Thinking in decibels — How Radio Works

Radio has a numbers problem, and it's worth seeing before we solve it. The wave leaving a Wi-Fi router carries about a tenth of a watt. The wave reaching your phone from a tower across town might carry a millionth of a millionth of that: written out, something like 0.000 000 000 000 1 watts. Both numbers live in the same conversation, often in the same sentence. Try doing arithmetic with them and the zeros swallow you whole.

Radio people talk about numbers like these all day, so they invented a kinder way to say them. It looks cryptic from the outside ("−60 dBm") and it is friendlier than almost anything else in engineering once you see the idea. The idea comes in two halves.

Count the tenfolds, not the zeros#

Here is the first half. Instead of writing a huge or tiny number, just say how many times you multiplied or divided by ten to get there. Call each of those a tenfold. A hundred is two tenfolds up from one. A thousandth is three tenfolds down. That whisper reaching your phone across town? Twelve tenfolds below the router's shout. One small counting number instead of a parade of zeros: that's the whole of it.

Multiplying becomes adding#

The second half is why it is beloved rather than merely cute. A radio signal's journey multiplies its strength, step after step: crossing the room divides it by a million, the wall divides it by ten, and then an amplifier in the receiver multiplies what's left by a hundred. Multiplying awkward numbers together is misery.

But count in tenfolds and watch what happens. Divided by a million: six tenfolds down. Divided by ten: one more down. Multiplied by a hundred: two back up. Six down, one down, two up: five tenfolds down, done. Counting tenfolds turns multiplication into adding and subtracting, and a signal's whole journey becomes a ladder you step down and up, one rung per tenfold. (This counting has a grown-up math name, the logarithm, and that is honestly all a logarithm is.)

Ten little steps: the decibel#

Whole tenfolds are coarse rungs, so engineers chopped each one into ten equal little steps. One step is a decibel, written dB (a tenth of a "bel", named for Alexander Graham Bell). So 10 dB is one full tenfold, 20 dB is two tenfolds (a hundredfold), and 30 dB is a thousandfold.

And one lovely accident makes the small steps sing: 3 dB is almost exactly a doubling. Double something ten times and you get 1,024, which is nearly 1,000, which is three tenfolds, which is 30 dB. So ten doublings is 30 dB, and one doubling is 3 dB. That's why "3 dB" is everywhere in radio: +3 dB means twice as strong, −3 dB means half, −6 dB a quarter. When you overhear it, hear "half" and "double" and you'll rarely be wrong.

A place to stand: dBm#

Decibels by themselves only compare: this is 40 dB weaker than that. To name one signal's actual strength you need something to compare against, and radio picked a homely anchor: one milliwatt, a thousandth of a watt. Decibels measured from that anchor are written dBm. So 0 dBm is one milliwatt exactly; +20 dBm is two tenfolds above it, the tenth of a watt a router shouts with; and −60 dBm is six tenfolds below it, about a billionth of a watt, which is a perfectly comfortable Wi-Fi whisper arriving across a room.

Two more landmarks and you can read the whole neighborhood. The ever-present grass of hiss from Where the hiss comes from sits somewhere around −94 dBm for a Wi-Fi-sized channel. And a GPS satellite's voice reaches you near −130 dBm, below the grass, a signal quieter than the hiss it swims in, which should sound impossible; Hiding in the noise is the chapter that explains how it's done anyway.

Now slide. The dBm number strolls along evenly, one polite step at a time, while the written-out wattage goes berserk by tenfolds. Feeling those two motions disagree is the whole lesson of this appendix.

−60 dBm

Written out, zeros and all: 0.000 000 001 watts
Said out loud: about a billionth of a watt
On the ladder: 6 tenfolds below one milliwatt

Signal strength −60 dBm

One ladder, three readouts. The slider moves calmly in dBm while the plain number sprouts and sheds zeros by the tenfold. The rungs are tenfolds; the labeled landmarks run from a shouting router down past the grass to a GPS satellite's whisper.

A journey in subtractions#

Here's the payoff, the everyday sum engineers call a link budget. Will the signal survive the trip? In watts that question is a multiplication nightmare. In decibels it's grade-school arithmetic: start with what the transmitter sends, subtract each toll along the way, and check that what's left still stands clear of the grass. Spreading across the room costs about 60 dB (the inverse-square fading of Fading away, priced in steps). The brick wall from What stops a wave takes about 10 dB more.

A link budget as a staircase: +20 dBm leaves the router, spreading across the room costs 60 dB, the wall costs 10 more, and −50 dBm arrives, still a comfortable 44 dB above the grass. No multiplication anywhere: just steps down a ladder.

That comfortable gap between what arrives and the grass has a name you've effectively already met: the signal-to-noise ratio, spoken in dB like everything else. A fat margin means the receiver can afford the crowded constellations of chapter 20; a thin one means retreating to sturdier stars. Every one of those trade-offs is negotiated in decibels, and The speed limit turns the whole negotiation into a single law.

The one thing to remember#

A decibel is not a mysterious unit; it's a way of counting. Ten dB is one tenfold, each dB is a tenth of a rung, and journeys that would multiply become sums you could do on your fingers. So when the wild reports "−3 dB", hear half. When a Wi-Fi menu shows −60 dBm, hear six tenfolds below a milliwatt: a comfortable whisper. And when someone says a signal sits 40 dB above the noise, picture the ladder: four sturdy rungs of daylight between the message and the grass.