Convolution and Fourier Time axis size

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mrish 2016-6-28

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https://ww2.mathworks.cn/matlabcentral/answers/292746-convolution-and-fourier-time-axis-size

Hello,

I have a question regarding convolution results and Fourier product results size.

I want to implement a simple optical telescope x2, for an input rectangle image defined with a scale vector x, according to Fourier optics operators. To do this I represent the rectangle as '1' in the center a matrix (all other element are zero). the rectangle is small so there is no zero padding issues.

The 'real life' dimensions are represent as an (x,y) scale vectors for the matrix. - so I need to update them during the calculations.

To perform the calculation, I need to multiply my input with the transfer function of the first lens L1, propagate the result by convolution with a quadratic phase, and multiply the results by transfer function of the second lens. because I use convolution the results should be a matrix of double size with a double size rectangle in it, defined over doubled size scale vector.

I can preform the same convolution by multiplying in the frequency domain and ifft the results. however the results will be as the the central part of the convolution only. and the scale vector does not change (as it transform to the frequency domain and back).

My question is - how is scaling x2 is expressed when preforming the calculation in the frequency domain?