LSB Dither RMS
RMS amplitude and quantization-noise floor for uniform (flat-PDF) dither in an ADC or DAC LSB.
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The engineering
Dither is a small controlled noise added before quantization so that quantization error stops correlating with the signal — it trades a distinct distortion tone for a smooth, benign noise floor. For a uniform (RPDF) dither spread over one LSB peak-to-peak the RMS is LSB/√12, exactly matching the quantizer's own noise; that's the amount that just decorrelates the error mean.
Reach for triangular (TPDF) dither, ~2 LSB p-p total, when you want both the mean and the variance of the error independent of the signal — it costs about 4.8 dB of noise floor but fully whitens the residual. Rule of thumb: 1-LSB RPDF fixes visible contouring; 2-LSB TPDF is the audiophile-grade fix. If your quantized signal shows harmonics that move with input level, you under-dithered.
The SNR row assumes a full-scale sine and the standard 6.02·N + 1.76 dB quantization ceiling degraded by the added dither power — use it as a sanity check, not a spec.
Where this math comes from
Dither was born in WWII: bomber navigation computers were mechanical, and crews noticed the analog integrators ran more accurately in flight than on the ground. The airframe vibration was jostling the gears past their stiction — engineers later added deliberate motors to reproduce it, the first intentional dither.
Bernard Widrow put the statistics on a rigorous footing in his 1956 MIT work on quantization as additive noise, and with István Kollár he wrote the definitive modern treatment, Quantization Noise (2008), which lays out exactly when RPDF versus TPDF dither decorrelates the moments of the error. Their quantizing theorem is why this card's √12 and √24 factors are not fudge numbers.
- 1945WWII navigation-computer crewsNotice airframe vibration improves mechanical integrator accuracy — accidental dither.
- 1956Bernard WidrowFormalizes quantization as additive noise and the quantizing theorem.
- 1984Lipshitz, Wannamaker & VanderkooyShow TPDF dither makes error mean and variance signal-independent.
- 2008Widrow & KollárPublish Quantization Noise, the standard reference for dither statistics.
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