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: