Active noise control (ANC) is achieved by introducing a canceling "antinoise" wave through an appropriate array of secondary sources. These secondary sources are interconnected through an electronic system using a specific signal processing algorithm for the particular cancellation scheme. ANC is an effective way to attenuate noise that is very difficult and expensive to control using passive means. It has application to a wide variety of problems in manufacturing, industrial operations, and consumer products. Many industrial companies and universities are currently engaged in ANC research and development. This book is intended primarily as a reference source for researchers and engineers, with emphasis on signal processing and implementation; however, it can also serve as a text or reference for senior- or graduate-level electrical, mechanical, and acoustical engineering courses that are being offered at more and more colleges to present the necessary background for ANC.

The current wave of interest in ANC is due largely to the confluence of digital signal processing (DSP) hardware and adaptive signal processing algorithms, both of which have come into prominence within the last decade. This book aims to introduce the basic concepts of ANC from the standpoint of these two widely held perspectives.

We have chosen to take a broad definition of ANC as being concerned with the reduction of any kind of undesirable disturbance or noise, whether it is borne by electrical, acoustic, vibration, or any other kind of media. The key attribute of ANC that distinguishes it from older adaptive noise cancellation techniques is the presence of a transfer function from the adaptive filter output to the error sensing node. In this book, we refer to this as the secondary path, which conveys the adaptive filter output signal through transducers and wave propagation to cancel the primary noise.

One of the authors originally introduced the notion of having a transfer function in the cancellation path of an adaptive filter in 1980 and suggested means for its compensation. Although that work was aimed at electrical problems and did not anticipate the more recent acoustic and vibration ANC applications, it did lay some of the groundwork for grappling with the problem of a transfer function in the secondary path, which, if ignored, can lead to instability. In particular, a means for compensation was suggested, whereby the reference signal is filtered by an estimate of the secondary-path transfer function before correlating with the error feedback signal. Shortly after, in 1981, Burgess independently conceived this method within the context of acoustic duct noise cancellation and rigorously justified its usage by explicitly calculating the gradient of the instantaneous error. A similar notion was also independently introduced in 1981 by Widrow in the context of adaptive control systems. In that work, the term "filtered-X least-mean-square" (FXLMS) algorithm was coined, which has since gained widespread usage. This algorithm forms the cornerstone of ANC.

The emphasis of this book is on the practical aspects of ANC systems in terms of signal processing and DSP implementation. Thus, the principles of adaptive signal processing are combined with experimental results and practical issues, including the implementation of these structures and algorithms using C programs and assembly programs on DSP chips (TMS320C25 and TMS320C30 from Texas Instruments). In the way of theory, the book features concise derivations and analyses of commonly used adaptive structures and algorithms for ANC applications. A compilation of C and assembly programs is included in a floppy disk along with this book, and can be used to implement many ANC systems. This software (source code) is in a form ready to be used by students, researchers, and engineers.

We have tried to select application examples not to dazzle the cognoscenti, but rather to motivate graduate students and nonexperts in the field, who may not completely follow all the theoretical development, but could at least get a flavor of the idea by seeing some concrete examples and results of actual experiments. The amply cited literature in this book is replete with more advanced and more fully developed application examples, many of which are still in a state of evolution.

In matters of notation and style, we have spent much time and thought to make the most efficient and parsimonious mathematical representations. Moreover, we have sought to employ conventional usage as much as possible within the fields of mathematics, digital signal processing, adaptive signal processing, and active noise control, which are the main disciplines represented in this book. Some compromises were necessary, however, in order to reconcile differences between the various fields.

Discrete-time signals are assumed throughout with time index in order to be compatible with most of the DSP literature. We use lowercase letters for time-domain signals and uppercase letters in the frequency domain, which is now standard practice. We have also adopted the preferred modern style of using bold lowercase for vectors and bold uppercase letters for matrices. (An occasional conflict occurs for representing vectors of frequency-domain quantities, in which case we defer to the uppercase.) When subscripts are employed to enumerate a set of quantities, we try to use the suggestive notation of a lowercase running index and an uppercase index of the same letter for the total number of items, for instance, .

In the adaptive signal processing literature, the use of for the adaptive weight vector is historic, starting with Widrow. In much of the literature, the number of adaptive weights is denoted by . However, numbering the weights as would be in conflict with the time index . Therefore the weight index was chosen and adaptive weights are enumerated as . Another convention that we have adopted is to order the adaptive weight update term as , putting the reference signal term first. This makes the ordering consistent with the multiple-channel case in which the update takes the untransposable matrix-vector form , and is also consistent with the usual mathematical operator order, where the signal flow is from right to left (as opposed to signal flow diagrams, which go from left to right).

We have also tried to develop a uniform notation within the framework of ANC systems. In order to be consistent with the use of for the primary path, we chose to represent the secondary path by rather than using as in some of the ANC literature, starting with Elliott. In representing the cancellation process, some devotees of ANC prefer to add the secondary path to the primary path. However, we prefer the convention of subtracting the secondary path from the primary path in order to be consistent with the vast body of adaptive filtering literature. The convention is rather arbitrary anyway, because one can always change the sign of the electrical signal feeding the secondary source to agree with whatever representation is desired.

In Chapter 1, we develop the basic philosophy of the ANC technique and explain why it is advantageous. Applications are cited in many fields. A general viewpoint of ANC is expressed that encompasses all types of noise media, such as air-acoustic, hydro-acoustic, and vibration.

In Chapter 2, we review adaptive filter theory and introduce commonly used algorithms that will carry over to ANC.

Chapter 3 covers broadband ANC. We discuss basic limitations due to coherence, derive the FXLMS algorithm, discuss feedback problems and solutions, and introduce the recursive filtered-U LMS algorithm. Practical system considerations are discussed throughout the development.

In Chapter 4, ANC techniques are specialized to the narrowband case, and the waveform synthesis method and adaptive notch filter are introduced.

Chapter 5 extends basic ANC techniques to multiple-channel algorithms and provides a unified and coherent treatment of the subject.

In Chapter 6, we first discuss classical nonadaptive feedback control to establish the background. Then, the concept of adaptive feedback ANC is developed from the standpoint of reference signal synthesis, thereby providing a link to the feedforward systems of Chapters 3 and 4. Finally, hybrid combinations of feedforward and feedback systems are considered.

Chapter 7 develops various on-line secondary-path modeling techniques. Attention is drawn to a fundamental bias problem, and several techniques for its solution are presented.

Chapter 8 introduces various special ANC algorithms and implementations such as the lattice ANC, frequency-domain ANC, recursive-least-squares (RLS) algorithm for ANC, subband ANC, and modal-based ANC.

Chapter 9 presents many examples of applications involving real and simulated experiments for air-acoustic and vibration problems.

As with any book attempting to capture the state of the art at a given time, there will necessarily be omissions, some intentional, some unintentional, that are necessitated by the rapidly evolving developments in this dynamic field of exciting theoretical and practical interest. We hope, at least, that this book will serve as a guide for what has already come and as an inspiration for what will follow.

S. M. KUO AND D. R. MORGAN