39
we can easily see the relation between FT and DFT
h(fn) ≈ Ahn.
(ATl)
The formula for the discrete inverse Fourier transform, which recovers the set of ∕ι⅛,s ex-
actly from the hn,s is
(A. 12)
where we have used ɪ ∑n e 2rri(j k,nlN = δjk. Let’s check out those terms which involve
DFT, for simplicity, here we only consider an ID case. The derivative related terms are
= -4πV-' [∕2≠ω] ≈ -4√∆z X = -4√1 2
4π2 J dff2∖φ{f)∖2 ≈ 4π2∆y £ f2n∖φ{fn)∖2 = 4π2- £ f2n ∖φn ∣2.
dφ(t)
For dipole-dipole interaction energy, we need to find D(t) = ʃ dt'V(t - t')φ(t') which is
alternatively
!D<tj) = ʃ dfe2πiftiV(f)φ(f) ≈ ∆z ∑ e2πW(∕nШ) = ɪ ∑ elmιl^Ql
this is exactly the inverse DFT. Since, in our case, we know the FT of V(t) analytically, we
use V(∕n) from FT rather than Vn from DFT.
Forthermore, fast Fourier transform (FFT), instead of DFT, is used for the numerical
computation. FFT is slightly different from DFT in the way how it sorts the transformed
data. In FFT, fn is ordered as
(A. 13)