In order to avoid operating the Power Amplifier (PA) in the nonlinear region, the input power can be reduced by an amount about equal to the PAR. However, two important facts related to this IBO amount can be observed from Figure 4.14. First, since the highest PAR values are uncommon, it might be possible to simply “clip” off the highest peaks in order to reduce the IBO and effective PAR, at the cost of some, ideally, minimal distortion of the signal. Second, and conversely, it can be seen that even for a conservative choice of IBO—say, 10dB—there is still a distinct possibility that a given OFDM symbol will have a PAR that exceeds the IBO and causes clipping. See Sidebar 4.2 for more discussion of how to predict the required backoff amount.
Clipping, sometimes called soft limiting, truncates the amplitude of signals that exceed the clipping level as
(4.31) where is the original signal, and is the output after clipping. The soft limiter can be equivalently thought of as a peak cancellation technique, like that shown in Figure 4.15. The soft- limiter output can be written in terms of the original signal and a canceling, or clipping, signal as (4.32) Figure 4.14 CCDF of PAR for QPSK OFDM system: L = 16, 64, 256, 1,024
2 4 6 8 10 12 14
10−5 10−4 10−3 10−2 10−1 100
PAR (dB)
CCDF
L= 16
L = 64
L = 256
L = 1,024
____Simulation -- - -- Results
x n Ae f x n A
x n f x n A
L
j x n
[ ] | [ ] |>
[ ], | [ ] |
[ ]
= ≤
⎧⎨
⎩
∠ ,
x n[ ] x n[ ]
…
x n[ ] = [ ]x n +c n[ ], for n = 0, ,L−1
where is the clipping signal defined by
(4.33) where ; that is, the phase of is out of phase with by 180°, and A is theclipping level, which is defined as
. (4.34)
In such a peak-cancellation strategy, an antipeak generator estimates peaks greater than clipping level. The clipped signal can be obtained by adding a time-shifted and scaled signal to the original signal . The exact clipping signal can be generated to reduce PAR, using a variety of techniques.
Obviously, clipping reduces the PAR at the expense of distorting the signal by the additive signal . The two primary drawbacks from clipping are (1) spectral regrowth—frequency- domain leakage—which causes unacceptable interference to users in neighboring RF channels, and (2) distortion of the desired signal. We now consider these two effects separately.
4.5.3.1 Spectral Regrowth
The clipping noise can be expressed in the frequency domain through the use of the DFT. The resulting clipped frequency-domain signal is
(4.35) where represents the clipped-off signal in the frequency domain. In Figure 4.16, the power- spectral density of the original (X), clipped ( ), and clipped-off (C) signals are plotted for dif- ferent clipping ratios of 3 dB, 5 dB, and 7 dB. The following deleterious effects are observed.
First, the clipped-off signal is strikingly increased as the clipping ratio is lowered from 7 dB to 3 dB. This increase shows the correlation between and inside the desired band at low clipping ratios, and causes the in-band signal to be attenuated as the clipping ratio is lowered.
Figure 4.15 A peak cancellation as a model of soft limiter when
Antipeak Generator
+
] [n c [n]
x x[n]
dB
=5 γ
γ= 5dB c n[ ]
c n A x n e if x n A
if x n A
j n
[ ] = | | [ ] || , | [ ] |>
0, | [ ] |
− [ ]
≤
⎧⎨
⎩
θ
ffor n = 0,...,NL− −1 3pt
θ[ ] =n arg(−x n[ ]) c n[ ] x n[ ]
γ E{| [ ] | }x nA 2 = Aεx
x n[ ] c n[ ]
x n[ ] c n[ ]
c n[ ]
X
…
Xk =Xk+Ck, = 0,k ,L−1, Ck
X γ
Ck
Xk Ck
Second, it can be seen that the out-of-band interference caused by the clipped signal is deter- mined by the shape of clipped-off signal . Even the seemingly conservative clipping ratio of 7 dB violates the specification for the transmit spectral mask of IEEE 802.16e-2005, albeit barely.
4.5.3.2 In-Band Distortion
Although the desired signal and the clipping signal are clearly correlated, it is possible, based on the Bussgang Theorem, to model the in-band distortion owing to the clipping process as the combination of uncorrelated additive noise and an attenuation of the desired signal [14, 15, 34]:
(4.38) Now, is uncorrelated with the signal , and the attenuation factor is obtained by
(4.39)
Si d e ba r 4 . 2 Q u a n t i f y i n g PAR: Th e C ub i c Me t r i c
Although the PAR gives a reasonable estimate of the amount of PA backoff required, it is not precise. That is, backing off on the output power by 3 dB may not reduce the effects of nonlinear distortion by 3 dB. Similarly, the pen- alty associated with the PAR does not necessarily follow a dB-for-dB rela- tionship. A typical PA gain can be reasonably modeled as
, (4.36)
where c1 and c2 are amplifier-dependent constants. The cubic term in the equation causes several types of distortion, including both in- and out-of-band distortion. Therefore, Motorola [29] proposed a “cubic metric” for estimating the amount of amplifier backoff needed in order to reduce the distortion effects by a prescribed amount. The cubic metric (CM) is defined as
, (4.37)
where is the signal of interest normalized to have an RMS value of 1, and is a low-PAR reference signal, usually a simple BPSK voice signal, also normalized to have an RMS value of 1. The constant c3 is found empirically through curve fitting; it was found that c3 ≈ 1.85 in [29].
The advantage of the cubic metric is that initial studies show that it very accurately predicts—usually within 0.1 dB—the amount of backoff required by the PA in order to meet distortion constraints.
νout( )t = c1vin( )t +c2(vin( )t )3
CM 20log10[ν–3]rms 20log10 νref–3
[ ]rms
– c3
---
= νref ν
X Ck
…
x n[ ] =αx n[ ]+d n[ ], for n = 0,1, ,L−1.
d n[ ] x n[ ] α
α= 1 γ πγ γ
2 ( ).
−e−2 + erfc
The attenuation factor is plotted in Figure 4.17 as a function of the clipping ratio . The attenuation factor is negligible when the clipping ratio is greater than 8dB, so high clipping ratios, the correlated clipped-off signal in Equation (4.33), can be approximated by uncorrelated noise . That is, as . The variance of the uncorrelated clip- ping noise can be expressed assuming a stationary Gaussian input as
(4.40) In WiMAX, the error vector magnitude (EVM) is used as a means to estimate the accuracy of the transmit filter and D/A converter, as well as the PA, nonlinearity. The EVM is essentially the average error vector relative to the desired constellation point and can be caused by a degra- dation in the system. The EVM over an OFDM symbol is defined as
, (4.41)
where is the maximum constellation amplitude. The concept of EVM is illustrated in Figure 4.18. In the case of clipping, a given EVM specification can be easily translated into an SNR requirement by using the variance of the clipping noise.
Figure 4.16 Power-spectral density of the unclipped (original) and clipped (nonlinearly distorted) OFDM signals with 2,048 block size and 64 QAM when clipping ratio ( ) is 3 dB, 5 dB, and 7 dB in soft limiter
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
–60 –50 –40 –30 –20 –10 0 10
Normalized Frequency
PowerSpectral Density (dB)
OriginalSignal Clipped-Off Signal Clipped Signal
: X
: C
~: X
dB 3
~
=7 γ
dB
=3 γ
dB
=5 γ
dB
=7 γ
Spectral Mask
γ
α γ
α γ
c n[ ] d n[ ] c n[ ]≈d n[ ] γ↑
x n[ ]
σd εx γ α
2 2 2
= (1−exp(− )− ).
EVM =
1 ( )
=1 =
2 2
2
2 2
N I Q
S S
k L
k k
∑ ∆ +∆ d
max max
σ
Smax
σd 2
Figure 4.17 Attenuation factor as a function of the clipping ratio
Figure 4.18 Illustrative example of EVM
0 2 4 6 8 10 12
0.75 0.8 0.85 0.9 0.95 1
Clipping Ratioγ (dB)
Attenuation Factorα
α γ
Imaginary
Reference Signal Measured
Signal
Error Vector
Real
It is possible to define the signal-to-noise-plus-distortion ratio (SNDR) of one OFDM sym- bol in order to estimate the impact of clipped OFDM signals over an AWGN channel under the assumption that the distortion is Gaussian and uncorrelated with the input and channel noise, which has variance :
(4.42) The bit error probability (BEP) can be evaluated for various modulation types by using the SNDR [14]. In the case of Multilevel-QAM and average power , the BEP can be approxi- mated as
(4.43) Figure 4.19 shows the BER for an OFDM system with L = 2,048 subcarriers and 64 QAM mod- ulation. As the SNR increases, the clipping error dominates the additive noise, and an error floor is observed. The error floor can be inferred from Equation (4.43) by letting the noise variance
.
A number of additional studies of clipping in OFDM systems have been completed in recent years [3, 4, 14, 32, 39]. In some cases, clipping may be acceptable, but in WiMAX systems, the margin for error is quite tight, as much of this book has emphasized. Hence, more aggressive and innovative techniques for reducing the PAR are being actively pursued by the WiMAX commu- nity in order to bring down the component cost and to reduce the degradation owing to the non- linear effects of the PA.