Geometric Signal Theory

In general, signals can be expanded as a linear combination
of *orthonormal basis signals*
[264]. In the
discrete-time case, this can be expressed as

where the coefficient of projection of onto is given by

(12.105) |

and the basis signals are

(12.106) |

The signal expansion (11.104) can be interpreted geometrically as a sum of orthogonal projections of onto , as illustrated for 2D in Fig.11.30.

A set of signals
is said to be
a *biorthogonal basis set* if any signal
can be represented
as

(12.107) |

where is some normalizing scalar dependent only on and/or . Thus, in a biorthogonal system, we project onto the signals and resynthesize in terms of the basis .

The following examples illustrate the Hilbert space point of view for various familiar cases of the Fourier transform and STFT. A more detailed introduction appears in Book I [264].

- Natural Basis
- Normalized DFT Basis for
- Normalized Fourier Transform Basis
- Normalized DTFT Basis
- Normalized STFT Basis
- Continuous Wavelet Transform
- Discrete Wavelet Transform
- Discrete Wavelet Filterbank
- Dyadic Filter Banks
- Dyadic Filter Bank Design
- Generalized STFT

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University