channel Model Performance of MC-CDMA System Performance of Overlapping Multicarrier DS-CDMA Systems Performance of Multicarrier DS-CDMA-I Systems Performance of AMC DS-CDMA Systems Performance of SFH/MC DS-CDMA Systems Chapter Summary and Conclusion 
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Overview of Multicarrier CDMA

Fractal behavior and long-range dependence have been observed in an astonishing number of physical, biological, geological, and socioeconomic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a measure of long-memory dependence. Both phenomena have been modeled and explained by self-affine random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-affinity implies a linear relationship between fractal dimension and Hurst coefficient and thereby links the two phenomena. This article introduces stochastic models that allow for any combination of fractal dimension and Hurst coefficient. Associated software for the synthesis of images with arbitrary, prespecified fractal properties and power-law correlations is available. The new models suggest a test for self-affinity that assesses coupling and decoupling of local and global behavior.

Stochastic Models That Separate Fractal Dimension and the Hurst Effect

The authors show that the application of the discrete wavelet transform (DWT) using the Haar basis to the increments of fractional Brownian motion (fBm), also known as discrete fractional Gaussian noise (DFGN), yields coefficients which are weakly correlated and have a variance that is exponentially related to scale Similar results were derived by Flandrin (1989), Tewfik, and Kim for a continuous-time fBm going through a continuous wavelet transform (CWT) The new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by Wornell and Oppenheim The performance of the new algorithm is compared with that of Wornell and Oppenheim's (see IEEE Trans Signal Processing, vol 40, p 611-623, Mar 1992) algorithm in numerical simulation >

Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis

The fractional Brownian motion (fBm) model has proven to be valuable in modeling many natural processes because of its persistence for large time lags. However, the model is characterized by one single parameter that cannot distinguish between short- and long-term correlation effects. This article investigates the idea of extending self-similarity to create a correlation model that generalizes discrete fBm referred to as asymptotic fBm (afBm). Namely, afBm is parameterized by variables controlling short- and long-term correlation effects. It proposes a fast parameter estimation algorithm for afBm based on the Haar transform, and demonstrates the performance of this parameter estimation algorithm with numerical simulations. >

/pdf/extending-self-similarity-for-fractional-brownian-motion-4spva58x90.pdf

Extending self-similarity for fractional Brownian motion

It is known that multicarrier code-division multiple-access (MC-CDMA) systems suffer from multiaccess interference (MAI) when the channel is frequency-selective fading. In this paper, we propose a Hadamard-Walsh code-based MC-CDMA system that achieves zero MAI over a frequency-selective fading channel. In particular, we will use appropriately chosen subsets of Hadamard-Walsh code as codewords. For a multipath channel of length L, we partition a Hadamard-Walsh code of size N into G subsets, where G is a power of two with GgesL. We will show that the N/G codewords in any of the G subsets yields an MAI-free system. That is, the number of MAI-free users for each codeword subset is N/G. Furthermore, the system has the additional advantage that it is robust to carrier frequency offset (CFO) in a multipath environment. It is also shown that the MAI-free property allows us to estimate the channel of each user separately and the system can perform channel estimation much more easily. Owing to the MAI-free property, every user can enjoy a channel diversity gain of order L to improve the bit error performance. Finally, we discuss a code priority scheme for a heavily loaded system. Simulation results are given to demonstrate the advantages of the proposed code and code priority schemes

https://ir.nctu.edu.tw:443/bitstream/11536/11938/1/000239775000027.pdf

MAI-Free MC-CDMA Systems Based on Hadamard&#8211;Walsh Codes

The solution of symmetric positive definite Toeplitz systems Ax=b by the preconditioned conjugate gradient (PCG) method was recently proposed by Strang (1986) and analyzed by Chan (1989) and Chan and Strang (1989). The convergence rate of the PCG method depends on the choice of preconditioners for the given Toeplitz matrices. The authors present a general approach to the design of Toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions. All resulting preconditioners can therefore be inverted via various fast transform algorithms with O(N log N) operations. For a wide class of problems, the PCG method converges in a finite number of iterations independent of N so that the computational complexity for solving these Toeplitz systems is O(N log N). >

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Design and analysis of Toeplitz preconditioners

The carrier frequency offset (CFO) effect of a new multiaccess orthogonal frequency division multiplexing (OFDM) transceiver is analyzed. We show that multiaccess interference (MAI) due to CFO can be mitigated by choosing a proper set of orthogonal codes. That is, if we use M/2 symmetric or antisymmetric codewords of the M Hadamard-Walsh codewords, MAI can be greatly reduced to a negligible amount. The new OFDM system has been shown to be approximately MAI-free when there is no CFO. The new result developed here shows that the system continues to be approximately MAI-free even in the presence of CFO. Furthermore, we derive a close form for the reduced MAI, which reveals an interesting property of the system. That is, when the number of users increases, the reduced MAI and the inter carrier interference (ICI) due to self-CFO will decrease. Finally, it is demonstrated by simulation that the proposed OFDM scheme with proper code selection outperforms the OFDMA and the multicarrier code division multiplexing access (MC-CDMA) systems in a CFO environment.

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C.-C.J. Kuo

Papers

Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis

Extending self-similarity for fractional Brownian motion

MAI-Free MC-CDMA Systems Based on Hadamard–Walsh Codes

Design and analysis of Toeplitz preconditioners

An approximately MAI-free multiaccess OFDM system in carrier frequency offset environment

C.-C.J. Kuo

Papers

Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis

Extending self-similarity for fractional Brownian motion

MAI-Free MC-CDMA Systems Based on Hadamard&#8211;Walsh Codes

Design and analysis of Toeplitz preconditioners

An approximately MAI-free multiaccess OFDM system in carrier frequency offset environment

MAI-Free MC-CDMA Systems Based on Hadamard–Walsh Codes