Implied symmetries for even/odd extensions of the inputs of discrete cosine transforms (DCTs), for the four most common DCT variants.

In each case, we look at a DCT of length N=11 of the same input data (shown as red dots), i.e. for an input index n=0..10. A DCT implies an even extension of this data set with respect to its left boundary, and either an even (for types I and II) or odd (for types (III and IV) extension with respect to its right boundary. The extension is shown as gray dots. Notice the subtle difference in the symmetry between types I and II, and similarly between types III and IV: in the first case (I/III), the data is symmetric around one of the samples, while in the second case (II/IV) it is symmetric around a point halfway between two samples.

Original illustration donated to Wikipedia under the GFDL. Copyright © 2006 —Steven G. Johnson.

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