Before this:Sampling & quantization
Decibels & dynamic range
Key takeaways A decibel (dB) is a logarithmic ratio: 10·log₁₀ of a power ratio, or 20·log₁₀ of an amplitude ratio. Logs turn multiplication into addition, so chained gains just add. For sampled signals we measure level in dBFS (relative to full scale), and the dynamic range a format can hold is set by its bit depth — roughly 6 dB per bit.
Radio spans a colossal range of signal strengths, and DSP code juggles gains at every stage. Decibels are how we keep that sane. This lesson builds on sampling & quantization and mirrors the RF path’s decibels lesson from the DSP side.
Why a logarithmic scale
The weakest signal you can still decode and the strongest one the front end can handle differ by factors of millions to trillions. Writing those as plain numbers is unwieldy, and gains multiply as they chain through a receiver. Taking a logarithm fixes both: it compresses a huge range into small numbers, and it turns each multiply into an add.
gain of 100x -> +20 dB
gain of 2x -> +3 dB (approx)
loss of 1/2 -> -3 dB
chain them: 100x then 2x -> 20 dB + 3 dB = 23 dB
Power vs amplitude: the 10 and the 20
There are two forms of the dB formula, and the difference trips people up:
| Quantity | Formula | Because… |
|---|---|---|
| Power ratio | dB = 10·log₁₀(P₁/P₀) | power is the base definition |
| Amplitude ratio | dB = 20·log₁₀(A₁/A₀) | power ∝ amplitude², and log of a square doubles the factor |
So doubling the amplitude of a signal is +6 dB, while doubling its power is +3 dB. Both describe the same physical change viewed through different quantities. A few values worth memorising:
+3 dB ~ 2x power +6 dB ~ 2x amplitude
+10 dB = 10x power +20 dB = 10x amplitude / 100x power
0 dB = no change (ratio of 1)
dBFS: the digital yardstick
For a signal that lives as numbers, the natural reference is the largest value the format can hold — full scale. Level measured against that is dBFS (dB relative to full scale). Full scale is 0 dBFS; every real sample sits below it, at a negative value. A signal peaking at −6 dBFS is using half the available amplitude; one that reaches 0 dBFS is on the edge of clipping, where samples slam into the limit and the waveform is destroyed — a failure mode covered in front-end overload.
Dynamic range and bit depth
Dynamic range is the span between the loudest signal a format can represent and the quantization-noise floor beneath it. It is set directly by bit depth: each extra bit doubles the number of levels, which is ~6 dB more range.
That is why an 8-bit SDR (about 48 dB) can be swamped by a strong nearby signal that a 12-bit unit (about 72 dB) handles, and why keeping the input well below 0 dBFS — but not so low it sinks into the noise floor — is the goal of gain staging.
Quick check: how much does adding one bit of depth increase dynamic range?
Recap
- A decibel is a log ratio: 10·log₁₀ of power, 20·log₁₀ of amplitude.
- Logs turn multiplied gains into added dB — the reason receiver budgets are done in dB.
- dBFS measures a sampled signal against full scale; 0 dBFS is the clipping edge.
- Dynamic range ≈ 6 dB per bit of depth, from full scale down to the noise floor.
Next up: the single most useful way to look at a signal — its frequency content.
Frequently asked questions
Why use decibels instead of plain ratios?
Signal strengths in radio span an enormous range — the strongest signal can be a trillion times more powerful than the weakest you can still decode. Decibels compress that range onto a manageable scale and turn multiplication into addition, so a gain of 100x becomes simply adding 20 dB. Both make signal-level bookkeeping far easier.
What is dBFS?
dBFS means decibels relative to full scale — the loudest level a sampled signal can represent before it clips. Full scale is 0 dBFS, and every real sample sits below it at a negative dBFS value. It is the natural yardstick for digital signals because it is referenced to the number format itself, not to any physical power.