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This may be an elementary question, but I don't know much about the discrete Fourier transforms (DFT).

Suppose I have a sequence $\{x_n\}_{n=0}^{N-1}$ of $n$ real numbers such that $x_0\geq |x_n|$ for all $n$. I take the DFT of the sequence as follows: $X_k=\sum_{n=0}^{N-1}x_n e^{-i2\pi\frac{k}{N}n}$.

I am wondering if the frequency-domain representation of this sequence has any special properties besides the symmetry associated with real-valued input. In particular, I am wondering whether anything useful can be said about the real parts of $\{X_k\}$ (s.t. whether they are positive).

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up vote 2 down vote accepted

Take $N=6$, sequence $3,0,2,2,2,0$, then $X_1=3+2a^2+2a^3+2a^4$ where $a=e^{-2\pi i/6}$. We have $a^3=-1$ and $a^2+a^4=-1$ so $X_1=-1$ has negative real part.

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Thanks! So $\{X_k\}$ doesn't really have any special properties? Now I am wondering what kind of $\{x_n\}$ would it take to yield $\{X_k\}$ with positive real parts, but I posted that as a separate question. –  M.B.M. May 23 '12 at 14:12

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