Let's say i have a real valued random noise $\eta(t)$, for which I took $N$ samples with $N$ even, i have therefore a vector of sampled values.
I want to compute the Discrete Fourier transform of the sampled values. By using the Fourier routine in Mathematica, i get a vector of length $N$ containing the the transformed values.
If i'm correct the zeroth bin of this vector tells me the information about the average value of the random noise and is, therefore, a real valued entry.
I also know that the values occuring after $N/2$ position are the complex conjugate of the ones occuring from $0$ to $N/2$.
What i'm getting however is that also the term appearing at position $N/2$ is real valued. is that expected? what information carries this term? is this related to the zeroth bin?
I hope i was clear, Thanks in advance
By the definition of the DFT you have
$$X_{N/2}=\sum_{n=0}^{N-1}x_ne^{-i\pi n}=\sum_{n=0}^{N-1}x_n(-1)^n$$
so if $x[n]$ is real, $X_{N/2}$ must also be real. This bin represents the highest possible frequency (called Nyquist frequency) of the signal.