Appendix 2
The Decibel Unit (dB)
The decibel (dB) is one tenth of the Bel unit, defined in honor of Alexander Graham Bell. It is a logarithmic unit of measurement of the ratio of two powers, voltages, currents, pressures, etc. It is particularly used in the fields of acoustics, physics and electronics, etc.
This unit of measurement (dB) is defined by the logarithmic ratio:
– of potential (V) or current (I):
– 20log ( Voutput / Vinput)
20log (Ioutput/Iinput)
– of power (P):
–10log (Poutput/Pinput)
If the result is positive, this is a case of amplification. If the result is negative, this is a case of attenuation.
NOTE:— The term “log” refers here to the base 10 logarithm.
There are different dB variants: dBm, dBW (dB watt), dBi, dBd, dBc, dBµV (dB microvolt), dBµVm1 (dB microvolt/m):
– dB: −10log10(Poutput/Pinput).
– dBm: −10log10(Poutput/1mW).
– dBW: −10log10(Poutput/1W) .
– dBi: the antenna’s gain versus the gain for an isotropic antenna. An isotropic antenna is an antenna that emits the same amount of energy in all directions. Such an antenna does not exist in reality.
– dBd: the antenna’s gain versus the gain for a half-wave dipole antenna. This antenna has a physical reality.
Note that catalogs often do not specify whether we are dealing with dBi or dBd. Yet, the difference is important. A 10 dBd antenna gain is a 12.15 dBi antenna gain. We see that the temptation is great to display dBi rather than dBd without explaining it.
– dBc: −10log10(Poutput/Pc). Here, an output power (Poutput) is compared to that of a carrier (Pc) where c is the carrier.
EXAMPLE: Amplitude difference between the fundamental signal and the various harmonics. Reduced to 1 Hz, we refer to dBc/Hz (dB relative to the carrier per Hz of bandwidth).
– dBµV: –20log10(voltage/1µW) . The reference voltage is equal to 1µV.
– dBµV/m: –10log10(field/1 ηVm–1) . The reference field is 1 µV/m1 .
The various operations on decibels are summarized below:
If
then
NOTE: The advantage of using the dB scale is that gains (amplifications) or attenuations are added (instead of multiplied).
The following table shows some correspondence between numerical values and their dB values.
| R | dB |
| 100,000 (105) | 50 |
| 10,000 (104) | 40 |
| 1,000 (103) | 30 |
| 100 (102) | 20 |
| 10 (101) | 10 |
| 1 (100) | 0 |
| 0.1 (10−1) | −10 |
| 0.01 (10−2) | −20 |
| 0.001 (10−3) | −30 |
| 0.0001 (10−4) | −40 |
| 0.00001 (10−5) | −50 |
| 0.000001 (10−6) | −60 |
| 0.0000001 (10−7) | −70 |
| 0.00000001 (10−8) | −80 |
| 0.000000001 (10−9) | −90 |
The dB values for some particular numerical values (10, 2, 0.1, and 0.5) are listed below:
– 10*log(10) = 10;
– 10*log(2) = 3;
– 10*log(1/10) = –10;
– 10*log(1/2) = –3.
As a result, the rules to remember are the following:
– Whenever we multiply by 10 in the current scale, we add 10 (dB) in the decibel scale.
– Whenever we multiply by 2 in the current scale, we add 3 (dB) in the decibel scale.
– Whenever we divide by 10 in the current scale, we subtract 10 (dB) in the decibel scale.
– Whenever we divide by 2 in the current scale, we subtract 3 (dB) in the decibel scale.
EXAMPLE A2.1:—
– 10 mW + 3 dB = 20 mW;
– 100 mW − 3 dB = 50 mW;
– 10 mW + 10 dB = 100 mW;
– 100 mW − 10 dB = 10 mW.