Basics of Fourier Analysis
1.1 FORWARD AND INVERSE FOURIER TRANSFORM
Fourier transform (FT) is a common and useful mathematical tool that is utilized in numerous applications in science and technology. FT is quite practical, especially for characterizing nonlinear functions in nonlinear systems, analyzing random signals, and solving linear problems. FT is also a very important tool in radar imaging applications as we shall investigate in the forthcoming chapters of this book. Before starting to deal with the FT and inverse Fourier transform (IFT), a brief history of this useful linear operator and its founders is presented.
1.1.1 Brief History of FT
Jean Baptiste Joseph Fourier, a great mathematician, was born in 1768 in Auxerre, France. His special interest in heat conduction led him to describe a mathematical series of sine and cosine terms that can be used to analyze propagation and diffusion of heat in solid bodies. In 1807, he tried to share his innovative ideas with researchers by preparing an essay entitled “On the Propagation of Heat in Solid Bodies.” The work was examined by Lagrange, Laplace, Monge, and Lacroix. Lagrange’s oppositions caused the rejection of Fourier’s paper. This unfortunate decision caused colleagues to wait for 15 more years to read his remarkable contributions on mathematics, physics, and, especially, signal analysis. Finally, his ideas were published in the book The Analytic Theory of Heat in 1822 .
Discrete Fourier transform (DFT) ...