I read Brockwell and Davis(2016), Shumway and Stoffer(2016), and Stoica and Moses(2004). However, none of them laid out clearly the reasoning behind the presumption of stationarity when conducting spectral analysis on a time series.
I understand we typically need to detrend the time series as necessary data pre-processing for spectral analysis, because otherwise the first cosine coefficient could distort the estimates/periodogram.(frequency close to zero could have very large spectrum).
But simple detrending doesn't make the process stationary. It may well have periodicity/seasonality built in, although it may not have a unit root (unit root stationarity tests are rendered useless in this case).
Based on above, how could we carry out a spectral analysis on a periodic time series which is not stationary when spectral analysis is only defined for stationary processes?