So far, we have treated what are known as analog signals and systems. Mathematically, analog signals are functions having continuous quantities as their independent variables, such as space and time. Discrete-time signals (Section 5.5) are functions defined on the integers; they are sequences. One of the fundamental results of signal theory (Section 5.3) will detail conditions under which an analog signal can be converted into a discrete-time one and retrieved without error. This result is important because discrete-time signals can be manipulated by systems instantiated as computer programs. Subsequent modules describe how virtually all analog signal processing can be performed with software.
As important as such results are, discrete-time signals are more general, encompassing signals derived from analog ones and signals that aren't. For example, the characters forming a text file form a sequence, which is also a discrete-time signal. We must deal with such symbolic valued (p. 180) signals and systems as well.
As with analog signals, we seek ways of decomposing real-valued discrete-time signals into simpler components. With this approach leading to a better understanding of signal structure, we can exploit that structure to represent information (create ways of representing information with signals) and to extract information (retrieve the information thus represented). For symbolic-valued signals, the approach is different: We develop a common representation of all symbolic-valued signals so that we can embody the information they contain in a unifed way. From an information representation perspective, the most important issue becomes, for both real-valued and symbolic-valued signals, efciency; What is the most parsimonious and compact way to represent information so that it can be extracted later.