Decibels are used throughout the radio communications and electronics industries, and are often misunderstood. This brief explanation should help the reader better understand decibels, how they are calculated and how they are used.

Logarithms. Numbers can be expressed in two forms. The first is the number itself. The second is how much the number 10 must be raised by an exponent to get that number. The examples below will help you understand this:

EXAMPLE

100 = 10^{2}

1000 = 10^{3}

0.0001 = 10^{-4}

2 = 10^{0.301}

3 = 10^{0.4771}

Logarithms are useful because extremely large and extremely small numbers are easier to express. The difference between the two numbers is not related to the numbers themselves. The advent of the scientific calculator has made the use of logarithms very easy. There are two types of logarithms, natural and common. The common logarithm is based on powers of 10 and is calculated using the key LOG and is different than the natural logarithm, which is calculated by using the LN key on the calculator. The LN key will not be used in these applications.

Gains of amplifiers, loss of power, audio levels, earthquakes and many other quantities are expressed in numbers based upon logarithms. Human senses, such as light intensity, hearing and pain intensity, also are measured logarithmically. The common level expressed in decibels is based upon a mathematical formula based upon logarithms.

Decibels

Decibels are the unit of relative measure used throughout the radio world. The following formulas are used for calculating decibels:

dB_{Power} = 10 x LOG (P_{OUT} / P_{IN})

dB_{Voltage} = 20 x LOG (V_{OUT} / V_{IN})

If the output level always is divided by the input level, the following always will hold true:

- A positive number represents GAIN
- A negative number represents LOSS

If the input and output are the same, the gain will be 0 dB.

Terminology. Now that we understand a little bit about decibels, let us now differentiate between the common uses and misuses of this term.

When working with radio signals, the levels can be extremely large or extremely small. As a result, the use of decibels is necessary to express these levels. The following sections will help in separating the uses of the term dB.

dB: When you use the term dB, you only are comparing one signal value to another signal value. When looking at a filter loss, amplifier gain, or antenna gain or loss, then the use of the term dB is the correct choice.

dBm: This is the value of a signal as referenced to 1 milliwatt. It is an absolute value. A 0 dBm signal equals 1 milliwatt. A 1 watt signal is equal to +30 dBm. If you had a 100-watt transmitter, that would equate to a +50.0 dBm level. If you had a 2 microvolt signal, that would equate to a -100 dBm signal. If you have a 0.5 microvolt (uV) signal, that would equate to -112 dBm.

Most two-way radio receivers operate in the range of -113 dBm to -124 dBm, while most wireless LAN receivers operate with a sensitivity level of -70 dBm to -100 dBm. Wi-Fi systems with extended range operate with a sensitivity of -130 dBm. GPS receivers have a sensitivity of -135 dBm. Most communications service monitors have a sensitivity of -100 dBm.

dBd: This is used when expressing antenna gain based on the difference between the signal of a dipole antenna verses the signal gain from the given antenna. Antennas derive gain by redirecting the signal from undesirable or non-useful directions to more useful directions. This gain can be measured or calculated, and is expressed in dBd.

dBi: As a ruse to make uninformed engineers and technicians think that an antenna has more gain than it really has, some manufacturers started using the term dBi to represent the gain over an ISOTROPIC point in space, which exists in theory only. There is a 2.1 dB difference between dBd and dBi. An example would be a 13.0 dBd gain antenna also would have a 15.1 dBi gain. In reality, it is the same antenna.

In parabolic reflector antennas that normally are used in microwave systems, the focus point is normally assumed to be a point in space as opposed to a dipole antenna, and the dBi term does meet the math formulas for parabolic reflectors. This is why microwave antennas do use the dBi term for the antenna gain. When making signal-strength measurements on a microwave system, you must subtract 4.2 dB from the theoretical link budget numbers to match the actual measured values of signal strength.

Practical uses. Remember, 0 dBm is referenced to 1 milliwatt. Likewise, 0 dBW is referenced to 1 watt and 0 dBk is referenced to 1000 watts (1 kW). Meanwhile, 0 dBc is referenced to a given carrier.

Using 0 dBm as a reference for 1mW, +30 dBm equals 1 watt, +60 dBm equals 1000 watts (or 1 kW) and +90 equals 1 million watts (or 1 MW). Using this same 0 dBm reference for 1mW, -10 dBm equals 100 microwatts (or 100 uW), -30 dBm equals 1 microwatt (or 1uW) and -60 dBm equals 1 nanowatt.

The abbreviation dBc frequently is found in radio subscriber gear specifications to indicate radiated spurious emission suppression in "so many dB" below the carrier. Thus, below the carrier is indicated as a negative number. Conversely, a positive number would indicate above the carrier. References to dBc also can be found in rejection figures on bandpass filters.

While there are many references to dB, the decibel alone with no reference has no meaning. For example, it is common to state that an amplifier has a gain of 3 dB when 5 watts is applied to the input of the amp and the output of the amp is 10 watts. With power, a +3 dB change is double the power and a -3 dB is half the power. Either the gain or the loss can be stated as simply a 3 dB change. Another everyday example is combiner insertion loss. If 100 watts is applied to the input of a combiner that is specified to have 6 dB of insertion loss at your operating frequencies, then you can expect 25 watts at the output of the combiner. Remember that, in this case, as we are discussing loss, 3 dB is half power. Thus, 100 watts less 3 dB equals 50 watts and 50 watts less 3 dB equals 25 watts.

Remember, if 0 dBm equals 1 mW, a +3 dB change equals 2 mW. Another +3 dB change equals 4 mW. Another +3 dB change equals 8 mW. Therefore 8 mW = 9 dBm. Remember in this case that every 3 dB is double power and if you follow this logic, +50 dBm equals 100 watts.

When discussing a loss of power, dB can be stated as so many “dB down.” For example, 50 watts is 3 dB down from 100 watts. This same change also can be specified as -3 dB; in other words, 100 watts less 3 dB equals 50 watts and 50 watts less 3 db equals 25 watts. Similarly, 25 watts less 3 dB equals 12.5 watts. Therefore, 12.5 watts is 9 dB down from 100 watts. One also can figure this example in a different fashion: 100 watts equal +50 dBm and 12.5 watts equal +41 dBm. Thus +50 dBm minus +41 dBm equals a 9 dB change.

Understanding decibels is crucial to anyone working in a field that uses logarithmic units of measure, especially radio communication personnel. Once understood, the decibel makes gains and losses in a system much easier to understand and often quick computations can be performed in one’s mind without the aid of a calculator. In its simplest form, the decibel allows like values to be added or subtracted with ease. Anyone who has ever developed a link budget, troubleshot a complex RF problem or designed an amplifier system with gains and losses understands and values the importance of the decibel.

Ira Wiesenfeld, P.E., has been involved with commercial radio systems since 1966, and has experience with land-mobile-radio, paging and military communications systems. He holds an FCC general radiotelephone operator’s license and is the author of Wiring for Wireless Sites, as well as many articles in various magazines. Wiesenfeld can be reached at iwiesenfel@aol.com.

Christopher Dalton has designed, staged and implemented virtually every kind of LMR system in his two-decade-long career, including conventional, trunked, simulcast, Project 25, single-site and multisite. He holds an FCC general radiotelephone operator’s license. Dalton can be reached at cdalton@fairpoint.net.