Electronics & General Electrical
ADC Quantization Error & SNR Calculator
Find the quantization step, the ±½ LSB error and the theoretical SNR of an ideal ADC, and see the code a given input voltage produces.
About this calculator
Quantization is the price of turning a continuous voltage into a finite number. A converter reports the nearest of its 2^N codes, so the value it gives back differs from the true input by up to half a step. That is the quantization error, and for an ideal converter it is the only error there is.
Half an LSB does not sound like much until it is compared with the signal. The standard figure for an ideal converter driven by a full-scale sine wave is SNR = 6.02·N + 1.76 dB — about 74 dB at 12 bits and 98 dB at 16 bits. Both terms are worth understanding. The 6.02·N says every extra bit halves the error and buys about 6 dB. The 1.76 dB is 10·log₁₀(3/2), and it comes specifically from comparing the RMS of a full-scale sinusoid against the RMS of a quantization error spread uniformly across one LSB. Quote that formula for a different signal or a different amplitude and it is simply the wrong number.
That last point matters more than it looks. Quantization noise does not shrink when the signal does — the step size is fixed by the reference and the bit depth, not by how hard you drive the input. So every dB of headroom you leave unused is a dB of signal-to-noise ratio given away. A 16-bit converter driven to a tenth of its range performs like a converter with about 3 bits fewer. Matching the sensor's span to the converter's range is one of the highest-value decisions in a measurement chain, and this calculator shows the cost of getting it wrong.
Enter an input voltage and the tool also reports the actual code, the voltage that code represents when converted back, and the real error for that specific input — which is somewhere between zero and half an LSB, not always the worst case.
Design notes & common mistakes
- Match the sensor's span to the converter's range. Quantization noise is set by the step size, not by the signal, so every dB of unused range is a dB of SNR thrown away — a 16-bit converter driven to a tenth of its range performs like a 13-bit one.
- The 1.76 dB term is not a fudge factor: it is 10·log₁₀(3/2), and it belongs specifically to a full-scale sine wave compared against error spread uniformly over one LSB. Quoting 6.02N + 1.76 for a DC measurement or a noise-like signal is quoting the wrong number.
- An input at exactly full scale clips. With the LSB defined as FSR/2^N there are 2^N codes numbered 0 to 2^N − 1, so the top code represents one LSB below full scale and there is no code for full scale itself.
- Dynamic range and SNR are different quantities and are routinely confused. Dynamic range is 20·log₁₀(2^N) — 72.2 dB at 12 bits — while ideal SNR is 6.02N + 1.76, which is 74.0 dB. The 1.76 dB gap is the sine-wave-versus-uniform-noise comparison.
- A truncating converter has an error of 0 to −1 LSB rather than ±½ LSB. The uncertainty is the same width, but it carries a half-LSB DC offset that rounding does not.
Assumptions
- The quantizer is ideal and uniform: every one of the 2^N steps is exactly the same width, and the converter rounds to the nearest code.
- Quantization error is treated as uniformly distributed across one LSB, which is the standard basis of the 6.02N + 1.76 dB figure.
- The SNR figure assumes a full-scale sine wave; the derated figure assumes a sine wave of the stated amplitude.
- The full-scale range is the reference voltage for a unipolar input, or twice it for a bipolar one.
When to use this calculator
Appropriate for
- Finding the quantization step and worst-case error of a converter at a given resolution and range
- Estimating the theoretical SNR ceiling a bit depth sets
- Quantifying how much SNR is lost when a signal does not fill the converter's range
- Checking what code a particular input voltage should produce
- Teaching or learning why matching sensor span to converter range matters
Not suitable for
- Predicting a real converter's performance, which needs its datasheet SNR, SINAD, THD and effective-number-of-bits figures rather than this ideal ceiling
- Budgeting overall measurement noise, which is a full study spanning sensor, amplifier, reference and converter
- Assessing performance at high signal frequencies, where aperture jitter dominates and this static model says nothing
- Analysing an oversampled or noise-shaped converter, whose in-band noise is deliberately not this figure
Engineering use
Intended use. Educational calculation of quantization error and theoretical signal-to-quantization-noise ratio for an ideal uniform quantizer.
Applicable for
- Quantization step size and worst-case ±½ LSB error, in volts and as a share of full scale
- The ideal 6.02N + 1.76 dB SNR figure for a full-scale sine wave
- The SNR penalty for a signal that does not fill the range
- The code, reconstructed voltage and actual error for a specific input
Does not account for
- Differential and integral non-linearity, which make real steps unequal
- Thermal, reference and amplifier noise ahead of the converter
- Aperture jitter, which dominates at high signal frequency
- Harmonic distortion and spurious tones, which SNR alone does not capture
- Oversampling and noise shaping, which move noise out of the band of interest
- Correlation between the signal and the quantization error at very small amplitudes
Verification required
- Confirm the device's datasheet SNR or SINAD before treating this ideal figure as achievable
- Confirm whether the converter rounds or truncates, which shifts the error window by half a step
- Confirm the input range is unipolar or bipolar, and the actual signal amplitude presented
What this calculator does not cover
- Gives the theoretical ceiling for a perfect converter. A real device never reaches it: differential and integral non-linearity make the steps unequal, and its effective number of bits is always below the nominal resolution.
- The uniform-error assumption breaks down for very small or highly periodic signals, where quantization error becomes correlated with the input and appears as harmonic distortion rather than as noise.
- Does not include thermal noise, reference noise or the noise of the amplifier stages ahead of the converter, any of which can exceed the quantization contribution entirely.
- Does not model aperture jitter, which dominates at high signal frequencies and is often the real limit on a fast converter's performance.
- Says nothing about distortion, spurious-free dynamic range or the effect of oversampling and noise shaping, all of which change the achievable figure in either direction.
- The 6.02N + 1.76 dB relation is specific to a full-scale sine wave and is not the correct figure for a DC measurement, a noise-like signal or any other waveform.
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
What is the SNR of a 12-bit ADC?
74.0 dB for an ideal converter driven by a full-scale sine wave, from SNR = 6.02·N + 1.76 dB. A 16-bit converter gives 98.1 dB on the same basis. These are ceilings — a real device's non-linearity and noise always put its effective number of bits, and therefore its SNR, below the nominal figure.
Where does the 1.76 dB in the SNR formula come from?
It is 10·log₁₀(3/2). The relation compares the RMS of a full-scale sine wave against the RMS of a quantization error assumed uniformly distributed across one LSB, and 1.76 dB is what that particular pairing produces. It is specific to a full-scale sinusoid, so quoting the formula for a DC measurement or a noise-like signal gives the wrong answer.
What is the quantization error of an ADC?
For a converter that rounds to the nearest code, up to half a step in either direction — ±½ LSB. At 12 bits on a 3.3 V reference the step is 0.806 mV, so the worst-case error is ±0.403 mV. A converter that truncates instead has an error of 0 to −1 LSB: the same width of uncertainty, offset by half a step.
Why does SNR drop when the signal doesn't fill the ADC range?
Because quantization noise is fixed by the step size, which depends on the reference and the bit depth rather than on the input. A smaller signal is measured against the same noise, so the ratio falls by 20·log₁₀ of the fraction used. Driving a converter to a tenth of its range costs 20 dB — roughly three bits of the resolution you paid for.
Is dynamic range the same as SNR for an ADC?
No, and they are often confused. Dynamic range is the ratio of full scale to one LSB, 20·log₁₀(2^N), which is 72.2 dB at 12 bits. Ideal SNR is 6.02·N + 1.76 dB, which is 74.0 dB. The 1.76 dB difference comes from comparing a sine wave's RMS against uniformly distributed quantization error rather than comparing two amplitudes directly.
References
- Standard uniform-quantisation relations: step size FSR/2^N, worst-case error ±½ LSB, and the ideal full-scale-sine SNR of 6.02·N + 1.76 dB, where 1.76 dB is 10·log₁₀(3/2).
- Walt Kester (ed.), The Data Conversion Handbook, Analog Devices.
- IEEE Std 1241, Standard for Terminology and Test Methods for Analog-to-Digital Converters, cited by identity for the definitions of SNR, SINAD and effective number of bits.
Part of a study
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