DCT and Convolution, [1994, Martucci]

I am really trying to digest this paper, but the more I read it again, the more I realize that I never fully understood it.

Basically, I am trying to get an example where I can check the similarities between (28) and (29) to see I am doing things properly.

Here are some important parts that I am try to follow:

• Eq 28 and Eq 29

• I am using DCT-II as $T_a$ and $T_b$ and iDCT-I as $T_c$. Thus according to the paper, I must use the extensions HSHS on both $x$ and $y$, them the output will be a circular convolution of their extended versions

• Then, we should be able to match this output by applying a multiplication operations on DCT representation of their unextended versions.

• However, when I get to reproduce this, I don’t see this resemblance.
  1. There is no window $mathcal{R}(n)$ I could extract from the first output bellow, that would match the second one.
  2. It could be that the output values using the transforms would needed to be scaled, somehow, but even doing so, it would not change their shape in the plot, which I am using to compare both outputs.
  3. According to the paper, I should get an extra sample on the approach using the transforms, since it outputs on the interval [-1, N-1], I am also not sure where that comes from.

In short: How to properly interpret and demonstrate this correspondence between (28) and (29) this combination of $T_a = T_b = text{DCT-II}$ and $T_c = text{DCT-I}$?