# Discrete Convolutions

1. Jul 19, 2011

### csand

I'm having trouble understanding the results of a discrete convolution. I have two functions:

1) High resolution spectra
2) Gaussian curve

The point of this operation for me is put the high resolution data in terms of the lower resolution represented by the guassian curve (filter)

I convolve one with the other, and the results seem reasonable. However I do not understand how to interpret my units.

It is my understanding that the final "length" or domain of my convoluted function will be:

len(1) + len(2) - 1.

So if I'm dealing in wavelengths or such units for my original function (1), how do I now map my new convoluted values to wavelengths? The domain of my convoluted function is bigger than either domain other the two original functions.

Do I "center" my convoluted data around the original data (1) and trim the values on the outside?

My understanding of this is limited so I hope the context to the question I've asked is understandable. I can try to elaborate if more information is required.

Chris

2. Jul 19, 2011

### pmsrw3

I'm going to assume that l1 is the length of your spectrum, l2 the length of your Gaussian kernel, and that l2 is odd. In that case, point i in your original spectrum corresponds to point i + (l2-1)/2 in the convolved spectrum. (So, yes, if you center the old on the new, you'll have the right correspondence.) Also, the length you give, l = l1 + l2 - 1, indicates that the real data are being padded at the ends when you do your convolution. You should not trust the end of the convolved spectrum. Only the central l1 - l2 + 1 values of the convolved data are free from contamination by the padding.

3. Jul 25, 2011