ABSTRACT
This chapter examines two important properties of the Fourier transform pair, linearity and duality. The Fourier transform is a linear operation. In other words, the Fourier transform of a linear combination of signals is equal to the same linear combination of their Fourier transforms. In Radiology, duality is most clearly expressed in nuclear magnetic resonance imaging. In nuclear magnetic resonance, the time domain signal is called the free induction decay. Its Fourier transform, the spectrum, is the frequency domain representation of the signal. One of the very important properties of a set of basis functions is orthogonality. The chapter explores that the imaginary exponentials all are orthogonal to each other. Mathematical texts discuss at great length the requirements that a Fourier transform of a function exists. The Fourier transform provides a method of representing all physical signals as linear combinations of the eigenfunctions of linear, time invariant systems.
