Rotational Properties of Vocal Tract Length Difference in Cepstral Space

In this paper, we prove that the direction of cepstrum vectors strongly depends on vocal tract length and that this dependency is represented as rotation in a cepstrum space. In speech recognition studies, vocal tract length normalization (VTLN) techniques are widely used to cancel ageand gender-difference. In VTLN, a frequency warping is often carried out and it can be modeled as a linear transform in a cepstrum space; ĉ=Ac. In this study, the geometric properties of this transformation matrix A are made clear using n dimensional geometry and it is shown that the matrix can be approximated as rotation matrix. Further, for better approximation, a new method is proposed. Namely, using eigenvalues of A, its quasi-rotational distortion is factorized into multiple true rotation operations and multiple magnification operations. This decomposition resolves the intrinsic ambiguity of the rotation angle based on the inner product, and it describes the detailed geometrical properties of the transformation caused by vocal tract length normalization. Experimental results using real and resynthesized speech samples demonstrate that the difference of cepstrum vectors extracted from different speakers is represented as rotation and magnification, and that the decomposition based on eigenvalues can capture it precisely.