SNR to Decibels⚠ unverified
Electrical / Signals · Convert a linear signal-to-noise ratio to decibels
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| snr | SNR | — | 100.0 | Linear SNR |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| snr_db | SNRdB | dB | SNR in decibels |
The science & history
Understanding the Parameters
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SNR (linear) — the dimensionless power ratio from Signal-to-Noise Ratio, $P_s/P_n$. Must be positive. Values $< 1$ mean noise stronger than signal and give negative dB.
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$\mathrm{SNR_{dB}}$ — logarithmic measure. Common anchors: $0\,\text{dB}$ → SNR = 1; $10\,\text{dB}$ → 10; $20\,\text{dB}$ → 100; $30\,\text{dB}$ → 1000. Each +3 dB is roughly ×2 in power; each +10 dB is ×10.
Derivation (Approaching a Proof)
The bel is defined so that a power ratio $P_2/P_1$ corresponds to $\log_{10}(P_2/P_1)$ bels. A decibel is one-tenth of a bel, hence
$$\left(\frac{P_2}{P_1}\right)_{\mathrm{dB}} = 10\log_{10}\left(\frac{P_2}{P_1}\right).$$
Identifying $P_2/P_1$ with the linear SNR $P_s/P_n$ gives
$$\mathrm{SNR_{dB}} = 10\log_{10}(\mathrm{SNR}).$$
Why not 20? If $P \propto V^2$ (fixed resistance), then $10\log_{10}(P_s/P_n) = 10\log_{10}(V_s^2/V_n^2) = 20\log_{10}(V_s/V_n)$. The 20 form is the voltage/current ratio map; it is equivalent only when both sides refer to the same impedance. This calculator’s input is already a power ratio, so the correct prefactor is 10.
Inverse: $\mathrm{SNR} = 10^{\mathrm{SNR_{dB}}/10}$.
History
The bel/decibel system came from early telephone engineering (Bell System, 1920s) to compress huge power ranges into manageable numbers. SNR in dB is now universal in RF, audio, ADC effective-number-of- bits estimates, and link budgets.
Related Concepts: Signal-to-Noise Ratio, THD to Decibels, Total Harmonic Distortion, Watt's Law
Notes: Registry calculator snr-db (unverified). Requires $\mathrm{SNR} > 0$. Domain of
$\log_{10}$ is not checked in the formula display — invalid at zero/negative input.