The standard type III discrete cosine transformation (DCT) is defined as follows:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} \right)} \right]\quad \quad k = 0, \ldots ,N - 1.$$
It transforms a list of $\{x_n\}$ into list of $\{X_k\}$
Now I encountered a summation in a paper like this:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{{\tilde N}}n\left( {k + \frac{1}{2}} \right)} \right]\quad \quad k = 0, \ldots ,\tilde N - 1.$$
The only difference is that now $\tilde N = 2N$.
The paper says this summation could exploit the power of fast DCT. However, it tooks me days, still can't figure out how to relate it to the standard DCT-III. How to do it?