Complete characterization of the solvability of PAPR reduction for OFDM by tone reservation

In this paper we analyze the peak-to-average power ratio (PAPR) reduction by tone reservation for orthogonal frequency division multiplexing (OFDM) schemes. In addition to the strong solvability of the PAPR reduction problem, where the PAPR has to be bounded by some constant, we consider a weaker form of solvability, where only the boundedness of the peak value of the signal is required. We show that for OFDM both forms of solvability are equivalent. Further, we show that in the case where the PAPR problem is not solvable, the set of input signals that lead to an unbounded OFDM signal is a residual set. As a consequence, if the upper density of the carriers, used for information transmission, is positive, the set of input signals that lead to a bounded OFDM signal is a meager set.