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.

Chapter 2 dealt with designing a single sequence with good auto-correlation properties. In a similar way, in many applications a set of sequences that have good correlation properties is desired, for example in MIMO (multi-input and multi-output) radar and CDMA (code division multiple access) systems. When transmitting orthogonal waveforms, a MIMO radar system can achieve a greatly increased virtual aperture compared with its phased array counterpart. This increased virtual aperture enables many of the MIMO radar advantages, such as better detection performance [Fishler et al. 2006], improved parameter identifiability [Li, Stoica, Xu & Roberts 2007], enhanced resolution [Bliss & Forsythe 2003] and direct applicability of adaptive array techniques [Xu et al. 2008].

In the case of waveform-set design, which is also referred to as multi-waveform design, both auto- and cross-correlations are involved. Good auto-correlation means that a transmitted waveform is nearly uncorrelated with its own time-shifted versions, while good cross-correlation indicates that any transmitted waveform is nearly uncorrelated with other time-shifted transmitted waveforms. Good correlation properties in the above sense reduce the risk that the received signal of interest is found in correlated multipath or clutter interference.

There is an extensive literature about multi-waveform design. In [Deng 2004] and [Khan et al. 2006], orthogonal waveforms are designed that have good auto- and cross-correlation properties, a topic that is directly tied to this chapter. More related to Chapters 13 and 14, [Fuhrmann & San Antonio 2008] and [Stoica et al. 2007] focus on optimizing the covariance matrix of the transmitted waveforms so as to achieve a given transmit beampattern and in [Stoica, Li & Zhu 2008] the waveforms are designed to approximate a given covariance matrix.

Among the tasks associated with cognitive radar [Haykin 2006], an important one is to adapt the spectrum of transmitted waveforms to the changing environment. In particular, the transmitted signal should not use certain frequency bands that have already been reserved, such as the bands for navigation and military communications; or there could exist strong emitters whose operating frequencies should be avoided. Therefore it is required that the spectral power of transmitted waveforms be small for certain frequency bands [Lindenfeld 2004][Salzman et al. 2001][Wang & Lu 2011][Headrick & Skolnik 1974].

The main focus in this chapter is on designing a discrete sequence whose spectral power is small in certain specified frequency bands. The designed sequence can be used in active sensing systems such as radar or sonar as a probing sequence. It can also be used as a spreading sequence in spread-spectrum applications such as CDMA (code division multiple access) systems.

Besides frequency notching, we also need to take into account the correlation properties of the designed sequence. As pointed out several times in previous chapters (e.g., in Chapter 1), in radar or sonar applications low auto-correlation of the probing sequence improves target detection when range compression is applied in the receiver. Furthermore, practical hardware components such as analog-to-digital converters and power amplifiers have a maximum signal-amplitude clip. In order to maximize the transmitted power that is available in the system, unimodular sequences are desirable.

There has been a long-standing interest in designing radar transmit signals and receive filters jointly for clutter/interference rejection; see, e.g., [Rummler 1967][DeLong Jr. & Hofstetter 1967][Spafford 1968][Stutt & Spafford 1968][Blunt & Gerlach 2006][Kay 2007] and [Stoica, Li & Xue 2008]. Clutter (or reverberation in sonar terminology) refers to unwanted echoes that are usually correlated with the transmitted signal, while interference is a term used for noise as well as (adverse) jamming signals. Since the negative impact from clutter and interference should be minimized at the receiver side, a natural criterion for designing transmit signals and receive filters would be to maximize the signal-to-clutter-plus-interference ratio (SCIR) of the receiver output at the time of target detection.

It is well known that a matched filter maximizes the signal-to-noise ratio (SNR) in the presence of additive white noise (see Chapter 1). The matched filter can be implemented as a correlator that multiplies the received signal with a delayed replica of the transmitted signal. The peak of the receiver output indicates the time delay of the target signal. If the received signal also has a Doppler frequency shift due to the relative movement between the target and the platform, a bank of filters is needed, each of which is matched to a specific Doppler frequency.

A matched filter is not able to take care of clutter or jamming suppression, a feature that is left to the transmit signal design.