ABSTRACT
This chapter introduces that the theory of frames, which was developed by Duffin and Schaeffer and popularized mostly through. Meanwhile frame theory, in particular the aspect of redundancy in signal expansions, has found numerous applications such as, for example, denoising, code division multiple access (CDMA), orthogonal frequency division multiplexing (OFDM) systems, coding theory, quantum information theory, analog-to-digital (A/D) converters, and compressive sensing. The chapter considers general signal expansions in finite-dimensional Hilbert spaces. There are two more properties of general bases in finite-dimensional spaces that carry over to the infinite-dimensional case, namely uniqueness of representation in the sense and biorthonormality between the frame and its canonical dual. There are two important classes of structured signal expansions that have found widespread use in practical applications, namely Weyl-Heisenberg expansions and affine expansions. Weyl-Heisenberg and wavelet expansions have been successfully used in signal detection, image representation, object recognition, and wireless communications.
