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Cyclostationary processes are an important class of nonstationary processes that have periodically varying correlation properties. They can model periodic phenomena occurring in science and technology, including communications (modulation, sampling, and multiplexing), meteorology, oceanography, climatology, astronomy (rotation of the Earth and other planets), and economics (seasonality). While cyclostationarity can manifest itself in statistics of arbitrary order, we will restrict our attention to phenomena in which the second-order correlation and complementary correlation functions are periodic in their global time variable.
Our program for this chapter is as follows. In Section 10.1, we discuss the spectral properties of harmonizable cyclostationary processes. We have seen in Chapter 8 that the second-order averages of a WSS process are characterized by the power spectral density (PSD) and complementary power spectral density (C-PSD). These each correspond to a single δ-ridge (the stationary manifold) in the spectral correlation and complementary spectral correlation. Cyclostationary processes have a (possibly countably infinite) number of so-called cyclic PSDs and C-PSDs. These correspond to δ-ridges in the spectral correlation and complementary spectral correlation that are parallel to the stationary manifold. In Section 10.2, we derive the cyclic PSDs and C-PSDs of linearly modulated digital communication signals. We will see that there are two types of cyclostationarity: one related to the symbol rate, the other to impropriety and carrier modulation.
Because cyclostationary processes are spectrally correlated between different frequencies, they have spectral redundancy. This redundancy can be exploited in optimum estimation.
One of the most important applications of probability in science and engineering is to the theory of statistical inference, wherein the problem is to draw defensible conclusions from experimental evidence. The three main branches of statistical inference are parameter estimation, hypothesis testing, and time-series analysis. Or, as we say in the engineering sciences, the three main branches of statistical signal processing are estimation, detection, and signal analysis.
A common problem is to estimate the value of a parameter, or vector of parameters, from a sequence of measurements. The underlying probability law that governs the generation of the measurements depends on the parameter. Engineering language would say that a source of information, loosely speaking, generates a signal x and a channel carries this information in a measurement y, whose probability law p(y∣x) depends on the signal. There is usually little controversy over this aspect of the problem because the measurement scheme generally determines the probability law. There is, however, a philosophical divide about the modeling of the signal x. Frequentists adopt the point of view that to assign a probability law to the signal assumes too much. They argue that the signal should be treated as an unknown constant and the data should be allowed to speak for itself. Bayesians argue that the signal should be treated as a random variable whose prior probability distribution is to be updated to a posterior distribution as measurements are made.
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).
Applied Speech and Audio Processing is a MATLAB-based, one-stop resource that blends speech and hearing research in describing the key techniques of speech and audio processing. This practically oriented text provides MATLAB examples throughout to illustrate the concepts discussed and to give the reader hands-on experience with important techniques. Chapters on basic audio processing and the characteristics of speech and hearing lay the foundations of speech signal processing, which are built upon in subsequent sections explaining audio handling, coding, compression, and analysis techniques. The final chapter explores a number of advanced topics that use these techniques, including psychoacoustic modelling, a subject which underpins MP3 and related audio formats. With its hands-on nature and numerous MATLAB examples, this book is ideal for graduate students and practitioners working with speech or audio systems.
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.