A kepstrum approach to filtering, smoothing and prediction with application to speech enhancement

Abstract

A kepstrum (or complex-cepstrum) approach to minimum-phase Wiener filtering of stationary scalar processes is proposed and solved for the case of signal plus coloured noise, where the noise possibly includes a white-noise component. A general solution is found in an innovations form. The spectral factorization of the noise model and of the signal-plus-noise model required for the solution are determined from data using the kepstrum technique with the fast Fourier transform. This approach avoids dependence on any form of multidimensional state-space or polynomial-based model and so avoids use of recursive parameter estimation or of Diophantine equations.

Identification of Non-Minimum Phase Transfer Function Components

“…Here we have used the word kepstrum rather than complex cepstrum since this work ki = F-(log(Si) +y), i = 0,1,2 .....,N -1 (12) There are N kepstrum coefficients satisfying the symmetry ki = kN-i i = 1, 2 ...... 1. The zeroth kepstrum coefficient must be halved thus ko ko / 2.…”

“…A convenient method of identifying the acoustic transfer functions Hk (s) (which is both the minimum phase and the reflected non-minimum phase term) is by using the complex cepstrum or kepstrum [9][10][11][12]. Here we have used the word kepstrum rather than complex cepstrum since this work ki = F-(log(Si) +y), i = 0,1,2 .....,N -1 (12) There are N kepstrum coefficients satisfying the symmetry ki = kN-i i = 1, 2 ...... 1.…”

Filtering, smoothing and prediction using a control‐loop spectral factorization method for coloured noise

“…When white-noise is added to the coloured noise model, then it too can have its own innovations model and we arrive back at the basic model used in Figure 1, but where the driving noise k is replaced by another white-noise innovations term. Hence, the model of Figure 1 is quite general if we assume in the same spirit as (10), but for the noise-alone transfer function, that W n .0/ D 1. It is important to use such innovation models, as without them the spectral factor is not unique and any constant can be absorbed into the driving noise variance.…”

“…Another approach is to use Kepstrum (complex cepstrum) analysis [10] or even Kalman filtering techniques [11]. With the exception of the Kesptrum method, most of these approaches do not distinguish the separate filtering, smoothing and prediction problems with one single approach.…”

“…where the estimate O y k D H.q 1 /s k and the Wiener estimator is found from the classical solution [10,12] …”