About the Project
30 Spheroidal Wave FunctionsApplications

§30.15 Signal Analysis

  1. §30.15(i) Scaled Spheroidal Wave Functions
  2. §30.15(ii) Integral Equation
  3. §30.15(iii) Fourier Transform
  4. §30.15(iv) Orthogonality
  5. §30.15(v) Extremal Properties

§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),
30.15.2 Λn=2γπ(Kn0(γ)An0(γ2))2;

see §30.11(v).

§30.15(ii) Integral Equation

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

§30.15(iii) Fourier Transform

30.15.4 eitωϕn(t)dt=(i)n2πτσΛnϕn(τσω)χσ(ω),
30.15.5 ττeitωϕn(t)dt=(i)n2πτΛnσϕn(τσω),


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=12πσσ|eitωf(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+1Λ01α)2.

The corresponding function f is given by

30.15.13 f(t) =aϕ0(t)χτ(t)+bϕ0(t)(1χτ(t)),
a =αΛ0,
b =1α1Λ0.

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).