Signal/Vector Reconstruction from Projections
We now arrive finally at the main desired result for this section:
Theorem: The projections of any vector
onto any orthogonal basis set
for
can be summed to reconstruct
exactly.
Proof: Let
denote any orthogonal basis set for
.
Then since
is in the space spanned by these vectors, we have
for some (unique) scalars
![$ \alpha_0,\ldots,\alpha_{N-1}$](http://www.dsprelated.com/josimages_new/mdft/img969.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ \sv_k$](http://www.dsprelated.com/josimages_new/mdft/img926.png)
![$\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})
$](http://www.dsprelated.com/josimages_new/mdft/img970.png)
![$ \sv_l$](http://www.dsprelated.com/josimages_new/mdft/img971.png)
![$ \sv_k$](http://www.dsprelated.com/josimages_new/mdft/img926.png)
![$ l\neq k$](http://www.dsprelated.com/josimages_new/mdft/img972.png)
![$\displaystyle {\bf P}_{\sv_k}(\sv_l) \isdef
\frac{\left<\sv_l,\sv_k\right>}{\l...
...ll}
\underline{0}, & l\neq k \\ [5pt]
\sv_k, & l=k. \\
\end{array} \right.
$](http://www.dsprelated.com/josimages_new/mdft/img973.png)
![$\displaystyle {\bf P}_{\sv_k}(x) = 0 + \cdots + 0 + \alpha_k{\bf P}_{\sv_k}(\sv_k) + 0 + \cdots + 0
= \alpha_k\sv_k.
$](http://www.dsprelated.com/josimages_new/mdft/img974.png)
![$ \sv_k$](http://www.dsprelated.com/josimages_new/mdft/img926.png)
![$ k=0,1,\ldots,
N-1$](http://www.dsprelated.com/josimages_new/mdft/img975.png)
![$ \sv_k$](http://www.dsprelated.com/josimages_new/mdft/img926.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$\displaystyle \sum_{k=0}^{N-1}
{\bf P}_{\sv_k}(x) = \sum_{k=0}^{N-1} \alpha_k \sv_k = x
$](http://www.dsprelated.com/josimages_new/mdft/img976.png)
![$ \,$](http://www.dsprelated.com/josimages_new/mdft/img131.png)
![$ \Box$](http://www.dsprelated.com/josimages_new/mdft/img362.png)
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Gram-Schmidt Orthogonalization
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General Conditions