Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions

Yu, Runyi; Özkaramanlı, Hüseyin

doi:10.1109/tsp.2005.859261

Cited by 49 publications

(20 citation statements)

References 7 publications

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“…The converse has been proven in [76], [122], making the condition necessary and sufficient. The necessary and sufficient conditions for the biorthogonal case were proven in [121].…”

Section: The Half-sample Delay Conditionmentioning

confidence: 98%

The dual-tree complex wavelet transform

Selesnick

Baraniuk

Kingsbury³

2005

IEEE Signal Process. Mag.

2,177

1,202

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[ A coherent framework for multiscale signal and image processing ] T he dual-tree complex wavelet transform (CWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher dimensions. It achieves this with a redundancy factor of only 2 d for d-dimensional signals, which is substantially lower than the undecimated DWT. The multidimensional (M-D) dual-tree CWT is nonseparable but is based on a computationally efficient, separable filter bank (FB). This tutorial discusses the theory behind the dual-tree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing. We use the complex number symbol C in CWT to avoid confusion with the often-used acronym CWT for the (different) continuous wavelet transform. BACKGROUNDThis article aims to reach two different audiences. The first is the wavelet community, many members of which are unfamiliar with the utility, convenience, and unique properties of complex wavelets. The second is the broader class of signal processing folk who work with applications where the DWT has proven somewhat disappointing, such as those involving complex or modulat-

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“…The converse has been proven in [76], [122], making the condition necessary and sufficient. The necessary and sufficient conditions for the biorthogonal case were proven in [121].…”

Section: The Half-sample Delay Conditionmentioning

confidence: 98%

The dual-tree complex wavelet transform

Selesnick

Baraniuk

Kingsbury³

2005

IEEE Signal Process. Mag.

2,177

1,202

View full text Add to dashboard Cite

show abstract

“…It has been demonstrated in [34] that an orthogonal wavelet basis is the Hilbert transform of another orthogonal wavelet basis if and only if the associated low-pass filter of the former is a half-sample delayed version of the low-pass filter of the latter, which is expressed as follows:…”

Section: Dual-tree Complex Wavelet Transformmentioning

confidence: 99%

A novel intelligent method for mechanical fault diagnosis based on dual-tree complex wavelet packet transform and multiple classifier fusion

2016

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“…In [29,30], it is stated that if the low-pass filter of second tree (g 0 (n)) is equal to the half sample delayed version of the low-pass filter of first tree (h 0 (n)), then the wavelet functions of DTCWT satisfy Hilbert transform pair condition and this condition can be shown as below in time domain.…”

Section: Dual Tree Complex Wavelet Transformmentioning

confidence: 99%