The Decibel

Intuition

A scale that counts steps

A camera's aperture is marked in stops. Each stop down halves the light reaching the sensor, each stop up doubles it, and a photographer setting an exposure thinks in steps instead of fractions. Two stops down is a quarter of the light, and the quarter never has to be worked out; it is two steps.

Signal levels behave the same way. An amplifier multiplies the level, a long cable divides it, a filter divides it again over part of its range. What arrives at the far end is the product of every factor along the path, and products of very large and very small numbers are awkward to carry in the head. The decibel replaces each factor with a step on a scale, so the chain becomes a running total.

The logarithm sets the size of the steps. Doubling the power is 3.01 dB. Ten times the power is 10.0 dB, and ten times again is another ten steps up the same scale. A power one millionth of another sits at -60.0 dB, a short figure standing in for six zeros.

Compression is what the scale buys. Hearing spans a range no linear axis handles comfortably, and so do radio reception and laboratory measurement. The decibel is not a unit of power, or of voltage, or of anything else physical. It reports how one quantity compares with another, and until that other quantity has been named, a figure in decibels is unfinished.

Practitioner

Ten for power, twenty for amplitude

The logarithm is base ten, and the factor of ten in front of it is what makes the unit a decibel and not a bel. A ratio above one gives a positive figure and a ratio below one gives a negative figure. Equal powers give zero. The sign carries the direction, so gain comes out positive and loss negative, although an attenuator is normally specified by the size of its loss with the sign left off.

Voltages and currents are amplitudes, and an amplitude ratio carries a different factor:

Doubling an amplitude comes to 6.02 dB, and multiplying one by ten comes to 20.0 dB. Both are twice the figure the same ratio produces in the power form, and that doubling holds right across the scale.

Most bench work runs on a small handful of these steps. 3.01 dB is a factor of two in power and 6.02 dB a factor of two in amplitude; 10.0 dB is a factor of ten in power and 20.0 dB a factor of ten in amplitude. Bigger figures get built by adding those together, which is quicker in the head than a fresh logarithm every time.

Since a decibel figure needs something to compare against, the working scales pin one side of the ratio to a fixed reference and write the reference into the name. dBm is the common one. Its zero is one milliwatt, so a level in dBm is an absolute power, even though the arithmetic underneath is still a ratio.

Because logarithms turn multiplication into addition, stage figures along a chain are added, with losses entered as negatives. A link budget is nothing more than that sum, and instrument front panels are calibrated in the same unit so their readings can join it.

Engineer

The condition hidden in the voltage form

The decibel is defined on a ratio of powers. The amplitude form is derived from that definition, and the derivation carries a condition that is easy to lose on the way.

Power in a resistance goes as the square of the voltage across it:

Take two voltages, turn each into a power through that expression, and divide one power by the other. Each voltage is squared. If the two resistances are equal they cancel, and what survives is the square of the voltage ratio. A square inside a logarithm comes out as a factor of two in front of it, so ten times the logarithm of the power ratio becomes twenty times the logarithm of the voltage ratio. The twenty is not a second convention. It is the ten, with a square lifted out of the logarithm.

That cancellation is the condition. Both voltages have to appear across the same impedance for the two forms to agree, and a great deal of electronics is not built that way.

Neither figure in that example is wrong. They answer different questions, and "how many decibels?" stays incomplete until it says which question was asked. The two expressions above are kept apart for that reason, one for a ratio of powers and one for a ratio of amplitudes, so that any calculation has to commit to a domain. A bare decibel figure with nothing said about which domain it belongs to is the error the separation is there to catch.

Where the condition does hold, one figure serves both readings. Radio and instrumentation work is built around a single system impedance, commonly 50 Ω, and every node in the chain meets that value, so a voltage ratio and a power ratio produce the same decibel figure and nobody has to say which was meant. Audio and most low-frequency electronics gave up matched impedances long ago. A source of a few tens of ohms drives an input of tens of kilohms on purpose, so that the voltage transfers and almost no power does, and the decibel figures quoted there are amplitude figures by convention.

The impedance condition is not the only thing the arithmetic takes for granted. The two quantities have to be measured the same way: on AC that means RMS values at both ends, since a peak reading at one end against an RMS reading at the other builds the gap between two conventions into the answer. The logarithm is undefined at zero and for negative arguments, so a decibel figure says nothing about polarity or phase; an inverting stage and a non-inverting one of the same size are identical in decibels, and what separates them shows up on the phase axis of a Bode plot. Adding stage figures assumes each stage still behaves as it did when its own figure was measured, and that assumption breaks the moment one stage loads the next.

Professional

Reference levels and the suffixes that name them

A suffix on the unit names the reference and turns a ratio into a reading. dBm counts from one milliwatt of power. dBV counts from one volt of amplitude. dBu counts from the voltage that would dissipate that same milliwatt in 600 Ω, which comes to 774.6 mV, a figure inherited from telephone practice and now used with no impedance implied at all. dBFS counts down from a converter's full-scale code and is never positive. dBµV is the form used in interference measurement, where the limit values belong to whichever standard applies to the product and the market it is sold into.

Turning a dBm figure into a voltage needs the impedance stated, and the answer moves with it. Zero on the dBm scale is 223.6 mV across 50 Ω and 774.6 mV across 600 Ω. The two scales most easily confused are dBu and dBV, which sit a fixed distance apart: zero on the dBu scale reads -2.22 dB on the dBV scale, and a console meter marked in one against a datasheet written in the other needs that shift applied before the numbers mean the same thing.

The round figures are approximations, and it pays to know by how much. A factor of two is 3.01 dB, not three, and the rounding accumulates once a long cascade is totalled. The same caution applies to the half-power point of a filter, which sits at the cutoff frequency and is quoted as a round three decibels almost everywhere.

Decibels also turn up with a denominator attached. A filter's skirt is quoted in decibels per decade or per octave, which is a slope on a Bode plot rather than a level, and an antenna's gain in dBi is referenced to an isotropic radiator. Reading either as a plain level misreads the graph it came from.

Instruments carry their own assumptions. A spectrum analyser displays dBm and arrives at that figure from a voltage at its input together with its own input impedance, so a reading taken through an unrecorded pad or an impedance the instrument does not expect is wrong by whatever those contribute. An oscilloscope shows volts, and converting a scope reading to dBm needs a load impedance supplied by the user. Record the reference and the impedance beside any decibel figure that will be quoted later.

Part of why the scale survives is that it suits perception. Loudness follows something closer to the logarithm of the stimulus than to the stimulus itself, so equal decibel steps sound like roughly equal changes. The smallest change a listener can detect is often quoted as about one decibel, but that figure moves with the programme material and with the listener, and it is a rough guide only.

Signal-to-noise ratio, noise figure and dynamic range are all decibel figures, which is how a receiver's noise floor and a transmitter's output can be compared on one axis. The decibel is a ratio and not an SI unit, and the conventions for writing and using it are set by standards that differ a little between fields; look up the one that governs your work before a figure goes into a report.