5 - Introducing the Decibel

We understand the problem. We are measuring our wireless power in milliwatts, yet milliwatts decrease exponentially as the wireless signal propagates. Therefore, we need a more linear way to measure it. The decibel (dB) will be our method for linearly measuring our wireless signal. However, what exactly is a decibel, and how are we going to use it? To fully grasp this, we'll need to delve deep into our math knowledge, perhaps revisiting high school or even college mathematics and recalling the concept of a logarithm.

Let's start by examining the logarithm as a foundation for understanding why we use decibels in Wi-Fi space. Now, the term 'decibel' is interesting because 'deci,' spelled as D-E-C-I, signifies one-tenth (1/10) of something, much like a decimeter. Although we more commonly use centimeters and millimeters, 'deci' means one-tenth. So, when we talk about a decibel, we're referring to one-tenth of a Bel. The 'Bel,' or B-E-L, is named after Alexander Graham Bell, credited with inventing the telephone.

You might have heard of decibels, often used, especially in sound-related contexts. This is because, just as a wireless transmitter propagates signals, an audio transmitter, headphones, or a speaker system emits sound radially, with energy levels like an access point's wireless signal.

Therefore, we can apply the same concepts whether discussing audio noise or wireless signal strength. The reason decibels are effective from a linear perspective is that they are based on a logarithmic function.

However, to comprehend the origins of decibels fully and properly, we must delve into the topic of logarithms. Essentially, a logarithm answers the question: How many times must we multiply a particular number by itself to obtain another number? Let's illustrate this concept with a straightforward example. Consider log base 3 of 27 (Log₃27). This question asks, 'How many times must we multiply 3 by itself to arrive at 27?' We can see those 27 results by multiplying 3 by 3, which is 9, and then by 3 again, resulting in 27 (3 x 3 x 3). Therefore, the answer to this mathematical equation is three instances of the number 3.

As we delve into networking, we frequently encounter binary systems. Consequently, we might find it more relatable to use log base 2 in our discussions. For instance, consider log base 2 of 8 (Log₂8). Knowing that 2 to the power of 3 (2³) equals 8, we conclude that we need three instances of the number 2 multiplied together to obtain 8. The answer here is also 3. Let's examine another binary number: log base 2 of 32 (Log₂32). Recognizing that 32 equals 2 to the fifth power (2⁵), we determine that the answer is 5.

The key point is this: as we progress in numbers, our linear count increases, while the background number grows exponentially. We transitioned from 8 to 32, a fourfold increase in value, yet our count went from 3 to 5 in a linear manner. For example, log base 2 of 16 equals 4, falling right in the middle. Thus, while 2 increases exponentially, our logarithmic function progresses from 3 to 4 to 5.

Now, as we return to the topic of wireless and decibels, we'll primarily be discussing a special type of logarithm known as log base 10.

So, log base 10 is something that's much more intuitive to us because we count in the base 10 mathematical system. If I say log base 10 of 100 (Log₁₀100), for example, how many 10s do I need to get to 100? The answer here is 2.

In fact, log base 10 is such a special and commonly used mathematical operation that we often omit the '10' part. For instance, if I were to ask, 'What is the log of 1,000?' Since I haven't specified the base, it's understood to be log base 10.

We know that we need three 10s (10 x 10 x 10) to reach 1,000. This demonstrates how the number can exponentially increase. For instance, 10,000 is 4, 100,000 is 5. Even though the starting number is small and grows exponentially larger, the logarithmic side of things also increases exponentially.

So, ultimately, why do we care? How will exponential increase affect us in the Wi-Fi world? Let's revisit the example we mentioned earlier: if my access point is sending out a signal, I might find that the signal is at 0.000001 milliwatts. This can be represented by a much more readable number: -60 dBm. It's negative because we've gone below one; anything below one is represented by a negative value. Now, if I have a stronger signal, let's say 0.0001, it would be -40 dBm. On the other hand, with a weaker signal at 0.00000001, we're at -80 dBm. So, we see a much more linear progression as we go from -40 to -60, and then from -60 to -80. This format is much more human-readable than dealing with all these fractions and numerous zeros.

Furthermore, this approach not only makes it more readable but also helps us understand how signal strength changes as we move farther away from an access point. Logarithms convert our exponential changes into something very linear. Decibels are a logarithmic function that allows us to measure our signals.

Happy Learning!!