There is this Wikipedia article on cycles in stock market data, which describes a 5-step process of finding dominant cycles in price data where step 2 reads:
Step 2: Subsequently, a cycles engine performs a spectral analysis based on an optimized Discrete Fourier Transform (DFT) and then isolates those cycles that are repetitive and have the largest amplitudes.
Now here is the issue: when looking at the amplitude spectrum of DFT of a time series it may make sense, but then what about the phase of the corresponding components? Let me illustrate my issue with the example. Let's say I calculate DFT of the signal from around 4k samples, so there is a very fine resolution of the high frequency/low period components. In the amplitude spectrum using numerical or visual methods I find a local maximum at 1.04 (row #349, column 'amp'):
coeff freq w amp ph c period 347 (-341.737858288+1419.41730018j) 0.079899 0.502018 0.672335 1.807059 0.0 12.515850 348 (385.34656135-2047.44320284j) 0.080129 0.503465 0.959425 -1.384764 0.0 12.479885 349 (-1137.33333723+1957.67508513j) 0.080359 0.504912 1.042630 2.097099 0.0 12.444126 350 (969.672144511-1569.38982154j) 0.080589 0.506358 0.849546 -1.017344 0.0 12.408571 351 (-914.401398452+857.116113024j) 0.080820 0.507805 0.577162 2.388520 0.0 12.373219
So there is a peak with the top at 1.04 corresponding to the cycle with the period of 12.44 samples (in this very case 1 sample = 1 day) and if I follow the guidelines from the article  I should select this cycle as (one of) the dominant in the price data. But what about its nearest neighbors in the amplitude spectrum? One can see that the 5 consecutive cycles with the top one in the middle that have significant amplitudes, but much different phases (the 'ph' column in the table above, values are in radians). If I select and plot the top one only I will get a simple sine wave, but if I add 2 or 4 neighbors the plot will be more complicated with amplitude reaching way above 1.04 up to the sum of all composite amplitudes when the cycle tops come in sync as pictured in the figure below where the top subplot is showing the top cycle, the middle one the sum of the top cycle and 2 nearest neighbors, and the bottom one the the sum of the top cycle and 4 nearest neighbors:
Apparently the middle and bottom plots are very different from the top one, so I'm wondering how such issue should be dealt with? Perhaps the envelope could be used as the dominant cycle instead? If the phase angles were not so spread out, maybe a cycle of an average phase could be fitted in, but what can be done about the cycles that have similar amplitudes and frequencies, but much different phases?