Web Exclusive

# Let’s Get Technical: The Math Behind the Mystical Decibel of Fiber Optics In my print column this month, “When a Loss Is Positive,” I discussed the confusing definition of decibel (dB) as used in various international fiber optic standards. Even though dB is used in practically every standard for fiber optics that includes measurements, it is frankly confusing to many fiber techs. I thought I might try to make it more understandable.

What is a decibel?

The term decibel is named for Alexander Graham Bell, the inventor of the telephone. It was developed as a measurement for audio signals. You have probably heard of noise levels expressed in decibels—conversations at about 60 dB, the sound of a jet plane passing overhead or a loud rock concert at around 120 dB. The same term works for audio power, radio signal power, optical power and many other measurements of signals that have large dynamic ranges.

The concept of a decibel is not, however, something our brains naturally understand. We don’t think in logarithms or decibels, although once you get the concept, it becomes easier (even if it’s not exactly second nature). So, let’s delve into the world of dB.

You have probably seen graphs like this that show the relationship of optical power measured in decibel-milliwatts (dBm), or dB in reference to 1 milliwatt (mW; 1/1,000 watt) power, to a linear measurement of power in milliwatts. Notice how the graph compresses a range of 0.001 mW (that’s 1 microwatt) up to 100 mW, a total range of 100,000 times, to simply -30 to +20 dBm or 50 dB. That’s what logarithms are good at—compressing data into a more usable form.

In the early days of fiber optics, measurement equipment was adapted from optics, where many light measurements were made in watts. But few fiber measurements are measurements of just power, mainly the output of laser sources in a transmitter or the input signal at a receiver. More measurements are made on the cable plant, splices and connectors to determine the loss. Loss measurements make no sense in watts, so the decibel, a term already used in many other fields, was adopted.

Decibels are great at expressing change and are even better for large changes. A decibel measures a ratio of powers and compresses it with logarithms. The graph above shows a range of power of 100,000 to 1, more easily expressed as 50 dB. That’s how logarithmic compression works. The equation for dB looks like this: As I explain below using math, a change of 10 dB means the signal was changed by 10 times. Add another 10 dB and you multiply the signal by 10, so 20 dB means the signal was changed by 100 times, 30 dB means it was changed by 1,000 times, and so on.

I can create a reference table of decibel-milliwatts, which is decibels calculated by the equation above, but with the reference power always being 1 mW—showing the actual values in mW in the graph above.

 dBm mW dBm mW 0 1.000 0 1.000 0.1 1.023 -0.1 0.977 0.2 1.047 -0.2 0.955 0.3 1.072 -0.3 0.933 0.4 1.096 -0.4 0.912 0.5 1.122 -0.5 0.891 0.6 1.148 -0.6 0.871 0.7 1.175 -0.7 0.851 0.8 1.202 -0.8 0.832 0.9 1.230 -0.9 0.813 1 1.259 -1 0.794 2 1.585 -2 0.631 3 1.995 -3 0.501 4 2.512 -4 0.398 5 3.162 -5 0.316 6 3.981 -6 0.251 7 5.012 -7 0.200 8 6.310 -8 0.158 9 7.943 -9 0.126 10 10 -10 0.1 20 100 -20 0.01 30 1,000 -30 0.001 40 10,000 -40 0.0001 50 100,000 -50 0.00001 60 1,000,000 -60 0.000001

But remember, most fiber optic measurements of decibels are measurements of optical loss. The table below is the same as the table above, but it shows the ratio of power for decibel differences in power.

 dB (gain) Power ratio dB (loss) Power ratio 0 1.000 0 1.000 0.1 1.023 -0.1 0.977 0.2 1.047 -0.2 0.955 0.3 1.072 -0.3 0.933 0.4 1.096 -0.4 0.912 0.5 1.122 -0.5 0.891 0.6 1.148 -0.6 0.871 0.7 1.175 -0.7 0.851 0.8 1.202 -0.8 0.832 0.9 1.230 -0.9 0.813 1 1.259 -1 0.794 2 1.585 -2 0.631 3 1.995 -3 0.501 4 2.512 -4 0.398 5 3.162 -5 0.316 6 3.981 -6 0.251 7 5.012 -7 0.200 8 6.310 -8 0.158 9 7.943 -9 0.126 10 10 -10 0.1 20 100 -20 0.01 30 1000 -30 0.001 40 10000 -40 0.0001 50 100000 -50 0.00001 60 1000000 -60 0.000001

Compare the positive and negative decibel values across the rows. The ratio of a positive dB is the inverse of the negative dB, e.g., +10 dB is a ratio of 10 times and -10 dB is a ratio of 1/10. Thus 10 dB is a ratio of 10 times: +10 dB means the power measured is 10 times greater than the reference power and -10 dB means the power is one-tenth as much. Some of the numbers are easy to remember and may be useful. For example, +3 dB is a factor of two in power and -3 dB is a factor of one-half.

When the two optical powers compared are equal, dB = 0, a convenient value that is easily remembered. If the measured power is higher than the reference power, dB will be a positive number, but if it is lower than the reference power, it will be negative. Thus, measurements of loss are properly expressed as negative numbers.

Now for the curious, let’s do some math

Let’s start with the equation that defines dB that should be familiar to most of you, the equation for attenuation in fiber optics: Let’s do some simplification. First manipulate the equation to get the 10 over to the left side of the equation by dividing both sides by 10. Now we need to deal with what is a logarithmic function (log). A logarithm is the exponent or “power” to which a base must be raised to yield a given number, for example: Based on that, we can further manipulate the equation above to get the equation expressed as 10 raised to the power of dB/10. So, if we convert 20 dB this way, showing it step by step. Thus 20 dB means the ratio of measured power to reference power is 100:1. Likewise, 10 dB is ratio of 10:1 and 30 dB is a ratio of 1,000:1.

Now there is one more thing to learn about logarithms: they can be positive or negative numbers. Consider this where dB is negative: So, if dB is negative, that means ratio of measured power to reference power is less than 1—the measured power is less than the reference power, or in fiber optic terms, we are measuring a loss. This is true unless, as has been done in some international standards, you reverse-measured any reference power in the equation so that loss is positive, and gain is negative. But why would you want to do that?