ABSTRACT
This chapter presents several of the properties of the Fourier transform. The single most important property of the Fourier transform is the mapping of the complex operation of convolution into the simple operation of multiplication. The Fourier transform maps convolution in one domain to multiplication in the other domain. This relation provides an efficient method for performing convolution numerically on a computer. The Fourier transform maps deconvolution in one domain into division in the other domain. Mapping of convolution into multiplication provides great numerical efficiency. The chapter explores that the shift of a signal in one domain results in the multiplication of its transform by an imaginary exponential. The real and imaginary parts of the Fourier transform of a real, causal signal are called quadrature signals. The uncertainty principle states that a signal and its transform cannot both be narrower than a certain combined limit.
