Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution
In this paper, a generalized notion of wide-sense α-stationarity for random signals is presented. The notion of stationarity is fundamental in the Fourier analysis of random signals. For this purpose, a definition of the fractional correlation between two random variables is introduced. It is shown...
- Autores:
- Tipo de recurso:
- Fecha de publicación:
- 2013
- Institución:
- Universidad Tecnológica de Bolívar
- Repositorio:
- Repositorio Institucional UTB
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.utb.edu.co:20.500.12585/9077
- Acceso en línea:
- https://hdl.handle.net/20.500.12585/9077
- Palabra clave:
- Fractional correlation
Fractional Fourier transformation
Fractional power spectral density
Random signals
Wiener-Khinchin theorem
Wigner-Ville distribution
Fractional correlation
Fractional Fourier Transformations
Fractional power spectral density
Random signal
Wiener-Khinchin theorem
Fourier optics
Power spectral density
Wigner-Ville distribution
Fourier analysis
- Rights
- restrictedAccess
- License
- http://creativecommons.org/licenses/by-nc-nd/4.0/
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Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution |
| title |
Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution |
| spellingShingle |
Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution Fractional correlation Fractional Fourier transformation Fractional power spectral density Random signals Wiener-Khinchin theorem Wigner-Ville distribution Fractional correlation Fractional Fourier Transformations Fractional power spectral density Random signal Wiener-Khinchin theorem Fourier optics Power spectral density Wigner-Ville distribution Fourier analysis |
| title_short |
Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution |
| title_full |
Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution |
| title_fullStr |
Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution |
| title_full_unstemmed |
Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution |
| title_sort |
Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution |
| dc.subject.keywords.none.fl_str_mv |
Fractional correlation Fractional Fourier transformation Fractional power spectral density Random signals Wiener-Khinchin theorem Wigner-Ville distribution Fractional correlation Fractional Fourier Transformations Fractional power spectral density Random signal Wiener-Khinchin theorem Fourier optics Power spectral density Wigner-Ville distribution Fourier analysis |
| topic |
Fractional correlation Fractional Fourier transformation Fractional power spectral density Random signals Wiener-Khinchin theorem Wigner-Ville distribution Fractional correlation Fractional Fourier Transformations Fractional power spectral density Random signal Wiener-Khinchin theorem Fourier optics Power spectral density Wigner-Ville distribution Fourier analysis |
| description |
In this paper, a generalized notion of wide-sense α-stationarity for random signals is presented. The notion of stationarity is fundamental in the Fourier analysis of random signals. For this purpose, a definition of the fractional correlation between two random variables is introduced. It is shown that for wide-sense α-stationary random signals, the fractional correlation and the fractional power spectral density functions form a fractional Fourier transform pair. Thus, the concept of α-stationarity plays an important role in the analysis of random signals through the fractional Fourier transform for signals nonstationary in the standard formulation, but α-stationary. Furthermore, we define the α-Wigner-Ville distribution in terms of the fractional correlation function, in which the standard Fourier analysis is the particular case for α=pi2, and it leads to the Wiener-Khinchin theorem. © 1991-2012 IEEE. |
| publishDate |
2013 |
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2013 |
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2020-03-26T16:32:54Z |
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2020-03-26T16:32:54Z |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
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info:eu-repo/semantics/article |
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info:eu-repo/semantics/publishedVersion |
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Artículo |
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publishedVersion |
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IEEE Transactions on Signal Processing; Vol. 61, Núm. 6; pp. 1555-1560 |
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1053587X |
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https://hdl.handle.net/20.500.12585/9077 |
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10.1109/TSP.2012.2236834 |
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Universidad Tecnológica de Bolívar |
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Repositorio UTB |
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56270896900 35094573000 |
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IEEE Transactions on Signal Processing; Vol. 61, Núm. 6; pp. 1555-1560 1053587X 10.1109/TSP.2012.2236834 Universidad Tecnológica de Bolívar Repositorio UTB 56270896900 35094573000 |
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https://hdl.handle.net/20.500.12585/9077 |
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eng |
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eng |
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http://purl.org/coar/access_right/c_16ec |
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http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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info:eu-repo/semantics/restrictedAccess |
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Atribución-NoComercial 4.0 Internacional |
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http://creativecommons.org/licenses/by-nc-nd/4.0/ Atribución-NoComercial 4.0 Internacional http://purl.org/coar/access_right/c_16ec |
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2020-03-26T16:32:54Z2020-03-26T16:32:54Z2013IEEE Transactions on Signal Processing; Vol. 61, Núm. 6; pp. 1555-15601053587Xhttps://hdl.handle.net/20.500.12585/907710.1109/TSP.2012.2236834Universidad Tecnológica de BolívarRepositorio UTB5627089690035094573000In this paper, a generalized notion of wide-sense α-stationarity for random signals is presented. The notion of stationarity is fundamental in the Fourier analysis of random signals. For this purpose, a definition of the fractional correlation between two random variables is introduced. It is shown that for wide-sense α-stationary random signals, the fractional correlation and the fractional power spectral density functions form a fractional Fourier transform pair. Thus, the concept of α-stationarity plays an important role in the analysis of random signals through the fractional Fourier transform for signals nonstationary in the standard formulation, but α-stationary. Furthermore, we define the α-Wigner-Ville distribution in terms of the fractional correlation function, in which the standard Fourier analysis is the particular case for α=pi2, and it leads to the Wiener-Khinchin theorem. © 1991-2012 IEEE.Recurso electrónicoapplication/pdfenghttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/restrictedAccessAtribución-NoComercial 4.0 Internacionalhttp://purl.org/coar/access_right/c_16echttps://www.scopus.com/inward/record.uri?eid=2-s2.0-84875015943&doi=10.1109%2fTSP.2012.2236834&partnerID=40&md5=8948c99af3dc2f6f9bcba86bcaee6a4dFractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distributioninfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArtículohttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_2df8fbb1Fractional correlationFractional Fourier transformationFractional power spectral densityRandom signalsWiener-Khinchin theoremWigner-Ville distributionFractional correlationFractional Fourier TransformationsFractional power spectral densityRandom signalWiener-Khinchin theoremFourier opticsPower spectral densityWigner-Ville distributionFourier analysisTorres R.Torres E.Namias, V., The fractional order Fourier transform and its application to quantum mechanics (1980) J. Inst. Math. Appl., 25, pp. 241-265Pellat-Finet, P., (2009) Optique de Fourier: Théorie Métaxiale et Fractionnaire, , Paris France: Springer VerlagLohmann, A.W., Image rotation, Wigner rotation, and the fractional Fourier transform (1993) J. Opt. Soc. Amer. A, 10, pp. 2181-2186. , OctOzaktas, H.M., Mendlovic, D., Fractional Fourier optics (1995) J. Opt. Soc. Amer. A, 12, pp. 743-751. , AprOzaktas, H.M., Zalevsky, Z., Kutay, M.A., (2001) The Fractional Fourier Transform with Applications in Optics and Signal Processing, , Chichester, U.K.: WileyBultheel, A., Martinez, H., Sulbaran, Recent developments in the theory of the fractional Fourier transforms and linear canonical transforms (2007) Bull. Belgian Math. Soc. Simon Stevin, 13, pp. 971-1005Mendlovic, D., Ozaktas, H.M., Lohmann, A.W., Fractional correlation (1995) Appl. Opt., 34, pp. 303-309. , JanAlmeida, L.B., Product and convolution theorems for the fractional Fourier transform (1997) IEEE Signal Process. Lett., 4, pp. 15-17Zayed, A.I., A convolution and product theorem for the fractional Fourier transform (1998) IEEE Signal Process. Lett., 5, pp. 101-103. , AprAkay, G.O., Boudreaux-Bartels, F., Fractional convolution and correlation via operator methods and an application to detection of linear FM signals (2001) IEEE Trans. Signal Process., 49, pp. 979-993. , MayTorres, R., Pellat-Finet, P., Torres, Y., Fractional convolution, fractional correlation and their translation invariance properties (2010) Signal Process., 90, pp. 1976-1984. , JunAlmeida, L.B., The fractional Fourier transform and time-frequency representations (1994) IEEE Trans. Signal Process., 42, pp. 3084-3091. , NovPei, S., Ding, J., Relations between fractional operations and timefrequency distributions, and their applications (2001) IEEE Trans. Signal Process., 49, pp. 1638-1655. , AugAlieva, T., Bastiaans, M.J., Stankovic, L., Signal reconstruction from two close fractional Fourier power spectra (2003) IEEE Trans. Signal Process., 51, pp. 112-123. , JanXia, X.-G., On bandlimited signals with fractional Fourier transform (1996) IEEE Signal Process. Lett., 3, pp. 72-74Candan, C., Ozaktas, H.M., Sampling and series expansion theorems for fractional Fourier and other transforms (2003) Signal Process., 83, pp. 2455-2457Torres, R., Pellat-Finet, P., Torres, Y., Sampling theorem for fractional bandlimited signals: A self-contained proof. Application to digital holography (2006) IEEE Signal Process. Lett., 13 (11), pp. 676-679. , NovMustard, D., The fractional Fourier transform and the Wigner distribution (1996) J. Aust. Math. Soc. Ser. B, 38, pp. 209-219Almeida, L.B., The fractional Fourier transform and time-frequency representations (1994) IEEE Trans. Signal Process., 42 (11), pp. 3084-3091Lohmann, A.W., Mendlovic, D., Zalevsky, Z., (1998) IV: Fractional Transformations in Optics Ser. Progress in Optics, 38, pp. 263-342. , E.Wolf, Ed. Amsterdam, The Netherlands: ElsevierDorsch, R.G., Lohmann, A.W., Bitran, Y., Mendlovic, D., Ozaktas, H.M., Chirp filtering in the fractional Fourier domain (1994) Appl. Opt., 33 (32), pp. 7599-7602. , NovOzaktas, H.M., Barshan, B., Mendlovic, D., Onural, L., Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms (1994) J. Opt. Soc. Amer. A, 11 (2), pp. 547-559. , FebTao, R., Zhang, F., Wang, Y., Fractional power spectrum (2008) IEEE Trans. Signal Process., 56, pp. 4199-4206. , SepPei, S.-C., Ding, J.-J., Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes (2010) Trans. Signal Process., 58 (8), pp. 4079-4092. , AugMartin, W., Time-frequency analysis of random signals (1982) Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), 7, pp. 1325-1328. , MayWigner, E., On the quantum correction for thermodynamic equilibrium (1932) Phys. Rev., 40, pp. 749-959. , JunVille, J., Théorie et application de la notion de signal analytique (French) (1948) Cables et Transmission, 1 (1), pp. 61-74The wigner-ville spectrum of nonstationary random signals (1997) The Wigner Distribution-Theory and Applications in Signal Processing, , Amsterdam, The Netherlands: ElsevierMcBride, A.C., Kerr, F.H., On Namias's fractional Fourier transforms (1987) IMA J. Appl. Math., 39 (2), pp. 159-175http://purl.org/coar/resource_type/c_6501THUMBNAILMiniProdInv.pngMiniProdInv.pngimage/png23941https://repositorio.utb.edu.co/bitstream/20.500.12585/9077/1/MiniProdInv.png0cb0f101a8d16897fb46fc914d3d7043MD5120.500.12585/9077oai:repositorio.utb.edu.co:20.500.12585/90772021-02-02 13:55:47.4Repositorio Institucional UTBrepositorioutb@utb.edu.co |