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In this paper, we present a shell mapping scheme to reduce the peak and average power of orthogonal frequency division multiplexing (OFDM) signals using quadrature amplitude modulation (QAM) by constellation shaping. Our approach can be considered as an extension of the shell mapping technique introduced originally in the V.34 recommendation of ITU-T. The shell mapping method uses multiple cost functions, each having the OFDM block length N as a parameter. The cost functions are presented in closed form for any N. The main advantage of the proposed scheme is that the peak and average power can be reduced without sacrificing the data rate, and no side information is needed at the receive side. We apply our constellation shaping technique to OFDM systems with N equal to 16, 32 and 128 subcarriers. The results show that the peak and average power decrease with increasing values of N. This advantage is partly offset by a slightly higher system complexity. . For a recent treatment of PAPR reduction of OFDM signals, see References [2-6]. There are also excellent books [7] available on this topic. Of special interest are PAPR reduction methods employing constellation shaping, because in this case the reduction of the PAPR is obtained without deleterious effects on other parameters, like the data rate and distortion. The idea was proposed in the one page conference paper of Kwok and Jones [8], in which the hypercube boundary traced out by the peak power bound in the time domain is mapped back to the frequency domain via the discrete Fourier transform (DFT), and a method is then constructed for indexing the OFDM constellation points. The authors claim an impressive PAPR reduction of over 20 dB. In Reference [9], the outer constellation points are extended to minimise the PAPR of the OFDM symbol. A similar idea is used in Reference [10]. In Reference [11], for 0 ≤ t ≤ T, where Re(w) denotes the real part of the complex number w. Equivalently, one can write theHaving these functions, the constellation-shaping algorithm can be applied for all N that are a power of two which can be as large as required, as described in for the particular case N = 8. Note, however, that the formulas above are more general, since they hold true for any positive integer N and not only for N being a power of two.

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