峰值因數

峰值因數又稱波峰因數（crest factor，簡寫 CF，又稱peak-to-average ratio，簡稱PAR）是和波形有關的無因次量，為波形的振幅再除以波形RMS （time-averaged）所得到的值。

${\displaystyle C={|x|_{\mathrm {peak} } \over x_{\mathrm {rms} }}}$

峰均功率比（peak-to-average power ratio，簡稱PAPR）是另一個相關的無因次量，定義為振幅平方（表示峰值功率）除以RMS平方（表示平均功率）的比值^[1]：

${\displaystyle {\mathit {PAPR}}={{|x|_{\mathrm {peak} }}^{2} \over {x_{\mathrm {rms} }}^{2}}=C^{2}}$

因一波形的振幅恆大於等於RMS值，因此峰值因數及峰均功率比的最小可能數值均為1，即峰均功率比 1:1 或 0dB。

常見波形的峰值因數

下表列出了常見波形的峰值因數。所有範例中的峰值都統一為 1。

波形名稱	波形圖	RMS 值	峰值因數	峰均功率比 (dB)
直流		1	1	0.0 dB
正弦波		${\displaystyle {1 \over {\sqrt {2}}}\approx 0.707}$[2]	${\displaystyle {\sqrt {2}}\approx 1.414}$	3.01 dB
正弦波經全波整流後		${\displaystyle {1 \over {\sqrt {2}}}\approx 0.707}$[2]	${\displaystyle {\sqrt {2}}\approx 1.414}$	3.01 dB
正弦波經半波整流後		${\displaystyle {1 \over 2}=0.5}$[2]	${\displaystyle 2\,}$	6.02 dB
三角波		${\displaystyle {1 \over {\sqrt {3}}}\approx 0.577}$	${\displaystyle {\sqrt {3}}\approx 1.732}$	4.77 dB
方波		1	1	0 dB
PWM 訊號 V(t) ≥ 0.0 V		${\displaystyle {\sqrt {\frac {t_{1}}{T}}}}$[2]	${\displaystyle {\sqrt {\frac {T}{t_{1}}}}}$	${\displaystyle 20\log {\mathord {\left({\frac {T}{t_{1}}}\right)}}}$ dB
QPSK		1	1	1.761 dB[3]
8PSK				3.3 dB[4]
π⁄4-DQPSK				3.0 dB[4]
OQPSK				3.3 dB[4]
8VSB				6.5–8.1 dB[5]
64QAM		${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle {\sqrt {\frac {7}{3}}}\approx 1.528}$	3.7 dB[6]
${\displaystyle \infty }$-QAM		${\displaystyle {1 \over {\sqrt {3}}}\approx 0.577}$	${\displaystyle {\sqrt {3}}\approx 1.732}$	4.8 dB[6]
WCDMA 下行載波				10.6 dB
OFDM			4	~12 dB
GMSK		1	1	0 dB
高斯噪声		${\displaystyle \sigma }$[7][8]	${\displaystyle \infty }$[9][10]	${\displaystyle \infty }$ dB
週期性啁啾(Chirp)		${\displaystyle {1 \over {\sqrt {2}}}\approx 0.707}$	${\displaystyle {\sqrt {2}}\approx 1.414}$	3.01 dB

附註:

1. 表中 QPSK, QAM, WCDMA 的峰值因數是進行可靠通訊時的典型數值，並非理論上的峰值因數數值，後者可能較高。

相關條目

• 波形因數 (Form Factor)

參考資料

1. Wireless 101: Peak to average power ratio (PAPR). [2013-03-17]. （原始内容存档于2013-02-21）. 
2. RMS and Average Values for Typical Waveforms. （原始内容存档于2010-01-23）. 
3. Palicot, Jacques; Louët, Yves. POWER RATIO DEFINITIONS AND ANALYSIS IN SINGLE CARRIER MODULATIONS (PDF). IETR/Supélec - Campus de Rennes. : 2 [2024-12-15]. （原始内容存档 (PDF)于2024-12-08）. 
4. Read steer_rf_chapter1.pdf. [2014-12-11]. （原始内容存档于2016-03-22）. 
5. Transitioning transmitters to COFDM. [2009-06-17]. （原始内容存档于2009-08-21）. 
6. R. Wolf; F. Ellinger; R.Eickhoff; Massimiliano Laddomada; Oliver Hoffmann. Periklis Chatzimisios , 编. Mobile Lightweight Wireless Systems: Second International ICST Conference, Mobilight 2010, May 10-12, 2010, Barcelona, Spain, Revised Selected Papers. Springer. 14 July 2011: 164 [13 December 2012]. ISBN 978-3-642-16643-3. 
7. Op Amp Noise Theory and Applications 互联网档案馆的存檔，存档日期2014-11-30. - 10.2.1 rms versus P-P Noise
8. Chapter 1 First-Order Low-Pass Filtered Noise （页面存档备份，存于互联网档案馆） - "The standard deviation of a Gaussian noise voltage is the root-mean-square or rms value of the voltage."
9. Noise: Frequently Asked Questions （页面存档备份，存于互联网档案馆） - "Noise theoretically has an unbounded distribution so that it should have an infinite crest factor"
10. Telecommunications Measurements, Analysis, and Instrumentation, Kamilo Feher, section 7.2.3 Finite Crest Factor Noise