Library / Signal processing and digital filters / DSP-05
Signal processing and digital filters
Ideal ADC SNR for a full-scale sine
Signal-to-quantisation-noise ratio of an ideal B-bit converter driven by a full-scale sine wave, about 6.02B + 1.76 dB.
- Ref
- DSP-05
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- SNR [dB]
An ideal B-bit converter has 2ᴮ levels and a rounding error uniform over one step, whose RMS is the step divided by √12. A full-scale sine has RMS amplitude 2ᴮ × step/(2√2), so the SNR is 20 log₁₀(2ᴮ√(3/2)), which is about 6.02B + 1.76 dB. Each extra bit adds 6.02 dB.
Signal-to-quantisation-noise ratio of an ideal B-bit converter driven by a full-scale sine wave, about 6.02B + 1.76 dB. It assumes uniform rounding error over one step, no overload, and a noise bandwidth equal to the Nyquist band.
Calculator
- Number of levels, 2ᴮ4,096
- SNR gain per extra bit, 20 log₁₀ 26.021 dB
- Step as a fraction of full scale, 1/2ᴮ2.441×10−4
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.
- An ideal 12-bit converter gives 74.01 dB of SNR on a full-scale sine, from 4,096 levels.
- Real converters fall short of this because of noise and distortion; the effective number of bits is the B that would give the measured SNR, (SNR − 1.76)/6.02.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (dB) | Local step | Change in result |
|---|---|---|---|---|---|---|
| B Number of bits | 1 to 32 bits | 7.782 | 194.4 | 186.6 | +1 bits | +6.02 dB |
Sweep of Number of bits
| Number of bits (bits) | Ideal SNR (dB) |
|---|---|
| 1 | 7.782 |
| 4.875 | 31.86 |
| 8.75 | 55.95 |
| 12.63 | 80.03 |
| 16.5 | 104.1 |
| 20.38 | 122.2 |
| 24.25 | 146.3 |
| 28.13 | 170.3 |
| 32 | 194.4 |
Sweep one variable
Variables and how each one contributes
Ideal SNR
What the equation gives you. Everything else feeds it.
Number of bits
The number of converter bits. Each extra bit adds 6.02 dB to the ideal SNR. Raising it by 10 % (to 13.2) moves the result by +8.14 %, from 74.01 to 80.03 dB.
Worked example with the values above
- Levels: 2^12 = 4,096.
- Factor: 2ᴮ√1.5 = 5,016.55.
- SNR = 20 log₁₀(5,016.55) = 74.008 dB.
Assumptions and limits
- Uniform sampling at a fixed, known rate with a clean, stable clock unless the page names a jitter term.
- Ideal, linear converters and a signal that stays inside full scale.
Common mistakes
- Using the formula for a signal that is not full scale, which lowers the SNR dB for dB.
- Counting the 6.02B + 1.76 dB as measured performance, when real converters have noise and distortion.
- Applying it when the signal bandwidth is less than Nyquist, which improves the in-band SNR.
Statistics and uncertainty
Real converters add offset, gain error, nonlinearity and thermal noise to the ideal quantisation and jitter limits, so a measured SNR is typically 3 dB to 10 dB below the ideal figure. Treat these formulas as upper bounds on performance.
Where it is used
- Generating test signals and predicting what a converter can resolve.
- Choosing a sampling rate, a bit depth and a clock source for an acquisition chain.
Maths you need
- Sampling theorem: a signal bandlimited below fs/2 is fully described by its samples
- Decibel: dB = 20 log₁₀ of an amplitude ratio
- Nyquist frequency: half the sampling rate