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1991issue C071-3

Closed-form phase for dominant-cycle spectra

A maximizing phase over one dominant-cycle length can be recovered from two orthogonal sums rather than a phase grid. The leftover amplitude is one Fourier bin, and a plot of that amplitude against the periods that fit the lookback is the measured spectrum.

  • A phase that maximizes a sine-weighted sum of prices over one dominant-cycle length can be recovered from two orthogonal sums. The two-argument inverse tangent replaces a 22.5-degree phase sweep.
  • Those sums rewrite the score as an amplitude times a cosine of residual phase, so the maximum occurs when residual phase is zero, after a half-cycle correction when the cosine-weight sum is negative.
  • The amplitude from the two sums is one discrete Fourier bin. Plotting it against trial periods that fit an integer number of cycles into the lookback is the measured spectrum.
  • A fixed window admits only integer-cycle periods such as 32, 16, and 10.67 bars on a 32-bar span. A two-cycle variable lookback can still estimate amplitude when a long sample is not stationary.
Entries in this reading3 entries
Current trackFourier analysis41 readingsSpectral analysis63 readingsDominant cycle detection180 readings

The historical construction recovers a maximizing phase from two precomputed sums, treats the leftover amplitude as one Fourier bin, and restricts trial periods to those that fit an integer number of cycles into the lookback. The steps below stay with that construction.

Recover phase from two sums

The dominant-cycle is the trial period used as the lookback when a sine-weighted match to an ordered price series is maximized. Phase is the offset that lines that sine up with the series. Once two orthogonal sums are in hand, that offset is obtained directly rather than by trying many offsets.

A phase that maximizes a sine-weighted sum of prices over one dominant-cycle length can be recovered from two precomputed sums instead of scoring many trial phases. The orthogonal-sums are those two totals of price against sine and cosine of the trial period. Together they give both the maximizing phase and an amplitude.

Those two sums rewrite the score as a single amplitude times a cosine of residual phase, so the maximum occurs when the residual phase is zero. A 22.5-degree phase sweep is unnecessary once the two sums exist, because the maximizing phase is the two-argument inverse tangent of those sums.

Correct negative and zero cosine weights

If the cosine-weight sum is negative, a half-cycle of 180 degrees must be added to the inverse-tangent result. If that cosine-weight sum is zero, the phase is 90 degrees when the sine-weight sum is positive and 270 degrees when it is negative. If both sums are zero, the price series is constant and a cycle search has no content.

Read amplitude as a Fourier bin

The amplitude formed from the two sums is one bin of a discrete Fourier transform. Fourier analysis, in this construction, is reading that amplitude as one discrete-transform bin at the trial period. Spectral analysis is plotting those amplitudes across the periods that fit an integer number of cycles into the chosen lookback. A plot of that amplitude against trial period is the measured spectrum.

Choose a lookback that fits whole cycles

Each amplitude sum is valid only when the lookback covers an integer number of cycles of the trial period. On a fixed 32-bar window the admissible periods include 32, 16, and 10.67 bars.

Stationarity is the assumption that the same cycle structure holds for the whole window. A long market sample often fails that assumption. Because market series are often not stationary over long spans, a two-cycle variable lookback can estimate each period’s amplitude. That variable-lookback is a window set to two cycles of each trial period so short, non-repeating spans can still be measured. A fixed-length window is still preferred when the goal is finer frequency resolution from less data.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
8 of 26 in the Fourier analysis track
19931-6 pp.Next on Fourier analysisEvaluating windowed spectral forecasts of index turning pointsA univariate spectral procedure treats an index as one series, splits its history into frequency components, and recombines those components when cycles align instead of extending past price movement.
All readings on this track · 26 readings
  1. 1982Critiquing Fourier, Elliott, and seasonal commodity models
  2. 1982Evaluating half-hour crossovers with cycle priors and stops
  3. 1984Building moving-window spectral cycle forecasts
  4. 1988Fourier cycles versus calendar seasonal regimes
  5. 1989Constructing analog cycle signatures from price lows
  6. 1990Curve fits fail the forecast test
  7. 1991Auditing moving-average length with Fourier spectra
  8. 1991Closed-form phase for dominant-cycle spectra
  9. 1993Evaluating windowed spectral forecasts of index turning points
  10. 1994Window length for moving-window spectral forecasts
  11. 1994Constructing WAMI power and phase spectra
  12. 1994Fourier-tuned momentum filter design
  13. 1994Aliasing, leakage, and dominant-cycle reads from sampled prices
  14. 1995Constructing indicator lookbacks from Fourier spectra
  15. 1996Constructing phase from a dominant-cycle sine and cosine pair
  16. 1999Sinusoidal pseudo-securities for indicator evaluation
  17. 2000Constructing dominant cycle length from Hilbert phase
  18. 2000Constructing adaptive channel breakouts from a measured cycle period
  19. 2005Embedding forex prices into a singular-spectrum forecast basis
  20. 2005Constructing recurrent forecasts from grouped spectral components
  21. 2007Constructing a dominant cycle from transformed Fourier spectra
  22. 2010Why cycle indicators fail when parameters shift
  23. 2019Constructing futures momentum as open-interest-scaled rate of change
  24. 2019Judging a dominant cycle and a trend filter by sample, lag and the move they keep
  25. 2020Constructing futures turnover time and urgency
  26. 2020Stock power ratio as a growth trend filter
By fieldCycles and spectral analysis129 readingsBy year1991166 readings
All 41 readings tagged Fourier analysis
Also on Fourier analysis5 readings
1984Building moving-window spectral cycle forecasts1989Constructing analog cycle signatures from price lows1991Auditing moving-average length with Fourier spectra1993Evaluating windowed spectral forecasts of index turning points1994Aliasing, leakage, and dominant-cycle reads from sampled prices