Module papr 

Peak-to-average power ratio (PAPR) of OFDM: its distribution, and the classical reduction techniques.

The PAPR of a block $x$ is $\max_{n} |x[n]|^{2} / \mathbb{E}|x[n]|^{2}$, measured on the oversampled signal (crate::ofdm::Ofdm with $L \ge 4$). Its complementary cumulative distribution (CCDF) $\Pr(\mathrm{PAPR} > \gamma)$ is the standard figure of merit: an OFDM signal is nearly Gaussian, and a peak is rare but large.

Distribution

• Nyquist-rate samples of $N$ independent Rayleigh amplitudes (van Nee & Prasad): ccdf_nyquist, $1 - (1 - e^{-\gamma})^{N}$;
• the continuous (oversampled) signal, for large $N$ (Ochiai & Imai 2001): ccdf_oversampled, $1 - \exp\left(-N \sqrt{\pi \gamma / 3}\, e^{-\gamma}\right)$;
• SLM with $U$ independent candidates: ccdf_slm (ccdf_slm_nyquist at the Nyquist rate), the CCDF raised to the power $U$.

Clipping

An envelope limiter acting on a complex Gaussian signal is, by Bussgang’s theorem, a gain plus an uncorrelated distortion, whose closed form (comnum_dsp::nonlinear::clipping_bussgang) gives the signal-to-distortion ratio, the error floor of clipping.

Reduction techniques

Technique	Principle	Distortion	Side information	Reference
Clip	envelope limiter	in band and out of band	none	Li & Cimini 1998
ClipFilter	repeated clipping and out-of-band filtering	in band	none	Armstrong 2002
Slm	best of $U$ phase-rotated copies	none	$\log_{2} U$ bits	Bäuml, Fischer & Huber 1996
Pts	best phase combination of $V$ subblocks	none	$(V - 1)\log_{2} W$ bits	Müller & Huber 1997
ToneReservation	peak-cancelling signal on reserved tones	none (rate loss)	none	Tellado 2000
Ace	outer constellation points pushed outwards	none (power increase)	none	Krongold & Jones 2003
Companding	μ-law or exponential compression of the envelope, expansion at the receiver	yes	none	Wang, Tjhung & Ng 1999; Jiang, Yang & Song 2005

DFT-spread OFDM (crate::ofdm::Ofdm::dft_spread, Myung, Lim & Goodman 2006) removes most of the PAPR at the source.

Examples

use comnum_waveform::ofdm::Ofdm;
use comnum_waveform::papr::{ccdf_oversampled, Slm};
use comnum_math::{papr_db, C64};
let ofdm = Ofdm::new(256, 256)?;
let qpsk: Vec<C64> = (0..256).map(|i| C64::new(if i % 3 == 0 { 1.0 } else { -1.0 }, if i % 5 < 2 { 1.0 } else { -1.0 }) / 2f64.sqrt()).collect();
let (x, side) = ofdm.modulate_with(&qpsk, &Slm { candidates: 16, seed: 1 })?;
assert!(papr_db(&x) <= papr_db(&ofdm.modulate(&qpsk)?));
assert_eq!(ofdm.demodulate_with(&x, &Slm { candidates: 16, seed: 1 }, &side)?.len(), 256);
assert!((ccdf_oversampled(256, 10.0) - 0.0369).abs() < 1e-3);

References

1. R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications, Artech House, 2000, ch. 6.
2. H. Ochiai and H. Imai, “On the distribution of the peak-to-average power ratio in OFDM signals,” IEEE Trans. Commun., 49(2), 282–289, 2001.
3. H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Trans. Commun., 50(1), 89–101, 2002.
4. D. Dardari, V. Tralli and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun., 48(10), 1755–1764, 2000.
5. J. J. Bussgang, “Crosscorrelation functions of amplitude-distorted Gaussian signals,” MIT RLE Tech. Rep. 216, 1952.
6. S. H. Han and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun., 12(2), 56–65, 2005.
7. X. Li and L. J. Cimini, “Effects of clipping and filtering on the performance of OFDM,” IEEE Commun. Lett., 2(5), 131–133, 1998.
8. J. Armstrong, “Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering,” Electron. Lett., 38(5), 246–247, 2002.
9. R. W. Bäuml, R. F. H. Fischer and J. B. Huber, “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,” Electron. Lett., 32(22), 2056–2057, 1996.
10. S. H. Müller and J. B. Huber, “OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences,” Electron. Lett., 33(5), 368–369, 1997.
11. J. Tellado, Multicarrier Modulation with Low PAR: Applications to DSL and Wireless, Kluwer, 2000.
12. B. S. Krongold and D. L. Jones, “PAR reduction in OFDM via active constellation extension,” IEEE Trans. Broadcast., 49(3), 258–268, 2003.
13. X. Wang, T. T. Tjhung and C. S. Ng, “Reduction of peak-to-average power ratio of OFDM system using a companding technique,” IEEE Trans. Broadcast., 45(3), 303–307, 1999.
14. T. Jiang, Y. Yang and Y.-H. Song, “Exponential companding technique for PAPR reduction in OFDM systems,” IEEE Trans. Broadcast., 51(2), 244–248, 2005.
15. H. G. Myung, J. Lim and D. J. Goodman, “Single carrier FDMA for uplink wireless transmission,” IEEE Veh. Technol. Mag., 1(3), 30–38, 2006.

Structs

Ace

Active constellation extension (Krongold & Jones 2003), clipping-based: the peaks are clipped, and the resulting change of every subcarrier is kept only where it moves an outer constellation point further out (which can only increase its distance to the decision boundaries); a few iterations converge. For square QAM.

Clip

Clipping of the envelope at ratio_db above the (unit) average power (Li & Cimini 1998): in-band distortion and out-of-band regrowth.

ClipFilter

Repeated clipping and frequency-domain filtering (Armstrong 2002): every iteration clips the envelope and removes the out-of-band components, which regrows some peaks; a few iterations reach the target PAPR without any out-of-band emission.

Pts

Partial transmit sequences (Müller & Huber 1997): the subcarriers are split into $V$ subblocks whose time signals are combined with phase factors from $\{e^{j 2\pi w / W}\}$, the first fixed to one; the combination of lowest PAPR is sent (exhaustive search over $W^{V-1}$ combinations) and its index is the side information.

Slm

Selected mapping (Bäuml, Fischer & Huber 1996): $U$ copies of the block rotated by known phase sequences (the first one is the original), the one of lowest PAPR is sent and its index is the side information.

ToneReservation

Tone reservation (Tellado 2000): a few subcarriers carry no data but a peak-cancelling signal. Every iteration projects the excess of the peaks above the target amplitude (the clipping noise) onto the reserved tones, by least squares, and subtracts it: a gradient descent on the peak excess that never touches the data tones.

Enums

Companding

Companding of the envelope at the transmitter and expansion at the receiver; the compressed signal keeps the unit average power.

Partition

How the subcarriers are split into the subblocks of Pts.

Functions

ccdf_nyquist

$\Pr(\mathrm{PAPR} > \gamma)$ of $n$ independent Rayleigh samples (Nyquist rate), $1 - (1 - e^{-\gamma})^{n}$ (van Nee & Prasad 2000; Ochiai & Imai 2001, eq. (11)).

ccdf_oversampled

$\Pr(\mathrm{PAPR} > \gamma)$ of the continuous OFDM signal of $n$ subcarriers, $1 - \exp(-n \sqrt{\pi\gamma/3}\, e^{-\gamma})$ (Ochiai & Imai 2001, eq. (27), the asymptotic form of the level-crossing analysis, within the resolution of the figures of $n \geq 100$).

ccdf_slm

The CCDF after SLM with candidates independent candidates: that of one candidate raised to the power $U$ (oversampled signal).

ccdf_slm_nyquist

The CCDF after SLM with candidates independent candidates at the Nyquist rate: ccdf_nyquist raised to the power $U$.