§30.15 Signal Analysis

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

30.15.1		$${\varphi }_n(t)=\sqrt{\frac{2n+1}{2\tau }}\sqrt{{\mathrm{\Lambda }}_n}{\mathsf{Ps}}_n^0(\frac{t}{\tau },{\gamma }^2),$$
$n=0,1,2,\mathrm{\dots }$,
ⓘ Defines: ${\varphi }_n(t)$: function (locally) Symbols: ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $n\ge m$: integer degree, $\tau >0$: parameter, $\mathrm{\Lambda }$ 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		$${\mathrm{\Lambda }}_n=\frac{2\gamma }{\pi }{\left(K_n^0(\gamma )A_n^0({\gamma }^2)\right)}^2;$$
ⓘ Defines: $\mathrm{\Lambda }$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $n\ge m$: 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).

30.15.3		$${\int }_{-\tau }^{\tau }\frac{\sin \sigma (t-s)}{\pi (t-s)}{\varphi }_n(s)ds={\mathrm{\Lambda }}_n{\varphi }_n(t).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $n\ge m$: integer degree, $\tau >0$: parameter, $\sigma >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$ Permalink: http://dlmf.nist.gov/30.15.E3 Encodings: TeX, pMML, png See also: Annotations for §30.15(ii), §30.15 and Ch.30

30.15.4		$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}t\omega }{\varphi }_n(t)dt={(-\mathrm{i})}^n\sqrt{\frac{2\pi \tau }{\sigma {\mathrm{\Lambda }}_n}}{\varphi }_n\left(\frac{\tau }{\sigma }\omega \right){\chi }_{\sigma }(\omega ),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, $\sigma >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$ 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		$${\int }_{-\tau }^{\tau }{\mathrm{e}}^{-\mathrm{i}t\omega }{\varphi }_n(t)dt={(-\mathrm{i})}^n\sqrt{\frac{2\pi \tau {\mathrm{\Lambda }}_n}{\sigma }}{\varphi }_n\left(\frac{\tau }{\sigma }\omega \right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, $\sigma >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$ 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
ⓘ 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 ${\varphi }_n$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma ,\sigma ]$.

30.15.7		${\displaystyle {\int }_{-\tau }^{\tau }}{\varphi }_k(t){\varphi }_n(t)dt$	$={\mathrm{\Lambda }}_n{\delta }_{k,n},$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$ 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		${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\varphi }_k(t){\varphi }_n(t)dt$	$={\delta }_{k,n}.$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $n\ge m$: integer degree and ${\varphi }_n(t)$: 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 ${\varphi }_n$, $n=0,1,2,\mathrm{\dots }$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^2(-\tau ,\tau )$.

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

30.15.9		$$\beta =\frac{1}{2\pi }{\int }_{-\sigma }^{\sigma }{\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}t\omega }f(t)dt\right|}^{2}d\omega$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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 $f\in L^2(-\mathrm{\infty },\mathrm{\infty })$ subject to

30.15.10		${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{|f(t)|}^{2}dt$	$=1,$
		${\displaystyle {\int }_{-\tau }^{\tau }}{|f(t)|}^{2}dt$	$=\alpha ,$
ⓘ 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 See also: Annotations for §30.15(v), §30.15 and Ch.30

for (fixed) , is given by

30.15.11		$$\arccos \sqrt{\mathrm{B}}+\arccos \sqrt{\alpha }=\arccos \sqrt{{\mathrm{\Lambda }}_0},$$
ⓘ Symbols: $\arccos z$: arccosine function, $\mathrm{\Lambda }$ 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		$$\mathrm{B}={\left(\sqrt{{\mathrm{\Lambda }}_0\alpha }+\sqrt{1-{\mathrm{\Lambda }}_0}\sqrt{1-\alpha }\right)}^2.$$
ⓘ Symbols: $\mathrm{\Lambda }$ 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		$f(t)$	$=a{\varphi }_0(t){\chi }_{\tau }(t)+b{\varphi }_0(t)(1-{\chi }_{\tau }(t)),$
		$a$	$=\sqrt{{\displaystyle \frac{\alpha }{{\mathrm{\Lambda }}_0}}},$
		$b$	$=\sqrt{{\displaystyle \frac{1-\alpha }{1-{\mathrm{\Lambda }}_0}}}.$
ⓘ Symbols: $\tau >0$: parameter, ${\varphi }_n(t)$: function, $\mathrm{\Lambda }$ 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 , 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).