The Fourier Transform in Biomedical Engineering

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.

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).

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.

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.

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.

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.