calibcant/main.bib: Don't worry about four-deriv citation
authorW. Trevor King <wking@tremily.us>
Tue, 11 Jun 2013 14:21:42 +0000 (10:21 -0400)
committerW. Trevor King <wking@tremily.us>
Tue, 11 Jun 2013 14:21:42 +0000 (10:21 -0400)
commit09f0410623bdd46e055a79d93d63e43cb374076a
tree608c65d5483c33ae99ef04e22685f8bda201f48d
parentab9618295c5b1c26836a88775ff0c8b8dd025db9
calibcant/main.bib: Don't worry about four-deriv citation

Derivation:

  F(d x(t) / dt) = F(x'(t))
    = 1/sqrt(2 π) \int_{-∞}^∞ x'(t) \exp{-i ω t} dt

Using integration by parts:

  \int v du = uv - \int u dv

with:

  v = \exp{-iωt}
  dv = -iω \exp{-iωt}
  du = x'(t) dt
  u = x(t)

gives:

  F(x'(t)) = 1/sqrt(2 π)  [
      x(t)\exp{-iωt}|_{-∞}^∞ -
      \int_{-∞}^{∞} x(t) iω\exp{-iωt} dt ]
    = iω ⋅ 1/sqrt(2 π) \int_{-∞}^∞ x(t) \exp{-iωt} dt
    = iω F(x(t))

Where the uv term drops out for any signal where x(t)→0 as |t|→∞.

For higher order differentials, just chain these together:

  F(d^n x(t) / dt^n)
    = iw F(d^{n-1} x(t) / dt^{n-1})
    = …
    = (iw)^n F(x(t))
src/calibcant/main.bib