In a hydroelasticity problem, I have used the temporal Fourier transform in order to solve the linear hydrodynamic equations. The system of interest consists of a solid particle moving close to an elastic membrane. Since the membrane shape depends on the history of the particle motion, a Fourier analysis is therefore needed. After resolution, I found that the interface elevation in the frequency space has a real odd part, and an imaginary even part. This means that the transformation is anti-hermitian, and the elevation back in the time space is purely imaginary. I think that this has no sense since the measured quantity should be real! Does anyone have an idea about how to explain that? Thank you!
Dear my Colleague,
Good Day,
For real-valued input data, however, the resulting DFT is hermitian—the real-part of the spectrum is an even function and the imaginary part is odd, such that X−k=Xk, where the bar represents complex conjugation. This means that all of the "negative" Fourier frequencies provide no new information. Both the k=0 and k=N/2 bins are real-valued, and there is a total of N/2+1 Fourier bins, so the total number of independent pieces of information (i.e. real and complex parts) is N, just as for the input time series. No information is created or destroyed by the DFT. please, see the attached website for detail.
http://www.cv.nrao.edu/course/astr534/FourierTransforms.html
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