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Signal processing is everywhere in modern technology. Its mathematical basis and many areas of application are the subject of this book, based on a series of graduate-level lectures held at the Mathematical Sciences Research Institute. Emphasis is on challenges in the subject, particular techniques adapted to particular technologies, and certain advances in algorithms and theory. The book covers two main areas: computational harmonic analysis, envisioned as a technology for efficiently analysing real data using inherent symmetries; and the challenges inherent in the acquisition, processing and analysis of images and sensing data in general [EMDASH] ranging from sonar on a submarine to a neuroscientist's fMRI study.
This article presents a simple version of Integrated Sensing and Processing (ISP) for statistical pattern recognition wherein the sensor measureIllents to be taken are adaptively selected based on task-specific metries. Thus the measurement space in which the pattern recognition task is ultimately addressed integrates adaptive sensor technology with the specific task for which the sensor is employed. This end-to-end optimization of sensor/ processor/exploitation subsystems is a theme of the DARPA Defense Sciences Office Applied and Computational Mathematics Program's ISP program. We illustrate the idea with a pedagogical example and application to the HyMap hyperspectral sensor and the Tufts University “artificial nose” chemical sensor.
1. Introduction
An important activity, common to many fields of endeavor, is the act of refining high order information (detections of events, classification of objects, identification of activities, etc.) from large volumes of diverse data which is increasingly available through modern means of measurement, communication, and processing. This exploitation function winnows the available data concerning an object or situation in order to extract useful and actionable information, quite often through the application of techniques from statistical pattern recognition to the data. This may involve activities like detection, identification, and classification which are applied to the raw measured data, or possibly to partially processed information derived from it.
When new data are sought in order to obtain information about a specific situation, it is now increasingly common to have many different measurement degrees of freedom potentially available for the task.
Underlying many of the current mathematical opportunities in digital signal processing are unsolved analog signal processing problems. For instance, digital signals for communication or sensing must map into an analog format for transmission through a physical layer. In this layer we meet a canonical example of analog signal processing: the electrical engineer's impedance matching problem. Impedance matching is the design of analog signal processing circuits to minimize loss and distortion as the signal moves from its source into the propagation medium. This paper works the matching problem from theory to sampled data, exploiting links between H∞ theory, hyperbolic geometry, and matching circuits. We apply J. W. Helton's significant extensions of operator theory, convex analysis, and optimization theory to demonstrate new approaches and research opportunities in this fundamental problem.
1. The Impedance Matching Problem
Figure 1 shows a twin-whip HF (high-frequency) antenna mounted on a superstructure representative of a shipboard environment. If a signal generator is connected directly to this antenna, not all the power delivered to the antenna can be radiated by the antenna. If an impedance mismatch exists between the signal generator and the antenna, some of the signal power is reflected from the antenna back to the generator. To effectively use this antenna, a matching circuit must be inserted between the signal generator and antenna to minimize this wasted power.
Figure 2 shows the matching circuit connecting the generator to the antenna. Port 1 is the input from the generator. Port 2 is the output that feeds the antenna.
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