Review of 1- D DCT

In the 1-D case the DCT is defined as

	(4.3.4)

for every N point signal having support The corresponding inverse transform, or IDCT, can be written as

	(4.3.5)

It turns out that this 1-D DCT can be understood in terms of the DFT of a symmetrically extended sequence,

	(4.3.6)

This is not the only way to symmetrically extend x , but this method results in the most widely used DCT sometimes called DCT-2 with support In fact, on defining the 2N point DFT we will show that the DCT can be alternatively expressed as

	(4.3.7)

Thus the DCT is just the DFT analysis of the symmetrically extended signal defined in (4.3.6):

Looking at this equation, we see that there is no overlap in its two components, which fit together without a gap. We can see that right after comes at position , which is then followed by the rest of the nonzero part of x in reverse order, upto , where sits .We can see a point of symmetry midway between and N, i.e., at .

If we consider its periodic extension we will also see a symmetry about the point . We thus expect that the 2N point will be real valued except for the phase factor . So the phase factor in eqn (4.3.7) is just what is needed to cancel out the phase term in Y and make the DCT real , as it must if the two equations, (4.3.1) and (4.3.7), are to agree for real valued inputs x.

To reconcile these two definitions, we start out with eqn (4.3.7), and proceed as follows:

the last line following from and Euler's relation, which agrees with the original definition, eqn (4.3.4).

The formula for the inverse DCT, can be established similarly, starting out from