Crest factor

Crest factoris a parameter of awaveform,such asalternating currentor 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 asdirect currentor asquare wave.Higher crest factors indicate peaks, for example sound waves tend to have high crest factors.

Crest factor is thepeak amplitudeof the waveform divided by theRMSvalue of the waveform.

Thepeak-to-average power ratio(PAPR) is the peak amplitude squared (giving the peakpower) divided by theRMSvalue squared (giving the averagepower).^[1]It is the square of the crest factor.

When expressed indecibels,crest factor and PAPR are equivalent, due to the way decibels arecalculated for power ratios vs amplitude ratios.

Crest factor and PAPR are thereforedimensionless quantities.While the crest factor is defined as a positivereal 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 indecibels (dB).The crest factor of the test signal is a fairly important issue inloudspeakertesting standards; in this context it is usually expressed in dB.^[2]^[3]^[4]

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

Examples

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

Wave type	RMSvalue	Crest factor	PAPR (dB)
DC	1	1	0.0 dB
Sine wave	${1 \over {\sqrt {2}}}\approx 0.707$ ^[5]	${\sqrt {2}}\approx 1.414$	3.01 dB
Full-wave rectifiedsine	${1 \over {\sqrt {2}}}\approx 0.707$ ^[5]	${\sqrt {2}}\approx 1.414$	3.01 dB
Half-wave rectifiedsine	${1 \over 2}=0.5$ ^[5]	$2\,$	6.02 dB
Triangle wave	${1 \over {\sqrt {3}}}\approx 0.577$	${\sqrt {3}}\approx 1.732$	4.77 dB
Square wave	1	1	0 dB
PWMsignal V(t) ≥ 0.0 V	${\sqrt {\frac {t_{1}}{T}}}$ ^[5]	${\sqrt {\frac {T}{t_{1}}}}$	$20\log {\mathord {\left({\frac {T}{t_{1}}}\right)}}$ dB
QPSK	1	1	1.761 dB^[6]
8PSK			3.3 dB^[7]
π⁄4-DQPSK			3.0 dB^[7]
OQPSK			3.3 dB^[7]
8VSB			6.5–8.1 dB^[8]
64QAM	${\sqrt {\frac {3}{7}}}$	${\sqrt {\frac {7}{3}}}\approx 1.528$	3.7 dB^[9]
$\infty$ -QAM	${1 \over {\sqrt {3}}}\approx 0.577$	${\sqrt {3}}\approx 1.732$	4.8 dB^[9]
WCDMAdownlink carrier			10.6 dB
OFDM		4	~12 dB
GMSK	1	1	0 dB
Gaussian noise	$\sigma$ ^[10]^[11]	$\infty$ ^[12]^[13]	$\infty$ dB
Periodic chirp	${1 \over {\sqrt {2}}}\approx 0.707$	${\sqrt {2}}\approx 1.414$	3.01 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 haveconstant envelopemodulation, 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 givenlinear amplifierhas 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 multiple xing(OFDM) is a very promising modulation technique; perhaps its biggest problem is its high crest factor.^[14]^[15]Many crest factor reduction techniques (CFR) have been proposed for OFDM.^[16]^[17]^[18]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 withlower-powerhardware (and therefore lower electricity costs^[19]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. ^[20]

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
Vibrationanalysis — for estimating the amount of impactwearin abearing^[21]
Radioandaudioelectronics — for estimating theheadroomrequired in a signal chain^[22]^[23]
- Musichas a widely varying crest factor. Typical values for a processed mix are around 4–8 (which corresponds to12–18 dBof headroom, usually involvingaudio level compression), and 8–10 for an unprocessed recording(18–20 dB).^[24]^[25]^[26]^[27]
Physiology— for analysing the sound ofsnoring^[28]

References

^"Wireless 101: Peak to average power ratio (PAPR)".
^JBL Speaker Power Requirements,which is applying theIECstandard 268-5, itself more recently renamed to 60268-5
^AES2-2012 standard, Annex B (Informative) Crest Factor, pp. 17-20 in the 2013-02-11 printing
^"Dr. Pro-Audio",Power handling,summarizes the various speaker standards
^^a ^b ^c ^d"RMS and Average Values for Typical Waveforms".Archived fromthe originalon 2010-01-23.
^Palicot, Jacques; Louët, Yves.POWER RATIO DEFINITIONS AND ANALYSIS IN SINGLE CARRIER MODULATIONS(PDF).IETR/Supélec - Campus de Rennes. p. 2.
^^a ^b ^c"Read steer_rf_chapter1.pdf".Archived fromthe originalon 2016-03-22.Retrieved2014-12-11.
^"Transitioning transmitters to COFDM".Archived fromthe originalon 2009-08-21.Retrieved2009-06-17.
^^a ^bR. 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.Retrieved13 December2012.
^Op Amp Noise Theory and Applications Archived2014-11-30 at theWayback Machine- 10.2.1 rms versus P-P Noise
^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."
^Noise: Frequently Asked Questions- "Noise theoretically has an unbounded distribution so that it should have an infinite crest factor"
^Telecommunications Measurements, Analysis, and Instrumentation, Kamilo Feher, section 7.2.3 Finite Crest Factor Noise
^"Crest Factor Reduction of an OFDM/WiMAX Network".
^"Low Crest Factor Modulation Techniques for Orthogonal Frequency Division Multiple xing (OFDM)"Archived2017-08-29 at theWayback Machine.
^R. Neil Braithwaite. "Crest Factor Reduction for OFDM Using Selective Subcarrier Degradation"Archived2018-08-06 at theWayback Machine.
^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 Archived2015-09-23 at theWayback Machine.
^S.C. Thompson, "Constant Envelope OFDM Phase Modulation," PhD Dissertation, UC San Diego, 2005.
^Nick Wells. "DVB-T2 in relation to the DVB-x2 Family of Standards"Archived2013-05-26 at theWayback Machine quote: "techniques which can reduce the PAPR,... could result in a significant saving in electricity costs."
^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.
^"What Is The" Crest Factor "And Why Is It Used?"(PDF).Archived fromthe original(PDF)on 2011-09-27.Retrieved2006-03-07.
^Crest factor analysis for complex signal processing Archived2006-04-27 at theWayback Machine
^PAPR simulation for 64QAM
^Crest factor definition—AESPro Audio Reference
^"Level Practices in Digital Audio".Archived fromthe originalon 2009-06-18.Retrieved2009-10-11.
^Gain Structure — Setting the System Levels Archived2007-09-28 at theWayback Machine,MackieMixer Tips
^Setting sound system level controls: The most expensive system set up wrong never performs as well as an inexpensive system set up correctly.
^Palatal snoring identified by acoustic crest factor analysis

General

This article incorporatespublic domain materialfromFederal Standard 1037C.General Services Administration.Archived fromthe originalon 2022-01-22.(in support ofMIL-STD-188).

External links

Definition ofpeak-to-average ratio– ATIS (Alliance for Telecommunications Industry Solutions) Telecom Glossary 2K
Definition ofcrest factor– ATIS (Alliance for Telecommunications Industry Solutions) Telecom Glossary 2K
Peak-to-average power ratio (PAPR) of OFDM systems - tutorial

[1] "Wireless 101: Peak to average power ratio (PAPR)".

[2] JBL Speaker Power Requirements,which is applying theIECstandard 268-5, itself more recently renamed to 60268-5

[3] AES2-2012 standard, Annex B (Informative) Crest Factor, pp. 17-20 in the 2013-02-11 printing

[4] "Dr. Pro-Audio",Power handling,summarizes the various speaker standards

[ness-5] "RMS and Average Values for Typical Waveforms".Archived fromthe originalon 2010-01-23.

[6] Palicot, Jacques; Louët, Yves.POWER RATIO DEFINITIONS AND ANALYSIS IN SINGLE CARRIER MODULATIONS(PDF).IETR/Supélec - Campus de Rennes. p. 2.

[modulation-7] "Read steer_rf_chapter1.pdf".Archived fromthe originalon 2016-03-22.Retrieved2014-12-11.

[8] "Transitioning transmitters to COFDM".Archived fromthe originalon 2009-08-21.Retrieved2009-06-17.

[ChatzimisiosVerikoukis2011-9] 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.Retrieved13 December2012.

[10] Op Amp Noise Theory and Applications Archived2014-11-30 at theWayback Machine- 10.2.1 rms versus P-P Noise

[11] 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."

[12] Noise: Frequently Asked Questions- "Noise theoretically has an unbounded distribution so that it should have an infinite crest factor"

[13] Telecommunications Measurements, Analysis, and Instrumentation, Kamilo Feher, section 7.2.3 Finite Crest Factor Noise

[14] "Crest Factor Reduction of an OFDM/WiMAX Network".

[15] "Low Crest Factor Modulation Techniques for Orthogonal Frequency Division Multiple xing (OFDM)"Archived2017-08-29 at theWayback Machine.

[16] R. Neil Braithwaite. "Crest Factor Reduction for OFDM Using Selective Subcarrier Degradation"Archived2018-08-06 at theWayback Machine.

[17] 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 Archived2015-09-23 at theWayback Machine.

[18] S.C. Thompson, "Constant Envelope OFDM Phase Modulation," PhD Dissertation, UC San Diego, 2005.

[19] Nick Wells. "DVB-T2 in relation to the DVB-x2 Family of Standards"Archived2013-05-26 at theWayback Machine quote: "techniques which can reduce the PAPR,... could result in a significant saving in electricity costs."

[20] 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.

[21] "What Is The" Crest Factor "And Why Is It Used?"(PDF).Archived fromthe original(PDF)on 2011-09-27.Retrieved2006-03-07.

[22] Crest factor analysis for complex signal processing Archived2006-04-27 at theWayback Machine

[23] PAPR simulation for 64QAM

[24] Crest factor definition—AESPro Audio Reference

[25] "Level Practices in Digital Audio".Archived fromthe originalon 2009-06-18.Retrieved2009-10-11.

[26] Gain Structure — Setting the System Levels Archived2007-09-28 at theWayback Machine,MackieMixer Tips

[27] Setting sound system level controls: The most expensive system set up wrong never performs as well as an inexpensive system set up correctly.

[28] Palatal snoring identified by acoustic crest factor analysis

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