Praise for Fundamentals of WiMAX

4.5 The Peak-to-Average Ratio 137Sidebar 4.2 Quantifying PAR: The Cubic MetricAlthough the PAR gives a reasonable estimate of the amount of PA backoffrequired, it is not precise. That is, backing off on the output power by 3 dBmay not reduce the effects of nonlinear distortion by 3 dB. Similarly, the penaltyassociated with the PAR does not necessarily follow a dB-for-dB relationship.A typical PA gain can be reasonably modeled asν out () t = c 1 v in () t + c 2 ( v in () t ) 3 , (4.36)where c 1 and c 2 are amplifier-dependent constants. The cubic term in theequation causes several types of distortion, including both in- and out-of-banddistortion. Therefore, Motorola [29] proposed a “cubic metric” for estimatingthe amount of amplifier backoff needed in order to reduce the distortioneffects by a prescribed amount. The cubic metric (CM) is defined asCM 20log 10 ν – 3– 3[ ]rms – 20log 10 [ ν ref ]= --------------------------------------------------------------------------------- rms, (4.37)c 3where ν is the signal of interest normalized to have an RMS value of 1, andν ref is a low-PAR reference signal, usually a simple BPSK voice signal, alsonormalized to have an RMS value of 1. The constant c 3 is found empiricallythrough curve fitting; it was found that c 3 ≈ 1.85 in [29].The advantage of the cubic metric is that initial studies show that it veryaccurately predicts—usually within 0.1 dB—the amount of backoff requiredby the PA in order to meet distortion constraints.Second, it can be seen that the out-of-band interference caused by the clipped signal ̃X is determinedby the shape of clipped-off signal C k. Even the seemingly conservative clipping ratio of7dB violates the specification for the transmit spectral mask of IEEE 802.16e-2005, albeitbarely.4.5.3.2 In-Band DistortionAlthough the desired signal and the clipping signal are clearly correlated, it is possible, based onthe Bussgang Theorem, to model the in-band distortion owing to the clipping process as thecombination of uncorrelated additive noise and an attenuation of the desired signal [14, 15, 34]:xn ̃ [ ] = α xn [ ] + dn [ ], forn= 0,1, …, L−1.Now, dn [ ] is uncorrelated with the signal xn [ ], and the attenuation factor α is obtained by(4.38)2− γα =1− e +πγerfc(γ).2(4.39)