Engineering and applied science rely heavily on complex variables and complex analysis to model and analyze real physical effects. Why should this be so? That is, why should real measurable effects be represented by complex signals? The ready answer is that one complex signal (or channel) can carry information about two real signals (or two real channels), and the algebra and geometry of analyzing these two real signals as if they were one complex signal brings economies and insights that would not otherwise emerge. But ready answers beg for clarity. In this chapter we aim to provide it. In the bargain, we intend to clarify the language of engineers and applied scientists who casually speak of complex velocities, complex electromagnetic fields, complex baseband signals, complex channels, and so on, when what they are really speaking of is the x- and y-coordinates of velocity, the x- and y-components of an electric field, the in-phase and quadrature components of a modulating waveform, and the sine and cosine channels of a modulator or demodulator.

For electromagnetics, oceanography, atmospheric science, and other disciplines where two-dimensional trajectories bring insight into the underlying physics, it is the complex representation of an ellipse that motivates an interest in complex analysis. For communication theory and signal processing, where amplitude and phase modulations carry information, it is the complex baseband representation of a real bandpass signal that motivates an interest in complex analysis.

Array signal processing or array processing has diverse applications including radar and sonar systems. An array of multiple sensors can provide spatial selectivity in receiving signals. In general, the more sensors, the better spatial selectivity can be achieved. Exploiting this spatial selectivity, we can have a clearer signal or less performance degradation due to interfering signals. An important application of array processing in cellular communication systems is smart antennas. The base stations equipped with antenna arrays for spatial selectivity can be considered as smart antenna systems. In cellular systems, since the performance is degraded by the inter-cell interference (which is the signal transmitted from adjacent cells where the same frequency band is used for communications), it is important to mitigate inter-cell interference. Smart antennas can mitigate inter-cell interference using the spatial selectivity and improve the performance of cellular systems.

In this chapter, we focus on signal combining and related techniques for array processing and discuss how array processing can be applied to smart antenna systems.

Antenna arrays

An antenna array is an array of antenna elements that allows spatial processing, which is also called array processing. For array processing, we need to take into account array configuration and spatial characteristics of signals. Array processing can be extended in both space and time domains. In this case, spatial and temporal processing is to be jointly performed. Array processing can be considered for both signal reception and transmission.

Statistical signal processing is a set of tools for dealing with random signals. As a set of tools, statistical signal processing has a broad range of applications from radars and sonars to speech and image processing. There are a number of books on this topic (e.g. (Scharf 1991) and (Orfanidis 1988)). In this book, instead of providing a comprehensive description of statistical signal processing with a broad range of applications, we focus on key approaches for communications. In particular, we attempt to present mainly signal detection and combining techniques in the context of wireless communications.

Applications in digital communications

The main aim of digital communications is to transmit a sequence of bits over a given channel to a receiver with minimum errors. In implementing digital communication systems, however, there are various constraints to be taken into account. For example, the transmission power is usually limited and the complexity of receiver is also limited. With practical implementation constraints including computational complexity, statistical signal processing plays a crucial role in designing a receiver for digital communications. Although there are a number of different roles that statistical signal processing can play, we confine ourselves to two main topics in this book: one is signal detection and the other is signal combining.

Signal detection has been well established as the main topic in communications. However, advances in multiuser detection have opened up a whole new approach for joint detection (Verdu 1998).

Statistical signal processing is a set of statistical techniques that have been developed to deal with random signals in a number of applications. Since it is rooted in detection and estimation theory, which are well established in statistics, the fundamentals are not changed although new applications have emerged. Thus, I did not have any strong motivation to write another book on statistical signal processing until I was convinced that there was a sufficient amount of new results to be put together with fundamentals of detection and estimation theory in a single book.

These new results have emerged in applying statistical signal processing techniques to wireless communications since 1990. We can consider a few examples here. The first example is smart antenna. Smart antenna is an application of array signal processing to cellular systems to exploit spatial selectivity for improving spectral efficiency. Using antenna arrays, the spatial selectivity can be used to mitigate incoming interfering signals at a receiver or control the transmission direction of signals from a transmitter to avoid any interference with the receivers which do not want to receive the signal. The second example is based on the development of code division multiple access (CDMA) systems for cellular systems. In CDMA systems, multiple users are allowed to transmit their signals simultaneously with different signature waveforms. The matched filter can be employed to detect a desired signal with its signature waveform. This detector is referred to as the single-user detector as it only detects one user's signal.

In various signal processing and communication applications, multiple signals can coexist and a receiver has to detect or estimate multiple signals (or possibly their features) simultaneously. For example, we can consider a multiple-speaker identification system that attempts to identify multiple speakers’ voices simultaneously. Another example is a multi-sensor system for multiple-signal classification in radar and sonar applications. The notion of signal combining in Chapter 4 can be extended to the case of multiple signals. Since other signals co-exist, the signal combiner plays a crucial role in not only combining multiple observations, but also mitigating the other signals.

In this chapter, we discuss optimal signal combining to estimate multiple signals simultaneously when multiple observations or received signals are available. Various well-known optimal combiners are introduced. In particular, we mainly focus on the MMSE combiner as it is widely used and has various crucial properties. In signal combining or estimation, no particular constraint on transmitted signals is imposed, while in Chapter 7 we will discuss signal detection (not estimation) for multiple signals under the assumption that each signal is an element of a signal alphabet or constellation, which becomes a crucial constraint in signal detection.

Systems with multiple signals

Suppose that there are K signal sources transmitting signals simultaneously through a common channel to a receiver as shown in Fig. 6.1. The signals generated from multiple sources could be correlated or not.

Statistical hypothesis testing is a process to accept or reject a hypothesis based on observations, where multiple hypotheses are proposed to characterize observations. Taking observations as realizations of a certain random variable, each hypothesis can be described by a different probability distribution of the random variable. Under a certain criterion, a hypothesis can be accepted for given observations. Signal detection is an application of statistical hypothesis testing.

In this chapter, we present an overview of signal detection and introduce key techniques for performance analysis. We mainly focus on fundamentals of signal detection in this chapter, while various signal detection problems and detection algorithms will be discussed in later chapters (e.g. Chapters 7, 8, and 9).

Elements of hypothesis testing

There are three key elements in statistical hypothesis testing, which are (i) observation(s); (ii) set of hypotheses; and (iii) prior information. With these key elements, the decision process or hypothesis testing can be illustrated as in Fig. 2.1.

In statistical hypothesis testing, observations and prior information are all important and should be taken into account. However, in some cases, no prior information is available or prior information could be useless. In this case, statistical hypothesis testing relies only on observations.

Suppose that there are M(≥ 2) hypotheses. Then, we have an M-ary hypothesis testing in which we will choose one of the M hypotheses that are proposed to characterize observations and prior information under a certain performance criterion. There are various hypothesis tests or decision rules depending on criteria.

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.