<i>Wavelets: Mathematics and Applications</i>

Benedetto, John J.; Frazier, Michael; Torrésani, Bruno

doi:10.1063/1.2808703

Cited by 132 publications

(60 citation statements)

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“…The evolution of f(x) in various scales can be probed through regulation dilation scale a. Detail of the concept, theory, applications and problems of WT can be referred to the above references, and related monographs and paper collections (Young, 1993;Schumaker and Webb, 1994;Foufoula-Geogiou and Kumar, 1994;Benedetto and Frazier, 1994;Kaiser, 1994). The 'Mexican hat' wavelet:…”

Section: Wavelet Transform (Wt)mentioning

confidence: 99%

Wavelet Analysis of Summer Rainfall over North China and India and SOI Using 1891-1992 Data

Nitta

1996

Journal of the Meteorological Society of Japan

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Localization of interannual and decadal variations of summer rainfall in North China and India and seasonal mean Southern Oscillation Index (SOI) from , and the favored time scales of the significant correlations among them are investigated with the wavelet transform (WT) analysis method. Strong localization and non-stationary evolution in the interannual and decadal variations of the rainfall in North China and India are demonstrated. The dominating time scales in the rainfall variations in North China and India are mainly located in two time scale bands: shorter than 10 years, and 14-28 years. The correlations between the rainfall variations in North China and India are time-scale dependent. The significant positive correlations are concentrated in two time-scale bands: shorter than 7 years and longer than 14 years. The correlations are insignificant on time scales of 7-14 years.El Nino and Southern Oscillation (ENSO) cycle is a nonstationary process. The dominant time scales for the variation of SOI are shorter than 5-7 years, and the decadal variations are obvious in 1891-1915 and 1970-1992. There is strong interaction between the Indian summer monsoon and the ENSO cycle. The significant positive correlations are mainly focused on the time scales shorter than 24-30 years, and the significant negative correlations on time scales longer than 40 years in summer and winter SOI cases. The correlation between the rainfall variations in North China and the ENSO cycle in various seasons are less significant, more scattered and complex, but there are some similarities in the correlation pattern compared with that of India, especially in the shorter time scales.The calculation shows that the summer rainfall variations in North China and India at different time scales are related with different general circulation anomalous patterns in middle and high latitudes of Eurasia and the western Pacific. The correlation patterns over Eurasia and the western Pacific between the summer rainfall variations in North China or India and the geopotential height at 500hPa are similar to the correlation patterns between the geopotential height and the WT results of the rainfall on time scales of about 5 years, but different from those on time scales of about 11 years.Therefore, the similar behavior between the rainfall variations in North China and India may be caused by the similar associations between the rainfall variations and the ENSO cycle and the general circulation anomalies in middle and high latitudes over Eurasia and the western Pacific, and the difference may be related to the different character of these associations.

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Section: Wavelet Transform (Wt)mentioning

confidence: 99%

Wavelet Analysis of Summer Rainfall over North China and India and SOI Using 1891-1992 Data

Nitta

1996

Journal of the Meteorological Society of Japan

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“…These references highlight that wavelet coefficients tend to be decorrelated provided that the decomposition level tends to infinity and the decomposition filters satisfy suitable properties. Second, results of the same order holds true for stationary random processes as shown in [15], [16], [17].…”

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confidence: 53%

Wavelet Packets of Fractional Brownian Motion: Asymptotic Analysis and Spectrum Estimation

Atto

Pastor

Mercier

2010

IEEE Trans. Inform. Theory

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This work provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional Brownian motion. It also discusses the convergence speed to the limit autocorrelation function, when the input random process is either a fractional Brownian motion or a wide-sense stationary second-order random process. The analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the Shannon paraunitary filters. From this analysis, we derive wavelet packet based spectrum estimation for fractional Brownian motions and wide-sense stationary random processes. Experimental tests show good results for estimating the spectrum of 1/f processes.

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“…Instead, we will discuss shortly the extension -the theory for frame multiresolution analysis -proposed by Benedetto and Li. For more information about wavelets in theory and practice we refer to the collections of research articles [2], [23].…”

Section: Wavelet Framesmentioning

confidence: 99%

Untitled

2001

Bull. Amer. Math. Soc.

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Abstract. This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in L 2 (R). The concept of an orthonormal basis {fn}, allowing every function f ∈ L 2 (R) to be written f = cnfn for suitable coefficients {cn}, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients {cn} are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research.

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Wavelets: Mathematics and Applications

Cited by 132 publications

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Wavelet Analysis of Summer Rainfall over North China and India and SOI Using 1891-1992 Data

Wavelet Analysis of Summer Rainfall over North China and India and SOI Using 1891-1992 Data

Wavelet Packets of Fractional Brownian Motion: Asymptotic Analysis and Spectrum Estimation

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