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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
![$\displaystyle \left<\varphi_k ,x\right> \; \isdef \sum_{n=-\infty}^{\infty}\varphi_k ^\ast(n) x(n) \qquad \qquad \hbox{(inner product)}$](img2274.png) |
(12.105) |
and the basis signals are orthonormal:
![$\displaystyle \left<\varphi_k ,\varphi_l \right> \eqsp \delta(k-l) \eqsp \left\{\begin{array}{ll} 1, & k=l \\ 0, & k\neq l \\ \end{array} \right. \qquad \hbox{(orthonormal)}$](img2275.png) |
(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
![$\displaystyle x = \sum_{k=1}^N \alpha_k\left<x,h_k\right>f_k$](img2279.png) |
(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].
Subsections
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