# APPENDIX

# G The Peak-Power Reduction Problem in OFDM

In Section 9.11 we discussed the multicarrier transmission technique, namely orthogonal frequency-division multiplexing (OFDM), which is of particular importance to wireless communications due to the computational benefits offered by the fast Fourier transform (FFT) algorithm. However, envelope variations are a frequently cited drawback of OFDM because of the *peak-power limited problem*. This problem arises due to the statistical possibility of a large number of independent subchannels in the OFDM becoming constructively superimposed, thereby resulting in high peaks. In the literature, the practical issue of envelope variations is described in terms of the *peak-to-average power ratio*, commonly abbreviated as PAPR.^{1}

In this section, we discuss the PAPR problem in wireless communications and how it can be reduced.

## G.1 PAPR Properties of OFDM Signals

Consider a single modulation interval, that is, a single symbol of OFDM, the duration of which is denoted by *T*_{s}. In its most basic form, the transmitted OFDM signal is described by

where the term Δƒ denotes the frequency separation between any two adjacent subchannels in the OFDM. By definition, the frequency separation Δƒ and symbol duration *T*_{s} are related by the time−bandwidth product:

This condition is required to satisfy the orthogonality requirement among the *N* subchannels of ...