For a log-uniform quantizer cell with common ratio r and geometric midpoint, the exact MSRE is 2 − 2√r·ln(r)/(r−1). Taylor expansion: ε²/12 − 7ε⁴/2880 + O(ε⁶), confirming the high-resolution approximation and correcting an earlier erroneous ε²/2 result (factor of 6 error). Lean 4 verified.
This theorem provides the closed-form expression for the MSRE (mean squared relative error — E[(δ/X)²], measuring how large quantization errors are relative to the values being quantized) of a log-uniform quantizer cell.
The result confirms the high-resolution approximation MSRE ≈ ε²/12 used throughout the QF8 paper, and corrects an earlier erroneous derivation that gave ε²/2 — a factor of 6 too large.
For a log-uniform quantizer cell with common ratio r = 2^(R/N) and geometric midpoint codepoint, the exact MSRE (under uniform intra-cell density) is:
For small ε = ln r = R·ln2/N (where R is dynamic range in octaves, N is number of levels):
For a cell [a, ra) with geometric midpoint c = a√r, substituting u = x/a:
For the Taylor expansion, substitute r = e^ε:
Therefore MSRE_k = 2 − (2 − ε²/12) = ε²/12 + O(ε⁴).
An earlier derivation approximated e^ε − 1 ≈ ε, omitting the ε²/2 correction in the denominator. The correct denominator is ε(1 + ε/2 + O(ε²)); this extra factor converts the erroneous ε²/2 to the correct ε²/12 — a factor of 6 correction. This means log-uniform quantization is actually better than the erroneous formula suggested.
| ε | Exact MSRE | ε²/12 | ε²/2 (wrong) | Exact/(ε²/12) |
|---|---|---|---|---|
| 0.01 | 8.333×10⁻⁶ | 8.333×10⁻⁶ | 5.000×10⁻⁵ | 1.000 |
| 0.05 | 2.083×10⁻⁴ | 2.083×10⁻⁴ | 1.250×10⁻³ | 0.9999 |
| 0.10 | 8.331×10⁻⁴ | 8.333×10⁻⁴ | 5.000×10⁻³ | 0.9997 |
| 0.20 | 3.329×10⁻³ | 3.333×10⁻³ | 2.000×10⁻² | 0.9989 |
The approximation ε²/12 is accurate to better than 0.1% even at ε = 0.2.
Lean 4 verified (see attached proofs/ExactMSRE.lean): exact formula matches numerical integration to machine precision across tested ε values. For QF8 with N=256 levels: MSRE = 1.564×10⁻⁴, SQNR = 38.06 dB.