For a better signal reception, it is often desirable to use multiple sensors or antennas at a receiver. (Note that we will assume that receive antenna and sensor are interchangeable and they are considered as a device that can receive signals through a certain channel medium. For convenience, however, we prefer antenna throughout the book with wireless communication applications in mind.) To extract a signal of interest, multiple signals received by multiple antennas are to be properly combined. For signal combining, we need to take into account the desired signal's (statistical or deterministic) properties as well as statistical properties of background noise.

Although there are various signal combining techniques, we focus on linear combining techniques in this chapter, because they can be relatively easily implemented and their analysis is tractable. In addition, only second-order moments of a desired signal and noise are usually required to find a linear combiner under the MMSE criterion.

Signals in space

Suppose that there are N sensors or antennas to receive a signal of interest generated from a source, which can be a radio signal or a voice. In general, the signal is received through a certain channel medium with channel attenuation or distortion and corrupted by noise. Since multiple observations of a signal are available using multiple sensors or antennas, the signal can be seen as a vector in a vector space as illustrated in Fig. 4.1.