Praise for Fundamentals of WiMAX

134 Chapter 4 • Orthogonal Frequency Division MultiplexingSimilarly, digital-signal processing (DSP) techniques developed to reduce the digital PAR maynot always have the anticipated effect on the analog PAR, which is what matters. In order tobring the analog PAR expression in Equation (4.27) and the digital PAR expression inEquation (4.28) closer together, oversampling can be considered for the digital signal. That is, afactor M additional samples can be used to interpolate the digital signal in order to better approximateits analog PAR.It can be proved that the maximum possible value of the PAR is L, which occurs when allthe subcarriers add up constructively at a single point. However, although it is possible to choosean input sequence that results in this very high PAR, such an expression for PAR is misleading.For independent binary inputs, for example, the probability of this maximum peak value occurringis on the order of 2 –L .Since the theoretical maximum (or similar) PAR value seldom occurs, a statistical descriptionof the PAR is commonly used. The complementary cumulative distribution function (CCDF= 1 – CDF) of the PAR is the most commonly used measure. The distribution of the OFDM PARhas been studied by many researchers [3, 32, 33, 44]. Among these, van Nee and de Wild [44]introduced a simple and accurate approximation of the CCDF for large L( ≥ 64) :βL⎛ Emax⎞CCDF( L, Emax ) =1 −G( L, Emax )=1 −F( L, Emax) =1− 1 − ( ) 22⎝⎜ expσ ⎠⎟βL,(4.29)E maxwhere is the peak power level and β is a pseudoapproximation of the oversampling factor,which is given empirically as β =2.8. Note that the PAR is E max/2σ2 and FL ( , E max) is thecummulative distribution function (CDF) of a single Rayleigh-distributed subcarrier withparameter σ 2 . The basic idea behind this approximation is that unlike a Nyquist-sampled signal,the samples of an oversampled OFDM signal are correlated, making it difficult to derive anexact peak distribution. The CDF of the Nyquist-sampled signal power can be obtained byLGL ( , E )= P( max || xt ( )|| ≤ E )= FL ( , E ) ,max max max(4.30)With this result as a baseline, the oversampled case can be approximated in a heuristic wayby regarding the oversampled signal as generated by βL Nyquist-sampled subcarriers. Note,however, that β is not equal to the oversampling factor M. This simple expression is quite effectivefor generating accurate PAR statistics for various scenarios, and sample results are displayedin Figure 4.14. As expected, the approximation is accurate for large L, and the PAR of OFDMsystem increases with L but not nearly linearly.5. Nyquist sampling means the minimum allowable sampling frequency without irreversible informationloss, that is, no oversampling is performed.