**Decibels: Definition, decibels that measure power and intensity**

While power is measured in watts, a widely-used measurement unit for relative amplitude, power, intensity, sound pressure at the listener, and comparative voltage from a microphone is the **decibel (dB)**. Named in honor of Alexander Graham Bell, the measurement was derived from a scale originally used to determine signal loss in telegraph and telephone lines, which resulted in the *bel*. To mitigate fractional values of the larger bel, a **decibel**, which is 1/10 of a bel, became the standard. A decibel is a *logarithmic* measurement that reflects the tremendous range of sound intensity our ears can perceive and closely correlates to the physiology of our ears and our perception of loudness. There are many different forms of decibel measurement, and it is not always clear which method of computation is being used, although a few labels exist, and you may see such qualifiers that suggest the method or reference value, such as dBm, dBV, dB SPL, dB SIL, dB SWL, dB FS, dB*u*, dBVU, or even scale weightings such as dB(A) and dB(C) which we use on our sound level meters and will examine later on.

*Ambiguity abounds in dB labeling*, so take care you know what is being measured and the reference benchmarks being used. The SI (International System of Units), ISO and others do not recognize any of the suffixes above for decibels, and in literature, the letter* 'p'* is often used in dB equations to represent power, pascals or pressure, which are NOT interchangeable, so mind your '*p*'s' if not '*q*'s'.

To get started, a quick look at **logarithms** (often misspelled by we musicians as *logarhythms*) is in order.

Be not intimidated by calculating logarithms-with cheap calculators to do the math (one previously used log tables), just a simple understanding of how they work is all that is necessary for decibel calculations (if you turn your iPhone or Android phone horizontally while using the onboard calculator, it will be happy to compute complex logs for you, but just be sure to use the log

Decibels are often used to measure very minute values, which can also be expressed by logs of decimals numbers or their negative power equivalents e.g.

For example,

If a log is expressed

**Decibels**

A *decibel* is a measurement used to compare the **ratio** of **power, intensity **or** amplitude** between two acoustic sounds or electronic signals. The ratio (R) of two signals expressed by their power in watts (W_{1} and W_{2}) is:

*A doubling of power equals an increase of +3 dB (try it out with the formula above).* When we study filters later on, you will notice that a *filter cut-off frequency* is defined as the *half-power point*, which is calculated as –3 dB.

**Decibels that measure power and intensity**

There are many different types of decibel measurements, so for the purpose of clarity, the above form, which measures power or intensity, is called **dBm **when a fixed reference value is used for the denominator. For the purpose of having a standardized *absolute* measurement of power in an **electronic circuit** (i.e., a comparison not to another signal, but to an industry-fixed value), the nominal reference wattage (W_{2}) has been defined as 1 milliwatt (0.001 watt). In absolute terms, a 1-watt signal, which has 1,000 times the power of the reference wattage, will be 30 dB, computed below:

dBm=10 log_{10} (1 watt/.001 watt)dBm =10 log _{10} (1000)dBm=10 x 3 [because log _{10} 1000 = 3]dBm=30 |

dBm is the form most commonly used to evaluate power in audio circuits.

For **acoustic** measurement of the total power radiated in all directions (referred to as Sound Power Level, or SWL), the same formula is used with a reference level correlating to threshold of audibility that we mentioned earlier, 10^{-12} watts (also called a *picowatt*). Note how loud a sound with 1 watt of total acoustic energy might be at 120 dB SWL—it is the reported sound of a typical jet plane at 500 feet (whaaat, those four engines produce only 1 acoustic watt of sound? Yeah.).

dB SWL=10 log_{10} (1 watt/10^{-12} watts)dB SWL =10 log _{10} (10^{12}) [that's some nifty math there]dB SWL=10 x 12 [because log _{10} 10^{12} = 12]dB SWL=120 |

Since** intensity (I)** at a fixed distance of measurement is directly proportional to power, a similar measurement can be made for intensity using a reference value of a

As a reminder, acoustic intensity is the measurement of power, or the rate at which energy passes through a defined area perpendicular to the direction of wave propagation. So a power measurement (watts) over an area measurement (meters^{2}) is needed, as demonstrated below.

A real-world example of intensity measured in decibels |

As you recall from the chart on the previous page, intensity is a measure of power present over or through an area, so let's figure out the acoustic intensity in dB SIL present at the surface of a loudspeaker with a cone radius of 0.5 meters receiving 1 watt of power. As stated, the common reference level for 10.* Note the example below contains some rounding for clarity.^{-12} W/m^{2}
1/0.78 W/m^{2} ≈ 1.28 W/m gives us the sound intensity at the surface of the speaker.^{2}
10 plugged into the dB SIL formula above: ^{-12} W/m^{2}So, 1 watt of power in terms of intensity *Thanks to David Howard/Jamie Angus |