The Complex Envelope of a Bandlimited OFDM Signal Converges Weakly to a Gaussian Random Process

Orthogonal frequency division multiplexing (OFDM) systems have been used extensively in wireless communications applications in recent years; thus, there is significant interest in analyzing the properties of the transmitted signal in such systems. In particular, a large amount of recent work has focused on analyzing the variation of the complex envelope of the transmitted signal and on designing methods to minimize this variation. In this paper, it is established that the complex envelope of a bandlimited uncoded OFDM signal converges weakly to a Gaussian random process as the number of subcarriers goes to infinity. This establishes that the properties of the OFDM signal will asymptotically approach those of a Gaussian random process over any finite time interval. The symbol length in a bandlimited OFDM system will eventually exceed any finite time interval as the number of subcarriers approaches infinity; however, practical interest is in how asymptotic approximations apply for a finite number of carriers, and, hence, the convergence proof is reasonable motivation for considering how the extremal value theory of Gaussian random processes might provide accurate approximations for the distribution of the peak-to-mean envelope power ratio (PMEPR) of practical OFDM systems. Indeed, numerical results are presented that indicate that the resulting simple expressions are accurate for a wide range of the distribution for moderate numbers of subcarriers. The important extensions of the analytical and numerical results to coded OFDM systems, as well as systems with unequal power allocation across subcarriers, are also presented.