Decibel Digression

Copyright © 2005 Dave Whitmore

In telecommunications in general and telephony in particular we worry a lot about signal strength. For example, a particular local loop should have both a high signal strength and a low noise figure. We express this relationship as the signal to noise ratio and we measure it in decibels. We could also talk about the gain of an amplifier and measure it in decibels. An amplifier is an active (powered) device and if it is working properly it will produce more output than input. We can also talk about the loss in a cable. A cable is passive and for various reasons rooted in physics the output will be less than the input. We would hope that the output is not too badly attenuated when compared to the input and we can measure the cable loss in decibels.

Mathematical Worries

Let's go back to the first example above. We could talk about the voltage or the wattage of the signal vs. the voltage or wattage of the noise noise and that's perfectly legitimate. However, there is usually a lot more signal than noise and the ratios can get pretty big. For example a typical local loop has a signal to noise ratio of about 1,000 to 1. The decibel scheme makes it easier to express these ratios because it is a logarithmic scale rather than a linear one. As you'll see the decibel ratios are smaller numbers and are easier to deal with.

Origins and Derivations

Part of the term "decibel" refers to Alexander Graham Bell of telephone system fame. It turns out that human hearing works on a logarithmic basis and there is a logarithmic relationship between acoustical power and your perception of loudness. We'll see how the decibel scale works to our advantage in just a minute.

The decibel is abbreviated dB. Note the large "B": that's in honor of Alexander Graham Bell. The decibel is based on a ratio of two power levels which are usually specified in milliwatts. (A milliwatt is 1/1000 of a watt and it's a very small amount of energy. However, the human ear is very sensitive and can easily detect a milliwatt signal.) Here's the calculation:

dB = 10 log ₁₀ (P₂/P₁)

where P₂ is the output power and P₁ is the input power. A decibel is a pure number and it has no units associated with it. For example, a decibel value is not expressed in watts, volts, or amperes.

Examples

Let's assume that we have an output signal that is only one half as big as the input signal. What difference does this represent in terms of decibels? The calculation looks like:

dB = 10 log ₁₀ (.5) = 10 (-.301) = -3.01 dB

This tells us that there is a 3 dB loss in this circuit. This is what we might find in a long cable in which the signal decays in strength because of the cable length.

How about an amplifier that produces twice as much output as input? Electrical engineers would say that this amplifier has a gain of 2. Here's the calculation:

dB = 10 log ₁₀ (2) = 10 (.301) = 3.01 dB

Therefore if our amplifier doubles the input we have a gain of 3.01 dB.

These little exercises show you two things. The first finding is that the sign of a dB value will tell you whether we have a gain or a loss. The second finding is that 3 dB represents roughly a doubling or halving of power levels. It's just a convenient number to keep in mind when you think about dB values and power ratios.

A Table

You can play with the above formula and derive dB values for any particular ratio. In fact most scientific calculators have a base 10 logarithm function so the calculations are pretty easy. Here's a table to give you a perspective on the dB values and on the associated power ratios.

To bring this back to telephony just a bit, the signal to noise ratio of the typical analog local loop is about 1,000 to 1 or 30 dB.

Again, note that the dB numbers are a little smaller than their accompanying power ratios and are easier to write and comprehend.

Decibels and the Milliwatt Standard

In telephony many comparisons are referenced to an arbitrary standard of 1 milliwatt or 1/1000 of a watt. It's important to have a standard so that we can have some uniform base value against which to make a comparison. Other branches of electrical engineering may choose other standards but for telephony and audio applications 1 milliwatt is the accepted standard. When I mentioned the 30 dB signal to noise ratio above it was based on a reference value of 1 milliwatt for the P₁ value in the formula. Decibel values referenced to 1 milliwatt are written as "dBm".

Here's an example of the use of the dBm measure. In telephone testing it's customary to attach a test set on one end of (say) a local loop and put a 1 milliwatt signal (0 dBm) onto the line. You measure the signal strength at the other end and you can figure the loss in the local loop referenced to the 1 milliwatt signal that was the input.

Let's compare some different power levels that are referenced to 1 milliwatt:

Naturally a local loop would be producing losses with negative dBm values. An amplified circuit, on the other hand, would produce positive dBm values.

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