This chapter unifies the principles and analyses of conventional signal processing algorithms for receive-side smart antennas, and compares their performance and numerical complexity. The chapter starts with a brief look at the traditional single-antenna optimum symbol-detector, continues with analyses of conventional smart antenna algorithms, i.e., statistical beamforming (BF) and maximal-ratio combining (MRC), and culminates with an assessment of their recently proposed superset known as eigencombining or eigenbeamforming. BF or MRC performance fluctuates with changing propagation conditions, although their numerical complexity remains constant. Maximal-ratio eigencombining (MREC) has been devised to achieve best (i.e., near-MRC) performance for complexity that matches the actual channel conditions. The authors derive MREC outage probability and average error probability expressions applicable for any correlation. Particular cases apply to BF and MRC. These tools and numerical complexity assessments help demonstrate the advantages of MREC versus BF or MRC in realistic scenarios.
General Perspective. Andrew Viterbi is credited with famously stating that “spatial processing remains as the most promising, if not the last frontier, in the evolution of multiple access systems” (Roy, 1998, p. 339). Multiple-antenna-transceiver communications systems, also known as single-input multiple-output (SIMO), multiple-input single-output (MISO), or multiple-input multiple-output (MIMO) systems, which exploit the spatial dimension of the radio channel, promise tremendous benefits over the traditional single-input single-output (SISO) transceiver concept, in terms of data rate, subscriber capacity, cell coverage, link quality, transmit power, etc. Such benefits can be achieved with smart antennas, i.e., SIMO, MISO, and MIMO systems that combine baseband signals for optimum performance (Paulraj, Nabar, & Gore, 2005).
Herein, we consider receive smart antennas (i.e., the SIMO case) deployed in noise-limited scenarios with frequency-flat multipath fading (El Zooghby, 2005, Section 3.3) (Jakes, 1974) (Vaughan & Andersen, 2003, Chapter 3), for which the following signal combining techniques have conventionally been proposed:
Statistical beamforming (BF), i.e., digitally steering a radio beam along the dominant eigenvector of the correlation matrix of the channel fading gain vector (S. Choi, Choi, Im, & Choi, 2002) (El Zooghby, 2005, Eqn. (5.23), p. 126, Eqns. (5.78–80), p. 148) (Vaughan & Andersen, 2003, Section 9.2.2). BF enhances vs. SISO the average, over the fading and noise, signal-to-noise ratio (SNR) by an array gain factor that is ultimately proportional to the antenna correlation and is no greater than the number of antenna elements. Since BF requires the estimation of only the projection of the channel gain vector onto the eigenvector mentioned above, it has low numerical complexity. However, BF is effective only for highly-correlated channel gains, i.e., when the intended signal arrives with narrow azimuth angle spread (AS).
Maximal-ratio combining (MRC), i.e., maximizing the output SNR conditioned on the fading gains (Brennan, 2003; Simon & Alouini, 2000). This SNR is computed by averaging over the noise only, i.e., conditioning on the channel gains. When the intended-signal AS is large enough to significantly reduce antenna correlation, MRC can greatly outperform BF as a result of diversity gain and array gain, at the cost of much higher numerical complexity incurred due to channel estimation for each antenna element.
Note that, for fully correlated (i.e., coherent) channel gains, both BF and MRC reduce to the classical notion of “beamforming” whereby a beam is formed towards the intended signal arriving from a discrete direction (Monzingo & Miller, 1980; Trees, 2002; Godara, 2004).
Statistical beamforming and diversity combining principles have traditionally been classified, studied, and applied separately, leading to disparate and limited performance analyses of BF and MRC. Furthermore, since BF and MRC optimize the average SNR and the conditioned SNR, respectively, they have opposing performance-maximizing spatial correlation requirements, as well as significantly different, correlation-independent, numerical complexities (Siriteanu, Blostein, & Millar, 2006; Siriteanu, 2006; Siriteanu & Blostein, 2007). Because correlation varies in practice due to variable AS (Algans, Pedersen, & Mogensen, 2002), BF or MRC performance fluctuates, whereas numerical complexity remains constant. Therefore, MRC can actually waste processing resources and power, whereas BF can often perform poorly (Siriteanu et al., 2006; Siriteanu, 2006; Siriteanu & Blostein, 2007).