Convolution of spectra to sensor response function?

I am really sorry if it is wrong place to ask this question but i couldn't not find any stackexchange website appropriate. i have some reflectance values of a material which are measured in 10.5 and 12.5 micrometers interval. This spectra is extracted from a spectral library. Then by using Kirchhoff's Radiation Law i calculated the emissivity values in each wavelength.

The problem is that, Here is the spectral response of Landsat5 TM thermal band. --> http://atmcorr.gsfc.nasa.gov/L5_handbook.rsp I have to convolve these emissivities to the spectral response above. Bu i dont have any idea. How can i do it?

Thanks alot for your helps

I found the solution after searching in articles, Here is the convolution = $$e_{i} = \frac{\int_{\lambda =\lambda _{1}}^{\lambda _{2}}f_{i}\varepsilon \left ( \lambda \right )\partial\lambda }{\int_{\lambda =\lambda _{1}}^{\lambda _{2}}f_{i}\partial\lambda}$$ Where i = channel number, $\lambda$ is wavelength, $\varepsilon \left ( \lambda \right )$ spectral emissivity optained from spectral library, $f_{i}$ sensors spectral function and $e_{i}$ is channel emissivity that we want to find. $\lambda _{1}$ and $\lambda _{2}$ are upper and lower bounds of thermal band.

• You could maybe explain the 2 datasets a bit more and the application involved or the expected Output type and given Input Types so that people can post a meaningful answer since I'm unable to understand what we're trying to convolve here. – Sudarsan Aug 4 '13 at 22:39
• Ok. The first dataset is a (for example) a reflectance data of a conifer tree measured by radiometers, here is the first dataset from ASTER JPL spectral library --> speclib.jpl.nasa.gov/speclibdata/… i will use this data only in 10.5 to 12.5 micrometer range and by applying kirchhoff's laf (e=1-r) where e=emissivity r=reflectance values. i will calculate emissivities. But to use this emissivities in my landsat thermal imagery for emissivity correction. i have to convolve it to thermal band's spectral response function above. – user5162 Aug 5 '13 at 0:20
• Sorry, for the spam but character limit. I am using landsat thermal images. And for vegetation in the landsat scene i have to make emissivity correction. The first spectral library data is measured in labs. I can't use the raw emissivities calculated from the ASTER library reflectance values. – user5162 Aug 5 '13 at 0:39
• @Geomatics It's OK to answer your own question (i.e. to move your question edit into a proper answer). – datageist Aug 5 '13 at 18:06