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

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

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

$${\bf P}_{\underline{s}_k}(x) = \alpha_0{\bf P}_{\underline{s}_k}(\underline{s}_0) + \alpha_1{\bf P}_{\underline{s}_k}(\underline{s}_1) + \cdots + \alpha_{N-1}{\bf P}_{\underline{s}_k}(\underline{s}_{N-1})$$

(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 $l\neq k$ :

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

We therefore obtain

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

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 $x$ :

$$\sum_{k=0}^{N-1} {\bf P}_{\underline{s}_k}(x) = \sum_{k=0}^{N-1} \alpha_k \underline{s}_k = x$$

by Eq.(5.3). $\Box$