30 Spheroidal Wave Functions Applications 30.14 Wave Equation in Oblate Spheroidal Coordinates 30.16 Methods of Computation

§30.15 Signal Analysis

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Keywords:: applications, scaled spheroidal wave functions, signal analysis, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.15
See also:: Annotations for Ch.30

§30.15(i) Scaled Spheroidal Wave Functions
§30.15(ii) Integral Equation
§30.15(iii) Fourier Transform
§30.15(iv) Orthogonality
§30.15(v) Extremal Properties

§30.15(i) Scaled Spheroidal Wave Functions

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Keywords:: scaled, spheroidal wave functions
Notes:: See Frieden (1971, p. 321) and Meixner et al. (1980, p. 114).
Referenced by:: item
Permalink:: http://dlmf.nist.gov/30.15.i
See also:: Annotations for §30.15 and Ch.30

Let $\tau$ and $\sigma$ be given. Set $\gamma=\tau\sigma$ and define

30.15.1 $\phi_{n}(t)=\sqrt{\frac{2n+1}{2\tau}}\sqrt{\Lambda_{n}}\mathsf{Ps}^{0}_{n}% \left(\frac{t}{\tau},\gamma^{2}\right),$ $n=0,1,2,\dots$ ,

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Defines:: $\phi_{n}(t)$ : function (locally)
Symbols:: $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $n\geq m$ : integer degree, $\tau>0$ : parameter, $\Lambda$ and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.15.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(i), §30.15 and Ch.30

30.15.2 $\Lambda_{n}=\frac{2\gamma}{\pi}\left(K_{n}^{0}(\gamma)A_{n}^{0}(\gamma^{2})% \right)^{2};$

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Defines:: $\Lambda$ (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ , $K_{n}^{m}(\gamma)$ : connection coefficient and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.15.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(i), §30.15 and Ch.30

see §30.11(v).

§30.15(ii) Integral Equation

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Keywords:: integral equation, integral equations, scaled spheroidal wave functions, spheroidal wave functions
Notes:: See Frieden (1971, p. 324), Papoulis (1977, p. 205), Slepian (1983).
Permalink:: http://dlmf.nist.gov/30.15.ii
See also:: Annotations for §30.15 and Ch.30

30.15.3 $\int_{-\tau}^{\tau}\frac{\sin\sigma(t-s)}{\pi(t-s)}\phi_{n}(s)\,\mathrm{d}s=% \Lambda_{n}\phi_{n}(t).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function, $n\geq m$ : integer degree, $\tau>0$ : parameter, $\sigma>0$ : parameter, $\phi_{n}(t)$ : function and $\Lambda$
Permalink:: http://dlmf.nist.gov/30.15.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(ii), §30.15 and Ch.30

§30.15(iii) Fourier Transform

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Keywords:: Fourier transform, bandlimited, bandlimited functions, scaled spheroidal wave functions, spheroidal wave functions
Notes:: See Frieden (1971, pp. 321, 322), Papoulis (1977, p. 208), Slepian (1983).
Permalink:: http://dlmf.nist.gov/30.15.iii
See also:: Annotations for §30.15 and Ch.30

30.15.4 $\int_{-\infty}^{\infty}e^{-\mathrm{i}t\omega}\phi_{n}(t)\,\mathrm{d}t=(-% \mathrm{i})^{n}\sqrt{\frac{2\pi\tau}{\sigma\Lambda_{n}}}\phi_{n}\left(\frac{% \tau}{\sigma}\omega\right)\chi_{\sigma}(\omega),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $n\geq m$ : integer degree, $\tau>0$ : parameter, $\sigma>0$ : parameter, $\phi_{n}(t)$ : function and $\Lambda$
Referenced by:: §30.15(iii)
Permalink:: http://dlmf.nist.gov/30.15.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(iii), §30.15 and Ch.30

30.15.5 $\int_{-\tau}^{\tau}e^{-\mathrm{i}t\omega}\phi_{n}(t)\,\mathrm{d}t=(-\mathrm{i}% )^{n}\sqrt{\frac{2\pi\tau\Lambda_{n}}{\sigma}}\phi_{n}\left(\frac{\tau}{\sigma% }\omega\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $n\geq m$ : integer degree, $\tau>0$ : parameter, $\sigma>0$ : parameter, $\phi_{n}(t)$ : function and $\Lambda$
Permalink:: http://dlmf.nist.gov/30.15.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(iii), §30.15 and Ch.30

where

30.15.6 $\chi_{\sigma}(\omega)=\begin{cases}1,&\mbox{$|\omega|\leq\sigma$},\\ 0,&\mbox{$|\omega|>\sigma$}.\end{cases}$

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Symbols:: $\sigma>0$ : parameter
Referenced by:: §30.15(iii)
Permalink:: http://dlmf.nist.gov/30.15.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(iii), §30.15 and Ch.30

Equations (30.15.4) and (30.15.6) show that the functions $\phi_{n}$ are $\sigma$ -bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma,\sigma]$ .

§30.15(iv) Orthogonality

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Keywords:: orthogonality, scaled spheroidal wave functions
Notes:: See Frieden (1971, p. 321, 324), Papoulis (1977, p. 207), Slepian (1983).
Permalink:: http://dlmf.nist.gov/30.15.iv
See also:: Annotations for §30.15 and Ch.30

30.15.7 $\int_{-\tau}^{\tau}\phi_{k}(t)\phi_{n}(t)\,\mathrm{d}t=\Lambda_{n}\delta_{k,n},$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $n\geq m$ : integer degree, $\tau>0$ : parameter, $\phi_{n}(t)$ : function and $\Lambda$
Permalink:: http://dlmf.nist.gov/30.15.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(iv), §30.15 and Ch.30

30.15.8 $\int_{-\infty}^{\infty}\phi_{k}(t)\phi_{n}(t)\,\mathrm{d}t=\delta_{k,n}.$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $n\geq m$ : integer degree and $\phi_{n}(t)$ : function
Permalink:: http://dlmf.nist.gov/30.15.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §30.15(iv), §30.15 and Ch.30

The sequence $\phi_{n}$ , $n=0,1,2,\dots$ forms an orthonormal basis in the space of $\sigma$ -bandlimited functions, and, after normalization, an orthonormal basis in $L^{2}(-\tau,\tau)$ .

§30.15(v) Extremal Properties

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Keywords:: applications, extremal properties, scaled spheroidal wave functions, signal analysis, spheroidal wave functions
Notes:: See Frieden (1971, §2.10), Papoulis (1977, p. 210), Slepian (1983).
Permalink:: http://dlmf.nist.gov/30.15.v
See also:: Annotations for §30.15 and Ch.30

The maximum (or least upper bound) $\mathrm{B}$ of all numbers

30.15.9 $\beta=\frac{1}{2\pi}\int_{-\sigma}^{\sigma}\left|\int_{-\infty}^{\infty}e^{-% \mathrm{i}t\omega}f(t)\,\mathrm{d}t\right|^{2}\,\mathrm{d}\omega$

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