I am wanting to use the discrete cosine transform to relate the autocovariance function of a process to its periodogram. Following Chris Chatfield's book (Time Series Analysis, p129), I am wanting to use the transformation in the form:
\begin{equation} I_{xx}\big[\omega_p\big] = \frac{1}{\pi}\left(c_0+2\sum_{k=1}^{N-1} c_{k}\cos\big[\omega_p k\big]\right) \end{equation}
In MATLAB, using the default function y = dct(acv), where acv is the autocovariance function of a time series 'x', gives me the same result as when I call periodogram(x). However, I am unable to reconcile either of these results with the expression given above. I am aware that there are different conventions for what is plotted and how the periodogram and spectra are displayed, but have so far been unable to figure out what MATLAB is actually doing, and why (how) it is different to Chatfield's result.
Any thoughts would be most appreciated.
