The fundamental role of general orthonormal bases in system identification

The purpose of this paper is threefold. Firstly, it is to establish that co ntrary to what might be expected, the accuracy of well-known and frequently used asymptotic variance results can depend on choices of fixed poles or z eros in the model structure, Secondly, it is to derive new variance express ions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros. This is achieved by employing ce rtain new results on generalized Fourier series and the asymptotic properti es of Toeplitz-like matrices in such a way that the new variance expression s presented here encompass pre-existing ones as special cases. Via this lat ter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification, Namely, that ort honormal bases are much more than an implementational option offering impro ved numerical properties. In fact, they are an intrinsic part of estimation since, as shown here, orthonormal bases quantify the asymptotic variabilit y of the estimates whether or not they are actually employed in calculating them.