Dr. Riki Morikawa, George Mason University
Updated 27May2015

Decibels

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What’s a decibel and why do we use them in telecommunications?

A decibel is a relative measure between two quantities (i.e., a decibel is dimensionless).  The formula for comparing two values in decibel format is:

                Decibel (dB) = 10 log₁₀(X/Y), where X and Y represent the two values you are comparing and log₁₀ refers to the fact that you are working in decimal (base 10) format.

In telecommunications, we typically reference transmitter power to either 1 watt (W) or 1 milliwatt (mW)[1].  Therefore, if your transmitter puts out 30 watts of power referenced to 1 watt, we can convert this into dBs:

                P_TX = 10 log₁₀(30W/1W) = 14.77 dBW

So what is the advantage of working with logs?  Consider the following mathematical relationship:

                log(xyz) = log(x) + log(y) + log(z)

In other words, using decibel values enables you to use simple addition and subtraction when determining values, such as power within a complex multi-stage telecommunication system.  In addition, since we are using logarithmic values, we can deal with very large or very small values more easily.  Consider the following example.

Example 1.  Figure 1 shows that the power input (P_in) into the system undergoes several levels of amplification (shown by the triangles), before it is transmitted through the air.  The signal experiences attenuation loss of 50% before it reaches the receiver.  Determine the power level in watts that reaches the receiver.

Figure 1. Power, P_in, goes through several amplification stages and attenuation prior to reaching the receiver at power P_rx

If P_in = 8.2 mW, what is the power received?

Answer (not taking advantage of decibel format):  P_rx = (((Pin x 2) x 10) x 3) x 50/100 =  246mW

As we see for this very simple example, we are required to multiply, divide and work with percentages.  A more realistic example would involve the inclusion of many more variables to determine signal power and quality (i.e., becomes mathematically complex quickly).

Example 2.  Same as example 2, but this time we work with decibels.  By working with logarithms, we simplify our calculations.

Answer: P_rx (dBmW) = 10log₁₀8.2 + 10log₁₀2 + 10log₁₀10 + 10log₁₀3 + 10log₁₀0.5

= 9.14dB + 3dB + 10dB + 4.77dB – 3dB = 23.91 dBmW

Once the independent variables are converted into decibel format, determining P_rx becomes a trivial addition or subtraction problem.

As a check, using the formula:  P (mW) = 10^(dBmW/10)

                P(mW) = 10^{(23.91dBmW/10)} = 246 mW, which is the same power we calculated in example 1 without using logarithms.

Now let’s say we wish to eliminate the x10 amplifier, and we find that the attenuation that our signal experiences is actually 78% vice 50%.  By using dBs, this becomes a trivial math problem:

                P_rx (dBmW) = 9.14dB + 3dB + 4.77dB + 10log₁₀0.22 =  10.33 dBmW

                or  P(mW) = 10^{(10.33dBmW/10)} = 10.79 mW

Besides simplifying link calculations, there are other reasons for using decibels in telecommunications such as the ability to view large variations on a logarithmic scale, and an ability to compare values to a reference.  The decibel is used widely in telecommunications, so an understanding of what it means is critical.