Let me start by saying that this question is motivated by data analysis that I am doing. Its part of a much larger set of signal processing operations, but in order to keep the question contained I'll talk only about what (to my best knowledge) is relevant.
The scenario is as follows: we start from some discrete dataset $u$ of size $N$ in the time domain, $u ={u_1,u_2,\dots,u_N}$. From observation one finds that this dataset has some special properties: every odd value is purely real, and every even value is purely imaginary. If I subsequently compute the discrete Fourier transform of this dataset given by $v ={v_1,v_2,\dots,v_N}$ where \begin{equation} v_s=\frac{1}{\sqrt{N}}\sum_{r=1}^{N}u_r\exp{\left[2\pi i (r-1)(s-1)/n\right]} \end{equation}
I find that I obtain a curious result: the outcome $v$ is almost purely real, and the even and odd values alternate in sign: $\mathrm{sgn}{(v_s)}=-\mathrm{sgn}{(v_{s+1})}$. With this first part I mean that the real data is $\mathcal{O}(10^{-4})$ and the imaginary part is $\mathcal{O}(10^{-12})$; I'd say is purely from numerical errors.
However, it is this second aspect that is not obvious to me; why do the real values alternate in sign? For those who appreciate some visual aspects, the real part of the Fourier transform ($\mathrm{Re}(v)$) looks like this
. Note that the first quarter is a mirror image of the second, and the third of the fourth; this is a different point that is not entirely relevant.
Note that an answer could also definitely be that, given the above, there is no explanation. I am not posting the entire dataset as its rather large and makes the question messy, but perhaps I am missing a crucial ingredient. In that case I will edit it to make it more complete.