442 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 2, FEBRUARY 2000 Bi-Iteration Multiple Invariance Subspace Tracking and Adaptive ESPRIT Peter Strobach, Senior Member, IEEE Abstract—A class of adaptive bi-iteration SVD (Bi-SVD) algo- rithms for tracking structured subspaces of the form . . . is introduced, where is an orthonormal core basis matrix, and is a set of subro- tors. Structured subspaces of this kind arise in all forms of mul- tiple invariance (MI) techniques, such as MI-ESPRIT. An adaptive MI-ESPRIT algorithm using Bi-SVD MI subspace tracking is de- veloped. Computer experiments validate the theoretical results. Index Terms—Author, please supply index terms. E-mail key- words@ieee.org for info. I. INTRODUCTION E STIMATION of signal parameters via rotational invari- ance techniques (ESPRIT) [1]–[3] has become a subject of widespread interest recently because the ESPRIT concept al- lows the computation of desired signal parameters without ex- plicit knowledge of the underlying array parameters. MI-ES- PRIT techniques are generalizations of ESPRIT to situations, where the temporal or spatial samples possess several displace- ment invariances. A general study of MI-ESPRIT is available in [4]. Various applications to two-dimensional (2-D) or az- imuth/elevation direction finding are described in [5]–[9]. There are many other situations in which the same data model will occur, for instance, joint angle/delay estimation [10]. The adaptive implementation of MI-ESPRIT requires a tracking of the signal subspace. Until now, only conventional unstructured subspace tracking has been considered in this context [22]. The problem of standard (unstructured) subspace tracking has been treated extensively in the numerical analysis and signal processing literature. See, for instance, [11]–[17], and the references listed therein. These subspace trackers, however, are not well matched to the problem at hand as they do not exploit the characteristic stack structure associated with the subspace bases for MI data. The usual practice is that a total least squares (TLS) smoothing is applied to the unstructured (“raw”) subspace basis Manuscript received June 17, 1998; revised July 13, 1999. The associate ed- itor coordinating the review of this paper and approving it for publication was Dr. Eric Moulines. The author is with the Department of Mathematics, University of Passau, Passau, Germany. Publisher Item Identifier S 1053-587X(00)00985-5. estimate to connect the core basis blocks. The method is called TLS-ESPRIT [1] and is suboptimal from a computational point of view. Indeed, we can establish a computationally much more sound procedure that estimates the desired structured subspace basis in closed form directly from the data. This is the idea behind the structured subspace trackers presented in this paper. We combine both subspace iteration and TLS principles into a closed-form solution for direct estimation of stack structured MI subspace bases. Our solution is therefore perfectly matched to the problem at hand. This paper is organized as follows. In Section II, we describe the problem and introduce the necessary basic relationships of MI subspaces that will be required later in the paper. In Section III, the Bi-SVD algorithms for MI structured subspace tracking are developed and summarized in two quasicode tables. In Sec- tion IV, an associated adaptive MI-ESPRIT algorithm is pre- sented. Simulation results with this adaptive MI-ESPRIT algo- rithm in connection with the MI-structured Bi-SVD subspace trackers are shown in Section V. II. MULTIPLE ROTATIONAL INVARIANCE AND STRUCTURED SUBSPACES In this section, we provide the necessary framework that will be required in the following sections. We introduce the MI data model and the MI data subspace. Linear invariance properties of structured and unstructured MI subspace bases and their con- nections are discussed. We show how bi-iteration concepts [15], [16] can be combined with structured subspace fitting to provide a conceptual basis for stack structured subspace tracking. A. The Rotational MI Data Model In this paper, we study sequences of signal snapshot vectors that satisfy the rotational MI data model (1) with (2) where array steering matrix; set of diagonal phase delay matrices; vector of incoherent signals; represents a realization of white noise. 1053-587X/00$10.00 © 2000 IEEE