# Converting to Logarithmic Units

On some instruments, you can display amplitude on either a linear scale or a decibel (dB) scale. The linear scale shows the amplitudes as they are. The decibel is a unit of ratio. The decibel scale is a transformation of the linear scale into a logarithmic scale.

The following equations define the decibel. Equation A defines the decibel in terms of power. Equation B defines the decibel in terms of amplitude.

dB = 10log10(P/Pr) | (A)

where P is the measured power, Pr is the reference power, and P/Pr is the power ratio.

dB = 20log10(A/Ar) | (B)

where A is the measured amplitude, Ar is the reference amplitude, and A/Ar is the voltage ratio.

Equations A and B require a reference value to measure power and amplitude in decibels. The reference value serves as the 0 dB level. Several conventions exist for specifying a reference value. You can use the following common conventions to specify a reference value for calculating decibels:

- Use the reference one volt-rms squared (1 Vrms2) for power, which yields the unit of measure dBV.
- Use the reference one volt-rms (1 Vrms) for amplitude, which yields the unit of measure dBV.
- Use the reference 1 mW into a load of 50 Ω for radio frequencies where 0 dB is 0.22 Vrms, which yields the unit of measure dBm.
- Use the reference 1 mW into a load of 600 Ω for audio frequencies where 0 dB is 0.78 Vrms, which yields the unit of measure dBm.

When using amplitude or power as the amplitude-squared of the same signal, the resulting decibel level is exactly the same. Multiplying the decibel ratio by two is equivalent to having a squared ratio. Therefore, you obtain the same decibel level and display regardless of whether you use the amplitude or power spectrum.

## Displaying Results on a Decibel Scale

Amplitude or power spectra usually are displayed on a decibel scale. Displaying amplitude or power spectra on a decibel scale allows you to view wide dynamic ranges and to see small signal components in the presence of large ones. For example, suppose you want to display a signal containing amplitudes from a minimum of 0.1 V to a maximum of 100 V on a device with a display height of 10 cm. Using a linear scale, if the device requires the entire display height to display the 100 V amplitude, the device displays 10 V of amplitude per centimeter of height. If the device displays 10 V/cm, displaying the 0.1 V amplitude of the signal requires a height of only 0.1 mm. Because a height of 0.1 mm is barely visible on the display screen, you might overlook the 0.1 V amplitude component of the signal. Using a logarithmic scale in decibels allows you to see the 0.1 V amplitude component of the signal.

The table below shows the relationship between the decibel and the power and voltage ratios.

dB | Power Ratio | Amplitude Ratio
+40 | 10,000 | 100
+20 | 100 | 10
+6 | 4 | 2
+3 | 2 | 1.4
0 | 1 | 1
–3 | 1/2 | 1/1.4
–6 | 1/4 | 1/2
–20 | 1/100 | 1/10
–40 | 1/10,000 | 1/100

The table above shows how you can compress a wide range of amplitudes into a small set of numbers by using the logarithmic decibel scale.