# Performing a Deconvolution

So I have the following data:

[[-344    13771   4600 ]
[-275.2  12478   6410 ]
[-206.4  19443    830 ]
[-137.6  69392   3830 ]
[ -68.8  143737  3780 ]
[   0    189278 16870 ]
[  68.8  184486  5090 ]
[ 137.6  188466  9380 ]
[ 206.4  185023 21680 ]
[ 275.2  128133  1460 ]
[ 344    51288   1950 ]
[ 412.8  10854   4290 ]]


First column is the x value (position in microns). Second value is the recorded data point (y), and the third column is the error (noise).

I am wondering how I can find the response of my device if the data I record is known to be a convolution of the response and a slit of width 83.6666 (plus noise).

My attempt was to use the convolution theorem of fourier transforms and use MatLab's fft() to solve for the desired function, but I could not figure out how to get everything the same length. I also thought about being blunt and using deconv() but that gave me something very strange, (with my rect function defined as

>> x=-344:1:412;
>> h=zeros(size(x));
>> h(300:386)=1;


So I used deconv(h,y) and got a very strange looking plot that was not a function.

So I was hoping someone could give me some advice on how to perform this mathematically (code or no code). Especially with the addition of noise.

Thanks!

• "the third column is the error (noise)." What does this mean, exactly? Is it a statistical error bound for each measurement? Or is it an actual error signal? – endolith Jun 25 '13 at 15:05
• It is the standard deviation of each measurement over a several measurements – yankeefan11 Jun 25 '13 at 16:54
• Ah, ok. I don't know how to use that information, though. So the slit is like a rectangular function with width 83.6? Like [0,0,0,1,1,1,1,1,0,0,0] but longer? – endolith Jun 25 '13 at 17:51
• Yes that is correct. It is then convolved with the response and I get the above data. I am not overly concerned with addressing the 'noise' – yankeefan11 Jun 26 '13 at 13:12
• Actually, should it be longer? Or should it just be [0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0] to be the same length as the original? – endolith Jun 26 '13 at 13:54

ifft(fft(y)./fft(h))