Convolution and Spectra

 

In human events, the word ``convoluted'' implies complexity. In science and engineering, ``convolution'' refers to a combining equation for signals, waves, or images. Although the combination may be complex, the convolution equation is an elementary one, ideally suited to be presented at the beginning of my long book on dissecting observations. Intimately connected to convolution are the concepts of pure tones and Fourier analysis.

Time and space are ordinarily thought of as continuous, but for the purposes of computer analysis we must discretize these axes. This is also called ``sampling'' or ``digitizing.'' You might worry that discretization is a practical evil that muddies all later theoretical analysis. Actually, physical concepts have representations that are exact in the world of discrete mathematics. In the first part of this book I will review basic concepts of convolution, spectra, and causality, while using and teaching techniques of discrete mathematics. By the time we finish with chapter , I think you will agree with me that many subtle concepts are easier in the discrete world than in the continuum.