Introduction

There is currently a heated debate among fund managers, academics and stakeholders alike concerning to what extent institutional investors like pension funds should take social, ethical and environmental (SEE) considerations into account in investment decisions. Proponents argue that “with great power comes great responsibility,” that is, that pension funds in industrialized nations – which today hold assets equivalent to (on average) 76 percent of the GDP of their respective countries (IFSL 2009) – for this reason also have a responsibility to care about the long-term prosperity and well-being of their countries’ populations (Sethi 2005; Viederman 2008). By integrating SEE concerns, it is argued that pension funds would incentivize companies listed on the stock market to promote sustainable development and more long-term growth (Kiernan 2009; Woods 2009). This is then, it is said, what a responsible pension fund ought to do. But is it really asset owners’ role to take a stand on such nonfinancial issues?

A central point of discussion in this context concerns how to understand the legal responsibilities owed by trustees to their beneficiaries; their so-called fiduciary duties. Pension trustees are under a legal obligation to manage their funds in the best interests of the ultimate recipients of the funds, that is, the future pensioners (Freshfields Bruckhaus Deringer 2005; Whitfield 2005). “Interests” have over time become equated with short term “financial interests,” as captured by prevailing finance theories, and has been thought to preclude fiduciaries from taking SEE concerns into account in investment decisions (Hawley et al. 2011; Hess 2007). However, this view is now being challenged on many fronts. A common argument in the debate is that “interests” can be understood more broadly to also include, for instance, beneficiaries’ ethical interests or welfare interests (Lydenberg 2012; Sandberg 2013). As Richardson (2007: 158–9) argues: “If beneficiaries share a moral objection to a particular form of investment, it could be construed as for their benefit if the trust avoided that investment, possibly even at the cost of a lower financial return.”

Signal parameter estimation, and specifically direction of arrival (DOA) estimation for sensor array data is encountered in a number of applications ranging from electronic surveillance to wireless communications. Subspace-based methods have shown to provide computationally as well as statistically efficient algorithms for DOA estimation. Estimator performance is ultimately limited by model disturbances like measurement noise and model errors. Herein, we review a recently proposed framework that allows the derivation of optimal subspace methods, taking both finite sample effects (noise) and model errors into account. We show how this generic estimator reduces to well-known techniques for cases when one disturbance completely dominates the other.

Introduction

Subspace-based techniques have been shown to be powerful tools in many signal processing applications where the observed data consist of low-rank signals in noise. Some examples include sensor array signal processing, harmonic retrieval, factor analysis, timing estimation, frequency offset estimation, image processing, system identification, and blind channel identification. By appropriate use of the underlying low-rank data model and the associated signal and noise characteristics, subspace estimation techniques can often be made computationally and/or statistically efficient.

This chapter focuses on subspace techniques for direction of arrival (DOA) estimation from data collected by a sensor array. This is quite a mature field of research by now; many tutorial papers and books have been presented, some detailing specific aspects and others giving broader views (e.g., Krim and Viberg (1996), Van Trees (2002), and the references therein). We have no ambition whatsoever to give a comprehensive account of the development of the field of DOA estimation in this chapter.