1 Introduction

In signal processing and control applications where the signals and transfer functions involved are time invariant and known at design time, designing fixed filters and controllers to achieve the desired design goals is sufficient. In many applications, however, signals, transfer functions, and the environment in which the system operates are time-varying. In some applications, such as in active noise and vibration control, it is the rule rather than the exception that the system to be controlled is unknown at design time. In such situations a self designing or self adjusting filter/controller is necessary to achieve the desired system function in a changing environment. This is usually done by adjusting the coefficients of a digital filter/controller on-line in the operation field by optimizing predefined quantities. Self adjusting filters, better known as adaptive filters, might have any underlying filter structure. The most widely used adaptive filter structure is the transversal structure due to the stability and simplicity of analysis of those filters. The linear combiner structure is a generalized version of the transversal structure, and is mainly used in array signal processing applications. Recursive adaptive filters have also found wide application in adaptive line enhancers, autoregressive signal modeling and channel equalization. A third structure which is widely used in adaptive linear prediction applications is the lattice structure. All above mentioned adaptive filter structures are well supported by the current release of the Adaptive Signal Processing Toolbox (ASPT). A short theoretical introduction to transversal filters, linear combiner filters, recursive filters, and lattice filters is given in Sections 2.2.1, 2.2.2, 2.2.3, and 2.2.4, respectively. Adaptive algorithms for adjusting the coefficients of those filter structures are documented in Chapters 4 to 7. This chapter also includes a brief review of the theory behind adaptive signal processing. Adaptive algorithms can roughly be divided into two main categories. The first based on statistical optimization which leads to the the Least Mean Squares (LMS) algorithm and its derivatives. The second is based on deterministic optimization which leads to the Recursive Least Squares (RLS) algorithm and its derivatives. The basic optimization problems for statistic and deterministic approaches and the model on which all ASPT functions are based is introduced in Section 2.3. A brief review of some common adaptive filters applications is given in Section 2.4. This is by far not a complete list of adaptive signal processing applications but gives the novice reader a good basis on which she can start building her own applications. System identification and forward modeling applications using adaptive filters are described in Section 2.4.1, equalization and inverse modeling in Section 2.4.2, adaptive linear prediction in Section 2.4.3, adaptive autoregressive spectrum analysis in Section 2.4.4, echo cancellation in Section 2.4.5, and finally adaptive interference canceling in Section 2.4.6