ABSTRACT
The concept of signal space has its roots in the mathematical theory of inner product spaces known as Hilbert spaces (Stakgold, 1967). Many books on linear systems touch on the subject of signal spaces in the context of Fourier series and transforms (Ziemer et al., 1998). The applications of signal space concepts in communication theory find their power in the representation of signal detection and estimation problems in geometrical terms, which provides much insight into signaling techniques and communication system design. The first person to have apparently exploited the power of signal space concept in communication theory was the Russian Kotel'nikov (1968), who presented his doctoral dissertation in January, 1947. Wozencraft and Jacobs (1965) expanded on this approach and their work is still today widely referenced. Arthurs and Dym (1962) made use of signal space concepts in the performance analysis of several digital modulation schemes. A one-chapter summary of the use of signal space methods in signal detection and estimation is provided in Ziemer and Tranter (2009). Another application of signal space concepts is in signal and image compression. Wavelet theory (Rioul and Vetterli, 1991) is currently finding use in these application areas. Finally, the application of signal space concepts to nonlinear filtering is discussed. In the next section, the fundamentals of generalized vector spaces are summarized, followed by an overview of several applications to signal representations.
