A Geometric Theory of Intersymbol Interference, Part I: Zero-Forcing and Decision-Feedback Equalization

The analysis of digital communication systems from a geometrical viewpoint--the viewing of waveforms as points in a signal space and the identification of cross-correlation with the formation of an inner product--is by now well established. To a large extent, this approach has been popularized by the book of Wozencraft and Jacobs. 1 However, when it comes to analyzing systems with intersymbol interference, frequency-domain techniques have almost exclusively been relied upon. The purpose of this paper is to consider pulse-amplitude modulation (PAM) systems with intersymbol interference from a geometric standpoint, and more specifically to develop a geometric theory of equalization. 1483 1484 THE B E L L S Y S T E M TECHNICAL JOURNAL, N O V E M B E R 1 9 7 3 Consideration of the geometric structure of intersymbol interference leads immediately to the observation of a striking correspondence to the theory of minimum mean-square error (MMSE) linear estimation of a wide-sense stationary discrete-parameter random process. The fact that the latter subject is almost exclusively treated by geometric methods2-3 is further impetus for this approach to equalization. The theories of linear zero-forcing equalization and decision-feedback equalization are well established. The properties of linear equalization