SC- and SS-wavelet transforms.

*(English)*Zbl 1151.42313Summary: A. M. Grigoryan [Fourier transform representation by frequency-time wavelets, IEEE Trans. Signal Process. 53 No. 7, 2489–2497 (2005; doi:10.1109/TSP.2005.849180)] has proposed an alternative representation of the Fourier transform, called A-wavelet transform. In that paper, the Cosine and Sine signals defined over one period have been used to develop the Cosine- and Sine-wavelet transforms and using those wavelet transforms the Fourier transform has been represented. For computing the Fourier transform at a given frequency, one does not require to compute the Cosine- and Sine-wavelet transforms at all time points in the time-frequency plane, but at specific time points that are separated out by \(2\pi /\omega , \omega \) is the frequency variable. In this paper, we propose SC- and SS-wavelet transforms that help representing the Fourier transform of a signal in a better way. The SC- and SS-wavelet transforms use the Cosine and Sine signals defined over the smaller intervals (of length \(2\pi /(m\omega ), m \geqslant 1\)) than that (of length \(2\pi /\omega \)) used in the A-wavelet transform. The SC- and SS-wavelet transforms not only give sharper time-frequency localization but also much more information in a better localized form than the A-wavelet transform.

##### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

65T50 | Numerical methods for discrete and fast Fourier transforms |

Full Text:
DOI

##### References:

[1] | B. Kieft, A Brief History of Wavelets (Internet website address: \langle http://www.gvsu.edu/math/wavelets/student_work/Kieft/Wavelets%20-%20Main%20Page.html\rangle ). |

[2] | Gabor, D.: Theory of communication, J. inst. Electr. eng. 93, 429-457 (1946) |

[3] | Mallat, S. G.: A theory for multiresolution signal decomposition: the wavelet representation, IEEE trans. Pattern anal. Mach intell. 11, No. 7, 674-693 (1989) · Zbl 0709.94650 · doi:10.1109/34.192463 |

[4] | Edward, T.: Discrete wavelet transforms: theory and implementation, (1991) |

[5] | Burrus, C. S.; Gopinath, R. A.; Guo, H.: Introduction to wavelets and wavelet transforms, a primer, (1998) |

[6] | Goswami, J. C.; Chan, A. K.: Fundamentals of wavelets: theory, algorithms, and applications, (1999) · Zbl 1209.65156 |

[7] | Y. Meyer, Wavelets: Algorithms and Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1993. · Zbl 0821.42018 |

[8] | I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992. · Zbl 0776.42018 |

[9] | Mallat, S. G.: A wavelet tour of signal processing, (1998) · Zbl 0937.94001 |

[10] | Grigoryan, A. M.: Representation of the Fourier transform by Fourier series, J. math imaging 25, 87-105 (2006) |

[11] | Grigoryan, A. M.: Fourier transform representation by frequency-time wavelets, IEEE trans. Signal process. 53, No. 7, 2489-2497 (2005) · Zbl 1370.42006 |

[12] | A.M. Grigoryan, S. Dursun, Multiresolution of the Fourier transform, in: Proceedings of Acoustics, Speech, and Signal Processing (ICASSP’05), vol. 4, 2005, pp. iv/577 – iv/580. |

[13] | Grigoryan, A. M.; Agaian, S. S.: Multidimensional discrete unitary transforms: representation, partitioning and algorithms, (2003) · Zbl 1114.94003 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.