Operator Notation

In this book, an operator is defined as a signal-valued function of a signal. Thus, for the space of length complex sequences, an operator is a mapping from to :

An example is the DFT operator:

The argument to an operator is always an entire signal. However, its output may be subscripted to obtain a specific sample, e.g.,

Some operators require one or more parameters affecting their definition. For example the shift operator (defined in §7.2.3 below) requires a shift amount :^7.3

A time or frequency index, if present, will always be the last subscript. Thus, the signal is obtained from by shifting it samples.

Note that operator notation is not standard in the field of digital signal processing. It can be regarded as being influenced by the field of computer science. In the Fourier theorems below, both operator and conventional signal-processing notations are provided. In the author's opinion, operator notation is consistently clearer, allowing powerful expressions to be written naturally in one line (e.g., see Eq.(7.8)), and it is much closer to how things look in a readable computer program (such as in the matlab language).

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.