ABSTRACT
Having established methods for digital representation of analog signals, consider how transformations of analog signals must be represented in computers. In the section “Basic Principles of Discrete Representation of Signal Transformations,” we formulate the basics principles of discrete representation of signal transformations and in the subsequent sections “Discrete Representation of the Convolution Integral,” “Discrete Representation of Fourier Integral Transform,” and “Discrete Representation of Fresnel Integral Transform,” we apply them to introduce digital convolution as discrete representation of the convolution integral, DFTs as discrete representation of the integral Fourier transform and discrete Fresnel transforms as discrete representation of the integral Fresnel transform. In the section “Discrete Representation of Kirchhoff Integral,” we introduce binary Hadamard and Walsh transforms as well as discrete wavelet transforms in their association with signal multiresolution analysis. This chapter is concluded with an extension of discrete transforms to their application in sliding window, which leads to signal “space-frequency” representation.
