An N-level quantizer on [a,b] with cell boundaries a·rᵏ where r = (b/a)^(1/N). The relative cell width w(x)/x is constant at ε = R·ln2/N (where R = log₂(b/a) is dynamic range in octaves), giving constant relative quantization error across all bins. Geometric midpoint codepoints: cₖ = a·r^(k+1/2).
A quantizer design where the ratio between consecutive representable values is constant, giving uniform precision in the logarithmic domain. This is the quantizer used by QF8 and the one proven minimax-optimal for NMSE in the QF8 paper.
The log-uniform quantizer with N levels over dynamic range [a, b] has cell boundaries a·rᵏ for k = 0, …, N, where:
and R = log₂(b/a) is the dynamic range in octaves.
With geometric midpoint codepoints:
The relative cell width is constant:
where ε = R·ln2/N is the bin width in the log domain.
This is the defining characteristic: unlike uniform quantizers (constant absolute cell width) or floating-point quantizers (piecewise-constant within binades), the log-uniform quantizer has truly constant relative precision everywhere.
Because w(x)/x is constant, the contribution to NMSE (normalized mean squared error — the fraction of signal power lost to quantization) from each region of the input space is proportional to the probability mass in that region weighted by x². The x² factors in numerator and denominator cancel, making NMSE independent of the source density. This density-independence is the equalization property that underlies minimax optimality.
| Quantizer | Cell width property | Precision distribution |
|---|---|---|
| Uniform | Constant absolute width | High precision for small values, low for large |
| IEEE floating-point (e.g. FP8 E4M3) | Constant within each binade, jumps at binade boundaries | Piecewise-constant, 8 levels per octave |
| Log-uniform | Constant relative width | Truly uniform in log domain, 16 levels per octave for QF8 |