Informational content in 2D discrete Fourier transform

[timelessmidgen]

When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire signal from just the first half after calculating its complex conjugate and stitching it together (right?)

I get confused when it comes to the two dimensional DFT. For the purposes of this conversation I'm assuming the result of the DFT is unshifted (IE all four corners in the two dimensional result correspond to low frequency signals, as opposed to the center as it would be in a shifted result.) When I think about the results I thought all the actual information would be contained within the lower left quadrant, and indeed I know that the numbers in the upper right quadrant will simply be complex conjugates of the numbers in the lower left quadrant. I also know that numbers in the upper left quadrant will be complex conjugates of the numbers in the lower right quadrant, but what is their relation to the numbers in the lower left quadrant (the 'important' quadrant in my mind). Can you recreate the upper left and lower right quadrants if all you have is the lower left quadrant? Or do they actually contain more information?

 

[marcusl]

Your first statement is true only for a purely real signal, and you have left off the symmetry condition. The correct statement is that the FT of a real signal is Hermitian, that is, complex conjugate symmetric.

 

[timelessmidgen]

Your first statement is true only for a purely real signal, and you have left off the symmetry condition. The correct statement is that the FT of a real signal is Hermitian, that is, complex conjugate symmetric.

Ah ok thanks, that's good to know. So in the case of a real signal do the top left and lower right quadrants contain additional information? Or are they derivable from the lower left quadrant?

 

[marcusl]

The upper right and lower left quadrants are complex conjugate symmetric, so can be derived from each other.

 

[timelessmidgen]

Thanks marcusl, but what I'm asking is if they (the top left and lower right quadrants) can be derived not from each other, but from the lower left quadrant. IE, for a purely real signal is the full informational content of the DFT contained within the lower left quadrant?

It's easy for me to physically interpret the lower left quadrant - it's the magnitude and phase of the frequencies present in the original signal. As we move right and up within the quadrant, the pixels correspond to progressively higher frequencies until we get to the edges of the quadrant (which are the centerlines of the full DFT grid) which correspond to the highest frequency. This sounds like a complete description of the information to me, hence the question of whether the top left and lower right quadrants contain additional information. If not, I would think that they must be derivable from the lower left quadrant by itself.

 

[marcusl]

You have the information needed to answer this question. What do you think?