Bark and ERB bilinear transforms

Abstract

Use of a bilinear conformal map to achieve a frequency warping nearly identical to that of the Bark frequency scale is described. Because the map takes the unit circle to itself, its form is that of the transfer function of a first-order allpass filter. Since it is a first-order map, it preserves the model order of rational systems, making it a valuable frequency warping technique for use in audio filter design. A closed-form weighted-equation-error method is derived that computes the optimal mapping coefficient as a function of sampling rate, and the solution is shown to be generally indistinguishable from the optimal least-squares solution. The optimal Chebyshev mapping is also found to be essentially identical to the optimal least-squares solution. The expression 0:8517 [arctan(0:06583fs)] 1=2 00:916 is shown to accurately approximate the optimal allpass coefficient as a function of sampling rate fs in kHz for sampling rates greater than 1 kHz. A filter design example is included that illustrates improvements due to carrying out the design over a Bark scale. Corresponding results are also given and compared for approximating the related "equivalent rectangular bandwidth (ERB) scale" of Moore and Glasberg using a first-order allpass transformation. Due to the higher frequency resolution called for by the ERB scale, particularly at low frequencies, the first-order conformal map is less able to follow the desired mapping, and the error is two to three times greater than the Bark-scale case, depending on the sampling rate.

A fixed dimension and perceptually based dynamic sinusoidal model of speech

“…In order to distinguish the smallest frequency difference that a listener could perceive, we adopted a perceptual sinusoidal model (PM) based on critical bands [14] in order to decrease and fix the number of parameters. The whole frequency band is divided into 21 critical bands [12]. Instead of using all harmonic components, only 21 sinusoids at the frequencies of critical band centres are used to represent speech, as illustrated in Fig.…”

“…Specifically, for a wideband speech signal (0Hz 8kHz), we assume 21 critical bands [12], and for each of these critical bands only one sinusoid is used. However, we found limiting the number of parameters in this way had some negative effects on speech quality which subsequently needed to be resolved.…”

Signal adaptive spectral envelope estimation for robust speech recognition

“…A good approximation of the mel-scale by the bilinear transformation is possible, if the warp factor is set accordingly. The optimal warp factor depends on the sampling frequency and can be found by different optimization methods [24]. Fig.…”

Uniform and warped low delay filter-banks for speech enhancement

“…The uniform and warped versions of these filter-banks 5 are each used. A real allpass coefficient of a = 0.4 is taken for the frequency warping, which provides a good approximation of the Bark scale for a sampling frequency of 8 kHz [16]. A transform size of M = 64 and prototype filter degree of L = 64 is used in all cases 6 .…”

“…Therefore, some authors have proposed the use of non-uniform AS FBs for noise reduction to obtain an improved (subjective) quality for the enhanced speech [8,9,10,11]. For such applications, the allpass transformed DFT filter-bank [12,13,14,15] is of special interest, because this frequency warped filter-bank achieves a very good approximation of the Bark scale [16]. This property is not only of interest for perceptual based speech enhancement, but also for speech and audio processing applications in general.…”