ADC quantization error, SQNR and ENOB explained

An analog-to-digital converter (ADC) rounds every sample to one of 2^N levels, so even a perfect converter is wrong by up to half a step. That rounding error, quantization error, sets the best signal-to-noise ratio an ideal N-bit converter can reach: 6.02N + 1.76 dB for a full-scale sine wave. This guide derives that number, shows where it breaks down, and explains how ENOB, oversampling and dither build on it.

Sampling, quantization and LSB size

Digitizing discretizes two axes. Sampling takes snapshots at a fixed rate f_s, discretizing time. Quantization maps each snapshot's amplitude onto a finite set of levels, discretizing amplitude. The two steps fail in different ways.

Sampling is reversible in principle: if the input has nothing above f_s/2, the samples hold everything needed to rebuild it. Quantization is not. Once 1.0003 V is stored as one code, the extra fraction is gone from that sample for good. For the general idea, which also underlies AI model compression, see what is quantization.

LSB size and the quantizer staircase

An N-bit converter splits its full-scale range (FSR) into 2^N equal steps. One step is the least significant bit, written LSB, q or Δ:

FSR is the span of input voltage the converter accepts, usually set by its reference. Because the codes run from 0 to 2^N − 1, the all-ones code sits one LSB below analog full scale.

Mid-tread and mid-rise quantizers

There are two standard ways to place the levels of a uniform quantizer:

A mid-tread quantizer has a flat step centred on zero, so inputs within ±Δ/2 read as exactly zero. With an even number of codes it cannot be symmetric: two's-complement output has one more negative code than positive. A mid-rise quantizer puts a step edge at zero, giving outputs of ±Δ/2, ±3Δ/2 and so on. It is symmetric but has no zero, so a near-silent input toggles between +Δ/2 and −Δ/2.

Quantization error and the uniform noise model

Define the error as e = x_Q − x. For any input inside the range, an ideal rounding quantizer keeps |e| ≤ Δ/2. Outside the range the converter clips and the error is unbounded; that is overload, a separate problem.

To predict the noise a converter adds, engineers model e as a random variable spread uniformly between −Δ/2 and +Δ/2. Its probability density is 1/Δ on that interval, so:

Analog Devices' tutorials reach the same result another way, by treating the error as a sawtooth and averaging its square over one period. The model also treats the noise as white, spread evenly from dc to f_s/2.

When the model is valid

The uniform, signal-independent model is an approximation. It holds when the input is "busy": it crosses many LSBs between samples and has no simple relationship to the sample clock. Analog Devices' MT-229 says it holds in most cases with N > 6 when the input frequency is not an exact submultiple of f_s. It breaks down for slowly changing or very small signals, and for tones locked to the sample rate. In those cases the error repeats with the signal and appears as harmonic distortion rather than a smooth noise floor. That failure is what dither fixes, as covered further down.

Deriving SQNR = 6.02N + 1.76 dB

Ideal signal-to-quantization-noise ratio (SQNR) is quoted for a sine that exactly fills the input range. The derivation takes five lines.

1. The full range is 2^NΔ peak to peak, so the sine's peak amplitude is A = 2^NΔ/2.
2. The RMS of a sine is its peak divided by √2: S_rms = 2^NΔ / (2√2).
3. From the noise model, N_rms = Δ/√12.
4. Divide them. Δ cancels: S_rms/N_rms = 2^N · √12/(2√2) = 2^N · √1.5.
5. Convert to decibels: SQNR = 20·log₁₀(2^N) + 20·log₁₀(√1.5) = N · 20·log₁₀2 + 10·log₁₀1.5.

6.02 = 20·log₁₀2 (each bit doubles the number of levels, halving the noise amplitude). 1.76 = 10·log₁₀1.5 (the ratio of a full-scale sine's power to the noise power at N = 0). The noise is measured over the full Nyquist band, dc to f_s/2.

ENOB: turning a measured SINAD into bits

Real converters add thermal noise, reference noise, clock jitter and distortion. Datasheets capture all of it in SINAD (signal to noise and distortion), measured with a near-full-scale sine. Solving the SQNR equation for N gives the effective number of bits:

If a 14-bit converter measures 70 dB SINAD, its ENOB is (70 − 1.76)/6.02 ≈ 11.3 bits. The extra 2.7 bits of resolution are real codes, but their information is buried in noise. When the test tone is below full scale, Analog Devices' MT-003 adds the shortfall back first: ENOB = (SINAD + 20·log₁₀(full scale / input amplitude) − 1.76)/6.02.

The gap widens at high resolution. With a 10 V range, a 24-bit LSB is 596 nV (MT-010), while a 2.2 kΩ resistor at room temperature produces about 0.6 µV RMS of Johnson noise in 10 kHz, √(4kTRB). Thermal noise alone fills an LSB, so no 24-bit converter reaches 146 dB.

Oversampling and noise shaping

The 6.02N + 1.76 dB figure counts noise from dc to f_s/2. If your signal occupies only a bandwidth BW, a digital low-pass filter can remove the noise above it. The gain, called process gain, is:

f_s/2BW is the oversampling ratio (OSR). Each doubling of OSR adds 10·log₁₀2 = 3.01 dB, which is half a bit.

Oversampling alone is expensive: one extra bit takes 4× the rate, four bits take 256×. MT-001's example samples at 65 MSPS for a 30 kHz channel. That is an OSR of about 1,083 and a process gain of 10·log₁₀(1,083) ≈ 30.3 dB.

Delta-sigma (sigma-delta) converters do better with noise shaping. A feedback loop around a coarse quantizer, often a 1-bit comparator, uses an integrator that passes the signal but pushes the quantization noise toward high frequencies. A digital decimation filter then removes most of that noise. MT-022 plots SNR against OSR: roughly 9 dB per doubling for a first-order loop and 15 dB per doubling for a second-order loop, compared with 3 dB for plain oversampling. Its example is an ideal second-order modulator at OSR 64, reaching about 80 dB (about 13 ENOB). Plain oversampling at the same ratio gains only 18 dB.

Dither: trading noise for resolution

Dither is a small random signal added before quantization so that the error behaves like noise again. Take the 12-bit, 3.3 V converter above with a steady input at 1,241.3 LSB. Without noise it reads 1,241 every time, and averaging leaves a permanent −0.3 LSB error. Add random noise of roughly half an LSB to one LSB RMS, and the output toggles among neighbouring codes: mostly 1,241, often 1,242, sometimes 1,240. The average of many readings moves toward 1,241.3. Each reading is noisier, but the average resolves fractions of an LSB.

Steven W. Smith's DSP Guide notes that the analog noise already present is often enough to act as dither. Elaborate systems use subtractive dither: a known random sequence is added before the ADC and subtracted digitally afterwards, keeping the linearizing benefit without most of the added noise. Audio uses dither for a different goal, removing audible distortion when reducing word length. That is covered in audio bit depth and dither.

To watch the staircase, error and SQNR change with bit depth, try the quantization playground.