Time-frequency localization operators: A geometric phase space approach

The author defines a set of operators which localize in both time and frequency. These operators are similar to but different from the low-pass time-limiting operator, the singular functions of which are the prolate spheroidal wave functions. The author's construction differs from the usual approach in that she treats the time-frequency plane as one geometric whole (phase space) rather than as two separate spaces. For disk-shaped or ellipse-shaped domains in time-frequency plane, the associated localization operators are remarkably simple. Their eigenfunctions are Hermite functions, and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions. >

[1]  Cornelis P. Janse,et al.  Time-Frequency Distributions of Loudspeakers: The Application of the Wigner Distribution , 1983 .

[2]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[3]  Miquel Bertran,et al.  Digital filtering and prolate functions , 1972 .

[4]  D. Tufts,et al.  Designing digital low-pass filters--Comparison of some methods and criteria , 1970 .

[5]  Mario Bertero,et al.  Resolution in diffraction-limited imaging, a singular value analysis: IV. The case of uncertain localization or non-uniform illumination of the object , 1984 .

[6]  O Brander,et al.  A generalisation of Slepian's solution for the singular value decomposition of filtered Fourier transforms , 1986 .

[7]  F. G. Tricomi,et al.  Asymptotische Eigenschaften der unvollständigen Gammafunktion , 1950 .

[8]  D. Thomson,et al.  Spectrum estimation and harmonic analysis , 1982, Proceedings of the IEEE.

[9]  D. Slepian,et al.  On bandwidth , 1976, Proceedings of the IEEE.

[10]  H. Pollak,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals , 1962 .

[11]  D. Slepian.  Prolate spheroidal wave functions, Fourier analysis and uncertainty — IV: Extensions to many dimensions; generalized prolate spheroidal functions , 1964 .

[12]  F. Alberto Grünbaum.  Differential Operators Commuting with Convolution Integral Operators , 1983 .

[13]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.

[14]  D. Slepian.  Some comments on Fourier analysis, uncertainty and modeling , 1983 .

[15]  F. Gori,et al.  Signal restoration for linear systems with weighted inputs. Singular value analysis for two cases of low-pass filtering , 1985 .