AuthorAgyo, Sanfo D.
KeywordPhase space methods; Coherent states; Bi-fractional coherent states; Bi-fractional Wigner function; Bi-fractional P−function; Bi-fractional Q−function; Bi-fractional Moyal star product; Bi-fractional Berezin formalism
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InstitutionUniversity of Bradford
DepartmentFaculty of Engineering and Informatics, School of Electrical Engineering and Computer Science
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AbstractThe displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.
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Novel Fractional Wavelet Transform with Closed-Form ExpressionAnoh, Kelvin O.O.; Abd-Alhameed, Raed A.; Jones, Steven M.R.; Ochonogor, O.; Dama, Yousef A.S. (2014-08)A new wavelet transform (WT) is introduced based on the fractional properties of the traditional Fourier transform. The new wavelet follows from the fractional Fourier order which uniquely identifies the representation of an input function in a fractional domain. It exploits the combined advantages of WT and fractional Fourier transform (FrFT). The transform permits the identification of a transformed function based on the fractional rotation in time-frequency plane. The fractional rotation is then used to identify individual fractional daughter wavelets. This study is, for convenience, limited to one-dimension. Approach for discussing two or more dimensions is shown.
Performance comparison of MIMO-DWT and MIMO-FrFT multicarrier systemsAnoh, Kelvin O.O.; Ali, N.T.; Migdadi, Hassan S.O.; Abd-Alhameed, Raed A.; Ghazaany, Tahereh S.; Jones, Steven M.R.; Noras, James M.; Excell, Peter S. (2013)In this work, we discuss two new multicarrier modulating kernels that can be adopted for multicarrier signaling. These multicarrier transforms are the fractional Forurier transform (FrFT) and discrete wavelet transforms (DWT). At first, we relate the transforms in terms of mathematical relationships, and then using numerical and simulation comparisons we show their performances in terms of bit error ratio (BER) for Multiple Input Multiple Output (MIMO) applications. Numerical results using BPSK and QPSK support that both can be applied for multicarrier signaling, however, it can be resource effective to drive the DWT as the baseband multicarrier kernel at the expense of the FrFT