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Signal/Vector Reconstruction from Projections

We now arrive finally at the main desired result for this section:



Theorem: The projections of any vector $ x\in{\bf C}^N$ onto any orthogonal basis set for $ {\bf C}^N$ can be summed to reconstruct $ x$ exactly.



Proof: Let $ \{\sv_0,\ldots,\sv_{N-1}\}$ denote any orthogonal basis set for $ {\bf C}^N$ . Then since $ x$ is in the space spanned by these vectors, we have

$\displaystyle x= \alpha_0\sv_0 + \alpha_1\sv_1 + \cdots + \alpha_{N-1}\sv_{N-1} \protect$ (5.3)

for some (unique) scalars $ \alpha_0,\ldots,\alpha_{N-1}$ . The projection of $ x$ onto $ \sv_k$ is equal to

$\displaystyle {\bf P}_{\sv_k}(x) = \alpha_0{\bf P}_{\sv_k}(\sv_0) +
\alpha_1{\bf P}_{\sv_k}(\sv_1) + \cdots + \alpha_{N-1}{\bf P}_{\sv_k}(\sv_{N-1})
$

(using the linearity of the projection operator which follows from linearity of the inner product in its first argument). Since the basis vectors are orthogonal, the projection of $ \sv_l$ onto $ \sv_k$ is zero for $ l\neq k$ :

$\displaystyle {\bf P}_{\sv_k}(\sv_l) \isdef
\frac{\left<\sv_l,\sv_k\right>}{\left\Vert\,\sv_k\,\right\Vert^2}\sv_k
= \left\{\begin{array}{ll}
\underline{0}, & l\neq k \\ [5pt]
\sv_k, & l=k. \\
\end{array} \right.
$

We therefore obtain

$\displaystyle {\bf P}_{\sv_k}(x) = 0 + \cdots + 0 + \alpha_k{\bf P}_{\sv_k}(\sv_k) + 0 + \cdots + 0
= \alpha_k\sv_k.
$

Therefore, the sum of projections onto the vectors $ \sv_k$ , $ k=0,1,\ldots,
N-1$ , is just the linear combination of the $ \sv_k$ which forms $ x$ :

$\displaystyle \sum_{k=0}^{N-1}
{\bf P}_{\sv_k}(x) = \sum_{k=0}^{N-1} \alpha_k \sv_k = x
$

by Eq.$ \,$ (5.3). $ \Box$


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA