- #1
krobben92
- 6
- 0
Hi guys,
Long story short, I need to compute an autocorrelation integral. Here's the problem:
There are two arbitrary gaussian pulses, one following the other by a fixed distance. By computing the autocorrelation over space(not time) and taking the derivative of the space-shift autocorrelation and setting it equal to zero, important information hopefully could be obtained.
The mathematics of this would be as followed:
[itex]\frac{\partial}{\partial\tau}\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(Ae^{-a(x-c-\tau)^2}+Be^{-b(x-d-\tau)^2})dx=0[/itex]
[itex]\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(a(x-c-\tau)Ae^{-a(x-c-\tau)^2}+b(x-d-\tau)Be^{-b(x-d-\tau)^2})dx=0[/itex]
I am NOT asking anyone to do this for me - I'll do it myself but I just need some ideas or directions on how to go about it.
I have experience in Fourier transforms, complex analysis and calculus of course. I've considered doing a complex contour integral but I'm not sure how reasonable that is after seeing how big of a pain the normal gaussian contour integral is. I've considered Fourier transforms a little - I didn't immediately see much help due to the Fourier transform of a gaussian just being another gaussian. I've thought about parametrization or even centering the integral about the center of the two gaussians but I don't know where to start I guess.
It's clearly a bound integral but is it just too impossibly hard to try?
Long story short, I need to compute an autocorrelation integral. Here's the problem:
There are two arbitrary gaussian pulses, one following the other by a fixed distance. By computing the autocorrelation over space(not time) and taking the derivative of the space-shift autocorrelation and setting it equal to zero, important information hopefully could be obtained.
The mathematics of this would be as followed:
[itex]\frac{\partial}{\partial\tau}\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(Ae^{-a(x-c-\tau)^2}+Be^{-b(x-d-\tau)^2})dx=0[/itex]
[itex]\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(a(x-c-\tau)Ae^{-a(x-c-\tau)^2}+b(x-d-\tau)Be^{-b(x-d-\tau)^2})dx=0[/itex]
I am NOT asking anyone to do this for me - I'll do it myself but I just need some ideas or directions on how to go about it.
I have experience in Fourier transforms, complex analysis and calculus of course. I've considered doing a complex contour integral but I'm not sure how reasonable that is after seeing how big of a pain the normal gaussian contour integral is. I've considered Fourier transforms a little - I didn't immediately see much help due to the Fourier transform of a gaussian just being another gaussian. I've thought about parametrization or even centering the integral about the center of the two gaussians but I don't know where to start I guess.
It's clearly a bound integral but is it just too impossibly hard to try?
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