cross correlation stuff
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A brief informal note by Phil dated 3.22.13, in the support folder of the Spectral Theory Book. It compares the Wolfram MathWorld and Wikipedia definitions of cross-correlation, finds the MathWorld statement incomplete, and relates correlation to convolution via f*(-t). It also recalls the convolution theorem (3.6) and autocorrelation (32.1) and looks for the Unicode star operator. Integrals are garbled in the extraction and the note ends mid-thought.
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Correlation stuff PhL 3.22.13
Wolfram: note there are two different symbols, star and asterisk.
http://mathworld.wolfram.com/Cross-Correlation.html
This is my general area of interest, but things are missing and wrong. First, the last claim would require that f be real and even, unless "even" has some special meaning for a complex function.
Wiki defines things like this
http://en.wikipedia.org/wiki/Cross-correlation
and again the star is used. OK, this agrees with Wolfram just take τ→-τ. So this looks the same as convolution but the sign of t is different and f* is appearing. Do I have a star character?
Do I have such a character? ⋆
⋆8902⋆22C6 STAR OPERATOR
My Lucida Unicode font has it. I have to make it larger to have it be right. Then
(f ⋆g )(t) = !Syntax Error, I dt' f*(t')g(t+t')dt'
(f*⋆g )(-t) = !Syntax Error, I dt' f(t')g(t' -t)dt' = (f * g)(t)
*******************
Recall the convolution theorem (3.6)
a(t) = !Syntax Error, I dt' b(t-t')c(t') A(ω) = B(ω) C(ω) (3.6)
The equation on the left is often written
a(t) = (b ∗ c)(t)
and one says that a is the convolution of b with c.
∗
[c(t) ⋆b(t)](t) = !Syntax Error, I dt' c*(t')b(t+t') = !Syntax Error, I dt" c*(-t")b(t-t") = [c*(-t) ∗ b(t)](t)
*******************************************
The Convolution Theorem:
a(t) = !Syntax Error, I dt' b(t-t')c(t') A(ω) = B(ω) C(ω) (3.6)
Autocorrelation Function:
rb(t) ≡ !Syntax Error, I dt' b(t') b(t' + t) = !Syntax Error, I dt' b(t') b(t' - t) (32.1)
Cross Correlation Function of b with itself
[b(t) ⋆b(t)](t) = !Syntax Error, I dt' b*(t')b(t+t')
This, if b is a real function,
If