A choice of some particular cutoff points is case dependent.
The integrand function may exhibit periodicity, as it can always be rewritten in terms of sines and cosines.
Implications of this fact are explored in detail later.
4.5. FFT
Numerical integration of complicated bivariate functions is computationally demanding. Fortunately, (9) is
a Fourier integral. It may be possible to apply very efficient FFT algorithms.
A symmetric Fourier transform pair of bivariate functions is:
OO OO
f (x,y) = j I F (kx,ky) e-^k.+yky 1dkχdky,
-∞ -∞
OO ∞
F (kx,ky) =∣ I f (x,y) e^k*x+k*y)dxdy.
-∞ -∞
FFT algorithms are applied to a discrete version of the above relationship.
b2 bl
asʃ .∣A -- ʃ)φ(ζι,ζ2) dζ 1dζ2,
a2 aι
Returning to equation (9), the goal is to approximate (2π)2 f (st , vτ )
where the cutoff points are sufficiently large in absolute value. Define:
Then:
∆1
&1 — ɑi
N
δ2 =
b2 — α2
N2
cjl = ai + ji^ ζj2 = a2 + J2δ2, Φ2ι,22 = φ (Cjl ,ζj2) , j1
= 0,...,N1, j2 =O,...,N2.
b. bi W2-1 Wι-1
I I e ' W1 ,C2) dζιdζ2 = ∆1∆2∑ ∑
a ai j'2=0 jl=0
- ζ>l+vτ ζχ2 4l,j-2.
Now, define :
2πk∣ 2πk∣
sτ>kl N1∆1 b1
2πk2
—, υτ,k2 = v ʌ = ,
a1 N2∆2 b2
2πk2
,
Λ2
where k1 and k2 are on the same grid as j1 and j2, respectively.
Then:
Æ2-1 ʌɪ-1
2 f (sτ,kl ,VT,k2 )= ∆1∆2 £ £ e Cu ■ ⅛ )φ.ι,.2 =
j2=0 jl=0
∆1∆2e-
i(sτ,kl αl+υτ,fe2 a2
Λ-2-1 Λl-1
) Σ Σ
e-i'2-(k. ⅛ +k2-H') φj
j l )J 2
j2=0 jl=0
The double summation term is by definition a discrete Fourier transform, for which FFT algorithms are
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