§30.15 Signal Analysis

Keywords:: scaled spheroidal wave functions, signal analysis, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.15

§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

Notes:: See Frieden (1971, p. 321) and Meixner et al. (1980, p. 114).
Keywords:: spheroidal wave functions
Referenced by:: §30.18(i)
Permalink:: http://dlmf.nist.gov/30.15.i

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

30.15.1 $\phi _{n}(t)=\sqrt{\frac{2n+1}{2\tau}}\sqrt{\Lambda _{n}}\mathop{\mathsf{Ps}^{{0}}_{{n}}\/}\nolimits\!\left(\frac{t}{\tau},\gamma^{2}\right),$ $n=0,1,2,\dots$ ,

Symbols:: $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $n\geq m$ : integer degree, $\tau>0$ : parameter, $\phi _{n}(t)$ : function, $\Lambda$ and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.15.E1
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30.15.2 $\Lambda _{n}=\frac{2\gamma}{\pi}\left(K_{n}^{0}(\gamma)A_{n}^{0}(\gamma^{2})\right)^{2};$

Symbols:: $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ , $K_{n}^{m}(\gamma)$ : connection coefficient, $\Lambda$ and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.15.E2
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see §30.11(v).

§30.15(ii) Integral Equation

Notes:: See Frieden (1971, p. 324), Papoulis (1977, p. 205), Slepian (1983).
Keywords:: integral equations, scaled spheroidal wave functions, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.15.ii

30.15.3 $\int _{{-\tau}}^{{\tau}}\frac{\mathop{\sin\/}\nolimits\sigma(t-s)}{\pi(t-s)}\phi _{n}(s)ds=\Lambda _{n}\phi _{n}(t).$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

§30.15(iii) Fourier Transform

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

30.15.4 $\int _{{-\infty}}^{{\infty}}e^{{-it\omega}}\phi _{n}(t)dt=(-i)^{n}\sqrt{\frac{2\pi\tau}{\sigma\Lambda _{n}}}\phi _{n}\left(\frac{\tau}{\sigma}\omega\right)\chi _{\sigma}(\omega),$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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

30.15.5 $\int _{{-\tau}}^{{\tau}}e^{{-it\omega}}\phi _{n}(t)dt=(-i)^{n}\sqrt{\frac{2\pi\tau\Lambda _{n}}{\sigma}}\phi _{n}\left(\frac{\tau}{\sigma}\omega\right),$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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

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
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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

Notes:: See Frieden (1971, p. 321, 324), Papoulis (1977, p. 207), Slepian (1983).
Keywords:: scaled spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.15.iv

30.15.7 $\int _{{-\tau}}^{{\tau}}\phi _{k}(t)\phi _{n}(t)dt=\Lambda _{n}\delta _{{k,n}},$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $dx$ : 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

30.15.8 $\int _{{-\infty}}^{{\infty}}\phi _{k}(t)\phi _{n}(t)dt=\delta _{{k,n}}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $dx$ : differential of $x$ , $\int$ : integral, $n\geq m$ : integer degree and $\phi _{n}(t)$ : function
Permalink:: http://dlmf.nist.gov/30.15.E8
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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

Notes:: See Frieden (1971, §2.10), Papoulis (1977, p. 210), Slepian (1983).
Keywords:: scaled spheroidal wave functions, signal analysis, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.15.v

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

30.15.9 $\beta=\frac{1}{2\pi}\int _{{-\sigma}}^{{\sigma}}\left|\int _{{-\infty}}^{{\infty}}e^{{-it\omega}}f(t)dt\right|^{2}d\omega$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\sigma>0$ : parameter and $f$ : function
Permalink:: http://dlmf.nist.gov/30.15.E9
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taken over all $f\in L^{2}(-\infty,\infty)$ subject to

30.15.10

$\int _{{-\infty}}^{{\infty}}|f(t)|^{2}dt=1,$

$\int _{{-\tau}}^{{\tau}}|f(t)|^{2}dt=\alpha,$

Symbols:: $dx$ : 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

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

30.15.11 $\mathop{\mathrm{arccos}\/}\nolimits\sqrt{\mathrm{B}}+\mathop{\mathrm{arccos}\/}\nolimits\sqrt{\alpha}=\mathop{\mathrm{arccos}\/}\nolimits\sqrt{\Lambda _{0}},$