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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\mathbb{C}^N$ onto any orthogonal basis set for $ \mathbb{C}^N$ can be summed to reconstruct $ x$ exactly.



Proof: Let $ \{\underline{s}_0,\ldots,\underline{s}_{N-1}\}$ denote any orthogonal basis set for $ \mathbb{C}^N$ . Then since $ x$ is in the space spanned by these vectors, we have

$\displaystyle x= \alpha_0\underline{s}_0 + \alpha_1\underline{s}_1 + \cdots + \alpha_{N-1}\underline{s}_{N-1} \protect$ (5.3)

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

$\displaystyle {\bf P}_{\underline{s}_k}(x) = \alpha_0{\bf P}_{\underline{s}_k}(\underline{s}_0) +
\alpha_1{\bf P}_{\underline{s}_k}(\underline{s}_1) + \cdots + \alpha_{N-1}{\bf P}_{\underline{s}_k}(\underline{s}_{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 $ \underline{s}_l$ onto $ \underline{s}_k$ is zero for $ l\neq k$ :

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

We therefore obtain

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

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

$\displaystyle \sum_{k=0}^{N-1}
{\bf P}_{\underline{s}_k}(x) = \sum_{k=0}^{N-1} \alpha_k \underline{s}_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 © 2016-05-31 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA