### The Decibel: A Ratio of Relative Strength

You know, it’s amazing to me that I have worked for years with people who, frankly, couldn’t have possibly understood what a decibel really was. They used the terminology but didn’t understand what it really meant. You’re not going to be like that. Your’ re going to understand the basic unit of measurement used in the FDDI world: the decibel.

The American Heritage dictionary provides us with a starting point for our understanding:

**de-ci-bel** (des’-i-bal) n. Abbr. dB. A unit used to express relative difference in power, usually between acoustic or electric signals, equal to ten times the common logarithm of the ratio of the two levels.

*common logarithm. A logarithm to the base 10*

Since FDDI defines so many things in terms of decibels and since decibels are defined based on common logarithms we’re going to have to make sure you understand the basics of logarithmic calculations before we can clearly define decibels! Tragically, we’re going to have to digress to the underlying math to make this concept clear.

When you consider the exponential powers of 10 you arrive at the following table:

100 = 1 (..any number to the zero power is equal to 1)

101 = 10

102 = 100

103 = 1000

We’ll get a math major on the telephone to explain LOGARITHMS to you. The same math major who explains logarithms will explain that any number raised to the zero power is equal to one and that any number raised to the first power is equal to itself. You already know (or you will now) that a number raised to the second power (“squared”) is the number multiplied by itself. You see, then, that 101 = 10 and 102 = 100. The exponent has changed from ‘1’ to ‘2’ and the resultant number has varied from 10 to 100. The mathematicians have come up with a way of expressing numbers by simply telling you the exponent value. This is the LOGARITHM of the number; an exponential representation of the power of a base. A logarithm references an exponential power of a BASE number; in our discussion the base is 10. The logarithm then consists of a value that is the exponent of the base that results in some particular number. As we just said, 10 raised to the second power is 100. “2” is the base ten (‘common’) logarithm that is equal to 100. When the base number used for the calculation is 10 the logarithm is called the COMMON LOGARITHM. We could rewrite the table above to look like this:

Log 1 = 0

Log 10 = 1

Log 100 = 2

Log 1000 = 3

We would say (if we wanted to impress our friends who were math majors) that, “The common logarithm of 100 is 2”

You should be able to extrapolate the concept that would assure you that there is some exponent, between 1 and 2, that would result in a number equal to, say, 47; or 93; or 22.5216; or any number between 10 and 100. In fact, you could get out your trusty table of common logarithms (..you still have one from your math class, don’t you?) or just whip out your handy scientific calculator, and you could arrive at the following table:

Log 47 = 1.67209

Log 93 = 1.96848

Log 22.5216 = 1.35259

You see, the closer the resultant number (47, 93, 22.5216) is to 100 (102), the closer the logarithm is to ‘2’, because Log 100 = 2. Logarithms have some interesting properties that engineers and scientists like (why do they like them; go find one and ask – if you dare!) For example, when you add the logarithms for two numbers the result is the same as you would get if you multiplied the two original values. For example (refer to the table above) an you’ll see:

47 * 93 = 4371

Log 47 + Log 93 = Log 4371

1.67209 + 1.96848 = 3.64057

Log 4371 = 3.64057

(If you’re in doubt, trust me! – or get out your calculator and prove it to yourself.) Here’s an easy and obvious problem to work through. You know that 1 + 2 = 3 (right?). If you take Log 10 (which is 1) plus Log 100 (which is 2) you end up with 3. What number has a log equal to 3? (Remember, 10 raised to the third power; 10*10*10 =???) Right! One Thousand. See, you add the logarithms to get the answer.

If you wanted to represent a number that was less than one you would use a negative logarithm. This might sound strange at first glance but, think about it; can you work this problem in your head: “I’ll lend you five dollars but you’re going to have to pay it back with 10% interest”. How do you compute 10% of $5.00. You shift the decimal point to the LEFT to divide by 10. Shifting the decimal point to the left is represented by a negative logarithm. Here is a continuation of a previous table:

Log .1 = -1

Log .01 = -2

Log .001 = -3

Log .0794 = -1.1

Log .00063095 = -3.2

Any number greater than zero can be represented by a logarithm. To apply your newfound understanding of logarithms, recall the definition of a decibel: A unit used to express relative difference in power, equal to ten times the common logarithm of the ratio of the two levels. A decibel is a RATIO between two numbers and this ratio is expressed as ten times the log of the number.

Suppose that the combined effects of transmission loss from a transmitter to a receiver results in a signal that is 12 times weaker at the destination that it was when it was first transmitted. This means that the signal was attenuated to 1/12 of its original strength. When you convert the fraction 1/12 to a decimal you get .0794. Look at the table above. What is the value of Log .0794? (The table shows -1.1). When represented in decibel units, the ratio 1-to-12 is -11dB (remember, dB is 10 times the log of the number).

It just so happens that -11dB is the measurement of the allowable attenuation when using Multimode FDDI fiber (MMF). The spec says “-11dB” and now you realize that this means 1/12, or 8% (7.94%) of the original value. (Why don’t they just say 8%? You’ll have to ask an engineer – they have their reasons.) You can convert any “dB” measurement into a percentage by dividing the dB number by 10 and using this as an exponent of 10. Here is the problem in reverse:

What percentage is represented by -11 dB?

-11 dB divided by 10 = -1.1

10 raised to the -1.1 power (use your scientific calculator) = 0.079432823

.079 = (roughly) 8%

Let’s do the same thing for a value of -32dB which is the allowable attenuation in Single Mode Fiber (SMF).

-32 dB divided by 10 = -3.2

10 raised to the -3.2 power = 6.309573445 X 10-4 = .0006309573445

.00063 = (roughly) .06%

OK, here’s a very important thing to remember about the specification for attenuation. When you see that the MMF attenuation is -11dB it means that the signal can be reduced to a level NO LOWER THAN 8% OF THE ORIGINAL VALUE. It does NOT mean that the resultant value has been attenuated BY 8% (which would make the resultant signal level be 92% OF the original – NO; that’s wrong.)

One way, as we’ve just seen, to use the decibel measurement is to simply reference a ratio between the output and input sides of a FDDI circuit. The engineers have to use a ratio (as opposed to an exact number; perhaps a specific number of milliwatts of power) because the power of the transmitted signal varies with the wavelength (color) of the light being transmitted. Higher frequency (shorter wavelength; violet) light has more energy that low frequency (longer wavelength; red) light. Whether a designer wants to use light at a wavelength of 1300 nanometers or at 850 nanometers; that’s up to the designer. The circuit, in any case, will have to guarantee attenuation of no more than 11 dB for Multimode Fiber. 11 dB, as a ratio of input to output, serves as a valid standard for measurement without regard for the specific energies (measured in milliwatts) of the signal.

When the power output of a station is referenced, or the power level of the light signal at any point is measured, the unit of measurement is the ratio of the measured signal as compared to a 1 MILLIVOLT REFERENCE POINT. Now, the fraction that results has the power level (in milliwatts) that is Measured divided by 1 milliwatt. The unit of measurement that results is called a “dBmw” (pronounced by saying the letters “d” and “B” and then the word “milliwatts”). You will notice that when measuring power, the “dBmw” unit of measurement is a representation of the numerator (the top part) of the fraction that produces the ratio. The effect is that a unit of measurement (the “dBmw”) was created that gave engineers a workable unit of measurement (as opposed to fractions like “.0007943 milliwatts”).

Multimode Fiber (MMF) transmitters output a light source at -20 dBmw, which translates to .01 milliwatts (because 10 raised to the -2 power = .01). An MMF receiver must be able to recognize power levels as low as -31 dBmw, which translates to .0007943 milliwatts. Do you recall that earlier we said that the allowable attenuation for MMF was 11 dB? Well, here’s where that ‘11″ comes from: -20 dBmw output at a transmitter which experiences 11 dB attenuation results in -31 dBmw at the receiver – and the receiver must be able to recognize these power levels.

The mathematical basis of decibel measurement is used in two different specifications. Signal attenuation is the decibel representation of the ratio between input and output voltage. Power level is the decibel representation (in DB-Milliwatts) of the ratio between power and a one milliwatt reference power.