Temporal and Spectral Processing Methods for Processing of Degraded Speech: A Review <s> MMSE Estimator <s> In this correspondence we derive a short-time spectral amplitude (STSA) estimator for speech signals which minimizes the mean-square error of the log-spectra (i.e., the original STSA and its estimator) and examine it in enhancing noisy speech. This estimator is also compared with the corresponding minimum mean-square error STSA estimator derived previously. It was found that the new estimator is very effective in enhancing the noisy speech, and it significantly improves its quality. <s> BIB001 </s> Temporal and Spectral Processing Methods for Processing of Degraded Speech: A Review <s> MMSE Estimator <s> Presents a study of the noise suppression technique proposed by Ephraim and Malah (1984,1985). This technique has been used previously for the restoration of degraded audio recordings because it is free of the frequently encountered 'musical noise' artifact. It is demonstrated how this artifact is actually eliminated without bringing distortion to the recorded signal even if the noise is only poorly stationary. > <s> BIB002 </s> Temporal and Spectral Processing Methods for Processing of Degraded Speech: A Review <s> MMSE Estimator <s> A speech pause detection algorithm is an important and sensitive part of most single-microphone noise reduction schemes for enhancement of speech signals corrupted by additive noise as an estimate of the background noise is usually determined when speech is absent. An algorithm is proposed which detects speech pauses by adaptively tracking minima in a noisy signal's power envelope both for the broadband signal and for the high-pass and low-pass filtered signal. In poor signal-to-noise ratios (SNRs), the proposed algorithm maintains a low false-alarm rate in the detection of speech pauses while the standardized algorithm of ITU G.729 shows an increasing false-alarm rate in unfavorable situations. These characteristics are found with different types of noise and indicate that the proposed algorithm is better suited to be used for noise estimation in noise reduction algorithms, as speech deterioration may thus be kept at a low level. It is shown that in connection with the Ephraim-Malah (1984) noise reduction scheme, the speech pause detection performance can even be further increased by using the noise-reduced signal instead of the noisy signal as input for the speech pause decision unit. <s> BIB003 </s> Temporal and Spectral Processing Methods for Processing of Degraded Speech: A Review <s> MMSE Estimator <s> A priori signal-to-noise ratio (SNR) estimation is of major consequence in speech enhancement applications. Recently, we introduced a noncausal recursive estimator for the a priori SNR based on a Gaussian speech model, and showed its advantage compared to using the decision-directed estimator. In particular, noncausal estimation facilitates a distinction between speech onsets and noise irregularities. In this paper, we extend our noncausal estimation approach to Gamma and Laplacian speech models. We show that the performance of noncausal estimation, when applied to the problem of speech enhancement, is better under a Laplacian model than under Gaussian or Gamma models. Furthermore, the choice of the specific speech model has a smaller effect on the enhanced speech signal when using the noncausal a priori SNR estimator than when using the decision-directed method. <s> BIB004
In spectral subtraction based methods, there were no specific assumptions made about the distribution of the spectral components of either speech or noise. Ephraim and Malah  have proposed a system that utilizes the MMSE criteria using models for the distribution of the spectral components of speech and noise signals. The MMSEshort time spectral amplitude (STSA) estimator for speech enhancement aims to minimize the mean square error between the short time spectral magnitude of the clean and enhanced speech signal. This method assumes that each of the Fourier expansion coefficients of the speech and of the noise process can be modeled as independent, zero-mean, Gaussian random variables  . In BIB001 , to incorporate perceptually significant information into  , the authors proposed a method to minimize the mean square error between the logarithm of the STSA of the clean and enhanced speech. This criterion of optimality gives good results in practice, with a noticeable reduction in musical noise. The MMSE log-spectral amplitude (MMSE-LSA) estimator for speech enhancement was also proposed by Ephraim and Malah in 1985 BIB001 . The aim of the authors, in their previous work on MMSE estimation of the STSA, was to enhance the speech by minimizing the error between the STSA of clean speech and enhanced speech. This optimality criterion does not consider any of the nonlinear characteristics observable in human perception  . To incorporate perceptually significant information into the algorithm, the authors proposed a method to minimize the mean square error between the logarithm of the STSA of the clean and enhanced speech. That is, the LSA estimator minimizes  E A e k e k where A k denotes the spectral speech amplitude, and Â k is its optimal estimator. A fundamental assumption made in the MMSE algorithms is that the real and imaginary parts of the clean DFT coefficients can be modeled by a Gaussian distribution. This Gaussian assumption might hold for the DFT coefficients of the noise, typically estimated using relatively short (20-30 ms) duration windows  . Based on this observation, a similar optimal MMSE-STSA estimator using non-Gaussian distributions is proposed. In particular, the Gamma BIB003 or the Laplacian  probability distributions are used to model the distributions of the real and imaginary parts of the DFT coefficients. All MMSE-based methods need the estimate of the a priori SNR, the SNR of the k th spectral component of the clean speech signal. Since the knowledge of clean signal is seldom available in practical systems, a decision-directed estimation and maximum likelihood (ML) estimation are used to compute a priori SNR  . Cappe BIB002 provided a more detailed analysis on the decision-directed estimation approach and proposed a lower limit to the estimate of the a priori SNR, in order to reduce the annoying musical tones. Cohen introduced causal and non-causal recursive estimators for the a priori SNR, which take into account the time-frequency correlation of speech signals BIB004 . The causal a priori SNR estimator is closely related to the decision-directed estimator. The non-causal a priori SNR estimator employs future spectral measurements to predict better the spectral variances of clean speech. Experimental results show that the non-causal estimator yields a higher improvement in the segmental SNR and lower log-spectral distortion than the decisiondirected method and the causal estimator BIB004 . Even though several improvements have been made in MMSE estimator, the algorithms proposed by the Ephraim and Malah  BIB001 are still considered the state of art algorithms for noisy speech enhancement.