Doug Kerr
Well-known member
Especially in connection with the concept of modulation transfer function (MTF), we often encounter reference to the Fourier transform. In fact, the modulation transfer function of an optical system (in the form that plots the modulation transfer coefficient against spatial frequency, not the form we usually see in lens specs) is the Fourier transform of the system point spread function.
If we are prepared to deal with the source and result functions expressed as discrete values at repetitive intervals, then it is the discrete Fourier transform (DFT) that is used.
There are many tools that will actually let us determine the DFT of a function. Perhaps the most widely accessible is the Fourier Analysis tool in the spreadsheet application Microsoft Excel.
But if we decide to try a test problem, we quickly encounter the matter of interpretation of the time scales. For example, suppose that our input and output functions descriptions are in terms of values at 16 points. We see that, for our system, the MTF drops to 10% of its "zero frequency" value, a level of interest, at point 7. But what is that in cycles per mm?
I have just released a new tutorial article, "The Fourier Transform Tool in Microsoft Excel", which explains how this works. It does so in the context of a brief discussion of the use of the tool itself, and first the article gives an explanation of the Fourier transform and the discrete Fourier transform themselves.
The article is available here:
http://doug.kerr.home.att.net/pumpkin/index.htm#Excel_Fourier
If we are prepared to deal with the source and result functions expressed as discrete values at repetitive intervals, then it is the discrete Fourier transform (DFT) that is used.
There are many tools that will actually let us determine the DFT of a function. Perhaps the most widely accessible is the Fourier Analysis tool in the spreadsheet application Microsoft Excel.
But if we decide to try a test problem, we quickly encounter the matter of interpretation of the time scales. For example, suppose that our input and output functions descriptions are in terms of values at 16 points. We see that, for our system, the MTF drops to 10% of its "zero frequency" value, a level of interest, at point 7. But what is that in cycles per mm?
I have just released a new tutorial article, "The Fourier Transform Tool in Microsoft Excel", which explains how this works. It does so in the context of a brief discussion of the use of the tool itself, and first the article gives an explanation of the Fourier transform and the discrete Fourier transform themselves.
The article is available here:
http://doug.kerr.home.att.net/pumpkin/index.htm#Excel_Fourier