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 $x\in{\bf C}^N$ from its projections onto different vectors $\sv_k\in{\bf C}^N$ , $k=0,1,2,\ldots,N-1$ . 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 has coordinates $(0,x_1,0,\ldots,0)$ , and so on. The original signal is then clearly the vector sum of its projections onto the coordinate axes:

$\begin{eqnarray*} x &=& (x_0,\ldots,x_{N-1})\\ &=& (x_0,0,\ldots,0) + (0,x_1,0,\ldots,0) + \cdots (0,\ldots,0,x_{N-1}) \end{eqnarray*}$

To make sure the previous paragraph is understood, let's look at the details for the case . We want to project an arbitrary two-sample signal 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 $\underline{e}_1=(0,\beta)$ , $\beta\neq 0$ . 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 onto $\underline{e}_0$ is, by definition,

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

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

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

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

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

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 to be applied to the coordinate vectors $\underline{e}_n$ in order to form an arbitrary vector $x\in{\bf C}^N$ as a linear combination of the coordinate vectors:

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

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.

Subsections

Order a Hardcopy of Mathematics of the DFT

Previous: Projection
Next: Changing Coordinates

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.

Sign in

Search Online Books

Free Online Books

Chapters

See Also

Mathematics of the DFT
Geometric Signal Theory
Signal Reconstruction from Projections