# Minimizing |FT(X)|_{\infty} by permutation of X_i - question on Fourier transform related to engineering problem (peak factor of OFDM system)

Consider vector X =( X_1 ... X_N), consider the discrete Fourier transform $Y=F(X)$.

I am interested to minimize $|Y|_{\infty}$, by permutation of numbers X_i, how to do it ?

Here $|Y|_{\infty}$ is infinity norm of the vector Y i.e. just the maximum of absolute values of components of Y.

More close to life problem is a little more complicated: my numbers X_i are splited at several subsequences such that |X|=const in each subsequence. And I am allowed to make "block" permutations of these subsequences. The goal is the same as to minimize $|Y|_{\infty}$

Background: roughly speaking the OFDM based ( = most advanced) radio telecommunication systems (LTE, WiMax, new WiFi) make the Fourier transform before transmitting data symbols to the space. Average power is fixed, but people care also about the maximal instant power, which they do not want to be big. Instant power is just the maximal component of the vector.

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I don't see how you can do anything non-trivial without additional restrictions on $X$. The trivial estimate is $|Y|{\infty} \leq |X|_{1}$ (assuming there is no normalizing factor in your definition of F, which, in any event, doesn't change the argument). However, taking $X=(1,1,\ldots 1)$ one sees that this is in fact an equality (for any reordering of the $X_i$'s).