# APPENDIX A FOURIER TRANSFORM

This appendix provides a brief review of the Fourier transform, and its properties, for functions of one and two variables.

## A.1 ONE-DIMENSIONAL FOURIER TRANSFORM

The harmonic function *F* exp(*j2πνt*) plays an important role in science and engineering. It has frequency *ν* and complex amplitude *F*. Its real part |*F*| cos(2*πνt* + arg{*F* }) is a cosine function with amplitude |*F*| and phase arg{*F* }. The variable *t* usually represents time; the frequency *ν* has units of cycles/s or Hz. The harmonic function is regarded as a building block from which other functions may be obtained by a simple superposition.

In accordance with the Fourier theorem, a complex-valued function *f* (*t*), satisfying some rather unrestrictive conditions, may be decomposed as a superposition integral of harmonic functions of different frequencies and complex amplitudes,

The component with frequency *ν* has a complex amplitude *F* (*ν*) given by

*F*(*ν*) is termed the **Fourier transform** of *f*(*t*), and *f*(*t*) is the **inverse Fourier transform** of *F*(*ν*). The functions *f*(*t*) and *F*(*ν*) form a **Fourier transform pair;** if one is known, the other may be determined.

In this book we adopt the convention that exp(*j2πνt*) is a harmonic function with positive frequency, ...

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