I am trying to understand the recursive implementation of the DCT on the input signal (used for LMS filtering) according to the pictures below.
The pictures and formulas are taken from the paper: "Adaptive Inverse Control: A Signal Processing Approach", by Bernard Widrow and Eugene Walach
Block diagram of DCT structure:
Normal Matrix-Vector DCT-formula
Recursive DCT-formula:
What bothers me is how can this recursive DCT implementation yield the same result as the normal computation by using matrix-vector multiplication when the recursive formula uses the signal $x_{k-n}$?
But on the other hand the normal computation only uses signals up to $x_{k-n+1}$.