Decibels (dB) are an important unit for dealing with electronic signals.  And even though their scaling properties and endless variants such as dBi, dBm, dBFS, and dBµ can make them seem inconvenient; we will quickly cover how to convert from decibels and back again.

Defining Decibels

Decibels were first reported in the Bell System Technical Journal in 1929.  The key to remember about them is that they are meant to compare the difference between two levels of power.

Today we’ll mostly compare the amount of power leaving a system compared to the power that went in.  But decibels can also compare a measurement to a standardized reference value.  So for example, when we talk about the loudness of sounds in decibels, we are describing how loud that sound is compared to the threshold for human hearing (which is the reference value).

When decibels were first used, their role was to compare the signal power entering a standardized cable to the amount of power exiting.  This became a convenient way to express signal losses.

Converting Ratios to Decibels

The following will convert this ratio to decibels:

…where P₁ is the input power and P₂ is the output power.  In other usages, the denominator term (P₁) will be a fixed reference value, and the numerator term (P₂) is a measurement. 

Converting Decibels to a Ratio

To do the reverse (i.e.: convert decibels to a power ratio), start with the number of decibels (x).  Then use the exponential formula as shown below.

Note this form is only appropriate for power.  It shouldn’t be used to find voltage ratios for example (more on why later below).

An Example

To practice converting, let’s look at a recent example from my severe weather tracking radio test.  Weather radio stations like the one plotted below send out forecasts and early warnings for dangerous threats such as tornados.  Notice the decibel scale to the left and the signal to noise ratio (SNR) to the right.

In Figure 1 above, the weather radio broadcast is depicted as a blue and white peak.  The base of the peak sits around -65 dbFS (decibels relative to full scale).  The peak (shown in the yellow circle) is very near -22 dBFS.  The upper edge of the plot is at 0 dBFS, which is the upper limit for the radio.  And finally: the SNR indicator on the right shows a reading of 43 dB.

Let start with the SNR value first by converting it from decibels to a ratio of power levels.  We will represent the noise power as P₁, the signal power as P₂, and the SNR (measured in decibels) as x.

So when the SNR is 43 dB, the result is a unitless value near 19,952.  This tells us the power of the received signals is ratiometrically 19,952 times more powerful than the noise, which is quite good.

Sanity Check: Converting Back

How about doing that in reverse?  In the above calculation, we used 10 as the base for our exponent.  So to reverse this process, we use a logarithm with base 10:

Entering P2 / P1 = 19952 yields the original 43 dB SNR.

From the table below, we can see that the decibel scale works well with our decimal counting system.  We could also say that a 3 dB increase corresponds to roughly a x2 increase in power; and a 3 dB decrease corresponds to a loss of roughly ½ power.

Table 1: Decibel Conversions by Decade

Decibels (dB)	Power Multiplier
+50 dB	100 000
+40 dB	10 000
+30 dB	1 000
+20 dB	100
+10 dB	10
+0 dB	1
-10 dB	0.1
-20 dB	0.01
-30 dB	0.001
-40 dB	0.000 1
-50 dB	0.000 01

Power vs. Voltage Decibels (Deception)

When expressing voltage gains in decibels instead of power gains, things work out a little differently.  The convention for converting voltage ratios is as follows…

Where |V₂| is the absolute value of an output voltage, and |V₁| is the absolute value of the input voltage.  For brevity, we will assume V₂/V₁ is always positive from this point forward, and we will omit the absolute value signs…

Table 2: Decibel Scaling for Power vs. Voltage

Power Decibels (dB)	Voltage Decibels (dB)	Multiplier
+50 dB	+100 dB	100 000
+40 dB	+80 dB	10 000
+30 dB	+60 dB	1 000
+20 dB	+40 dB	100
+10 dB	+20 dB	10
+0 dB	0 dB	1
-10 dB	-20 dB	0.1
-20 dB	-40 dB	0.01
-30 dB	-60 dB	0.001
-40 dB	-80 dB	0.000 1
-50 dB	-100 dB	0.000 01

At first, this may look like a double standard, where power calculations are ultimately multiplied by 10 and voltage calculations are multiplied by 20.  But truly, there is no double-standard.  The starting point was 10 ∙ log₁₀(A/B) the whole time!

To see why: recall that decibels were originally meant to compare power (measured in watts).  Voltage is not wattage.  However, we can use the Power Law to calculate the wattage if the voltage (V) and the resistance (R) (or impedance) are both known…

Substituting this into the decibel conversion formula yields…

This can be simplified using two log properties:

• Product Rule for Logarithms: log₁₀(A*B) = log₁₀(A) + log₁₀(B)
• Quotient Rule for Logarithms: log₁₀(A/C) = log₁₀(A) – log₁₀(C)

This lets us separate the voltage and resistance terms.

We use the log exponent rule to simplify the first logarithm…

• Log₁₀(A^B) = B log₁₀(A)

If R₁ = R₂, then we can simplify things further, since log₁₀(1) = 0.  Even if R₁ and R₂ are unknown, power is still proportional to V² if the resistance or impedance are constant.  So we can approximate the gain as…

This explains why “20” is associated with voltages.  This also applies to other units such as electric fields (measured in volts per meter).

To convert a voltage representation x from decibels to a voltage ratio, we use an expression like the one before, only the decibels are divided by 20…

Why Decibels?

This begs the question: why not simply express the difference as P2/P1?  What do we gain by introducing a logarithm?

This is something we’ll cover in more detail shortly.  For now, I’ll note:

• Decibels can convey extremely large and small values in only a few digits.
• Decibels are great for values that approach zero, but aren’t zero exactly.
• Decibels are additive in a Communications Link Budget.

And in special situations, decibels can be less deceptive than percentages.

References

[1]	D. Liu and H. C. Roberts, “The Physics of sound and hearing,” in Environmental Engineers’ Handbook, Boca Raton, FL, CRC Press, 1997, pp. 452-454.
[2]	W. H. Martin, “Decibel – The name for the transmission unit,” Journal of the A.I.E.E, vol. 48, no. 3, p. 223, 1929.
[3]	M. Mardiguian, “Generalities in Radiated Interference,” in Controlling Radiated Emissions by Design, 3rd ed., Springer Science + Business Media, 2014, p. 4.

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