Exponentially Modified Gaussian

[James_1978]

I am fitting a detector response using Matlab. I have been asked to fit the spectrum using a Exponentially Modified Gaussian. This is as follows.

[ tex ] a^x_n [ /tex ]

h(t) = [ tex ] \frac{A}{\sqrt{2 \pi} \sigma} e^{-\frac{(t-t_{R})^{2}}{2\sigma^{2}} [ /tex ]

f(t) = [ tex ] \frac{1}{\tau}e^{-\frac{t}{tau}}[ /tex ]

Using the convolution y(t) = [ tex ] \int^{\infinity}_{0} h(t^{\prime})f(t-t^{prime})dt^{\prime}[ /tex ]

This gives y(t) = [ tex ] \frac{A}{\tau \sigma \sqrt{2 \pi}} \int^{\infinity}_{0}} e^{-{\frac{(t-t_{R} - t^{\prime})^{2}}{2\sigma^{2}}e^{\frac{-t^{\prime}}{tau}} [ /tex ]

This is the convolution of h(t)*f(t) to give y(t)

The answer is

y(t) = [ tex ] \frac{A}{2 \tau}[1-erf(\frac{\sigma}{\sqrt{2}\tau} - \frac{t-t_{R}}{\sqrt{2}\sigma]e^{\frac{\sigma^{2)}{\sqrt{2}\tau} - \frac{t-t_{R}}{\tau}}[ /tex ]

I have been unable to get the answer. Has anyone ever worked this out to get y(t)? Also this is my first post and I am not sure how to get the latex to appear?

 

[coki2000]

I am fitting a detector response using Matlab. I have been asked to fit the spectrum using a Exponentially Modified Gaussian. This is as follows.

h(t) = \frac{A}{\sqrt{2 \pi} \sigma} e^{-\frac{(t-t_{R})^{2}}{2\sigma^{2}}

f(t) = \frac{1}{\tau}e^{\frac{t}{\tau}}

Using the convolution y(t) = \int^{\infinity}_{0} h(t^{\prime})f(t-t^{\prime})dt^{\prime}

This gives y(t) = \frac{A}{\tau \sigma \sqrt{2 \pi}} \int^{\infinity}_{0} e^{-{-\frac{(t-t_{R} - t^{\prime})^{2}}{2\sigma^{2}}e^{\frac{t^{\prime}}{\tau}}

This is the convolution of h(t)*f(t) to give f(t)

The answer is

y(t) = \frac{A}{2 \tau}[1-erf(\frac{\sigma}{\sqrt{2}\tau} - \frac{t-t_{R}}{\sqrt{2}\sigma})]e^{\frac{\sigma^{2}}{\sqrt{2}\tau} - \frac{t-t_{R}}{\tau}}

I have been unable to get the answer. Has anyone ever worked this out to get y(t)? Also this is my first post and I am not sure how to get the latex to appear?

You must write your equations between

 

[mathman]

You need to prescribe the limits of integration on t'.

 

[James_1978]

Yes...but I believe one must complete the square for the terms in the exponent. This gives you the z for the erf(z). However, I am unable to complete the square correctly. I am close and still working on it.