The Fourier Transform
of the
Quadratic Kubo-Bass Term



Robert Kessel
Code 8123
Naval Research Laboratory



January 18, 2003

Abstract

This calculation yields a relatively simple form for the Fourier transform of the quadratic, or 2nd order, Kubo-Bass term.

Key words: Fourier transform, quadratic Kubo-Bass term, 2nd order Kubo-Bass term

The Fourier transform of the quadratic Kubo-Bass term





The behavior given by the first two terms of the Kubo-Bass series is

B(t) = ∫ R(t') G1(t-t') dt' + ∫ ∫ R(t') R(t'') G2(t-t', t'-t'') dt' dt''    .
(1)
While the Fourier transform of the linear term is well known, r(f)g1(f), the quadratic term has rarely been used. In the frequency domain the 2nd order term is given by
b2(f) = ∫ e2πi ft dt ∫ ∫ R(t') R(t'') G2(t-t', t'-t'') dt' dt''    .
(2)
Replacement of R(t') and R(t'') by their inverse Fourier transform representation[1] yields
b2(f)
=
∫ e2πi ftdt ∫ ∫ [ ∫ r(f') e-2πi f't' df' ] [ ∫ r(f'') e-2πi f''t'' df'' ] G2(t-t', t'-t'') dt' dt''    ,
(3)
=
∫ dt ∫ dt' ∫ dt'' ∫ df' ∫ df'' e2πi (ft - f't' - f''t'') G2(t - t', t' - t'') r(f') r(f'')    . 
(4)
Set v = t'- t'' or t'' = t'- v and dt' = - dv, so that upon substitution and rearrangement of the integration order
b2(f)
=
- ∫ dt ∫ dt' ∫ df' ∫ df'' e2πi ft [ ∫ dv  e-2πi (f't' + f''(t'- v)) G2(t-t',v) ] r(f') r(f'')    ,
(5)
=
- ∫ dt ∫ dt' ∫ df' ∫ df'' e2πi ft [ ∫ dv  e-2πi (f't' + f''t' - f''v) G2(t-t',v) ] r(f') r(f'')    ,
(6)
=
- ∫ dt ∫ dt' ∫ df' ∫ df'' e2πi ft e-2πi (f' + f'')t' [ ∫ dv  e2πi f''v G2(t-t',v) ] r(f') r(f'')    .
(7)
Next set u = t - t' or t' = t - u and dt' = - du, hence upon substitution and further rearrangement
b2(f)
=
∫ dt ∫ df' ∫ df'' ∫ du  e2πi ft e-2πi (f'+ f'')(t - u) [ ∫ dv  e2πi f''v G2(u,v) ] r(f') r(f'')    ,
(8)
=
∫ dt ∫ df' ∫ df'' ∫ du  e2πi (ft - f't + f'u - f''t + f''u) [ ∫ dv  e2πi f''v G2(u,v) ] r(f') r(f'')    ,
(9)
=
∫ dt ∫ df' ∫ df'' ∫ du  e2πi (f - f'- f'')t e2πi (f' +f'')u [ ∫ dv  e2πi f''v G2(u,v) ] r(f') r(f'')    ,
(10)
=
∫ df' ∫ df'' ∫ dt  e2πi (f - f'- f'')t [ ∫ ∫ du dv  e2πi (f'+f'')u e2πi f''v G2(u,v) ] r(f') r(f'')    .
(11)
The factor in the square parentheses is a two-dimensional Fourier transform, so Equation 11 can be rewritten in the simpler form of
b2(f) = ∫ df' ∫ df''
∫ dt  e2πi (f - f'- f'')t
g2(f'+f'', f'') r(f')r(f'')    .
(12)
Recognizing the underlined factor in Equation 12 as the Fourier transform representation of ((f-f') - f'') [2] and, that from symmetry it is also equal to (f'' - (f-f')), yields upon substitution and evaluation of the integral over df''
b2(f) = ∫ df' ∫ df'' (f'' - (f-f')) g2(f' + f'', f'') r(f') r(f'')    ,
(13)
=
∫ df' g2(f' + (f-f'), f-f') r(f') r(f-f')    ,
(14)
=
∫ df' g2(f, f-f') r(f') r(f-f')    .
(15)
All that remains are the two small tasks of developing a stable numerical technique to estimate g2(f, f-f') from data and an experimental test of whether Equation 15 can outperform linear term.

References

[1]
Ron Bass both showed the value of replacing R(t') and R(t'') by their inverse Fourier transform representation and developed the general outline of this calculation in the early 1980's. This calculation is just a recovery of his original result.

[2]
Arfken, G. (1970). Mathematical Methods for Physicists, (pp. 671-673). New York:Academic Press.

File translated from LATEX by TTH, version 2.21.
On 16 Jan 2003, 09:49.
(with later hand modifications of the equations and such)

The calculation given on this web-page is also available in Adobe Acrobat format at 2nd_order_freq.pdf. Note the .pdf will render a bit fuzzy on most browsers, but when printed, a hard copy will have full and sharp resolution. The LATEX source file used as the basis of the both the .html and .pdf versions of this calculation is available at 2nd_order_freq.tex.