View source for Crest factor

{{Short description|Peak divided by the Root mean square (RMS) of the waveform}}
'''Crest factor''' is a parameter of a [[waveform]], such as [[alternating current]] or sound, showing the ratio of peak values to the effective value. In other words, crest factor indicates how extreme the peaks are in a waveform. Crest factor 1 indicates no peaks, such as [[direct current]] or a [[square wave]]. Higher crest factors indicate peaks, for example sound waves tend to have high crest factors.

Crest factor is the [[peak amplitude]] of the waveform divided by the [[root mean square|RMS]] value of the waveform.

The '''peak-to-average power ratio''' ('''PAPR''') is the peak amplitude squared (giving the peak ''power'') divided by the [[root mean square|RMS]] value squared (giving the average ''power'').<ref>{{cite web |title=Wireless 101: Peak to average power ratio (PAPR) |url=http://www.eetimes.com/design/microwave-rf-design/4017754/Wireless-101-Peak-to-average-power-ratio-PAPR-}}</ref> It is the square of the crest factor.

When expressed in [[decibel]]s, crest factor and PAPR are equivalent, due to the way decibels are [[Decibel#Definition|calculated for power ratios vs amplitude ratios]].

Crest factor and PAPR are therefore [[dimensionless quantities]].  While the crest factor is defined as a positive [[real number]], in commercial products it is also commonly stated as the ratio of two whole numbers, e.g., 2:1. The PAPR is most used in signal processing applications. As it is a power ratio, it is normally expressed in [[Decibel|decibels (dB)]].<!-- the next ref applies only to speakers--> The crest factor of the test signal is a fairly important issue in [[loudspeaker]] testing standards; in this context it is usually expressed in dB.<ref>[https://jblpro.com/en/site_elements/speaker-power-requirements-faq   JBL Speaker Power Requirements], which is applying the [[International Electrotechnical Commission|IEC]] standard 268-5, itself more recently renamed to 60268-5</ref><ref>[[Audio Engineering Society|AES]]2-2012 standard, Annex B (Informative) Crest Factor, pp. 17-20 in the 2013-02-11 printing</ref><ref>"Dr. Pro-Audio", [http://www.doctorproaudio.com/doctor/temas/powerhandling.htm Power handling], summarizes the various speaker standards</ref>

The minimum possible crest factor is 1, 1:1 or 0&nbsp;dB.

== Examples ==
This table provides values for some [[Normalisable wave function|normalized]] [[waveform]]s. All peak magnitudes have been normalized to 1.

{| class="wikitable" style="text-align:center;"
! Wave type
! Waveform
! [[Root mean square|RMS]] value
! Crest factor
! PAPR (dB)
|-
| [[Direct current|DC]] ||  || 1 || 1 || 0.0&nbsp;dB
|-
| [[Sine wave]] || [[Image:Simple sine wave.svg|100px]] || <math>{1 \over \sqrt{2}} \approx 0.707</math><ref name=ness>{{cite web|title=RMS and Average Values for Typical Waveforms |url=http://www.nessengr.com/techdata/rms/rms.html |archive-url=https://web.archive.org/web/20100123085330/http://www.nessengr.com/techdata/rms/rms.html |archive-date=2010-01-23 |url-status=dead }}</ref> || <math>\sqrt{2} \approx 1.414</math> || 3.01&nbsp;dB
|-
| [[Rectifier#Full-wave rectification|Full-wave rectified]] sine || [[Image:Simple full-wave rectified sine.svg|100px]] || <math>{1 \over \sqrt{2}} \approx 0.707</math><ref name=ness/> || <math>\sqrt{2} \approx 1.414</math> || 3.01&nbsp;dB
|-
| [[Rectifier#Half-wave rectification|Half-wave rectified]] sine || [[Image:Simple half-wave rectified sine.svg|100px]] || <math>{1 \over 2 } = 0.5</math><ref name=ness/> || <math>2 \,</math> || 6.02&nbsp;dB
|-
| [[Triangle wave]] || [[Image:Triangle wave.svg|100px]] || <math>{1 \over \sqrt{3}} \approx 0.577</math> || <math>\sqrt{3} \approx 1.732</math> || 4.77&nbsp;dB
|-
| [[Square wave]] || [[Image:Square wave.svg|100px]] || 1 || 1 || 0&nbsp;dB
|-
| [[Pulse-width modulation|PWM]] signal <br/>V(t) ≥ 0.0&nbsp;V|| [[Image:Pulse wide wave.svg|100px]] || <math>\sqrt{ \frac{t_1}{T}}</math><ref name=ness/> || <math>\sqrt{\frac{T}{t_1}}</math> || 
<math>20\log\mathord\left(\frac{T}{t_1}\right)</math>&nbsp;dB
|-
| [[QPSK]] || || 1 || 1 || 1.761&nbsp;dB<ref>{{Cite book|url=http://kilyos.ee.bilkent.edu.tr/~signal/defevent/papers/cr1037.pdf|title=POWER RATIO DEFINITIONS AND ANALYSIS IN SINGLE CARRIER MODULATIONS|last1=Palicot|first1=Jacques|last2=Louët|first2=Yves|publisher=IETR/Supélec - Campus de Rennes|pages=2}}</ref>
|-
| [[8PSK]] || || || || 3.3&nbsp;dB<ref name=modulation>{{Cite web | url=http://www.readbag.com/ece-ucsb-yuegroup-teaching-ece594bb-lectures-steer-rf-chapter1 | title=Read steer_rf_chapter1.pdf | access-date=2014-12-11 | archive-date=2016-03-22 | archive-url=https://web.archive.org/web/20160322185955/http://www.readbag.com/ece-ucsb-yuegroup-teaching-ece594bb-lectures-steer-rf-chapter1 | url-status=dead }}</ref>

|-
| [[Phase-shift keying#π/4-QPSK|{{frac|π|4}}-DQPSK]] || || || || 3.0&nbsp;dB<ref name=modulation />
|-
| [[OQPSK]] || || || || 3.3&nbsp;dB<ref name=modulation />
|-
| [[8VSB]] || || || || 6.5–8.1&nbsp;dB<ref>{{cite web |url=http://broadcastengineering.com/mag/broadcasting_transitioning_transmitters_cofdm/ |title=Transitioning transmitters to COFDM |access-date=2009-06-17 |url-status=dead |archive-url=https://web.archive.org/web/20090821020320/http://broadcastengineering.com/mag/broadcasting_transitioning_transmitters_cofdm/ |archive-date=2009-08-21 }}</ref>
|-
| [[64QAM]] || || <math>\sqrt{ \frac{3}{7} }</math> || <math>\sqrt{ \frac{7}{3} } \approx 1.528</math> || 3.7&nbsp;dB<ref name="ChatzimisiosVerikoukis2011">{{cite book|url=https://books.google.com/books?id=tSwKZxtx82gC|title=Mobile Lightweight Wireless Systems: Second International ICST Conference, Mobilight 2010, May 10-12, 2010, Barcelona, Spain, Revised Selected Papers|author1=R. Wolf|author2=F. Ellinger|author3=R.Eickhoff|author4=Massimiliano Laddomada|author5=Oliver Hoffmann|date=14 July 2011|publisher=Springer|isbn=978-3-642-16643-3|editor=Periklis Chatzimisios|page=164|access-date=13 December 2012}}</ref>
|-
| <math>\infty</math>-QAM || || <math>{1 \over \sqrt{3}} \approx 0.577</math> || <math>\sqrt{3} \approx 1.732</math> || 4.8&nbsp;dB<ref name="ChatzimisiosVerikoukis2011"/>
|-
| [[WCDMA]] downlink carrier || || || || 10.6&nbsp;dB
|-
| [[OFDM]] || || || 4 || ~12&nbsp;dB
|-
| [[GMSK]] || || 1 || 1 || 0&nbsp;dB
|-
| [[Gaussian noise]] || || [[standard deviation|<math>\sigma</math>]]<ref>[http://www.ti.com/lit/ml/sloa082/sloa082.pdf Op Amp Noise Theory and Applications] {{webarchive|url=https://web.archive.org/web/20141130224524/http://www.ti.com/lit/ml/sloa082/sloa082.pdf |date=2014-11-30 }} - 10.2.1 rms versus P-P Noise</ref><ref>[http://users.ece.gatech.edu/mleach/ece6416/Labs/exp01.pdf 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."</ref> || <math>\infty</math><ref>[http://noisewave.com/faq.pdf Noise: Frequently Asked Questions] - "Noise theoretically has an unbounded distribution so that it should have an infinite crest factor"</ref><ref>Telecommunications Measurements, Analysis, and Instrumentation, Kamilo Feher, section 7.2.3 Finite Crest Factor Noise</ref>  || <math>\infty</math> dB
|-
| [[Chirp|Periodic chirp]] || || <math>{1 \over \sqrt{2}} \approx 0.707</math> || <math>\sqrt{2} \approx 1.414</math> || 3.01&nbsp;dB
|}

Notes:
# 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:
# 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;
# the average power of the signal is the peak output power divided by the crest factor;
# 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.<ref>[http://www.apc.org/en/spectrum/pubs/crest-factor-reduction-ofdmwimax-network "Crest Factor Reduction of an OFDM/WiMAX Network"].
</ref><ref>[http://www.ubicc.org/files/pdf/UBICC_Eltholth_cf_115.pdf "Low Crest Factor Modulation Techniques for Orthogonal Frequency Division Multiplexing (OFDM)"] {{Webarchive|url=https://web.archive.org/web/20170829193818/http://www.ubicc.org/files/pdf/UBICC_Eltholth_cf_115.pdf |date=2017-08-29 }}.</ref> Many crest factor reduction techniques (CFR) have been proposed for OFDM.<ref>R. Neil Braithwaite.
[http://wwwen.zte.com.cn/endata/magazine/ztecommunications/2011Year/no4/articles/201202/t20120202_283037.html "Crest Factor Reduction for OFDM Using Selective Subcarrier Degradation"] {{Webarchive|url=https://web.archive.org/web/20180806221454/http://wwwen.zte.com.cn/endata/magazine/ztecommunications/2011Year/no4/articles/201202/t20120202_283037.html |date=2018-08-06 }}.</ref><ref>K. T. Wong, B. Wang & J.-C. Chen,
[http://www.eie.polyu.edu.hk/~enktwong/ktw/WongKT_EL0111.pdf "OFDM PAPR Reduction by Switching Null Subcarriers & Data-Subcarriers," Electronics Letters, vol. 47, no. 1, pp. 62-63 January, 2011] {{Webarchive|url=https://web.archive.org/web/20150923234513/http://www.eie.polyu.edu.hk/~enktwong/ktw/WongKT_EL0111.pdf |date=2015-09-23 }}.</ref><ref>S.C. Thompson,
[https://web.archive.org/web/20080706134849/http://zeidler.ucsd.edu/students/thesis_sthompson.pdf "Constant Envelope OFDM Phase Modulation," PhD Dissertation, UC San Diego, 2005].
</ref> 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 [[low-power electronics|lower-power]] hardware (and therefore lower electricity costs<ref>Nick Wells.
[http://www.atsc.org/cms/pdf/pt2/Wells_ATSC_paper_on_T2.pdf "DVB-T2 in relation to the DVB-x2 Family of Standards"] {{webarchive|url=https://web.archive.org/web/20130526003113/http://www.atsc.org/cms/pdf/pt2/Wells_ATSC_paper_on_T2.pdf |date=2013-05-26 }}
quote: "techniques which can reduce the PAPR, ... could result in a significant saving in electricity costs."
</ref> 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.
<ref>{{cite conference |last1=Huleihel |first1=Yara |last2=Ben-Dror|first2=Eilam |last3=Permuter |first3=Haim H. |title=Low PAPR Waveform Design for OFDM Systems Based on Convolutional Autoencoder |conference=2020 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS) |date=2020 |pages=1–6.}}</ref>
{{buzzword|date=August 2023}}

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

== Applications ==
* [[Electrical engineering]] — for describing the quality of an AC power waveform
* [[Oscillation|Vibration]] analysis — for estimating the amount of impact [[wear]] in a [[Bearing (mechanical)|bearing]]<ref>{{Cite web |url=http://www.dliengineering.com/downloads/crest%20factor.pdf |title=What Is The "Crest Factor" And Why Is It Used? |access-date=2006-03-07 |archive-date=2011-09-27 |archive-url=https://web.archive.org/web/20110927001050/http://www.dliengineering.com/downloads/crest%20factor.pdf |url-status=dead }}</ref>
* [[Radio]] and [[audio frequency|audio]] electronics — for estimating the [[headroom (audio signal processing)|headroom]] required in a signal chain<ref>[http://rfdesign.com/mag/radio_crest_factor_analysis/ Crest factor analysis for complex signal processing] {{webarchive|url=https://web.archive.org/web/20060427062603/http://rfdesign.com/mag/radio_crest_factor_analysis/ |date=2006-04-27 }}</ref><ref>[http://www.rfconsult.uk/papr/ PAPR simulation for 64QAM]</ref>
** [[Music]] has a widely varying crest factor.  Typical values for a processed mix are around 4–8 (which corresponds to {{nowrap|12–18 dB}} of headroom, usually involving [[audio level compression]]), and 8–10 for an unprocessed recording {{nowrap|(18–20 dB).}}<ref>[http://www.aes.org/par/c/#cps Crest factor definition]  — [[Audio Engineering Society|AES]] Pro Audio Reference</ref><ref>{{Cite web |url=http://www.digido.com/level-practices-part-1.html |title=Level Practices in Digital Audio |access-date=2009-10-11 |archive-url=https://web.archive.org/web/20090618130742/http://www.digido.com/level-practices-part-1.html |archive-date=2009-06-18 |url-status=dead }}</ref><ref>[http://www.mackie.com/pdf/CMRefGuide/Tips_Ch4.pdf#page=2 Gain Structure — Setting the System Levels] {{Webarchive|url=https://web.archive.org/web/20070928044203/http://www.mackie.com/pdf/CMRefGuide/Tips_Ch4.pdf#page=2 |date=2007-09-28 }}, [[Mackie (company)|Mackie]] Mixer Tips</ref><ref>[http://digitalcontentproducer.com/mag/avinstall_setting_sound_system/index.html Setting sound system level controls: The most expensive system set up wrong never performs as well as an inexpensive system set up correctly.]</ref>
* [[Physiology]] — for analysing the sound of [[snoring]]<ref>[http://www.iop.org/EJ/abstract/0967-3334/20/2/306 Palatal snoring identified by acoustic crest factor analysis]</ref>

== See also ==
* [[Clipping (signal processing)]]
* [[Form factor (electronics)]]

== References ==
{{Reflist}}

===General===
{{FS1037C MS188}}

== External links ==
* Definition of [https://web.archive.org/web/20080628071722/http://www.atis.org/glossary/definition.aspx?id=2773 peak-to-average ratio] – ATIS (Alliance for Telecommunications Industry Solutions) Telecom Glossary 2K
* Definition of [https://web.archive.org/web/20120318202105/http://www.atis.org/glossary/definition.aspx?id=6766 crest factor] – ATIS (Alliance for Telecommunications Industry Solutions) Telecom Glossary 2K
* [http://www.etti.unibw.de/labalive/tutorial/papr/ Peak-to-average power ratio (PAPR) of OFDM systems - tutorial]

{{Waveforms}}

[[Category:Waveforms]]