There is one Fourier transform only, that's mathematics (see textbook or Wikipedia).

When programming a discrete transform, you make many choices.

Instead of an integral from - infinity to +infinity you limit the scope. To avoid unwanted border effects, you filter your analysis window (weighting, convoluting...). Then you discretise quantify, that's a number of things different from the single mathematical object called a Fourier transform.

This is true for discrete cosine transform, wavelet transform, etc.

You will find this described in classical Signal Processing textbooks from the years 1980 to today, no need of the latest, it's old and well-known by the Ancients (LOL, sometimes knowledge gets assumed, and then forgotten, and discovered again)

Dear Kadem,

I second the colleague Renaud and i want add some clarifying points:

The Fourier transform of well behaved waveform is unique.In the sense there is one and only one Fourier transform expression.

The matter is different with discrete fourier transform and fast Fourier transform.

The waveform is quantized in time and converted into time sequence. This form represents only the signal in a specific time instants which are called sampling instants. Varying the sampling instants will change the observed values of the sequence. But if the sampling theory is preserved and applied on any sampling process, the resulting sequence will be a representation of the waveform which means after their interpolation will resume the original waveform with noise added to them which is the quantification noise.

Equally well in the frequency domain, the DFT and FFT are the discretization of the Fourier transform of the function. So, the sequence which you get represents a sampled version of the Fourier transform. The values of the sequence will depend on the sampling frequencies and hence will be different. However all of them will represent the same Fourier transform.

I hope i could explain the issue raised by you by a simple and understandable method.

Best wishes

Although I'm not an expert in this topic, I suggest you to try involve your problem in Mathematica 8 (9, 10 or 11) via GPU Enhanced Fast Fourier Transforms (for this Software Development in Wolfram Mathematica 8,see https://www.wolfram.com/mathematica/new-in-8/cuda-and-opencl-support/gpu-enhanced-fast-fourier-transforms.html; related command can be find in Mathematica 9 writting “Fast Fourier Transform” in Help of FUNCTION NAVIGATOR).

Fourier transform is unitary transform so for each function there is one FT

I agree with advice from colleagues above, to be more concrete (math), look at this lecture on discrete Fourier Transform:

https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l7.pdf&ved=2ahUKEwji46Kx9KnhAhVGTxUIHdkPDZkQFjAHegQIDRAh&usg=AOvVaw3puYA898L6TQ3mkuhl2ktP