From: W. Trevor King <wking@tremily.us>
Date: Tue, 11 Jun 2013 14:21:42 +0000 (-0400)
Subject: calibcant/main.bib: Don't worry about four-deriv citation
X-Git-Tag: v1.0~111
X-Git-Url: http://git.tremily.us/?a=commitdiff_plain;h=09f0410623bdd46e055a79d93d63e43cb374076a;p=thesis.git

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))
---

diff --git a/src/calibcant/main.bib b/src/calibcant/main.bib
index d27d06f..e4b95b2 100644
--- a/src/calibcant/main.bib
+++ b/src/calibcant/main.bib
@@ -20,12 +20,12 @@
 
 
 @Misc{four-deriv,
-  note = "Hmm, it is suprisingly difficult to find an `official' reference for this.
-    I obviously need to get a spectral analysis book :p.
-    See Wikipedia's currently excellent page (Feb 15th, 2008) \\
-    \url{http://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships},\\
-    or derive it for yourself in about three lines :p.",
+  note = "See Wikipedia's currently excellent page (Feb 15th, 2008) \\
+    \url{http://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships}, \\
+    or derive it for yourself in about three lines.",
   year = 2008,
+  month = feb,
+  day = 15,
 }
 
 @Inbook{parseval,