Statistics of nonlinear averaging spectral estimators and a novel distance measure for HMMs with application to speech quality estimation

Although numerous advanced signal processing methods are available for spectral estimation, many modern test and measurement instruments such as sampling oscilloscopes typically utilize only non-parameterc FFT-based approaches. One such approach averages the FFT magnitudes rather than averaging the squared magnitudes, as is done in the well-known Welch averaged periodogram. Another takes the geometric mean rather than the arithmetic mean. Since these simple but non-conventional methods are common in industry, an understanding of their performance is important. The first part of this research investigates the asymptotic biases and variances of the above two non-conventional spectral estimators. The mean and variance analysis for finite data length and quasi-stationary signals is also carried out. It is shown that both averaging methods provide biased spectral estimates for wide sense stationary (WSS) signals. For quasi-stationary signals, they provide less biased spectral estimates than the Welch's method.; We also propose a novel approach to approximate the Kullback-Leibler distance rate (KLDR) between two hidden Markov models (HMMs). The proposed approximation is based on the stationary observation distribution and the first order observation-transition probability distribution. It gives a closed form approximation of the KLDR in terms of the model parameters of the given HMMs. The performance of the proposed method matches well with commonly used Monte Carlo approaches as well as other previously proposed methods in the literature, and provides some computational advantages. We also investigate the use of hidden Markov models in output-based speech quality estimation, in which long random speech records are used to train relatively low order HMMs. This is different from common uses of hidden Markov models in speech processing. This new approach to hidden Markov models has many potentially useful applications. We propose adaptation of the above distance measure for objective speech quality estimation and discuss the "hiddenness" of a hidden Markov model. Experimental results show that the above techniques work successfully for objective speech quality estimation.