# Crest factor

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 RMS value of the waveform. This is equivalent to the ratio of the L norm to the L2 norm of the function of the waveform:

$C={\frac {|x_{\mathrm {peak} }|}{x_{\mathrm {rms} }}}={\frac {\|x\|_{\infty }}{\|x\|_{2}}}$ ${C}_{\mathrm {dB} }=20\log _{10}{|x_{\mathrm {peak} }| \over x_{\mathrm {rms} }}.$ The peak-to-average power ratio (PAPR) is the peak amplitude squared (giving the peak power) divided by the RMS value squared (giving the average power). It is the square of the crest factor:

${\mathit {PAPR}}={\frac {{|x_{\mathrm {peak} }|}^{2}}{{x_{\mathrm {rms} }}^{2}}}=C^{2}$ ${\mathit {PAPR}}_{\mathrm {dB} }=10\log _{10}{\frac {{|x_{\mathrm {peak} }|}^{2}}{{x_{\mathrm {rms} }}^{2}}}=C_{\mathrm {dB} }.$ When expressed in decibels, crest factor and PAPR are equivalent, due to the way decibels are 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 decibels (dB). 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.

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

## Examples

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

Wave type Waveform RMS value Crest factor PAPR (dB)
DC 1 1 0.0 dB
Sine wave   ${1 \over {\sqrt {2}}}\approx 0.707$  ${\sqrt {2}}\approx 1.414$  3.01 dB
N superimposed sine waves
(same amplitudes, different frequencies)[dubious ]
${\sqrt {\frac {N}{2}}}$  ${\sqrt {2N}}$

$10\log 2N$  dB

Full-wave rectified sine   ${1 \over {\sqrt {2}}}\approx 0.707$  ${\sqrt {2}}\approx 1.414$  3.01 dB
Half-wave rectified sine   ${1 \over 2}=0.5$  $2\,$  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
PWM Signal
V(t) $\geq$  0.0 V
${\sqrt {\frac {t_{1}}{T}}}$  ${\sqrt {\frac {T}{t_{1}}}}$

$10\log {\frac {T}{t_{1}}}$  dB

QPSK 1 1 1.761 dB
8PSK 3.3 dB
π/4DQPSK 3.0 dB
OQPSK 3.3 dB
8VSB 6.5–8.1 dB
64QAM ${\sqrt {\frac {3}{7}}}$  ${\sqrt {\frac {7}{3}}}\approx 1.542$  3.7 dB
$\infty$ -QAM ${1 \over {\sqrt {3}}}\approx 0.577$  ${\sqrt {3}}\approx 1.732$  4.8 dB
WCDMA downlink carrier 10.6 dB
OFDM 4 ~12 dB
GMSK 1 1 0 dB
Gaussian noise $\sigma$  $\infty$  $\infty$  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.

## Digital multimeters

Crest factor is an important parameter to understand when trying to take accurate measurements of low frequency signals. For example, given a certain digital multimeter with an AC accuracy of 0.03% (always specified for sine waves) with an additional error of 0.2% for crest factors between 1.414 and 5, then the total error for measuring a triangular wave (crest factor = 1.73) is 0.03% + 0.2% = 0.23%.

## Acoustics and audio engineering

In acoustics and audio engineering, crest factor is usually expressed in decibels, so it's defined as the level difference between the RMS and the peak value of the waveform. For example, for a sine wave the 1.414 ratio is 20 log(1.414) or 3 dB. Most ambient noise has a crest factor of around 10 dB while impulsive sounds such as gunshots can have crest factors of over 30 dB.[citation needed]

## Peak-to-average ratio (PAR) meter

A peak-to-average ratio meter (Par meter) is a device used to measure the ratio of the peak power level to the time-averaged power level in an electrical circuit. This quantity is known as the peak-to-average ratio (p/a r or PAR). Such meters are used as a quick means to identify degraded telephone channels.

Par meters are very sensitive to envelope delay distortion. They may also be used for idle channel noise, nonlinear distortion, and amplitude-distortion measurements.

The peak-to-average ratio can be determined for many signal parameters, such as voltage, current, power, frequency, and phase.

## 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. Many crest factor reduction techniques (CFR) have been proposed for OFDM. 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 costs and less expensive hardware), or both.

### Crest factor reduction methods

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