Optimal Bark Warping

Figure E.1 illustrates the surprisingly good match between the allpass transformation and a Bark frequency warping when the map parameter is properly chosen. In the following, a simple direct-form expression is developed for the map parameter giving the best least-squares fit to a Bark scale for a chosen sampling rate. As Fig.E.1 shows, the error is so small that the solution is also very close to the optimal Chebyshev fit. In fact, the optimal warping is within 0.04 Bark of the optimal warping. Since the experimental uncertainty when measuring critical bands is on the order of a tenth of a Bark or more [162,165,229,274], we consider the optimal Chebyshev and least-squares maps to be equivalent psychoacoustically.

---

Subsections

• Computing
• Optimal Frequency Warpings
• Relative Bandwidth Mapping Error
• Error Significance
• Arctangent Approximations for
• Filter Design Example

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: The Bilinear Transform
Next: Computing

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.