DFT: What is the physical meaning of the symmetry about the nyquist frequency?

[jsparger]

When I take the fft of a set of data and plot it, there is a reflection around the nyquist. Everybody knows this, but I would like to know what the physical meaning of the second half (the reflected half) is.

The real component is the same as the first half, and the imaginary component has the opposite sign.

I can use the first half of the data to reconstruct my signal, and I understand how it relates to frequency and phase angle, but I am not clear on the second half. Can I reconstruct my signal from this half of the data as well? Why do the frequencies have the opposite phase angle? These frequencies correspond with frequencies that could be aliasing if you extend the frequencies with the same spacing (for example, I am not clear whether you should take the info to be arranged as:

0 up to near the Nyquist, back down to near zero (so that the second half is some other expression of the first half of the data, i.e. it corresponds to the same frequencies;

or

0 up to near the Nyquist, rest is junk; (second half means nothing)

or

0 up to near the Nyquist (first half), then up to near the samplingFrequency (second half), as in the second set corresponds to higher frequencies that may be aliasing.

where k is the frequency spacing = samplingFrequency/numberOfSamples

Maybe this is just an artifact of the DFT. If not, could somebody explain to me what it means? Feel free to ignore the blather above, since I really have no clue what I am talking about. Just would like to know what the reflection corresponds to physically, and why the imaginary part is opposite.

Thanks
--John

 

[fresh_42]

Maybe this answers your question: https://jackschaedler.github.io/circles-sines-signals/sampling2.html

 

[RPinPA]

When I take the fft of a set of data and plot it, there is a reflection around the nyquist. Everybody knows this, but I would like to know what the physical meaning of the second half (the reflected half) is.

The real component is the same as the first half, and the imaginary component has the opposite sign.

It's an "artifact" of starting with a real signal. In order for a complex FT to represent a real signal, the coefficient of -f must be the complex conjugate of the coefficient of +f. And there's a periodicity in a FT, repeating the spectrum every $f_{Max}$, i.e. every $n$ points.

Putting these things together, for an $n$ point digital FT, the value at $(n/2)+m$ is the same as the value at $(n/2) + m - n = -(n/2) + m = -[(n/2) - m]$ from the periodicity, and this is the complex conjugate of the value at $(n/2) - m$. Thus the values at $m$ above the midpoint and $m$ below the midpoint are conjugates.

It's common practice to swap the upper and lower halves for plotting, so that the corresponding frequency values go from -(n/2) to (n/2) instead of 0 to n. MATLAB has a built-in function FFTSHIFT that does this.