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## Über dieses Buch

In 1994, in my role as Technical Program Chair for the 17th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, I solicited proposals for mini-symposia to provide delegates with accessible summaries of important issues in research areas outside their particular specializations. Terry Peters and his colleagues submitted a proposal for a symposium on Fourier Trans­ forms and Biomedical Engineering whose goal was "to demystify the Fourier transform and describe its practical application in biomedi­ cal situations". This was to be achieved by presenting the concepts in straightforward, physical terms with examples drawn for the parti­ cipants work in physiological signal analysis and medical imaging. The mini-symposia proved to be a great success and drew a large and appreciative audience. The only complaint being that the time allocated, 90 minutes, was not adequate to allow the participants to elaborate their ideas adequately. I understand that this feedback helped the authors to develop this book.

## Inhaltsverzeichnis

### Chapter 1. Introduction to the Fourier Transform

Abstract
We begin this chapter with an introduction to basic Fourier principles and the notation used, and follow in succeeding chapters with specific applications in the various areas in biomedical engineering. For the nonmathematical readers, we first introduce the basic concepts of sine and cosine waves, their representation in terms of complex numbers, and their role in Fourier transforms.
T. M. Peters

### Chapter 2. Linear Systems and the 1-D Fourier Transform

Abstract
In this chapter we consider systems that receive a single input and produce a single output in response to the input. If the input and output are only able to assume fixed values that do not vary, then we consider that the system is static. In general, however, the input and output signals will vary with time t, in which case we must consider the system to be dynamic. We may thus represent such a system as a “black box.” We can probe this system only by examining the relationships between its inputs x(t) and its corresponding outputs y (t) (Fig. 2.1).
J. H. T. Bates

### Chapter 3. The 2-D Fourier Transform

Abstract
Chapter 1 introduced the concept of Fourier transforms and demonstrated that 1-D signals and 2-D images could be completely specified by Fourier components.
T. M. Peters

### Chapter 4. Multidimensional Fourier Transforms in Magnetic Resonance Imaging

Abstract
In this chapter, Fourier transforms of dimension greater than one (i.e., 2-D, 3-D, etc.) are considered, and the specific application of magnetic resonance imaging is employed to demonstrate their use and relevance in biomedical sciences. First, a brief introduction to the physical basis of nuclear magnetic resonance (NMR) is presented. This should provide an intuitive understanding of basic events in an NMR experiment. Next, the concept of magnetic field gradients is developed, and their use in magnetic resonance imaging (MRI) described. The relationship between the received signal in MRI and the Fourier transform of the object being imaged is the focus of this section. Having established the Fourier view of MRI, this chapter concludes with a discussion of two advanced topics, magnetic resonance spectroscopic imaging (MRSI) and motion effects in MRI. The purpose of these sections is to illustrate the simplifying and unifying powers of a Fourier perspective in MRI.
G. B. Pike

### Chapter 5. Beyond Fourier: The Wavelet Transform

Abstract
Wavelet analysis is a relatively recent signal processing tool that has been successfully used in a number of fields. This chapter presents a general overview of wavelet analysis by emphasizing its relationship to the Fourier transform. Although formulas are used to support important concepts, mathematical rigor is left aside in the interest of simplicity and clarity. The reader may refer to the bibliography for a more complete and rigorous description of the subject. The continuous wavelet transform can be introduced through the concept of time-frequency analysis. In chapter 2 we discussed the concept of windows to isolate short records of a long sequence. A generalization of this technique is the windowed Fourier transform. Because the window may be placed anywhere in the signal, the windowed Fourier transform is a time-frequency extension of the usual Fourier transform. Is is used here to link the concepts of the Fourier transform and wavelet transform. Multiresolution analysis is then introduced and to leads to an efficient algorithm for computing the wavelet transform of a discrete signal. Finally, some applications of wavelet analysis in the biomedical domain are discussed.
P. Munger

### Chapter 6. The Discrete Fourier Transform and the Fast Fourier Transform

Abstract
The preceding chapters have made extensive mention of the Fourier transform (FT), the discrete Fourier transform (DFT), and the fast Fourier transform (FFT). This chapter examines the relationship between the FT and the DFT, discusses the FFT algorithm as a means of computing the DFT much more rapidly than can be achieved with the DFT algorithm directly, and presents some practical guidelines for using the FFT.
T. M. Peters, J. H. T. Bates

### Backmatter

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