Chapter 2 Signal Processing Transforms

In a broader scope, a transformation in mathematics can broadly be defined as the operation which takes its input and “represents” it in a different form. Such a definition immediately implies preserving the essential characteristics of the input through several conservation rules or laws. In order to outline the complete input–output relationship in an abstract transformation, input to and output of the transformation should be characterized as well. Therefore, mathematically speaking, a transformation can be viewed as a special function (or a correspondence) whose input and output can be a single value or another function. In this context, signals are considered to be both input and output functions of the transformation of interest. 1 Depending both on the transform and the input signal characteristics, different types of transforms exist in the literature. For instance, there are transforms which assume deterministic input (e. g., Fourier transform), whereas there are transforms operating on stochastic input (e. g., Karhunen–Loéve transform). A different way of classifying transforms is to consider the nature of the input signal such as continuous and discrete transforms, which assume continuous and discrete signals as input, respectively. It is worth mentioning at this point that some of the transformations yield output in a different domain – which is sometimes called “transform domain”– from that of the input. A very well–known example of this is a time–frequency transformation, which operates on a signal in the time domain and yields an output signal in the frequency domain or vice versa. However, such a domain change is not always observed for every transformation.