Decibel (dB)

Chen Weixiang @ Dec 14, 2018 | TAG: Telecom

This article introduce the Decibel (dB) used in communication.

Overview

The decibel (symbol: dB) is a unit of measurement used to express the ratio of one value of a physical property to another on a logarithmic scale. It can be used to express a change in value (e.g., +1 dB or −1 dB) or an absolute value. In the latter case, it expresses the ratio of a value to a reference value; when used in this way, the decibel symbol should be appended with a suffix that indicates the reference value, or some other property. For example, if the reference value is 1 volt, then the suffix is V (e.g., 20 dBV), and if the reference value is one milliwatt, then the suffix is m (e.g., 20 dBm).

As description in The Decibel Scale, the decibel is a unit representing one tenth (deci-) of a bel. Therefore, a figure reported in decibels is ten times the value reported in bels. The expression for a proportionality ratio expressed in decibel units (symbol dB) is:

\[dB ≡ 10 ∙ log_{10} (\frac {power_1} {power_0})\]

dBm and dBW

When \(P_{ref} = 1 mW\), definition of decibel-milliwatts (dBm) is:

\[dBm = 10 ∙ log_{10} (\frac {P} {P_{ref}}) = 10 ∙ log_{10} (\frac {P} {1 mW})\]

When \(P_{ref} = 1 W\), definition of decibel-Watt (dBW) is:

\[dBW = 10 ∙ log_{10} (\frac {P} {P_{ref}}) = 10 ∙ log_{10} (\frac {P} {1 W})\]

The relationship between decibel-milliwatts (dBm) and decibel-Watt (dBW) is:

\[P_{dBm} = 10 ∙ log_{10} (\frac {P} {1mW}) = 10 ∙ log_{10} (\frac {P ∙ 1000} {1W}) = 10 ∙ log_{10} (\frac {P} {1W}) + 30 = P_{dBW} + 30\]

That’s,

\[P_{dBm} = P_{dBW} + 30\] \[P_{dBW} = P_{dBm} - 30\]

dBm <-> Watt

Here is conversion table of dBW to dBm and Watt:

Power (dBW) Power (dBm) Power (Watt)
-130 dBW -100 dBm 0.1 pW
-120 dBW -90 dBm 1 pW
-110 dBW -80 dBm 10 pW
-100 dBW -70 dBm 100 pW
-90 dBW -60 dBm 1 nW
-80 dBW -50 dBm 10 nW
-70 dBW -40 dBm 100 nW
-60 dBW -30 dBm 1 μW
-50 dBW -20 dBm 10 μW
-40 dBW -10 dBm 100 μW
-30 dBW 0 dBm 1 mW
-20 dBW 10 dBm 10 mW
-10 dBW 20 dBm 100 mW
-1 dBW 29 dBm 0.79 W
0 dBW 30 dBm 1 W
1 dBW 31 dBm 1.26 W
10 dBW 40 dBm 10 W
20 dBW 50 dBm 100 W
30 dBW 60 dBm 1 kW
40 dBW 70 dBm 10 kW
50 dBW 80 dBm 100 kW
60 dBW 90 dBm 1 MW
70 dBW 100 dBm 10 MW
80 dBW 110 dBm 100 MW
90 dBW 120 dBm 1 GW
100 dBW 130 dBm 10 GW

dBm 转换为 Watt 的口算规律是要先记住1个基准和2个原则：

1个基准

  • 30dBm = 1W

2个原则

  • +3dBm，功率乘2倍; -3dBm，功率除2
33dBm = 30dBm + 3dBm = 1W * 2 = 2W
27dBm = 30dBm - 3dBm = 1W / 2 = 0.5W
  • +10dBm，功率乘10倍; －10dBm，功率乘1/10
40dBm = 30dBm + 10dBm = 1W * 10 = 10W
20dBm = 30dBm - 10dBm = 1W / 10 = 0.1W

几乎所有整数的 dBm 都可用以上的 1个基准 和 2个原则 转换为 Watt，例如：

44dBm = 30dBm + 10dBm + 10dBm - 3dBm  - 3dBm
      = 1W    * 10    * 10    / 2     / 2
      = 25 (W)

32dBm = 30dBm + 3dBm + 3dBm + 3dBm + 3dBm - 10dBm
      = 1W    * 2    * 2    * 2    * 2    / 10
      = 1.6 (W)

+1dBm 和 +2dBm 的计算技巧:

P + 1dBm = P + 10dBm - 3dBm - 3dBm - 3dBm
         = P * 10    / 2    / 2    / 2
　　　    = P * 1.25 (W)

P + 2dBm = P - 10dBm + 3dBm + 3dBm + 3dBm + 3dBm
         = P / 10    * 2    * 2    * 2    * 2
　　　    = P × 1.6 (W)

-1dBm 和 -2dBm 的计算技巧:

P - 1dBm = P - 10dBm + 3dBm + 3dBm + 3dBm
         = P / 10    * 2    * 2    * 2
         = P * 0.8 (W)

P - 2dBm = P - 3dBm + 1dBm
         = P / 2    * 1.25
         = P * 0.625 (W)

dBFS

Decibels relative to full scale (symbol: dBFS) is a unit of measurement for amplitude levels in digital systems, such as pulse-code modulation (PCM), which have a defined maximum peak level. The unit is similar to the units dBov and dBO.

The level of 0 dBFS is assigned to the maximum possible digital level. For example, a signal that reaches 50% of the maximum level has a level of -6 dBFS, which is 6 dB below full scale. Conventions differ for Root Mean Square (RMS) measurements, but all peak measurements smaller than the maximum are negative levels.

For instance, if the full scale value is:

\[FullScale = 2^{32} = 4294967296\]

Then, the \(Value = 1900000\) in dBFS is:

\[10 ∙ log (\frac {Value} {FullScale}) = 10 ∙ log_{10} (\frac {1900000} {4294967296}) = -33.54 dBFS\]

Although the decibel (dB) is permitted for use alongside units of the International System of Units (SI), the dBFS is not.

dBc

Decibels relative to the carrier (dBc) is the power ratio of a signal to a carrier signal, expressed in decibels. For example, phase noise is expressed in dBc/Hz at a given frequency offset from the carrier. dBc can also be used as a measurement of Spurious-Free Dynamic Range (SFDR) between the desired signal and unwanted spurious outputs resulting from the use of signal converters such as a digital-to-analog converter (DAC) or a frequency mixer.

If the dBc figure is positive, then the relative signal strength is greater than the carrier signal strength. If the dBc figure is negative, then the relative signal strength is less than carrier signal strength.

Although the decibel (dB) is permitted for use alongside the International System of Units (SI), the dBc is not.

References