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Crest factor
Peak divided by the Root mean square (RMS) of the waveform

Crest factor is a key parameter of a waveform, such as alternating current or sound, representing the ratio of the peak amplitude to the RMS value. A crest factor of 1 indicates no peaks, seen in direct current or a square wave, while higher values indicate more extreme peaks, common in sound waves. The related peak-to-average power ratio (PAPR) is the peak amplitude squared divided by the RMS squared, essentially the crest factor squared. Both quantities are dimensionless and often expressed in decibels, especially in applications like loudspeaker testing. The minimum crest factor is 1, or 0 dB.

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Examples

This table provides values for some normalized waveforms. All peak magnitudes have been normalized to 1.

Wave typeWaveformRMS valueCrest factorPAPR (dB)
DC110.0 dB
Sine wave 1 2 ≈ 0.707 {\displaystyle {1 \over {\sqrt {2}}}\approx 0.707} 5 2 ≈ 1.414 {\displaystyle {\sqrt {2}}\approx 1.414} 3.01 dB
Full-wave rectified sine 1 2 ≈ 0.707 {\displaystyle {1 \over {\sqrt {2}}}\approx 0.707} 6 2 ≈ 1.414 {\displaystyle {\sqrt {2}}\approx 1.414} 3.01 dB
Half-wave rectified sine 1 2 = 0.5 {\displaystyle {1 \over 2}=0.5} 7 2 {\displaystyle 2\,} 6.02 dB
Triangle wave 1 3 ≈ 0.577 {\displaystyle {1 \over {\sqrt {3}}}\approx 0.577} 3 ≈ 1.732 {\displaystyle {\sqrt {3}}\approx 1.732} 4.77 dB
Square wave110 dB
PWM signal V(t) ≥ 0.0 V t 1 T {\displaystyle {\sqrt {\frac {t_{1}}{T}}}} 8 T t 1 {\displaystyle {\sqrt {\frac {T}{t_{1}}}}}

20 log ⁡ ( T t 1 ) {\displaystyle 20\log {\mathord {\left({\frac {T}{t_{1}}}\right)}}}  dB

QPSK111.761 dB9
8PSK3.3 dB10
π⁄4-DQPSK3.0 dB11
OQPSK3.3 dB12
8VSB6.5–8.1 dB13
64QAM 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} 7 3 ≈ 1.528 {\displaystyle {\sqrt {\frac {7}{3}}}\approx 1.528} 3.7 dB14
∞ {\displaystyle \infty } -QAM 1 3 ≈ 0.577 {\displaystyle {1 \over {\sqrt {3}}}\approx 0.577} 3 ≈ 1.732 {\displaystyle {\sqrt {3}}\approx 1.732} 4.8 dB15
WCDMA downlink carrier10.6 dB
OFDM4~12 dB
GMSK110 dB
Gaussian noise σ {\displaystyle \sigma } 1617 ∞ {\displaystyle \infty } 1819 ∞ {\displaystyle \infty } dB
Periodic chirp 1 2 ≈ 0.707 {\displaystyle {1 \over {\sqrt {2}}}\approx 0.707} 2 ≈ 1.414 {\displaystyle {\sqrt {2}}\approx 1.414} 3.01 dB

Notes:

  1. Crest factors specified for QPSK, QAM, WCDMA are typical factors needed for reliable communication, not the theoretical crest factors which can be larger.

Crest factor reduction

Many modulation techniques have been specifically designed to have constant envelope modulation, i.e., the minimum possible crest factor of 1:1.

In general, modulation techniques that have smaller crest factors usually transmit more bits per second than modulation techniques that have higher crest factors. This is because:

  1. any given linear amplifier has some "peak output power"—some maximum possible instantaneous peak amplitude it can support and still stay in the linear range;
  2. the average power of the signal is the peak output power divided by the crest factor;
  3. the number of bits per second transmitted (on average) is proportional to the average power transmitted (Shannon–Hartley theorem).

Orthogonal frequency-division multiplexing (OFDM) is a very promising modulation technique; perhaps its biggest problem is its high crest factor.2021 Many crest factor reduction techniques (CFR) have been proposed for OFDM.222324 The reduction in crest factor results in a system that can either transmit more bits per second with the same hardware, or transmit the same bits per second with lower-power hardware (and therefore lower electricity costs25 and less expensive hardware), or both. Over the years, numerous model-driven approaches have been proposed to reduce the PAPR in communication systems. In recent years, there has been a growing interest in exploring data-driven models for PAPR reduction as part of ongoing research in end-to-end communication networks. These data-driven models offer innovative solutions and new avenues of exploration to address the challenges posed by high PAPR effectively. By leveraging data-driven techniques, researchers aim to enhance the performance and efficiency of communication networks by optimizing power utilization. 26

Crest factor reduction methods

Various methods for crest factor reduction exist, such as peak windowing, noise shaping, pulse injection and peak cancellation.

Applications

See also

General

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

References

  1. "Wireless 101: Peak to average power ratio (PAPR)". http://www.eetimes.com/design/microwave-rf-design/4017754/Wireless-101-Peak-to-average-power-ratio-PAPR- ↩

  2. JBL Speaker Power Requirements, which is applying the IEC standard 268-5, itself more recently renamed to 60268-5 https://jblpro.com/en/site_elements/speaker-power-requirements-faq ↩

  3. AES2-2012 standard, Annex B (Informative) Crest Factor, pp. 17-20 in the 2013-02-11 printing /wiki/Audio_Engineering_Society ↩

  4. "Dr. Pro-Audio", Power handling, summarizes the various speaker standards http://www.doctorproaudio.com/doctor/temas/powerhandling.htm ↩

  5. "RMS and Average Values for Typical Waveforms". Archived from the original on 2010-01-23. https://web.archive.org/web/20100123085330/http://www.nessengr.com/techdata/rms/rms.html ↩

  6. "RMS and Average Values for Typical Waveforms". Archived from the original on 2010-01-23. https://web.archive.org/web/20100123085330/http://www.nessengr.com/techdata/rms/rms.html ↩

  7. "RMS and Average Values for Typical Waveforms". Archived from the original on 2010-01-23. https://web.archive.org/web/20100123085330/http://www.nessengr.com/techdata/rms/rms.html ↩

  8. "RMS and Average Values for Typical Waveforms". Archived from the original on 2010-01-23. https://web.archive.org/web/20100123085330/http://www.nessengr.com/techdata/rms/rms.html ↩

  9. Palicot, Jacques; Louët, Yves. POWER RATIO DEFINITIONS AND ANALYSIS IN SINGLE CARRIER MODULATIONS (PDF). IETR/Supélec - Campus de Rennes. p. 2. http://kilyos.ee.bilkent.edu.tr/~signal/defevent/papers/cr1037.pdf ↩

  10. "Read steer_rf_chapter1.pdf". Archived from the original on 2016-03-22. Retrieved 2014-12-11. https://web.archive.org/web/20160322185955/http://www.readbag.com/ece-ucsb-yuegroup-teaching-ece594bb-lectures-steer-rf-chapter1 ↩

  11. "Read steer_rf_chapter1.pdf". Archived from the original on 2016-03-22. Retrieved 2014-12-11. https://web.archive.org/web/20160322185955/http://www.readbag.com/ece-ucsb-yuegroup-teaching-ece594bb-lectures-steer-rf-chapter1 ↩

  12. "Read steer_rf_chapter1.pdf". Archived from the original on 2016-03-22. Retrieved 2014-12-11. https://web.archive.org/web/20160322185955/http://www.readbag.com/ece-ucsb-yuegroup-teaching-ece594bb-lectures-steer-rf-chapter1 ↩

  13. "Transitioning transmitters to COFDM". Archived from the original on 2009-08-21. Retrieved 2009-06-17. https://web.archive.org/web/20090821020320/http://broadcastengineering.com/mag/broadcasting_transitioning_transmitters_cofdm/ ↩

  14. R. Wolf; F. Ellinger; R.Eickhoff; Massimiliano Laddomada; Oliver Hoffmann (14 July 2011). Periklis Chatzimisios (ed.). Mobile Lightweight Wireless Systems: Second International ICST Conference, Mobilight 2010, May 10-12, 2010, Barcelona, Spain, Revised Selected Papers. Springer. p. 164. ISBN 978-3-642-16643-3. Retrieved 13 December 2012. 978-3-642-16643-3 ↩

  15. R. Wolf; F. Ellinger; R.Eickhoff; Massimiliano Laddomada; Oliver Hoffmann (14 July 2011). Periklis Chatzimisios (ed.). Mobile Lightweight Wireless Systems: Second International ICST Conference, Mobilight 2010, May 10-12, 2010, Barcelona, Spain, Revised Selected Papers. Springer. p. 164. ISBN 978-3-642-16643-3. Retrieved 13 December 2012. 978-3-642-16643-3 ↩

  16. Op Amp Noise Theory and Applications Archived 2014-11-30 at the Wayback Machine - 10.2.1 rms versus P-P Noise http://www.ti.com/lit/ml/sloa082/sloa082.pdf ↩

  17. Chapter 1 First-Order Low-Pass Filtered Noise - "The standard deviation of a Gaussian noise voltage is the root-mean-square or rms value of the voltage." http://users.ece.gatech.edu/mleach/ece6416/Labs/exp01.pdf ↩

  18. Noise: Frequently Asked Questions - "Noise theoretically has an unbounded distribution so that it should have an infinite crest factor" http://noisewave.com/faq.pdf ↩

  19. Telecommunications Measurements, Analysis, and Instrumentation, Kamilo Feher, section 7.2.3 Finite Crest Factor Noise ↩

  20. "Crest Factor Reduction of an OFDM/WiMAX Network". http://www.apc.org/en/spectrum/pubs/crest-factor-reduction-ofdmwimax-network ↩

  21. "Low Crest Factor Modulation Techniques for Orthogonal Frequency Division Multiplexing (OFDM)" Archived 2017-08-29 at the Wayback Machine. http://www.ubicc.org/files/pdf/UBICC_Eltholth_cf_115.pdf ↩

  22. R. Neil Braithwaite. "Crest Factor Reduction for OFDM Using Selective Subcarrier Degradation" Archived 2018-08-06 at the Wayback Machine. http://wwwen.zte.com.cn/endata/magazine/ztecommunications/2011Year/no4/articles/201202/t20120202_283037.html ↩

  23. K. T. Wong, B. Wang & J.-C. Chen, "OFDM PAPR Reduction by Switching Null Subcarriers & Data-Subcarriers," Electronics Letters, vol. 47, no. 1, pp. 62-63 January, 2011 Archived 2015-09-23 at the Wayback Machine. http://www.eie.polyu.edu.hk/~enktwong/ktw/WongKT_EL0111.pdf ↩

  24. S.C. Thompson, "Constant Envelope OFDM Phase Modulation," PhD Dissertation, UC San Diego, 2005. https://web.archive.org/web/20080706134849/http://zeidler.ucsd.edu/students/thesis_sthompson.pdf ↩

  25. Nick Wells. "DVB-T2 in relation to the DVB-x2 Family of Standards" Archived 2013-05-26 at the Wayback Machine quote: "techniques which can reduce the PAPR, ... could result in a significant saving in electricity costs." http://www.atsc.org/cms/pdf/pt2/Wells_ATSC_paper_on_T2.pdf ↩

  26. Huleihel, Yara; Ben-Dror, Eilam; Permuter, Haim H. (2020). Low PAPR Waveform Design for OFDM Systems Based on Convolutional Autoencoder. 2020 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS). pp. 1–6. ↩

  27. "What Is The "Crest Factor" And Why Is It Used?" (PDF). Archived from the original (PDF) on 2011-09-27. Retrieved 2006-03-07. https://web.archive.org/web/20110927001050/http://www.dliengineering.com/downloads/crest%20factor.pdf ↩

  28. Crest factor analysis for complex signal processing Archived 2006-04-27 at the Wayback Machine http://rfdesign.com/mag/radio_crest_factor_analysis/ ↩

  29. PAPR simulation for 64QAM http://www.rfconsult.uk/papr/ ↩

  30. Crest factor definition — AES Pro Audio Reference http://www.aes.org/par/c/#cps ↩

  31. "Level Practices in Digital Audio". Archived from the original on 2009-06-18. Retrieved 2009-10-11. https://web.archive.org/web/20090618130742/http://www.digido.com/level-practices-part-1.html ↩

  32. Gain Structure — Setting the System Levels Archived 2007-09-28 at the Wayback Machine, Mackie Mixer Tips http://www.mackie.com/pdf/CMRefGuide/Tips_Ch4.pdf#page=2 ↩

  33. Setting sound system level controls: The most expensive system set up wrong never performs as well as an inexpensive system set up correctly. http://digitalcontentproducer.com/mag/avinstall_setting_sound_system/index.html ↩

  34. Palatal snoring identified by acoustic crest factor analysis http://www.iop.org/EJ/abstract/0967-3334/20/2/306 ↩