# Decibels Demystified, Part 1

TO UNDERSTAND DECIBLES, A BIT OF MATH IS ALL THAT'S REQUIRED
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Illustration: Jack Desrocher

You probably have some idea of the notion that decibels measure signal levels. However, most people don't understand exactly what decibels are or how they are used in the audio world. Even audio professionals are often a bit fuzzy about the precise nature of decibels.

That is understandable; decibels can be quite confusing. There are many types of decibels, and manufacturers use them in their specifications with reckless abandon. To clear away the fog surrounding the essential concept of decibels, I'll start with some basic math. It's important to understand the material from “Square One Classics: The Shocking Truth” in the June 2001 issue, so try to have a copy of that issue handy.

EXPONENTS

Thanks to high school math teachers, exponents and logarithms frighten many people, but they're really not that complicated. Exponents provide a way to simply and elegantly represent the result of multiplying the same number several times. For example, consider the following equation:

2 × 2 × 2 × 2 × 2 = 25 = 32

In that example, the 2 is called the base, and the 5 is called the exponent.

FIG. 1: If you raise a specific number to different exponents, the result grows quickly as the exponent increases (a). In this case, y = 10x. On the other hand, if you take the log of ­different numbers, the result grows very slowly as the initial numbers increase dramatically (b). In this case, y = log x. Notice that the graph resembles that of an audio compressor''s performance.

Exponents also help express very large numbers with relatively few digits. For example, 10,000,000 = 107. You can even use fractional exponents; for instance, 52.3 = 40.52. In fact, you can make a graph of the relationship between exponents and the value they generate for a given base (see Fig. 1a).

Exponents also help express mathematical formulas more elegantly. For example, take a look at the DC form of Joule's Law defining electrical power (in the following equations, recall that P = power, V = voltage, I = current, and R = DC resistance):

P = V × I

From Ohm's Law relating current, voltage, and resistance, you know that I = V/R. If you substitute V/R for I in Joule's Law, you get:

P = V(V/R) = (V × V)/R = V2/R

You can make a similar substitution for V, which equals I × R:

P = (I × R)I = (I × I)R = I2 × R

Now you have three equivalent expressions of Joule's Law:

P = V × I
P = V2/R
P = I2 × R

LOGARITHMS

Logarithms (or logs) are simply the opposite of exponents. In sound applications, the base is assumed to be 10, and logs are defined thus:

If a = 10b then b = log a

The following is difficult to translate to English, but I'll give it a try. Logarithms identify the exponent (b) to which you would raise 10 in order to obtain the number you are taking the logarithm of (a).

For example, 100 = 102, so log 100 = 2. That also works with fractional exponents. For example, 20 = 101.301, so log 20 = 1.301. If you create a graph of the relationship between numbers and the logs of those numbers, you see that the graph is identical to the exponent graph flipped across a diagonal (see Fig. 1b). To calculate logs, use a calculator with a log function.

Logs help you manipulate large numbers more easily. They also help you manipulate large ranges of numbers, which is why they're used in decibels: audio signal levels encompass a large range of possible values. In fact, logs act like “mathematical compressors.” Just as an audio compressor accepts a large range of input levels and outputs a smaller range of levels, logarithms accept a large range of numbers and return a much smaller range. The graph even resembles the graph of a compressor's input versus output.

I covered voltage, current, impedance, and power in the June 2001 “Square One Classics,” and I discussed exponents and logarithms here, so it's time to look at decibels. For now I'll stick with electrical decibels; I'll apply the same principles to acoustic decibels in a future column.

The following information is pretty dense. At first it might seem highly theoretical, but have patience; I'll include some practical examples in the next column.

Many people use the term decibel as if it were an absolute unit for measuring the amplitude of electrical audio signals. However, that is not correct. When used with electrical audio signals, decibels express the ratio of two values. Scientists at Bell Labs invented a unit of measurement to compare two power values and called it the bel in honor of Alexander Graham Bell. By definition:

Number of bels = log (P1/P0)

P1 and P0 are quantities of power in watts, and P0 is usually a reference power value to which another power value (P1) is compared.

There are several reasons to work with a power ratio's log instead of the ratio itself. As mentioned previously, logs help you work with large ranges of numbers more easily, and audio ratios can encompass a very large range. For example, the ratio of the loudest sound you can stand to the softest sound you can hear is approximately one trillion to one. Logs act as mathematical compressors, reducing a large range of values to more manageable proportions.

In addition, the sensitivity of human hearing to amplitude is generally logarithmic. You perceive equal changes in the percentage of amplitude, not in amplitude itself. For example, if one sound seems twice as loud as another sound, the louder sound's amplitude is more than three times the softer sound's amplitude, not twice the amplitude. That is why logarithmic potentiometers are used in most audio gear instead of linear pots.

As it turns out, bels “compress” power-ratio values too much to be useful in audio circuits. As a result, audio engineers use the decibel, which is equal to one-tenth of a bel (that is, there are ten decibels to one bel) and is abbreviated dB. By definition:

Number of decibels = 10 log (P1/P0)

FIG. 2: If you graph decibels with respect to the ­corresponding power ratios using a linear scale (left), each increase of 10 dB takes ten times more vertical space. However, if you use a logarithmic scale (right), each increase of 10 dB takes the same vertical space, and the curve becomes a straight line.

Decibels are often plotted on a graph with dB on the x-axis and the power ratio from which they arise on the y-axis. As you can see in Fig. 2, that can be done in two ways. If the graph uses a linear scale, the curve is relatively complex and takes ten times as much physical space to represent the ratio 100:10 as it does for the ratio of 10:1, even though both ratios are equivalent. You would need a very large piece of graph paper to represent a ratio of 1,000,000,000,000:1.

If you use a logarithmic scale, however, the curve becomes a straight line and equivalent ratios (for example, 100:10 and 10:1) occupy the same physical space on the graph. That makes it easy to see that a difference of 10 dB is the same percentage of change in the power ratio, regardless of the power values. It also makes it easier to chart large ranges of numbers in less physical space.

VOLTAGE AND CURRENT

Decibels are also used to compare voltage values, especially with circuits that exhibit high impedance and let little current flow. However, the equation is slightly different. Without going into the mathematical derivation, the formula for voltage-referenced decibels is:

Number of decibels = 20 log (V1/V0)

By the same reasoning, you can apply the same equation to current.

Number of decibels = 20 log (I1/I0)

Decibels are rarely applied to current. If a circuit draws more than a negligible current from a voltage source, units of power are used instead. If a circuit draws very little current (that is, impedance is high and load is small), volts are used.

Now that the basic concepts are out of the way, I can present some more practical information and examples of how decibels are used. But that will have to wait until next time, so stay tuned.

EM technical editor Scott Wilkinson has been zapped more than once after carelessly touching the poles of an AC wall outlet.