A brief overview of speech enhancement with linear filtering
---
paper_title: Noise power spectral density estimation based on optimal smoothing and minimum statistics

paper_content:

We describe a method to estimate the power spectral density of nonstationary noise when a noisy speech signal is given. The method can be combined with any speech enhancement algorithm which requires a noise power spectral density estimate. In contrast to other methods, our approach does not use a voice activity detector. Instead it tracks spectral minima in each frequency band without any distinction between speech activity and speech pause. By minimizing a conditional mean square estimation error criterion in each time step we derive the optimal smoothing parameter for recursive smoothing of the power spectral density of the noisy speech signal. Based on the optimally smoothed power spectral density estimate and the analysis of the statistics of spectral minima an unbiased noise estimator is developed. The estimator is well suited for real time implementations. Furthermore, to improve the performance in nonstationary noise we introduce a method to speed up the tracking of the spectral minima. Finally, we evaluate the proposed method in the context of speech enhancement and low bit rate speech coding with various noise types.
---
paper_title: Dual-Channel Speech Enhancement by Superdirective Beamforming

paper_content:

In this contribution, a dual-channel input-output speech enhancement system is introduced. The proposed algorithm is an adaptation of the well-known superdirective beamformer including postfiltering to the binaural application. In contrast to conventional beamformer processing, the proposed system outputs enhanced stereo signals while preserving the important interaural amplitude and phase differences of the original signal. Instrumental performance evaluations in a real environment with multiple speech sources indicate that the proposed computational efficient spectral weighting system can achieve significant attenuation of speech interferers while maintaining a high speech quality of the target signal.
---
paper_title: Speech enhancement using a minimum mean-square error log-spectral amplitude estimator

paper_content:

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.
---
paper_title: Reduction of broad-band noise in speech by truncated QSVD

paper_content:

We consider an algorithm for reduction of broadband noise in speech based on signal subspaces. The algorithm is formulated by means of the quotient singular value decomposition (QSVD). With this formulation, a prewhitening operation becomes an integral part of the algorithm. We demonstrate that this is essential in connection with updating issues in real-time recursive applications. We also illustrate by examples that we are able to achieve a satisfactory quality of the reconstructed signal.
---
paper_title: Codebook-Based Bayesian Speech Enhancement for Nonstationary Environments

paper_content:

In this paper, we propose a Bayesian minimum mean squared error approach for the joint estimation of the short-term predictor parameters of speech and noise, from the noisy observation. We use trained codebooks of speech and noise linear predictive coefficients to model the a priori information required by the Bayesian scheme. In contrast to current Bayesian estimation approaches that consider the excitation variances as part of the a priori information, in the proposed method they are computed online for each short-time segment, based on the observation at hand. Consequently, the method performs well in nonstationary noise conditions. The resulting estimates of the speech and noise spectra can be used in a Wiener filter or any state-of-the-art speech enhancement system. We develop both memoryless (using information from the current frame alone) and memory-based (using information from the current and previous frames) estimators. Estimation of functions of the short-term predictor parameters is also addressed, in particular one that leads to the minimum mean squared error estimate of the clean speech signal. Experiments indicate that the scheme proposed in this paper performs significantly better than competing methods
---
paper_title: Speech enhancement using a soft-decision noise suppression filter

paper_content:

One way of enhancing speech in an additive acoustic noise environment is to perform a spectral decomposition of a frame of noisy speech and to attenuate a particular spectral line depending on how much the measured speech plus noise power exceeds an estimate of the background noise. Using a two-state model for the speech event (speech absent or speech present) and using the maximum likelihood estimator of the magnitude of the speech spectrum results in a new class of suppression curves which permits a tradeoff of noise suppression against speech distortion. The algorithm has been implemented in real time in the time domain, exploiting the structure of the channel vocoder. Extensive testing has shown that the noise can be made imperceptible by proper choice of the suppression factor.
---
paper_title: Suppression of acoustic noise in speech using spectral subtraction

paper_content:

A stand-alone noise suppression algorithm is presented for reducing the spectral effects of acoustically added noise in speech. Effective performance of digital speech processors operating in practical environments may require suppression of noise from the digital wave-form. Spectral subtraction offers a computationally efficient, processor-independent approach to effective digital speech analysis. The method, requiring about the same computation as high-speed convolution, suppresses stationary noise from speech by subtracting the spectral noise bias calculated during nonspeech activity. Secondary procedures are then applied to attenuate the residual noise left after subtraction. Since the algorithm resynthesizes a speech waveform, it can be used as a pre-processor to narrow-band voice communications systems, speech recognition systems, or speaker authentication systems.
---
paper_title: A signal subspace approach for speech enhancement

paper_content:

A perceptually based linear signal estimator for enhancing speech signals degraded by uncorrelated additive noise is developed. The estimator is designed by minimizing the signal distortion while maintaining the residual noise level below some given threshold. The estimator is shown to be a Wiener filter with adjustable input noise level. This level is determined by the threshold of the permissible residual noise. The estimator is implemented using the signal subspace approach. The vector space of the noisy signal is decomposed into a signal subspace and complementary orthogonal noise subspace. Estimation is performed from vectors in the signal subspace only, since the orthogonal subspace does not contain signal information. The proposed estimator is shown to be a refinement of a version of the spectral subtraction signal estimator. The latter estimator is shown to be asymptotically optimal for stationary signal and noise in the linear minimum mean square error sense. >
---
paper_title: Noise Correlation Matrix Estimation for Multi-Microphone Speech Enhancement

paper_content:

For multi-channel noise reduction algorithms like the minimum variance distortionless response (MVDR) beamformer, or the multi-channel Wiener filter, an estimate of the noise correlation matrix is needed. For its estimation, it is often proposed in the literature to use a voice activity detector (VAD). However, using a VAD the estimated matrix can only be updated in speech absence. As a result, during speech presence the noise correlation matrix estimate does not follow changing noise fields with an appropriate accuracy. This effect is further increased, as in nonstationary noise voice activity detection is a rather difficult task, and false-alarms are likely to occur. In this paper, we present and analyze an algorithm that estimates the noise correlation matrix without using a VAD. This algorithm is based on measuring the correlation of the noisy input and a noise reference which can be obtained, e.g., by steering a null towards the target source. When applied in combination with an MVDR beamformer, it is shown that the proposed noise correlation matrix estimate results in a more accurate beamformer response, a larger signal-to-noise ratio improvement and a larger instrumentally predicted speech intelligibility when compared to competing algorithms such as the generalized sidelobe canceler, a VAD-based MVDR beamformer, and an MVDR based on the noisy correlation matrix.
---
paper_title: New insights into the noise reduction Wiener filter

paper_content:

The problem of noise reduction has attracted a considerable amount of research attention over the past several decades. Among the numerous techniques that were developed, the optimal Wiener filter can be considered as one of the most fundamental noise reduction approaches, which has been delineated in different forms and adopted in various applications. Although it is not a secret that the Wiener filter may cause some detrimental effects to the speech signal (appreciable or even significant degradation in quality or intelligibility), few efforts have been reported to show the inherent relationship between noise reduction and speech distortion. By defining a speech-distortion index to measure the degree to which the speech signal is deformed and two noise-reduction factors to quantify the amount of noise being attenuated, this paper studies the quantitative performance behavior of the Wiener filter in the context of noise reduction. We show that in the single-channel case the a posteriori signal-to-noise ratio (SNR) (defined after the Wiener filter) is greater than or equal to the a priori SNR (defined before the Wiener filter), indicating that the Wiener filter is always able to achieve noise reduction. However, the amount of noise reduction is in general proportional to the amount of speech degradation. This may seem discouraging as we always expect an algorithm to have maximal noise reduction without much speech distortion. Fortunately, we show that speech distortion can be better managed in three different ways. If we have some a priori knowledge (such as the linear prediction coefficients) of the clean speech signal, this a priori knowledge can be exploited to achieve noise reduction while maintaining a low level of speech distortion. When no a priori knowledge is available, we can still achieve a better control of noise reduction and speech distortion by properly manipulating the Wiener filter, resulting in a suboptimal Wiener filter. In case that we have multiple microphone sensors, the multiple observations of the speech signal can be used to reduce noise with less or even no speech distortion
---
paper_title: Noise Reduction in Speech Processing

paper_content:

Noise is everywhere and in most applications that are related to audio and speech, such as human-machine interfaces, hands-free communications, voice over IP (VoIP), hearing aids, teleconferencing/telepresence/telecollaboration systems, and so many others, the signal of interest (usually speech) that is picked up by a microphone is generally contaminated by noise. As a result, the microphone signal has to be cleaned up with digital signal processing tools before it is stored, analyzed, transmitted, or played out. This cleaning process is often called noise reduction and this topic has attracted a considerable amount of research and engineering attention for several decades. One of the objectives of this book is to present in a common framework an overview of the state of the art of noise reduction algorithms in the single-channel (one microphone) case. The focus is on the most useful approaches, i.e., filtering techniques (in different domains) and spectral enhancement methods. The other objective of Noise Reduction in Speech Processing is to derive all these well-known techniques in a rigorous way and prove many fundamental and intuitive results often taken for granted. This book is especially written for graduate students and research engineers who work on noise reduction for speech and audio applications and want to understand the subtle mechanisms behind each approach. Many new and interesting concepts are presented in this text that we hope the readers will find useful and inspiring.
---
paper_title: Speech Enhancement: Theory and Practice

paper_content:

With the proliferation of mobile devices and hearing devices, including hearing aids and cochlear implants, there is a growing and pressing need to design algorithms that can improve speech intelligibility without sacrificing quality. Responding to this need, Speech Enhancement: Theory and Practice, Second Edition introduces readers to the basic problems of speech enhancement and the various algorithms proposed to solve these problems. Updated and expanded, this second edition of the bestselling textbook broadens its scope to include evaluation measures and enhancement algorithms aimed at improving speech intelligibility. Fundamentals, Algorithms, Evaluation, and Future Steps Organized into four parts, the book begins with a review of the fundamentals needed to understand and design better speech enhancement algorithms. The second part describes all the major enhancement algorithms and, because these require an estimate of the noise spectrum, also covers noise estimation algorithms. The third part of the book looks at the measures used to assess the performance, in terms of speech quality and intelligibility, of speech enhancement methods. It also evaluates and compares several of the algorithms. The fourth part presents binary mask algorithms for improving speech intelligibility under ideal conditions. In addition, it suggests steps that can be taken to realize the full potential of these algorithms under realistic conditions. Whats New in This Edition Updates in every chapter A new chapter on objective speech intelligibility measures A new chapter on algorithms for improving speech intelligibility Real-world noise recordings (on accompanying CD) MATLAB code for the implementation of intelligibility measures (on accompanying CD) MATLAB and C/C++ code for the implementation of algorithms to improve speech intelligibility (on accompanying CD) Valuable Insights from a Pioneer in Speech Enhancement Clear and concise, this book explores how human listeners compensate for acoustic noise in noisy environments. Written by a pioneer in speech enhancement and noise reduction in cochlear implants, it is an essential resource for anyone who wants to implement or incorporate the latest speech enhancement algorithms to improve the quality and intelligibility of speech degraded by noise. Includes a CD with Code and Recordings The accompanying CD provides MATLAB implementations of representative speech enhancement algorithms as well as speech and noise databases for the evaluation of enhancement algorithms.
---
paper_title: New insights into the noise reduction Wiener filter

paper_content:

The problem of noise reduction has attracted a considerable amount of research attention over the past several decades. Among the numerous techniques that were developed, the optimal Wiener filter can be considered as one of the most fundamental noise reduction approaches, which has been delineated in different forms and adopted in various applications. Although it is not a secret that the Wiener filter may cause some detrimental effects to the speech signal (appreciable or even significant degradation in quality or intelligibility), few efforts have been reported to show the inherent relationship between noise reduction and speech distortion. By defining a speech-distortion index to measure the degree to which the speech signal is deformed and two noise-reduction factors to quantify the amount of noise being attenuated, this paper studies the quantitative performance behavior of the Wiener filter in the context of noise reduction. We show that in the single-channel case the a posteriori signal-to-noise ratio (SNR) (defined after the Wiener filter) is greater than or equal to the a priori SNR (defined before the Wiener filter), indicating that the Wiener filter is always able to achieve noise reduction. However, the amount of noise reduction is in general proportional to the amount of speech degradation. This may seem discouraging as we always expect an algorithm to have maximal noise reduction without much speech distortion. Fortunately, we show that speech distortion can be better managed in three different ways. If we have some a priori knowledge (such as the linear prediction coefficients) of the clean speech signal, this a priori knowledge can be exploited to achieve noise reduction while maintaining a low level of speech distortion. When no a priori knowledge is available, we can still achieve a better control of noise reduction and speech distortion by properly manipulating the Wiener filter, resulting in a suboptimal Wiener filter. In case that we have multiple microphone sensors, the multiple observations of the speech signal can be used to reduce noise with less or even no speech distortion
---
paper_title: A generalized subspace approach for enhancing speech corrupted by colored noise

paper_content:

A generalized subspace approach is proposed for enhancement of speech corrupted by colored noise. A nonunitary transform, based on the simultaneous diagonalization of the clean speech and noise covariance matrices, is used to project the noisy signal onto a signal-plus-noise subspace and a noise subspace. The clean signal is estimated by nulling the signal components in the noise subspace and retaining the components in the signal subspace. The applied transform has built-in prewhitening and can therefore be used in general for colored noise. The proposed approach is shown to be a generalization of the approach proposed by Y. Ephraim and H.L. Van Trees (see ibid., vol.3, p.251-66, 1995) for white noise. Two estimators are derived based on the nonunitary transform, one based on time-domain constraints and one based on spectral domain constraints. Objective and subjective measures demonstrate improvements over other subspace-based methods when tested with TIMIT sentences corrupted with speech-shaped noise and multi-talker babble.
---
paper_title: Speech enhancement from noise: a regenerative approach

paper_content:

Abstract In this paper a speech enhancement technique is proposed based on principal component analysis and a new criterion for the selection of the parsimonious number of components for noise-free signal regeneration. Both isolated phonemes and continuous speech experiments are presented. The results have been evaluated by informal listening and SNR computations, which show that the methodology has an improved performance compared to existing techniques.
---
paper_title: GSVD-based optimal filtering for single and multimicrophone speech enhancement

paper_content:

A generalized singular value decomposition (GSVD) based algorithm is proposed for enhancing multimicrophone speech signals degraded by additive colored noise. This GSVD-based multimicrophone algorithm can be considered to be an extension of the single-microphone signal subspace algorithms for enhancing noisy speech signals and amounts to a specific optimal filtering problem when the desired response signal cannot be observed. The optimal filter can be written as a function of the generalized singular vectors and singular values of a speech and noise data matrix. A number of symmetry properties are derived for the single-microphone and multimicrophone optimal filter, which are valid for the white noise case as well as for the colored noise case. In addition, the averaging step of some single-microphone signal subspace algorithms is examined, leading to the conclusion that this averaging operation is unnecessary and even suboptimal. For simple situations, where we consider localized sources and no multipath propagation, the GSVD-based optimal filtering technique exhibits the spatial directivity pattern of a beamformer. When comparing the noise reduction performance for realistic situations, simulations show that the GSVD-based optimal filtering technique has a better performance than standard fixed and adaptive beamforming techniques for all reverberation times and that it is more robust to deviations from the nominal situation, as, e.g., encountered in uncalibrated microphone arrays.
---
paper_title: Reduction of broad-band noise in speech by truncated QSVD

paper_content:

We consider an algorithm for reduction of broadband noise in speech based on signal subspaces. The algorithm is formulated by means of the quotient singular value decomposition (QSVD). With this formulation, a prewhitening operation becomes an integral part of the algorithm. We demonstrate that this is essential in connection with updating issues in real-time recursive applications. We also illustrate by examples that we are able to achieve a satisfactory quality of the reconstructed signal.
---
paper_title: A signal subspace approach for speech enhancement

paper_content:

A perceptually based linear signal estimator for enhancing speech signals degraded by uncorrelated additive noise is developed. The estimator is designed by minimizing the signal distortion while maintaining the residual noise level below some given threshold. The estimator is shown to be a Wiener filter with adjustable input noise level. This level is determined by the threshold of the permissible residual noise. The estimator is implemented using the signal subspace approach. The vector space of the noisy signal is decomposed into a signal subspace and complementary orthogonal noise subspace. Estimation is performed from vectors in the signal subspace only, since the orthogonal subspace does not contain signal information. The proposed estimator is shown to be a refinement of a version of the spectral subtraction signal estimator. The latter estimator is shown to be asymptotically optimal for stationary signal and noise in the linear minimum mean square error sense. >
---
paper_title: A generalized subspace approach for enhancing speech corrupted by colored noise

paper_content:

A generalized subspace approach is proposed for enhancement of speech corrupted by colored noise. A nonunitary transform, based on the simultaneous diagonalization of the clean speech and noise covariance matrices, is used to project the noisy signal onto a signal-plus-noise subspace and a noise subspace. The clean signal is estimated by nulling the signal components in the noise subspace and retaining the components in the signal subspace. The applied transform has built-in prewhitening and can therefore be used in general for colored noise. The proposed approach is shown to be a generalization of the approach proposed by Y. Ephraim and H.L. Van Trees (see ibid., vol.3, p.251-66, 1995) for white noise. Two estimators are derived based on the nonunitary transform, one based on time-domain constraints and one based on spectral domain constraints. Objective and subjective measures demonstrate improvements over other subspace-based methods when tested with TIMIT sentences corrupted with speech-shaped noise and multi-talker babble.
---
paper_title: Noise power spectral density estimation based on optimal smoothing and minimum statistics

paper_content:

We describe a method to estimate the power spectral density of nonstationary noise when a noisy speech signal is given. The method can be combined with any speech enhancement algorithm which requires a noise power spectral density estimate. In contrast to other methods, our approach does not use a voice activity detector. Instead it tracks spectral minima in each frequency band without any distinction between speech activity and speech pause. By minimizing a conditional mean square estimation error criterion in each time step we derive the optimal smoothing parameter for recursive smoothing of the power spectral density of the noisy speech signal. Based on the optimally smoothed power spectral density estimate and the analysis of the statistics of spectral minima an unbiased noise estimator is developed. The estimator is well suited for real time implementations. Furthermore, to improve the performance in nonstationary noise we introduce a method to speed up the tracking of the spectral minima. Finally, we evaluate the proposed method in the context of speech enhancement and low bit rate speech coding with various noise types.
---
paper_title: A perspective on multichannel noise reduction in the time domain

paper_content:

Conventional multichannel noise reduction techniques are formulated by splitting the processed microphone observations into two terms: filtered noise-free speech and residual additive noise. The first term is treated as desired signal while the second is a nuisance. Then, the objective has typically been to reduce the nuisance while keeping the filtered speech as similar as possible to the clean speech. It turns out that this treatment of the overall filtered speech as the desired signal is inappropriate as will become clear soon. In this paper, we present a new study of the multichannel time-domain noise reduction filters. We decompose the noise-free microphone array observations into two components where the first is correlated with the target signal and perfectly coherent across the sensors while the second consists of residual interference. Then, well-known time-domain filters including the minimum variance distortionless response (MVDR), the space–time (ST) prediction, the maximum signal-to-noise ratio (SNR), the linearly constrained minimum variance (LCMV), the multichannel tradeoff, and Wiener filters are derived. Besides, the analytical performance evaluation of these time-domain filters is provided and new insights into their functioning are presented. Numerical results are finally given to corroborate our study.
---
paper_title: Single-channel noise reduction using optimal rectangular filtering matrices

paper_content:

This paper studies the problem of single-channel noise reduction in the time domain and presents a block-based approach where a vector of the desired speech signal is recovered by filtering a frame of the noisy signal with a rectangular filtering matrix. With this formulation, the noise reduction problem becomes one of estimating an optimal filtering matrix. To achieve such estimation, a method is introduced to decompose a frame of the clean speech signal into two orthogonal components: One correlated and the other uncorrelated with the current desired speech vector to be estimated. Different optimization cost functions are then formulated from which non-causal optimal filtering matrices are derived. The relationships among these optimal filtering matrices are discussed. In comparison with the classical sample-based technique that uses only forward prediction, the block-based method presented in this paper exploits both the forward and backward prediction as well as the temporal interpolation and, therefo...
---
paper_title: Time-domain noise reduction based on an orthogonal decomposition for desired signal extraction.

paper_content:

This paper addresses the problem of noise reduction in the time domain where the clean speech sample at every time instant is estimated by filtering a vector of the noisy speech signal. Such a clean speech estimate consists of both the filtered speech and residual noise (filtered noise) as the noisy vector is the sum of the clean speech and noise vectors. Traditionally, the filtered speech is treated as the desired signal after noise reduction. This paper proposes to decompose the clean speech vector into two orthogonal components: one is correlated and the other is uncorrelated with the current clean speech sample. While the correlated component helps estimate the clean speech, it is shown that the uncorrelated component interferes with the estimation, just as the additive noise. Based on this orthogonal decomposition, the paper presents a way to define the error signal and cost functions and addresses the issue of how to design different optimal noise reduction filters by optimizing these cost functions. Specifically, it discusses how to design the maximum SNR filter, the Wiener filter, the minimum variance distortionless response (MVDR) filter, the tradeoff filter, and the linearly constrained minimum variance (LCMV) filter. It demonstrates that the maximum SNR, Wiener, MVDR, and tradeoff filters are identical up to a scaling factor. It also shows from the orthogonal decomposition that many performance measures can be defined, which seem to be more appropriate than the traditional ones for the evaluation of the noise reduction filters.
---
paper_title: New insights into the noise reduction Wiener filter

paper_content:

The problem of noise reduction has attracted a considerable amount of research attention over the past several decades. Among the numerous techniques that were developed, the optimal Wiener filter can be considered as one of the most fundamental noise reduction approaches, which has been delineated in different forms and adopted in various applications. Although it is not a secret that the Wiener filter may cause some detrimental effects to the speech signal (appreciable or even significant degradation in quality or intelligibility), few efforts have been reported to show the inherent relationship between noise reduction and speech distortion. By defining a speech-distortion index to measure the degree to which the speech signal is deformed and two noise-reduction factors to quantify the amount of noise being attenuated, this paper studies the quantitative performance behavior of the Wiener filter in the context of noise reduction. We show that in the single-channel case the a posteriori signal-to-noise ratio (SNR) (defined after the Wiener filter) is greater than or equal to the a priori SNR (defined before the Wiener filter), indicating that the Wiener filter is always able to achieve noise reduction. However, the amount of noise reduction is in general proportional to the amount of speech degradation. This may seem discouraging as we always expect an algorithm to have maximal noise reduction without much speech distortion. Fortunately, we show that speech distortion can be better managed in three different ways. If we have some a priori knowledge (such as the linear prediction coefficients) of the clean speech signal, this a priori knowledge can be exploited to achieve noise reduction while maintaining a low level of speech distortion. When no a priori knowledge is available, we can still achieve a better control of noise reduction and speech distortion by properly manipulating the Wiener filter, resulting in a suboptimal Wiener filter. In case that we have multiple microphone sensors, the multiple observations of the speech signal can be used to reduce noise with less or even no speech distortion
---
paper_title: Time-domain noise reduction based on an orthogonal decomposition for desired signal extraction.

paper_content:

This paper addresses the problem of noise reduction in the time domain where the clean speech sample at every time instant is estimated by filtering a vector of the noisy speech signal. Such a clean speech estimate consists of both the filtered speech and residual noise (filtered noise) as the noisy vector is the sum of the clean speech and noise vectors. Traditionally, the filtered speech is treated as the desired signal after noise reduction. This paper proposes to decompose the clean speech vector into two orthogonal components: one is correlated and the other is uncorrelated with the current clean speech sample. While the correlated component helps estimate the clean speech, it is shown that the uncorrelated component interferes with the estimation, just as the additive noise. Based on this orthogonal decomposition, the paper presents a way to define the error signal and cost functions and addresses the issue of how to design different optimal noise reduction filters by optimizing these cost functions. Specifically, it discusses how to design the maximum SNR filter, the Wiener filter, the minimum variance distortionless response (MVDR) filter, the tradeoff filter, and the linearly constrained minimum variance (LCMV) filter. It demonstrates that the maximum SNR, Wiener, MVDR, and tradeoff filters are identical up to a scaling factor. It also shows from the orthogonal decomposition that many performance measures can be defined, which seem to be more appropriate than the traditional ones for the evaluation of the noise reduction filters.
---
paper_title: Speech Enhancement in the STFT Domain

paper_content:

This work addresses this problem in the short-time Fourier transform (STFT) domain. We divide the general problem into five basic categories depending on the number of microphones being used and whether the interframe or interband correlation is considered. The first category deals with the single-channel problem where STFT coefficients at different frames and frequency bands are assumed to be independent. In this case, the noise reduction filter in each frequency band is basically a real gain. Since a gain does not improve the signal-to-noise ratio (SNR) for any given subband and frame, the noise reduction is basically achieved by liftering the subbands and frames that are less noisy while weighing down on those that are more noisy. The second category also concerns the single-channel problem. The difference is that now the interframe correlation is taken into account and a filter is applied in each subband instead of just a gain. The advantage of using the interframe correlation is that we can improve not only the long-time fullband SNR, but the frame-wise subband SNR as well. The third and fourth classes discuss the problem of multichannel noise reduction in the STFT domain with and without interframe correlation, respectively. In the last category, we consider the interband correlation in the design of the noise reduction filters. We illustrate the basic principle for the single-channel case as an example, while this concept can be generalized to other scenarios. In all categories, we propose different optimization cost functions from which we derive the optimal filters and we also define the performance measures that help analyzing them.
---
paper_title: Single-channel noise reduction in the STFT domain based on the bifrequency spectrum

paper_content:

This paper studies the problem of noise reduction in the short-time Fourier transform (STFT) domain. Traditionally, the STFT coefficients in different frequency bands are assumed to be independent. This assumption holds when the signals are stationary and the fast Fourier transform(FFT) length is sufficiently large. In practice, however, speech is nonstationary and also the FFT length cannot be very large due to practical reasons. So, there always exists some correlation between STFT coefficients from neighboring frequency bands. An important question then arises: how the interband correlation can be used to optimize noise reduction performance? This paper addresses this issue. We discuss two solutions in the framework of the bifrequency spectrum. One considers the cross-correlation between all the frequency bands and the other takes into account only the cross-correlation between neighboring bands. While the former is optimal from a theoretical perspective, the latter is more practical as it is more immune to the error in correlation matrix estimation.
---
paper_title: A minimum variance distortionless response filter based on the bifrequency spectrum for single-channel noise reduction

paper_content:

a b s t r a c t Article history: Available online 1 July 2014 This paper deals with the problem of single-channel noise reduction in the short-time Fourier transform (STFT) domain. Many algorithms have been developed to solve this important problem, most of which generally assume that the STFT coefficients in different frequency bands are uncorrelated, so the noise reduction is achieved by applying a gain function to the STFT of the noisy speech in each frequency band. However, this assumption is not accurate and the STFT coefficients of speech signals between neighboring frequency bands are correlated in practice due to the use of small lengths of the fast Fourier transform (FFT) and overlap add/save techniques in implementation. This paper formulates the noise reduction problem by taking into account the interband correlation using the so-called bifrequency spectrum. Based on this formulation, a single-channel minimum variance distortionless response (MVDR) filter is derived, which is shown to be able to significantly improve the signal-to-noise ratio (SNR) and meanwhile maintain the desired speech not much distorted. Simulations are presented to justify the claimed merits of the developed MVDR filter.
---
paper_title: Noise Reduction in Speech Processing

paper_content:

Noise is everywhere and in most applications that are related to audio and speech, such as human-machine interfaces, hands-free communications, voice over IP (VoIP), hearing aids, teleconferencing/telepresence/telecollaboration systems, and so many others, the signal of interest (usually speech) that is picked up by a microphone is generally contaminated by noise. As a result, the microphone signal has to be cleaned up with digital signal processing tools before it is stored, analyzed, transmitted, or played out. This cleaning process is often called noise reduction and this topic has attracted a considerable amount of research and engineering attention for several decades. One of the objectives of this book is to present in a common framework an overview of the state of the art of noise reduction algorithms in the single-channel (one microphone) case. The focus is on the most useful approaches, i.e., filtering techniques (in different domains) and spectral enhancement methods. The other objective of Noise Reduction in Speech Processing is to derive all these well-known techniques in a rigorous way and prove many fundamental and intuitive results often taken for granted. This book is especially written for graduate students and research engineers who work on noise reduction for speech and audio applications and want to understand the subtle mechanisms behind each approach. Many new and interesting concepts are presented in this text that we hope the readers will find useful and inspiring.
---
paper_title: A Perspective on Single-Channel Frequency-Domain Speech Enhancement

paper_content:

Abstract This book focuses on a class of single-channel noise reduction methods that are performed in the frequency domain via the short-time Fourier transform (STFT). The simplicity and relative effectiveness of this class of approaches make them the dominant choice in practical systems. Even though many popular algorithms have been proposed through more than four decades of continuous research, there are a number of critical areas where our understanding and capabilities still remain quite rudimentary, especially with respect to the relationship between noise reduction and speech distortion. All existing frequency-domain algorithms, no matter how they are developed, have one feature in common: the solution is eventually expressed as a gain function applied to the STFT of the noisy signal only in the current frame. As a result, the narrowband signal-to-noise ratio (SNR) cannot be improved, and any gains achieved in noise reduction on the fullband basis come with a price to pay, which is speech distortion...
---
paper_title: A perspective on multichannel noise reduction in the time domain

paper_content:

Conventional multichannel noise reduction techniques are formulated by splitting the processed microphone observations into two terms: filtered noise-free speech and residual additive noise. The first term is treated as desired signal while the second is a nuisance. Then, the objective has typically been to reduce the nuisance while keeping the filtered speech as similar as possible to the clean speech. It turns out that this treatment of the overall filtered speech as the desired signal is inappropriate as will become clear soon. In this paper, we present a new study of the multichannel time-domain noise reduction filters. We decompose the noise-free microphone array observations into two components where the first is correlated with the target signal and perfectly coherent across the sensors while the second consists of residual interference. Then, well-known time-domain filters including the minimum variance distortionless response (MVDR), the space–time (ST) prediction, the maximum signal-to-noise ratio (SNR), the linearly constrained minimum variance (LCMV), the multichannel tradeoff, and Wiener filters are derived. Besides, the analytical performance evaluation of these time-domain filters is provided and new insights into their functioning are presented. Numerical results are finally given to corroborate our study.
---
paper_title: Multichannel signal enhancement using non-causal, time-domain filters

paper_content:

In the vast amount of time-domain filtering methods for speech enhancement, the filters are designed to be causal. Recently, however, it was shown that the noise reduction and signal distortion capabilities of such single-channel filters can be improved by allowing the filters to be non-causal. While non-causal filters require knowledge of the future, they can be implemented in practice by introducing a short delay. In this paper, we generalize the idea of exploiting non-causality in optimal filter designs to the multichannel scenario. More specifically, a set of optimal, non-causal, multichannel filters for enhancement based on an orthogonal decomposition is proposed. The evaluation shows that there is a potential gain in noise reduction and signal distortion by introducing non-causality. Moreover, experiments on real-life speech show that we can improve the perceptual quality.
---
paper_title: Microphone Arrays Signal Processing Techniques and Applications

paper_content:

I. Speech Enhancement.- 1 Constant Directivity Beamforming.- 2 Superdirective Microphone Arrays.- 3 Post-Filtering Techniques.- 4 Spatial Coherence Functions for Differential Microphones in Isotropic Noise Fields.- 5 Robust Adaptive Beamforming.- 6 GSVD-Based Optimal Filtering for Multi-Microphone Speech Enhancement.- 7 Explicit Speech Modeling for Microphone Array Speech Acquisition.- II. Source Localization.- 8 Robust Localization in Reverberant Rooms.- 9 Multi-Source Localization Strategies.- 10 Joint Audio-Video Signal Processing for Object Localization and Tracking.- III. Applications.- 11 Microphone-Array Hearing Aids.- 12 Small Microphone Arrays with Postfilters for Noise and Acoustic Echo Reduction.- 13 Acoustic Echo Cancellation for Beamforming Microphone Arrays.- 14 Optimal and Adaptive Microphone Arrays for Speech Input in Automobiles.- 15 Speech Recognition with Microphone Arrays.- 16 Blind Separation of Acoustic Signals.- IV. Open Problems and Future Directions.- 17 Future Directions for Microphone Arrays.- 18 Future Directions in Microphone Array Processing.
---
paper_title: Microphone Array Signal Processing

paper_content:

Classical Optimal Filtering.- Conventional Beamforming Techniques.- On the Use of the LCMV Filter in Room Acoustic Environments.- Noise Reduction with Multiple Microphones: a Unified Treatment.- Noncausal (Frequency-Domain) Optimal Filters.- Microphone Arrays from a MIMO Perspective.- Sequential Separation and Dereverberation: the Two-Stage Approach.- Direction-of-Arrival and Time-Difference-of-Arrival Estimation.- Unaddressed Problems.
---
paper_title: Speech Enhancement in the STFT Domain

paper_content:

This work addresses this problem in the short-time Fourier transform (STFT) domain. We divide the general problem into five basic categories depending on the number of microphones being used and whether the interframe or interband correlation is considered. The first category deals with the single-channel problem where STFT coefficients at different frames and frequency bands are assumed to be independent. In this case, the noise reduction filter in each frequency band is basically a real gain. Since a gain does not improve the signal-to-noise ratio (SNR) for any given subband and frame, the noise reduction is basically achieved by liftering the subbands and frames that are less noisy while weighing down on those that are more noisy. The second category also concerns the single-channel problem. The difference is that now the interframe correlation is taken into account and a filter is applied in each subband instead of just a gain. The advantage of using the interframe correlation is that we can improve not only the long-time fullband SNR, but the frame-wise subband SNR as well. The third and fourth classes discuss the problem of multichannel noise reduction in the STFT domain with and without interframe correlation, respectively. In the last category, we consider the interband correlation in the design of the noise reduction filters. We illustrate the basic principle for the single-channel case as an example, while this concept can be generalized to other scenarios. In all categories, we propose different optimization cost functions from which we derive the optimal filters and we also define the performance measures that help analyzing them.
---
paper_title: On Optimal Frequency-Domain Multichannel Linear Filtering for Noise Reduction

paper_content:

Several contributions have been made so far to develop optimal multichannel linear filtering approaches and show their ability to reduce the acoustic noise. However, there has not been a clear unifying theoretical analysis of their performance in terms of both noise reduction and speech distortion. To fill this gap, we analyze the frequency-domain (non-causal) multichannel linear filtering for noise reduction in this paper. For completeness, we consider the noise reduction constrained optimization problem that leads to the parameterized multichannel non-causal Wiener filter (PMWF). Our contribution is fivefold. First, we formally show that the minimum variance distortionless response (MVDR) filter is a particular case of the PMWF by properly formulating the constrained optimization problem of noise reduction. Second, we propose new simplified expressions for the PMWF, the MVDR, and the generalized sidelobe canceller (GSC) that depend on the signals' statistics only. In contrast to earlier works, these expressions are explicitly independent of the channel transfer function ratios. Third, we quantify the theoretical gains and losses in terms of speech distortion and noise reduction when using the PWMF by establishing new simplified closed-form expressions for three performance measures, namely, the signal distortion index, the noise reduction factor (originally proposed in the paper titled ldquoNew insights into the noise reduction Wiener filter,rdquo by J. Chen (IEEE Transactions on Audio, Speech, and Language Processing, Vol. 15, no. 4, pp. 1218-1234, Jul. 2006) to analyze the single channel time-domain Wiener filter), and the output signal-to-noise ratio (SNR). Fourth, we analyze the effects of coherent and incoherent noise in addition to the benefits of utilizing multiple microphones. Fifth, we propose a new proof for the a posteriori SNR improvement achieved by the PMWF. Finally, we provide some simulations results to corroborate the findings of this work.
---
paper_title: Binaural Noise Reduction in the Time Domain With a Stereo Setup

paper_content:

Binaural noise reduction with a stereophonic (or simply stereo) setup has become a very important problem as stereo sound systems and devices are being more and more deployed in modern voice communications. This problem is very challenging since it requires not only the reduction of the noise at the stereo inputs, but also the preservation of the spatial information embodied in the two channels so that after noise reduction the listener can still localize the sound source from the binaural outputs. As a result, simply applying a traditional single-channel noise reduction technique to each channel individually may not work as the spatial effects may be destroyed. In this paper, we present a new formulation of the binaural noise reduction problem in stereo systems. We first form a complex signal from the stereo inputs with one channel being its real part and the other being its imaginary part. By doing so, the binaural noise reduction problem can be processed by a single-channel widely linear filter. The widely linear estimation theory is then used to derive optimal noise reduction filters that can fully take advantage of the noncircularity of the complex speech signal to achieve noise reduction while preserving the desired signal (speech) and spatial information. With this new formulation, the Wiener, minimum variance distortionless response (MVDR), maximum signal-to-noise ratio (SNR), and tradeoff filters are derived. Experiments are provided to justify the effectiveness of these filters.
---
paper_title: Binaural noise-reduction hearing aid scheme with real-time processing in the frequency domain.

paper_content:

A binaural noise-reduction algorithm for suppressing lateral noise sources was implemented in real-time and tested with normal and hearing-impaired subjects. With this algorithm, frequency bands with interaural time and amplitude differences characteristic for the "target" direction of the desired speech sound reach both ears without any attenuation. However, frequency bands with an interaural time and amplitude difference deviating from these desired values are attenuated. Speech quality assessments as well as speech intelligibility tests demonstrate a significant noise reduction of up to about 5 dB in signal-to-noise ratio for a dummy-head recorded "cocktail-party situation" in an anechoic chamber. For two additional acoustical situations, however, the subjective speech quality from the algorithm degrades with increasing reverberation, while the output from a real-time "dereverberation" algorithm was ranked higher with increasing reverberation. The results with hearing-impaired listeners suggest a combination of both algorithms, while stressing the importance of binaural listening and binaural hearing aids for the rehabilitation of impaired listeners in noisy environments.
---

Title: A brief overview of speech enhancement with linear filtering

Section 1: Introduction
Description 1: This section introduces the problem of speech enhancement and noise reduction, discusses historical approaches and their limitations, and outlines the scope and structure of the paper.

Section 2: Signal model and problem formulation
Description 2: This section presents the general signal model, key assumptions, and the problem statement related to speech enhancement using linear filters.

Section 3: Speech enhancement with the conventional approach
Description 3: This section explains the principles behind the conventional approach for speech enhancement and describes various performance measures and optimal filters that can be derived using this approach.

Section 4: Speech enhancement with the orthogonal decomposition of the speech signal vector
Description 4: This section introduces the orthogonal decomposition approach for speech enhancement, details its principles, and demonstrates how to derive optimal filters.

Section 5: Application examples
Description 5: This section provides practical applications of both approaches in different contexts, including single-channel noise reduction in the time and time-frequency domains, multichannel noise reduction in the time and frequency domains, and binaural noise reduction.

Section 6: Conclusions
Description 6: This section summarizes the key findings of the paper, comparing the two approaches in terms of performance, and highlights the applications and implications of the results.

Section 7: Endnotes
Description 7: This section contains additional remarks and clarifications related to the content discussed in the paper.