Signal Reconstruction from Projections

We now know how to project a signal onto other signals. We now need to learn how to reconstruct a signal $\underline{x}\in{\bf C}^N$ from its projections onto $N$ different vectors $\underline{s}_k\in{\bf C}^N$, $k=0,1,2,\ldots,N-1$. This will give us the inverse DFT operation (or the inverse of whatever transform we are working with).

As a simple example, consider the projection of a signal $x\in{\bf C}^N$ onto the rectilinear coordinate axes of ${\bf C}^N$. The coordinates of the projection onto the 0th coordinate axis are simply $(x_0,0,\ldots,0)$. The projection along coordinate axis $1$ has coordinates $(0,x_1,0,\ldots,0)$, and so on. The original signal $x$ is then clearly the vector sum of its projections onto the coordinate axes:

$$x = (x_0,\ldots,x_{N-1}) = (x_0,0,\ldots,0) + (0,x_1,0,\ldots,0) + \cdots (0,\ldots,0,x_{N-1})$$

To make sure the previous paragraph is understood, let's look at the details for the case $N=2$. We want to project an arbitrary two-sample signal $\underline{x}= (x_0,x_1)$ onto the coordinate axes in 2D. A coordinate axis can be generated by multiplying any nonzero vector by scalars. The horizontal axis can be represented by any vector of the form $\underline{e}_0=(\alpha,0)$, $\alpha\neq0$ while the vertical axis can be represented by any vector of the form $\underline{e}_1=(0,\beta)$, $\beta\neq0$. For maximum simplicity, let's choose

$$\begin{eqnarray*} \underline{e}_0 &\isdef & (1,0), \\ \underline{e}_1 &\isdef & (0,1). \end{eqnarray*}$$

The projection of $\underline{x}$ onto $\underline{e}_0$ is, by definition,

$$\begin{eqnarray*} {\bf P}_{\underline{e}_0}(\underline{x}) &\isdef & \frac{\left... ...0}) \underline{e}_0 = x_0 \underline{e}_0\\ [5pt] &=& [x_0,0]. \end{eqnarray*}$$

Similarly, the projection of $\underline{x}$ onto $\underline{e}_1$ is

$$\begin{eqnarray*} {\bf P}_{\underline{e}_1}(\underline{x}) &\isdef & \frac{\left... ...1}) \underline{e}_1 = x_1 \underline{e}_1\\ [5pt] &=& [0,x_1]. \end{eqnarray*}$$

The reconstruction of $x$ from its projections onto the coordinate axes is then the vector sum of the projections:

$$\underline{x}= {\bf P}_{\underline{e}_0}(\underline{x}) + {\bf P}... ...\underline{e}_0 + x_1 \underline{e}_1 \isdef x_0 (1,0) + x_1 (0,1) = (x_0,x_1)$$

The projection of a vector onto its coordinate axes is in some sense trivial because the very meaning of the coordinates is that they are scalars $x_n$ to be applied to the coordinate vectors $\underline{e}_n$ in order to form an arbitrary vector $\underline{x}\in{\bf C}^N$ as a linear combination of the coordinate vectors:

$$\underline{x}\isdef x_0 \underline{e}_0 + x_1 \underline{e}_1 + \cdots + x_{N-1} \underline{e}_{N-1}$$

Note that the coordinate vectors are orthogonal. Since they are also unit length, $\Vert\underline{e}_n\Vert=1$, we say that the coordinate vectors $\{\underline{e}_n\}_{n=0}^{N-1}$ are orthonormal.

What's more interesting is when we project a signal $\underline{x}$ onto a set of vectors other than the coordinate set. This can be viewed as a change of coordinates in ${\bf C}^N$. In the case of the DFT, the new vectors will be chosen to be sampled complex sinusoids.