Almost every number on an RF datasheet or on a spectrum analyzer screen is written in some kind of dB. The plain dB is a ratio, and the letters after it name the reference that turns the ratio into an absolute level. On this page I'll start from the definition and show why a log scale is useful. Then I'll go through the common suffixes, and finish with the calculation rules that make dB so convenient in RF work.

• How Is the dB Defined?
• Why log scale ?
• dBm, dBc, dBi etc
• How Do You Calculate with dB Values?

How Is the dB Defined?

Let's start with the definition, because every other dB unit on this page is built from it. A dB always compares two quantities of the same kind, so a plain dB value has no physical unit by itself.

dB is one of the most common measurement unit in RF area. The methematical definition of dB is as follows. It is just a number presented in log scale. So there is not many things I can say in terms of definition.  It is the value of Power in log scale.

The name comes from the bel, which is log₁₀ of a power ratio. A decibel is one tenth of a bel, and that is where the factor 10 in front of the log comes from. A positive dB value means the measured quantity is larger than the reference, 0 dB means the two are equal, and a negative value means it is smaller. So a loss to half the power is -3.01 dB, and a loss to one tenth is -10 dB. The two formulas below show the definition in its two forms.

or

Figure 1. The two forms of the dB definition. The 10 log form applies to a power ratio, and the 20 log form applies to an amplitude ratio such as a voltage ratio.

• 10 log is the form for power : 10 log₁₀(2) = 3.01, so doubling the power adds about 3 dB, and ten times the power adds exactly 10 dB.
• 20 log is the form for amplitude : the right-hand formula writes P inside the 20 log form, but 20 log belongs to an amplitude such as a voltage or a current. Applied to a power ratio, 20 log would double every dB value.
• Both forms give the same dB for the same change : power is proportional to the square of the voltage when both are measured across the same impedance. So twice the voltage is 20 log₁₀(2) = 6.02 dB, and four times the power is 10 log₁₀(4) = 6.02 dB.

Why log scale ?

An RF system handles power levels that are very far apart. A transmitter can put out 1 W, while a receiver may have to detect a signal of about 1 fW, which is -120 dBm in the table further down this page. That is a ratio of 10¹⁵, and a linear axis cannot show both ends of such a range at once.

My personal question when I first see this unit was "Why they invented this kind of Log scale unit ?" (Actually you would have seen many other Log scale units in other area as well.. pH in chemistry is another example of log scale unit)

To me, Log is seen as a magic lens that make small thing big and big thing small as illustrated below.

Figure 2. The log as a lens. It shrinks large values and enlarges small values, so both fit on one axis.

Then next question is "Why we need this kind of magic lens ?". The answer to this question is that we can visualize a set of data which has both very small values and large values at the same time. Let me give you an example as shown below. On the top graph, you see a very smooth curve throughout the whole range. But if you magnify the bottom part of the graph, it is fluctuating as shown at the bottom graph. However, this fluctuation is not noticed if we plot the graph in the linear scale as in the top graph.

Figure 3. A linear axis hides the small values. The fluctuation between x = -1 and x = 1 only appears when that region is magnified.

However, if you plot this data in a log scale. you would notice the area with unnoticed fluctuation gets magnified and become very obvious as shown in (b), (c). If you compare the graph (a) - linear scale with (b), you would notice how the small value in (a) get magnified in (b) and how the big value in (a) get shrinked in (b). By this way, the log scale shows the details of both small number range and large number range at the same time.

Figure 4. The same data on one linear axis and two log axes. On the log axes the fluctuation and the large values at the edges are visible together.

• The log compresses the large values : the curve reaches about 250000 at x = -2 and x = 2. In (b) this becomes log₁₀(250000) = 5.4, the top of the green lines.
• The log expands the small values : the minima of the fluctuation are close to 0.1, and log₁₀(0.1) = -1 is the lower red line in (b). The ripple that was a flat line in (a) now fills a large part of the axis.
• Graph (c) is graph (b) multiplied by 20 : the shape is identical and only the axis changes, now running to about 108.
• For a power, the dB value is 10 log : if p(x) is a power, its value in dB is 10 log₁₀ p(x), which is half of the values in (c). The 20 log form in (c) is the one for an amplitude.

In many cases, we use dB scale to represent a ratio.

The empty box after dB in this formula is where the subscript goes. The subscript names the Reference, so dBm means that the Reference is 1 mW, and dBW means that the Reference is 1 W.

Depending on what kind of ratio it represents, we represent the dB with various subscript as examples shown below.

dBm, dBc, dBi etc

A plain dB value tells you how much bigger one quantity is than another. It does not tell you how big either of them is. A suffix fixes the reference, and with a fixed reference the dB value becomes an absolute level or a ratio against a named signal.

dBm represents the ratio of measured power and 1 mW, meaning the measured power with reference to 1 mW.

dBc represents the ratio of the measured power at a specific frequency point and the power at the center frequency.

dBi represents the ratio of radiation power in a certain direction (the direction with the strongest radiation power) and the imaginary isotropic radiation power. See Antenna pages for the details of dBi.

Let's put numbers on dBc. The c stands for carrier, and for a single carrier the carrier sits at the center frequency. Suppose the carrier is at 20 dBm and a spurious signal is at -40 dBm. The spur is then 60 dB below the carrier, and you write it as -60 dBc. A dBc value does not change when the whole signal is scaled up or down, so harmonic and spurious levels on a datasheet are often given in dBc.

dBi has a close relative, dBd, which uses a half-wave dipole as the reference instead of the isotropic radiator. A half-wave dipole has a directivity of 1.64 against the isotropic radiator, so 0 dBd = 10 log₁₀(1.64) = 2.15 dBi. Check which of the two an antenna datasheet uses before you compare two gains.

Here goes some examples of different dB scales.

Figure 5. Common dB forms. Each line keeps the 10 log or 20 log form of its quantity, and the level forms only add a fixed reference.

• The ratio forms follow Figure 1 : a voltage ratio takes 20 log and a power ratio takes 10 log.
• dBμV is a voltage level and needs 20 log : the dBμV line in the image uses 10 log, but the correct form is 20 log₁₀(V / 1 μV), like the voltage ratio above it. With the correct form, 0 dBm across 50 ohm is 0.2236 V, which is 107.0 dBμV.
• dBm and dBW differ only in the reference : 1 mW against 1 W is a factor of 1000, which is 30 dB. So a level in dBm is always 30 more than the same level in dBW.

Following is a table shows precalculated values of W, dBW, dBm. You may create your own table like this depending on your need.

NOTE : Simply put, dBm is equal to dBW + 30

Two of the watt values in the table are rounded. 0.032 W is 31.6 mW and 0.003 W is 3.16 mW, because 10^-1.5 = 0.0316 and 10^-2.5 = 0.00316. The dB columns are exact, and you can always go back to watts with P = 10^dBW/10 W.

• dBm is the level against 1 mW : 0 dBm is 1 mW, 30 dBm is 1 W and -120 dBm is 1 fW.
• dBc is a ratio against the carrier : a spur at -40 dBm next to a 20 dBm carrier is -60 dBc.
• dBi and dBd are antenna gains with different references : 0 dBd is 2.15 dBi.
• dBm = dBW + 30 : the two scales run in parallel, 30 dB apart.

How Do You Calculate with dB Values?

The main reason engineers use dB is not the display. It is the arithmetic. A chain of gains and losses is a chain of multiplications in linear units, and the log turns every multiplication into an addition. But the rule only works when you keep ratios and levels apart.

Let's follow a signal through a short chain. A source puts out 20 dBm, which is 100 mW. A cable loses 2 dB, an amplifier adds 15 dB and an attenuator removes 10 dB. In dB, the output is 20 - 2 + 15 - 10 = 23 dBm. In linear units, the same chain is 100 mW x 0.631 x 31.6 x 0.1 = 199.5 mW, and 10 log₁₀(199.5) = 23.0 dBm. The two paths give the same answer, but the dB path needs only additions.

The rule is simple once you see the units. A level plus a ratio is a level, so dBm + dB = dBm. A ratio plus a ratio is a ratio, so dB + dB = dB. The difference of two levels is a ratio, so 23 dBm - 20 dBm = 3 dB. But two levels do not add in dB. Two uncorrelated signals of 10 dBm each carry 10 mW + 10 mW = 20 mW, which is 13.01 dBm and not 20 dBm. A 10 dBm signal and a 0 dBm signal together carry 11 mW, which is 10.41 dBm. To add two levels, convert them to mW, add, and convert back.

A few ratios come up so often that it helps to know them by heart. The table below lists them.

The same rules give the thermal noise floor, which appears in every receiver calculation. The noise power density kT at 290 K is 1.38 x 10^-23 J/K x 290 K = 4.00 x 10^-21 W/Hz, which is -174.0 dBm/Hz. For a 1 MHz bandwidth you add 10 log₁₀(10⁶) = 60 dB and get -114.0 dBm. Here a level per hertz plus a bandwidth ratio gives a level, exactly as the rule above says.

• Gains and losses add in dB : 20 dBm - 2 dB + 15 dB - 10 dB = 23 dBm, the same as 199.5 mW in linear units.
• A level minus a level is a ratio : 23 dBm - 20 dBm is 3 dB, a factor of 2 in power.
• Levels never add directly in dB : 10 dBm plus 10 dBm is 13.01 dBm, and 10 dBm plus 0 dBm is 10.41 dBm.
• The thermal noise floor is -174 dBm/Hz at 290 K : add 10 log₁₀ of the bandwidth in Hz to get the noise power in that bandwidth.