Thinking in Decibels

Why Wireless Refuses to Use Plain Numbers

Your phone transmits at roughly 0.2 watts. The weakest signal it can still turn back into a phone call is about 0.0000000000001 watts — a tenth of a picowatt. Those two numbers describe the same radio, and they differ by a factor of two trillion. Try writing a page of engineering calculations in which some quantities have twelve zeros in front of them and others have none, and you will make an arithmetic slip before you reach the bottom of the page.

That is the first reason wireless engineers stopped using plain numbers. The second reason is better. A radio link is not a sum of powers — it is a product of ratios. The amplifier multiplies the power by 4. The cable divides it by 1.6. Free space divides it by ten billion. The receive antenna multiplies it by 2. To find out what arrives, you multiply five numbers, three of which are awkward.

Logarithms convert multiplication into addition. That is their whole job, and it is exactly the job wireless needs done. Express every gain and every loss as a logarithm and the five multiplications become five additions you can do in your head. The unit that carries that logarithm is the decibel.

The name honours Alexander Graham Bell. The original unit, the bel, was simply log₁₀ of a power ratio, and it turned out to be inconveniently large: almost every useful number came out as a fraction. So the working unit became one tenth of a bel — a decibel — which is where the factor of 10 in the formula comes from. One dB is a power ratio of about 1.26, roughly the smallest change in loudness a person notices, which is a pleasant accident rather than a design goal.

The Other Formula, and Why the Factor Differs

You will also meet dB written with a 20 in front of the logarithm instead of a 10. This trips up more beginners than any other part of the subject, and the reason is not arbitrary convention. It follows from one fact you already know from Lesson 1 of this module: power goes as amplitude squared.

If you are handed a ratio of voltages, or field strengths, or any other amplitude, then the corresponding power ratio is that number squared. And the logarithm of a square is twice the logarithm. So the 20 is not a second definition of the decibel — it is the same definition with the squaring already carried out.

Check it with numbers. Double a voltage and the power quadruples. Via the amplitude formula: 20 log₁₀ 2 = 6.02 dB. Via the power formula: 10 log₁₀ 4 = 6.02 dB. The same physical change, the same answer. If your two routes disagree by a factor of two in dB, you have used the wrong formula for the quantity in your hand.

A Decibel Is a Ratio, Not a Quantity

This is the idea to hold on to, and it is the one most often lost. A decibel value never tells you how much of anything there is. It only tells you how much more, or how much less, than something else. There is a division sign inside the formula and it never goes away.

So "30 dB" means "a thousand times" — it does not and cannot mean "a thousand watts". If an amplifier is described as having 30 dB of gain, that describes what it does to whatever you feed it: a microwatt in becomes a milliwatt out, and a watt in becomes a kilowatt out. The gain figure alone tells you nothing about the power level, and a spec sheet that quotes only a gain has not told you how big the signal is.

This is also why some dB figures are negative and nothing is wrong. A negative dB value simply means the ratio was less than one: the output is smaller than the input. Cable loss, path loss and filter attenuation are all naturally negative gains, and writing them as negative numbers is what lets you add everything up in one pass instead of tracking which terms multiply and which divide.

Giving the Ratio a Reference

If a decibel is only ever a ratio, how does a data sheet manage to quote an absolute power in dB? By fixing the denominator and saying so in the name of the unit. Once P₂ is pinned to a specific, agreed quantity, the ratio pins down a real number — and the letter tacked onto "dB" tells you which quantity was chosen.

dBm is the one you will meet most. Its reference is 1 milliwatt, which suits the power levels of handsets and Wi-Fi radios. A phone transmitting at 23 dBm is putting out about 200 mW; a Wi-Fi access point at 20 dBm is putting out 100 mW; a receiver quoted as sensitive to −100 dBm can work with a tenth of a picowatt. The transmit and receive figures of a single phone therefore span 123 dB, which is that two-trillion-to-one ratio from the opening paragraph written in four characters.

dBW uses 1 watt instead. Broadcast and satellite engineers prefer it because their power levels are large, and converting is trivial: 1 W is 1000 mW, and 1000 is 30 dB, so dBm = dBW + 30. A 50 kW FM transmitter is 47 dBW, or 77 dBm.

dBi is the one Module 7 is built on, and its reference is not a power but an antenna. An isotropic radiator is an imaginary antenna that spreads power perfectly evenly in every direction — it cannot be built, which is exactly why it makes a good yardstick: everybody's dBi figures are measured against the same fiction. An antenna quoted at 6 dBi concentrates power four times more densely, in its best direction, than an isotropic radiator fed the same power. The plain half-wave dipole, the simplest antenna worth building, comes out at 2.15 dBi.

Six Anchors Worth Memorising

Because decibels add, six memorised values will get you through most of a conversation without a calculator. Combine them by adding: ×20 is ×2 followed by ×10, so it is 3 + 10 = 13 dB. ×4 is two doublings, so 6 dB. ×5 is ×10 then ÷2, so 10 − 3 = 7 dB.

One honest caveat about the most popular of those six. Doubling is not exactly 3 dB. It is 10 log₁₀ 2 = 3.0103 dB, and the difference matters as soon as you stack the approximation. Use "3 dB is double" four times in a row and you arrive at ×16 where the true answer for 12 dB is ×15.85 — roughly 1% out. That is entirely fine for a sanity check on a whiteboard and not fine for a specification. The 10 dB and 20 dB anchors carry no such error: they are exact, because the decibel is defined by base-10 logarithms.

Cashing In: The dB Figures You Have Already Seen

WRL-101 has been quoting decibels since Module 2, Lesson 1. Every one of them now decodes.

The "3 dB bandwidth" of a signal or a filter, mentioned in M2-L1, is the width of the band measured between the two points where the power has fallen to half of its peak. Half power is −3 dB, which is why the definition is stated that way rather than as "the half-power bandwidth". In amplitude terms the same points sit at 1/√2 ≈ 0.707 of the peak voltage, and 20 log₁₀ 0.707 = −3.01 dB — the two formulas agreeing again, as they must.

The SNR range of 0 to 30 dB quoted in M2-L3 says that practical cellular links run with the wanted signal anywhere from equal to the noise power (0 dB, a ratio of 1) up to a thousand times the noise power (30 dB). Simple modulation survives near the bottom of that range; the dense constellations that carry high data rates need to be near the top. Module 9 turns this into a design constraint.

The 18.75 dB figure in M4-L2 is the clearest worked example in the course. FM's noise advantage over AM is a power ratio of 3β², and for broadcast FM with β = 5 that is 3 × 25 = 75. So the improvement in dB is 10 log₁₀ 75. You can get there with the anchors alone: ×100 is 20 dB, and 75 is three quarters of 100, which costs 10 log₁₀ 0.75 = −1.25 dB. That gives 20 − 1.25 = 18.75 dB, which is the printed figure to the last digit.

The 3 dB capture advantage in M4-L3 means that one FM signal only needs twice the power of a competing one for the limiter to suppress the weaker completely. Written as a ratio it sounds like a lot; written as 3 dB it sounds like the small margin it really is.

A Link Budget in One Line

Here is the payoff, and it is the reason this lesson comes before Modules 7, 8 and 9. Take a short radio link: a transmitter putting out 20 dBm, a feed cable that loses 2 dB, a transmit antenna with 6 dBi of gain, 100 dB of path loss between the two ends, and a receive antenna with 3 dBi of gain.

That is the entire link budget: −73 dBm arrives at the receiver, and you can do it in your head. Now do the same sum in watts. Start with 100 mW, divide by 1.585 for the cable, multiply by 3.981 for the antenna, divide by 10¹⁰ for the path, multiply by 1.995 for the receive antenna, and you get 5.01 × 10⁻⁸ mW — that is 50 picowatts, which is the same answer. It took five multiplications, four irrational-looking constants, and eleven decimal places to reach a number that the dB version reached with four additions.

Notice how the units behaved in that sum. dBm plus and minus a pile of plain dB values gives dBm, because the reference never changed — only the ratio did. Adding two dBm values together, by contrast, is meaningless: it would divide by a milliwatt twice. This is a genuinely useful sanity check on any link budget. Exactly one term should carry a reference letter, and it should be the one you started from.