Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
19.6 Peak-to-Average Power Ratio
19.6.1 Origin of the Peak-to-Average Ratio Problem
One of the major problems of OFDM is that the peak amplitude of the emitted signal can be consid- erably higher than the average amplitude. ThisPeak-to-Average Ratio(PAR) issue originates from the fact that an OFDM signal is the superposition ofN sinusoidal signals on different subcarriers.
On average the emitted power is linearly proportional toN. However, sometimes, the signals on the subcarriers add up constructively, so that theamplitudeof the signal is proportional toN, and the power thus goes withN2. We can thus anticipate the (worst case) power PAR to increase linearly with the number of subcarriers.
We can also look at this issue from a slightly different point of view: the contributions to the total signal from the different subcarriers can be viewed as random variables (they have quasi-random phases, depending on the sampling time as well as the value of the symbol with which they are modulated). If the number of subcarriers is large, we can invoke the central limit theorem to show that the distribution of the amplitudes of in-phase components is Gaussian, with a standard deviation σ =1/√
2 (and similarly for the quadrature components) such that mean power is unity. Since both in-phase and quadrature components are Gaussian, the absolute amplitude is Rayleigh distributed (see Chapter 5 for details of this derivation). Knowing the amplitude distribution, it is easy to compute the probability that the instantaneous amplitude will lie above a given threshold, and similarly for power. For example, there is a exp(−106/10)=0.019 probability that the peak power is 6 dB above the average power. Note that the Rayleigh distribution can only be an approximation for the amplitude distribution of OFDM signals: an actual OFDM signal has a bounded amplitude (N∗ amplitude of signal on one subcarrier), while realizations of a Rayleigh distribution can take on arbitrarily large values.
There are three main methods to deal with the Peak-to-Average Power Ratio (PAPR):
1. Put a power amplifier into the transmitter that can amplify linearly up to the possible peak value of the transmit signal. This is usually not practical, as it requires expensive and power-consuming class-A amplifiers. The larger the number of subcarriersN, the more difficult this solution becomes.
2. Use a nonlinear amplifier, and accept the fact that amplifier characteristics will lead to distortions in the output signal. Those nonlinear distortions destroy orthogonality between subcarriers, and also lead to increased out-of-band emissions (spectral regrowth – similar to third-order inter- modulation products – such that the power emitted outside the nominal band is increased). The first effect increases the BER of the desired signal (see Figure 19.12), while the latter effect causes interference to other users and thus decreases the cellular capacity of an OFDM system (see Figure 19.13). This means that in order to have constant adjacent channel interference we can trade off power amplifier performance against spectral efficiency (note that increased carrier separation decreases spectral efficiency).
3. Use PAR reduction techniques. These will be described in the next subsection.
10–1 10–2 10–3 10–4 10–5
BER
0 2 4 6 8 10 12 14 16 18 20
Average channel SNR (dB)
BO = 9 dB BO = 3 dB BO = 2 dB BO = 1 dB BO = 0 dB
Figure 19.12 Bit error rate as a function of the signal-to-noise ratio, for different backoff levels of the trans- mit amplifier.
Reproduced with permission from Hanzo et al. [2003]©J. Wiley & Sons, Ltd
–10
–20
–30
–40
1.0 1.2 1.4 1.6 1.8 2.0
Carrier separation (B)
Interference power (dB)
BO = –6 dB BO = –3 dB BO = 0 dB BO = 3 dB BO = 6 dB BO = 9 dB
Figure 19.13 Interference power to adjacent bands (OFDM users), as a function of carrier separation, for different values of backoff of the transmit amplifier.
Reproduced with permission from Hanzo et al. [2003]©J. Wiley & Sons, Ltd.
Orthogonal Frequency Division Multiplexing (OFDM) 431
19.6.2 Peak-to-Average Ratio Reduction Techniques
A wealth of methods for mitigating the PAR problem has been suggested in the literature. Some of the promising approaches are as follows:
1. Coding for PAR reduction: under normal circumstances, each OFDM symbol can represent one of 2N codewords (assuming BPSK modulation). Now, of these codewords only a subset of size 2K is acceptable in the sense that its PAR is lower than a given threshold. Both the transmitter and the receiver know the mapping between a bit combination of lengthK, and the codeword of lengthN that is chosen to represent it, and which has an admissible PAR. The transmission scheme is thus the following: (i) parse the incoming bitstream into blocks of lengthK; (ii) select the associated codeword of lengthN; (iii) transmit this codeword via the OFDM modulator.
The coding scheme can guarantee a certain value for the PAR. It also has some coding gain, though this gain is smaller than for codes that are solely dedicated to error correction.
2. Phase adjustments: this scheme first defines an ensemble of phase adjustment vectors φl, l= 1, . . . , L, that are known to both the transmitter and receiver; each vector hasN entries{φn}l. The transmitter then multiplies the OFDM symbol to be transmittedcn by each of these phase vectors to get
{ˆcn}l=cnexp[j (φn)l] (19.14) and then selects
lˆ=arg min
l (P AR({ˆcn}l)) (19.15) which gives the lowest PAR. The vector{ˆcn}lˆis then transmitted, together with the indexl. Theˆ receiver can then undo phase adjustment and demodulate the OFDM symbol. This method has the advantage that the overhead is rather small (at least as long asLstays within reasonable bounds); on the downside, it cannot guarantee to keep the PAR below a certain level.
3. Correction by multiplicative function: another approach is to multiply the OFDM signal by a time-dependent function whenever the peak value is very high. The simplest example for such an approach is the clipping we mentioned in the previous subsection: if the signal attains a level sk> A0, it is multiplied by a factorA0/sk. In other words, the transmit signal becomes
ˆ
s(t )=s(t )
1−
k
max
0,|sk| −A0
|sk|
(19.16) A less radical method is to multiply the signal by a Gaussian function centered at times when the level exceeds the threshold:
ˆ
s(t )=s(t )
1−
n
max
0,|sk| −A0
|sk|
exp
− t2 2σt2
(19.17) Multiplication by a Gaussian function of varianceσt2 in the time domain implies convolution with a Gaussian function in the frequency domain with variance σf2=1/(2π σt2). Thus, the amount of out-of-band interference can be influenced by the judicious choice ofσt2. On the downside, we find that the ICI (and thus BER) caused by this scheme is significant.
4. Correction by additive function: in a similar spirit, we can choose an additive, instead of a multiplicative, correction function. The correction function should be smooth enough not to introduce significant out-of-band interference. Furthermore, the correction function acts as addi- tional pseudo noise, and thus increases the BER of the system.
When comparing the different approaches to PAR reduction, we find that there is no single “best”
technique. The coding method can guarantee a maximum PAR value, but requires considerable overhead, and thus reduced throughput. The phase adjustment method has a smaller overhead (depending on the number of phase adjustment vectors), but cannot give a guaranteed performance.
Neither of these two methods leads to an increase in either ICI or out-of-band emissions. The correction by multiplicative functions can guarantee performance – up to a point (subtracting the Gaussian functions centered at one point might lead to larger amplitudes at another point). Also, it can lead to considerable ICI, while out-of-band emissions are fairly well controlled.