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Operator Notation

In this book, an operator is defined as a signal-valued function of a signal. Thus, for the space of length $ N$ complex sequences, an operator $ \hbox{\sc Op}$ is a mapping from $ {\bf C}^N$ to $ {\bf C}^N$:

$\displaystyle \hbox{\sc Op}(x) \in{\bf C}^N\, \forall x\in{\bf C}^N

An example is the DFT operator:

$\displaystyle \hbox{\sc DFT}(x) = X

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

$\displaystyle \hbox{\sc DFT}_k(x) = X(k)

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 $ \Delta\in{\bf Z}$:

$\displaystyle \hbox{\sc Shift}_{\Delta,n}(x) \isdef x(n-\Delta)

A time or frequency index, if present, will always be the last subscript. Thus, the signal $ \hbox{\sc Shift}_{\Delta}(x)$ is obtained from $ x$ by shifting it $ \Delta$ 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 mathematical 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. (7.10)), and it is much closer to how things look in a maximally readable computer program (such as in the matlab language).

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``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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