## ABSTRACT

This chapter considers the problem of adaptively identifying a linear time-invariant causal system by processing only the input and output signals. The total system response to any set of initial conditions and excitation signal can be completely described by the proper weighted sum of modes. The system response can be seen as a finite subset of the infinite number of possibilities of signals. Casting an error signal as an orthogonal projection by definition guarantees the minimum error as the shortest distance between a plane and a point in space is a line normal to the plane and intersecting the point. In the age of signal processing tools such as spectrum analyzers and MATLAB, it is useful to have audio band curves to quickly convert a narrowband sound spectrum to an A-weighted level. The subspace spanned by the observations of the system input signal form the basis of a projection operator.