Frequency Warping for VTLN and Speaker Adaptation by Linear Transformation

Vocal Tract Length Normalization (VTLN) for standard filterbank-based Mel Frequency Cepstral Coefficient (MFCC) features is usually implemented by warp-ing the center frequencies of the Mel filterbank, and the warping factor is estimated using the maximum likelihood score (MLS) criterion (Lee and Rose, 1998). A linear transform (LT) equivalent for frequency warping (FW) would enable more efficient MLS estimation (Umesh et al., 2005). We recently proposed a novel LT to perform FW for VTLN and model adaptation with standard MFCC features (Panchapage-san, 2006). In this paper, we present the mathematical derivation of the LT and give a compact formula to calculate it for any FW function. We also show that our LT is very closely related to previously proposed LTs for FW (McDonough, 2000; Pitz et al., 2001; Umesh et al., 2005), and these LTs for FW are all found to be numerically almost identical for the sine-log all-pass transform (SLAPT) warping functions. Our formula for the transformation matrix is, however, computationally simpler and un-like other previous linear transform approaches to VTLN with MFCC features (Pit