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 $\tau$ $(>0)$ and $\sigma$ $(>0)$ be given. Set $\gamma=\tau\sigma$ and define

30.15.1		$\phi_{n}(t)=\sqrt{\frac{2n+1}{2\tau}}\sqrt{\Lambda_{n}}\mathsf{Ps}^{0}_{n}% \left(\frac{t}{\tau},\gamma^{2}\right),$
$n=0,1,2,\dots$ ,
ⓘ 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		$\Lambda_{n}=\frac{2\gamma}{\pi}\left(K_{n}^{0}(\gamma)A_{n}^{0}(\gamma^{2})% \right)^{2};$
ⓘ 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		$\int_{-\tau}^{\tau}\frac{\sin\sigma(t-s)}{\pi(t-s)}\phi_{n}(s)\,\mathrm{d}s=% \Lambda_{n}\phi_{n}(t).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : 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		$\int_{-\infty}^{\infty}e^{-\mathrm{i}t\omega}\phi_{n}(t)\,\mathrm{d}t=(-% \mathrm{i})^{n}\sqrt{\frac{2\pi\tau}{\sigma\Lambda_{n}}}\phi_{n}\left(\frac{% \tau}{\sigma}\omega\right)\chi_{\sigma}(\omega),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : 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		$\int_{-\tau}^{\tau}e^{-\mathrm{i}t\omega}\phi_{n}(t)\,\mathrm{d}t=(-\mathrm{i}% )^{n}\sqrt{\frac{2\pi\tau\Lambda_{n}}{\sigma}}\phi_{n}\left(\frac{\tau}{\sigma% }\omega\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : 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$}.\end{cases}$
ⓘ 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	$\displaystyle\int_{-\tau}^{\tau}\phi_{k}(t)\phi_{n}(t)\,\mathrm{d}t$	$\displaystyle=\Lambda_{n}\delta_{k,n},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : 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	$\displaystyle\int_{-\infty}^{\infty}\phi_{k}(t)\phi_{n}(t)\,\mathrm{d}t$	$\displaystyle=\delta_{k,n}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : 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		$\beta=\frac{1}{2\pi}\int_{-\sigma}^{\sigma}\left\|\int_{-\infty}^{\infty}e^{-% \mathrm{i}t\omega}f(t)\,\mathrm{d}t\right\|^{2}\,\mathrm{d}\omega$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sigma>0$ : parameter and $f$ : function Permalink: http://dlmf.nist.gov/30.15.E9 Encodings: TeX, pMML, png See also: Annotations for §30.15(v), §30.15 and Ch.30

taken over all $f\in L^{2}(-\infty,\infty)$ subject to

30.15.10	$\displaystyle\int_{-\infty}^{\infty}\|f(t)\|^{2}\,\mathrm{d}t$	$\displaystyle=1,$
30.15.10	$\displaystyle\int_{-\tau}^{\tau}\|f(t)\|^{2}\,\mathrm{d}t$	$\displaystyle=\alpha,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\tau>0$ : parameter and $f$ : function Permalink: http://dlmf.nist.gov/30.15.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.15(v), §30.15 and Ch.30

for (fixed) $\Lambda_{0}<\alpha\leq 1$ , is given by

30.15.11		$\operatorname{arccos}\sqrt{\mathrm{B}}+\operatorname{arccos}\sqrt{\alpha}=% \operatorname{arccos}\sqrt{\Lambda_{0}},$
ⓘ Symbols: $\operatorname{arccos}\NVar{z}$ : arccosine function, $\Lambda$ and $\mathrm{B}$ : maximum bound Permalink: http://dlmf.nist.gov/30.15.E11 Encodings: TeX, pMML, png See also: Annotations for §30.15(v), §30.15 and Ch.30

or equivalently,

30.15.12		$\mathrm{B}=\left(\sqrt{\Lambda_{0}\alpha}+\sqrt{1-\Lambda_{0}}\sqrt{1-\alpha}% \right)^{2}.$
ⓘ Symbols: $\Lambda$ and $\mathrm{B}$ : maximum bound Permalink: http://dlmf.nist.gov/30.15.E12 Encodings: TeX, pMML, png See also: Annotations for §30.15(v), §30.15 and Ch.30

The corresponding function $f$ is given by

30.15.13	$\displaystyle f(t)$	$\displaystyle=a\phi_{0}(t)\chi_{\tau}(t)+b\phi_{0}(t)(1-\chi_{\tau}(t)),$
	$\displaystyle a$	$\displaystyle=\sqrt{\frac{\alpha}{\Lambda_{0}}},$
	$\displaystyle b$	$\displaystyle=\sqrt{\frac{1-\alpha}{1-\Lambda_{0}}}.$
ⓘ Symbols: $\tau>0$ : parameter, $\phi_{n}(t)$ : function, $\Lambda$ and $f$ : function Permalink: http://dlmf.nist.gov/30.15.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §30.15(v), §30.15 and Ch.30

If $0<\alpha\leq\Lambda_{0}$ , then $\mathrm{B}=1$ .

For further information see Frieden (1971), Lyman and Edmonson (2001), Papoulis (1977, Chapter 6), Slepian (1983), and Slepian and Pollak (1961).

30.14 Wave Equation in Oblate Spheroidal Coordinates 30.16 Methods of Computation

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