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30 Spheroidal Wave FunctionsApplications

§30.15 Signal Analysis

Contents

§30.15(i) Scaled Spheroidal Wave Functions

Let τ (>0) and σ (>0) be given. Set γ=τσ and define

30.15.1 ϕn(t)=2n+12τΛnPsn0(tτ,γ2),
n=0,1,2,,
30.15.2 Λn=2γπ(Kn0(γ)An0(γ2))2;

see §30.11(v).

§30.15(ii) Integral Equation

30.15.3 -ττsinσ(t-s)π(t-s)ϕn(s)ds=Λnϕn(t).

§30.15(iii) Fourier Transform

30.15.4 -e-itωϕn(t)dt=(-i)n2πτσΛnϕn(τσω)χσ(ω),
30.15.5 -ττe-itωϕn(t)dt=(-i)n2πτΛnσϕn(τσω),

where

30.15.6 χσ(ω)={1,|ω|σ,0,|ω|>σ.

Equations (30.15.4) and (30.15.6) show that the functions ϕn are σ-bandlimited, that is, their Fourier transform vanishes outside the interval [-σ,σ].

§30.15(iv) Orthogonality

30.15.7 -ττϕk(t)ϕn(t)dt =Λnδk,n,
30.15.8 -ϕk(t)ϕn(t)dt =δk,n.

The sequence ϕn, n=0,1,2, forms an orthonormal basis in the space of σ-bandlimited functions, and, after normalization, an orthonormal basis in L2(-τ,τ).

§30.15(v) Extremal Properties

The maximum (or least upper bound) B of all numbers

30.15.9 β=12π-σσ|-e-itωf(t)dt|2dω

taken over all fL2(-,) subject to

30.15.10 -|f(t)|2dt =1,
-ττ|f(t)|2dt =α,

for (fixed) Λ0<α1, is given by

30.15.11 arccosB+arccosα=arccosΛ0,

or equivalently,

30.15.12 B=(Λ0α+1-Λ01-α)2.

The corresponding function f is given by

30.15.13 f(t) =aϕ0(t)χτ(t)+bϕ0(t)(1-χτ(t)),
a =αΛ0,
b =1-α1-Λ0.

If 0<αΛ0, then B=1.

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