Introduction

In verbal communication, the presence of background noise, such as the sound of a passing car or an air vent, can impact the quality of the speech signal in a detrimental way, something that affects the listener and thus also the communication in several negative ways. Not only may the perceived quality of the speech be harmed, but also its intelligibility may be degraded. Even if only the perceived quality of the speech is affected, this may have a severe impact on the ability of the users to communicate, as exposure to noisy signals may cause listener fatigue. The presence of noise in signals is, though, not only a problem for humans. In speech processing systems, background noise causes additional problems, as such systems often comprise components that are designed under the assumption that only one, clean speech signal is present at any given time. This is, for example, the case for automatic speech recognizers and speech coders. This is typically done to simplify the design of these components, as the underlying statistical models then do not have to account for all possible noise types. Not only does this simplify the training of such models, it also, generally speaking, leads to faster algorithms; but it also renders these components vulnerable to noise.

As we have argued, the presence of background noise is problematic for humans and computers alike, and the problem of dealing with it, which is called speech enhancement or noise reduction, is an important and long-standing problem in signal processing (see, e.g., [1] and [2] for recent surveys), and the search for new and better methods continues today. Speech enhancement algorithms are important components in many systems, where speech plays a part, including telephony, hearing aids, voice over IP, and automatic speech recognizers. Speech enhancement is generally concerned with the problem of enhancing the quality of speech signals. This can, of course, mean many things, but it is often associated with the specific problem of reducing the impact of additive noise, which is also what we are concerned with in the present book. Additive noise occurs naturally in acoustic environments when multiple sources are present, and examples of common noise types are street, car, and babble. Moreover, it can also be caused by intrinsic noise in the sensor system, i.e., from the electrical components. To be more precise, the purpose of speech enhancement is to minimize the impact of the background noise while preserving the speech signal. Hence, there are two performance measures by which the efficiency of speech enhancement methods is compared: speech distortion and noise reduction [3]. These two measures are often conflicting, meaning that if we want to achieve the highest possible noise reduction, then we must accept speech distortion and, similarly, that if we cannot accept any speech distortion, then our ability to perform noise reduction will be hampered. An extreme example of this is the maximum signal-to-noise ratio filter [1] which achieves the highest possible noise reduction but at the cost of severe speech distortion.

The history of noise reduction can be traced back to the work of Wiener [4], i.e., to the very early days of signal processing. Due to the importance of the problem in particular in speech applications, many different solutions have been proposed over the years, and much time and effort is still devoted to the problem today. The problem is often broken into two sub-problems, namely the problem of finding a function to be applied to the observed signal so as to extract the desired signal, i.e., the speech signal, and the problem of finding the information that this function depends on. If we restrict ourselves to linear filters, then the first sub-problem is the problem of finding the optimal filter, i.e., a filter design problem. If the criterion for optimality is the mean-square error, then the so-called Wiener filter is the solution. This filter requires knowledge of the noise statistics (or the speech statistics), and the second sub-problem is then that of finding those statistics, often in the form of the noise correlation matrix or its power spectral density. In the past decade, most work seems to have focused on the second sub-problem, e.g., [5–9], under difficult conditions when the noise is nonstationary. This book is, however, concerned with the first sub-problem, which is determining the function that should be applied to the observed signal. This problem has, though, also seen some important new contributions regarding optimal filtering in the past few years, including [3,10,11].

In the literature, one can find many (seemingly) different attempts at solving the problem of speech enhancement, and at the time a new method is published, it is often not clear how exactly it relates to other, existing methods, often because it is either not clear exactly what problems are being solved, or that the problems are stated in different ways whose relation is difficult to ascertain. In fact, it appears that the retrospective process of relating methods may take decades, if it ever occurs. When listing existing classes of methods for speech enhancement, spectral subtraction, (optimal) linear filtering, statistical model-based approaches, and subspace methods are typically mentioned. Indeed, these are also the names of the chapters in the book [2]. The focus in the present book is on the class of methods generally known as optimal filtering, of which the classical Wiener filter is a special case. However, in this book, we will show how speech enhancement using the principles of subspace-based methods can be cast as an optimal filtering problem. As such, the present book unifies what has previously been considered two competing principles of speech enhancement in one framework. As a consequence, it is both possible to combine the benefits of the subspace methods and optimal filtering methods and to analyze and compare the performance of the various approaches analytically.