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
from its projections
onto different vectors
,
. 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
onto the
rectilinear *coordinate axes* of . The coordinates of the
projection onto the 0th coordinate axis are simply
.
The projection along coordinate axis has coordinates
, and so on. The original signal is then clearly
the *vector sum* of its projections onto the coordinate axes:

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 , while the vertical axis can be represented by any vector of the form , . For maximum simplicity, let's choose

The projection of onto is, by definition,

Similarly, the projection of onto is

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

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*
in
order to form an arbitrary vector
as a *linear combination*
of the coordinate vectors:

- Changing Coordinates

- Projection onto Linearly Dependent Vectors
- Projection onto Non-Orthogonal Vectors
- General Conditions
- Signal/Vector Reconstruction from Projections
- Gram-Schmidt Orthogonalization

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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