Signal/Vector Reconstruction from Projections

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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 exactly.

Proof: Let $\{\sv_0,\ldots,\sv_{N-1}\}$ denote any orthogonal basis set for ${\bf C}^N$ . Then since 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

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 $l\neq k$ :

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

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

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

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

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
    Geometric Signal Theory
       Signal Reconstruction from Projections
          Signal/Vector Reconstruction from Projections