The goal of an active sensing system, such as radar or sonar, is to determine useful properties of the targets or of the propagation medium by transmitting certain waveforms toward an area of interest and analyzing the received signals. For example, a land-based surveillance radar sends electromagnetic waves in the direction of the sky, where objects such as airplanes can reflect a (usually very tiny) fraction of the transmitted signal back to the radar. By measuring the round-trip time delay, the distance (called the range) between the radar and the target can be estimated since the speed of propagation for radio waves is known (3 × 108 m/s). Additional target properties can be obtained by performing further processing at the receiver side; e.g., the speed of a target can be estimated by measuring the Doppler frequency shift of the received signal.

In 1904, a German engineer named Christian Hülsmeyer carried out the first radar experiment using his “telemobiloscope” to detect ships in dense fog by means of radio waves. As to sonar, Reginald Fessenden, a Canadian engineer, demonstrated this in 1914 using a sound echo device, though not successfully, for iceberg detection off the east coast of Canada. It was amongst several other experiments and patents said to be prompted by the 1912 Titanic disaster.

Radar and sonar underwent considerable development during the two world wars and later on spread into diverse fields including weather monitoring, flight control and underwater sensing.

Achieving reliable communication over underwater acoustic (UWA) channels is a challenging problem owing to the scarce bandwidth available and the double spreading phenomenon, i.e., spreading in both time (multipath delay spread) and frequency domains (Doppler spread) [Kilfoyle & Baggeroer 2000]. Delay and Doppler spreading is inherent to many practical communication channels, but are considerably amplified in UWA environments [Stojanovic et al. 1994]. Double spreading complicates the receiver structure and makes it difficult to extract the desired symbols from the received signals.

Telemetry systems adopting direct-sequence spread-spectrum (DSSS) based modulation techniques are conventionally referred to as operating at low data rates. Existing literature regarding low data rate UWA communications include [Palmese et al. 2007][Stojanovic et al. 1998][Hursky et al. 2006][Yang & Yang 2008][Stojanovic & Freitag 2000][Blackmon et al. 2002][Ritcey & Griep 1995][Stojanovic & Freitag 2004][Sozer et al. 1999][Tsimenidis et al. 2001][Iltis & Fuxjaeger 1991]. By sacrificing the data rate, DSSS techniques exploit frequency diversity in the frequency-selective UWA channel and benefit from spreading gain to allow many co-channel users. At the receiver side, decentralized reception schemes encompass nonlinear equalization including hypothesis-feedback equalization [Stojanovic & Freitag 2000], and linear equalization including RAKE receivers [Tse & Viswanath 2005]. Performance comparisons of hypothesis-feedback equalization and RAKE reception are presented in [Blackmon et al. 2002].

In this chapter, we consider a single user scenario with a coherent RAKE reception scheme (the noncoherent schemes, which do not require a channel estimation, will be discussed in Chapter 19).

The development of breast cancer imaging techniques such as microwave imaging [Meaney et al. 2000][Guo et al. 2006], ultrasound imaging [Szabo 2004][Kremkau 1993], thermal acoustic imaging [Kruger et al. 1999] and MRI has improved the ability to visualize and accurately locate a breast tumor without the need for surgery [Gianfelice et al. 2003]. This has led to the possibility of noninvasive local hyperthermia treatment of breast cancer. Many studies have been performed to demonstrate the effectiveness of local hyperthermia on the treatment of breast cancer [Falk & Issels 2001][Vernon et al. 1996]. A challenge in the local hyperthermia treatment is heating the malignant tumors to a temperature above 43C for about thirty to sixty minutes, while at the same time maintaining a low temperature in the surrounding healthy breast tissue region.

There are two major classes of local hyperthermia techniques: microwave hyperthermia [Fenn et al. 1996] and ultrasound hyperthermia [Diederich & Hynynen 1999]. The penetration of microwave in biological tissues is poor. Moreover, the focal spot generated by microwave is unsatisfactory at the normal/cancerous tissue interface because of the long wavelength of the microwave. Ultrasound can achieve much better penetration depths than microwave. However, because its acoustic wavelength is very short, the focal spot generated by ultrasound is very small (millimeter or submillimeter in diameter) compared with a large tumor (centimeter in diameter on the average). Thus, many focal spots are required for complete tumor coverage, which results in a long treatment time and missed cancer cells.

The focus of this book is on developing computational algorithms for transmit waveform design in active sensing applications, such as radar, sonar, communications and medical imaging. Waveforms are designed to achieve certain desired properties, which are divided into three categories corresponding to the three main parts in the book, namely good aperiodic correlations, good periodic correlations and beampattern matching. The principal approach is based on formulating practical problems mathematically and then solving the problems using optimization techniques. Particular attention is paid to making the developed algorithms computationally efficient. Theoretical analysis that describes performance lower bounds or limitations is provided. Various application examples using the newly designed waveforms are presented, including radar imaging, channel estimation, an ultrasound system for medical treatment and covert underwater spread spectrum communications.

This book is a research monograph. Its backbone is a series of innovative waveformdesign algorithms that we have developed in recent years. These algorithms address different specific problems of waveform design, yet the topics discussed are all centered around active sensing applications and the optimization techniques share similar ideas (e.g., iterative and cyclic procedures, incorporation of fast Fourier transforms, etc.). Notably, all these algorithms are computational approaches that reply on the implementation of computer programs, as opposed to classic waveform design approaches that are mostly analytical. By stitching these algorithms together in a book, we are able to tell a detailed story on various aspects of waveform design, within a consistent framework highlighting computational approaches.

Chapter 13 focused on the optimization of the waveform covariance matrix R for transmit beampattern design. While designing R is an important step, the final goal is the waveform design and we might therefore think of designing directly the probing signals by optimizing a given performance measure with respect to the matrix X of the signal waveforms. However, compared with optimizing the same performance measure with respect to the covariance matrix R of the transmitted waveforms, optimizing directly with respect to X is a more complicated problem. This is so because X has more unknowns than R and the dependence of various performance measures on X is more intricate than the dependence on R (as R is a quadratic function of X).

In this chapter we consider the following problem: with R obtained in a previous (optimization) stage, we want to determine a signal waveform matrix X whose covariance matrix is equal or close to R, and which also satisfies some practically motivated constraints (such as constant-modulus or low PAR constraints). We present a cyclic algorithm for the synthesis of such an X. We also consider the case where the synthesized waveforms are required to have good auto- and cross-correlation properties in time. Several numerical examples are provided to demonstrate the effectiveness of the proposed methodology.

Problem formulation

Consider an active sensing system equipped with M transmit antennas. Let the columns of X ∈ ℂN×M be the transmitted waveforms, where N denotes the number of samples in each waveform.

Antenna array beampattern design has been a well-studied topic and there is a considerable literature available from classic analytical design [Mailloux 1982][Dolph 1946][Elliott 1975][Ward et al. 1996][Van Trees 2002] to more recent works that resort to numerical optimization [Lebret & Boyd 1997][Scholnik & Coleman 2000][Cardone et al. 2002][San Antonio & Fuhrmann 2005][Li, Xie, Stoica, Zheng & Ward 2007]. The predominant problem considered in the literature refers to the receive beampattern design, which is concerned with designing weights for the received signal so that the signal component impinging from a particular direction is reinforced while those from other directions are attenuated, a way in which certain signal properties (e.g., the signal power or direction-of-arrival) can be estimated. Such a problem usually boils down to the design of an FIR (finite-impulse-response) filter in the narrowband case or a set of FIR filters in the wideband case.

The transmit beampattern design, on the other hand, is concerned with designing the probing signals to approximate a desired transmit beampattern (i.e., an energy distribution in space and frequency). It has been often stated that the receive and transmit beampattern designs are essentially equivalent, which is partly true in the sense that the two scenarios bear similar problem formulations and that the FIR filter taps obtained via receive pattern design can be used theoretically as the probing signal to achieve an identical transmit pattern. In practice, however, the transmit beampattern design problem appears to be much harder because of the energy and peak-to-average power ratio (PAR) constraints on the transmit waveforms.