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Knowing that DFT of n-values real signal in 1d consists of n/2+1 different values where the second half of the spectrum is complex conjugate of the first one (Hermitian symmetry).

However in the spectrum of 2d signal that I've made in Octave I see complex conjugates in both dimensions. However the FFTW manual states that we can take a use of the symmetry in only one dimension.

What is the truth? Is Hermitian symmetry only useful in one of multidimensional transform or could it be used in other dimensions?

[EDIT]

Ok, so since the Hermitian symmetry occurs in each of the directions then why the transformed vector in FFTW is diminished only by half (only one dimension values are stored on N/2+1 complex numbers)? Why isn't each direction diminished?

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Yes, multidimensional DFT's have complex conjugate symmetry in each dimension

I think you might be misinterpreting the FFTW manual which is referring to internal storage formats.

Edit:

why the transformed vector in FFTW is diminished only by half (only one dimension values are stored on N/2+1 complex numbers)? Why isn't each direction diminished?

From the FFTW manual:

The multi-dimensional transforms of FFTW [...] compute simply the separable product of the given 1d transform along each dimension of the array

So for example if you provide a 256x256 array of real numbers the result will be 256 1 dimensional DFT's. As they mention at the bottom of that documentation this is not a multi-dimensional DFT.

see http://www.fftw.org/fftw3_doc/Complex-Multi_002dDimensional-DFTs.html

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  • $\begingroup$ Thanks for the answer! I have edited the question, as I still don't know if this property couldn't be used in diminishing the size of the DFT by more than half. $\endgroup$ – Paweł J Mar 31 at 9:59

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