A Short Course on Frame Theory

Hilbert spaces [1, Def. 3.1-1] and the associated concept of orthonormal bases are of fundamental importance in signal processing, communications, control, and information theory. However, linear independence and orthonormality of the basis elements impose constraints that often make it difficult to have the basis elements satisfy additional desirable properties. This calls for a theory of signal decompositions that is flexible enough to accommodate decompositions into possibly nonorthogonal and redundant signal sets. The theory of frames provides such a tool. This chapter is an introduction to the theory of frames, which was developed by Duffin and Schaeffer [2] and popularized mostly through [3–6]. Meanwhile frame theory, in particular the aspect of redundancy in signal expansions, has found numerous applications such as, e.g., denoising [7, 8], code division multiple access (CDMA) [9], orthogonal frequency division multiplexing (OFDM) systems [10], coding theory [11, 12], quantum information theory [13], analog-to-digital (A/D) converters [14–16], and compressive sensing [17–19]. A more extensive list of relevant references can be found in [20]. For a comprehensive treatment of frame theory we refer to the excellent textbook [21].