Library / Circuits, electronics and noise / ELN-50
Circuits, electronics and noise
ADC quantisation noise
RMS error from rounding in an ideal converter under the uniform-error model, with the error spread evenly over one quantisation step Δ.
- Ref
- ELN-50
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- vQ,rms [V]
An ideal analogue-to-digital converter rounds each sample to the nearest level, so the error is between −Δ/2 and +Δ/2. If it is uniformly distributed and uncorrelated with the signal, its RMS value is Δ/√12 ≈ 0.289 Δ. This is the quantisation noise floor of the converter. For a very small or exactly repeating input the error is not random and the formula does not hold.
RMS error from rounding in an ideal converter under the uniform-error model, with the error spread evenly over one quantisation step Δ. For a coherent small input the error can depend on the signal.
Calculator
- Largest error, Δ/25×10−4 V
- Full-scale for 12 bits, 4096 Δ4.096 V
- Ideal SNR for a full-scale sine and N bits, 6.02N + 1.76 (12 bits)74 dB
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A step of 0.001 V gives 2.887×10−4 V RMS (0.2887 mV) of quantisation noise, with a largest error of 5×10−4 V.
- Each extra bit halves Δ and so halves the noise, which improves the ideal signal-to-noise ratio by 6.02 dB.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Δ Quantisation step (1 LSB) | 1e-6 to 1 V | 2.887e-7 | 0.2887 | 0.2887 | +10% | +10 % |
Sweep of Quantisation step (1 LSB)
| Quantisation step (1 LSB) (V) | RMS quantisation noise (V) |
|---|---|
| 1e-6 | 2.887e-7 |
| 5.623e-6 | 1.623e-6 |
| 3.162e-5 | 9.129e-6 |
| 1.778e-4 | 5.133e-5 |
| 0.001 | 2.887e-4 |
| 0.005623 | 0.001623 |
| 0.03162 | 0.009129 |
| 0.1778 | 0.05133 |
| 1 | 0.2887 |
Sweep one variable
Variables and how each one contributes
RMS quantisation noise
What the equation gives you. Everything else feeds it.
Quantisation step (1 LSB)
The quantisation step Δ (one LSB), in V. Raises the RMS error in direct proportion. Raising it by 10 % (to 0.0011) moves the result by +10 %, from 2.887×10−4 to 3.175×10−4 V.
Worked example with the values above
- Step: Δ = 0.001 V.
- Divide by √12 = 3.4641.
- vQ,rms = 2.8868×10−4 V.
Assumptions and limits
- White noise with a uniform spectral density over the stated one-sided bandwidth.
- Independent, uncorrelated noise sources, so powers (not voltages) add.
Common mistakes
- Adding noise voltages directly instead of adding their squares.
- Using a two-sided bandwidth where a one-sided one is needed.
- Forgetting that real filters pass more noise than their −3 dB bandwidth: use the noise bandwidth.
Statistics and uncertainty
The formula is an average for a busy, uncorrelated signal. A very small or perfectly periodic signal gives a different error pattern, and dither is often added to make the error random.
Where it is used
- Estimating the noise floor, resolution and signal-to-noise ratio of sensor and measurement circuits.
- Choosing amplifiers for a given source resistance.
Maths you need
- Spectral density: power or voltage squared per hertz
- RMS: the root of the mean square
- Boltzmann’s constant: 1.380649 × 10⁻²³ J/K