Time-Frequency Analysis and Synthesis of Linear Signal Spaces: Time-Frequency Filters, Signal Detection and Estimation, and Range-Doppler Estimation

F. Hlawatsch

Boston, MA: Kluwer, 1998.

**Abstract**

Linear signal spaces are of fundamental importance in signal and system theory, communication theory, and modern signal processing. This book proposes a time-frequency analysis of linear signal spaces that is based on two novel time-frequency representations called the “Wigner distribution of a linear signal space” and the “ambiguity function of a linear signal space.”

Besides being a useful display and analysis tool, the Wigner distribution of a linear signal space allows the design of high-resolution time-frequency filtering methods. This book develops such methods and applies them to the enhancement, decomposition, estimation, and detection of noisy deterministic and stochastic signals. Formulation of the filtering (estimation, detection) methods in the time-frequency plane yields a direct interpretation of the effect of adding or deleting information, changing parameters, etc. In a sense, the prior information and the signal processing tasks are brought to life in the time-frequency plane.

The ambiguity function of a linear signal space, on the other hand, is closely related to a novel maximum-likelihood multipulse estimator of the range and Doppler shift of a slowly fluctuating point target – an estimation problem that is important in radar and sonar. Specifically, the ambiguity function of a linear signal space is relevant to the problem of optimally designing a set of radar pulses.

**Table of contents**

Preface. 1. Introduction and Outline. 2. The Wigner Distribution of a Linear Signal Space. 3. Time-Frequency Localization of Linear Signal Spaces. 4. Time-Frequency Synthesis of Linear Signal Spaces. 5. Time-Frequency Filters and Expansions. 6. Signal Estimation and Signal Detection. 7. The Ambiguity Function of a Linear Signal Space. 8. Range-Doppler Estimation. 9. Conclusions. References. Index.

**Detailed outline**

Chapter 1 is of introductory nature. It contains brief reviews of linear signal spaces and time-frequency (TF) analysis as well as an outline of the subsequent chapters.

Chapter 2 introduces and studies the Wigner distribution (WD) of a linear signal space. Similar to the WD of a signal, the WD of a linear signal space describes the space’s energy distribution over the TF plane. We derive simple expressions for the WD of a signal space in terms of the space’s orthogonal projection operator and orthonormal bases. We study important properties and the “energetic” interpretation of the WD of a signal space, and we consider the results obtained for some specific spaces. The cross-WD of two signal spaces and a discrete-time WD version are introduced. We discuss the minimization and maximization of WD integrals. We also show how any arbitrary quadratic signal representation or quadratic signal parameter can be extended to linear signal spaces.

Chapter 3 uses the WD of a linear signal space for an investigation into the TF localization of linear signal spaces. We discuss the geometric properties (or “shape”) of the WD of a space. A distinction between “sophisticated” and “simple” spaces is drawn. We discuss the “TF disjointness” and “TF affiliation” of two spaces, and of a signal and a space. Furthermore, two quantities measuring the overall TF concentration of a space are introduced, and concentration bounds are formulated that can be viewed as uncertainty relations for signal spaces. The spaces attaining these concentration bounds (i.e., the spaces with maximum TF concentration) are derived. Finally, two related quantities describing the TF concentration and localization of a space in a given TF region are proposed, and bounds for these quantities are derived.

The optimum TF synthesis or TF design of linear signal spaces is considered in Chapter 4. We present a method for constructing a space that, loosely speaking, “comprises all signals located in a given TF region.” This can be made mathematically precise by using the WD of a space. Specifically, the optimum space is defined as the space whose WD is closest to the indicator function of the given TF region. This optimum space is shown to be an “eigenspace” of the TF region, and some properties of eigenspaces are discussed. The design method is then extended to include a signal subspace constraint, and it is reformulated in a discrete-time setting. Finally, it is shown that the spaces with maximum TF concentration considered in Chapter 3 are also optimum in the context of TF synthesis.

Chapter 5 considers two signal processing applications of the design methods discussed in Chapter 4, namely, TF projection filters and TF signal expansions. A TF projection filter is a linear, time-varying filter with a specified “TF pass region,” i.e., the filter passes all signals located in the given TF pass region and suppresses all signals located outside the TF pass region. A TF signal expansion allows the parsimonious representation of signals located in a given TF support region. Both methods permit the effective suppression of noise and interfering signals. The concept of TF projection filters is extended to TF filter banks possessing the perfect reconstruction property. The performance of TF projection filters and TF filter banks is demonstrated using computer simulation.

The application of TF projection filters to statistical signal processing is considered in Chapter 6. We derive the optimum projection systems for estimating (enhancing) and detecting a nonstationary random process corrupted by noise. We then show how these statistically optimum projection systems can be approximated by simple TF filter designs that require less statistical a priori knowledge than the optimum systems. The satisfactory performance of these approximate TF projection filters is verified using computer simulation. A second type of problems considered is the estimation and detection of signals located in a given TF region.

While Chapters 2 through 6 are dedicated to the WD of a linear signal space and to its various applications, i.e., to the “energetic” type of TF analysis, Chapters 7 and 8 consider the “correlative” TF analysis of linear signal spaces. Chapter 7 introduces and studies the ambiguity function (AF) of a linear signal space. Simple expressions in terms of the space’s orthogonal projection operator and orthonormal bases are given, important interpretations and properties of the AF of a space are discussed, and the results obtained for some specific spaces are studied. Since the AF and WD of a space are a Fourier transform pair, this chapter is largely analogous to Chapter 2 (discussing the WD of a space). The AF of a space is also shown to allow a simple characterization and interpretation of sophisticated spaces.

Chapter 8 considers a potential application of the AF of a linear signal space, namely, the radar/sonar problem of jointly estimating the range and radial velocity of a slowly fluctuating point target. After a brief review of the classical maximum-likelihood single-pulse estimator, the maximum-likelihood multipulse estimator is derived under the assumption that a number of pulses can be transmitted and received independently of each other, and the Cramér-Rao lower bounds are calculated for this situation. It is shown that a global performance measure of the multipulse estimator is optimized if the transmitted pulses are orthogonal and have equal energies. In this case, the estimator’s performance is characterized by the AF of the linear signal space spanned by the transmitted pulses. It is shown that the “thumbtack shape” of the AF of a linear signal space (related to the estimator’s performance) can be made arbitrarily good by using a sufficiently high number of pulses. This is verified experimentally by studying the AFs of some specific spaces.

Finally, Chapter 9 contains concluding remarks that summarize and discuss the material presented and indicate possible extensions and suggestions for future research.