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1996issue C111-2

Constructing phase from a dominant-cycle sine and cosine pair

Phase is assembled from one-cycle cosine and sine sums. After a quadrant correction and a sine-wave reference, the wrapped angle is the phase reading, and the same sums also identify a linear slope.

  • Over one dominant cycle, a cosine-weighted sum of prices forms the real component and a sine-weighted sum over the same window forms the imaginary component.
  • The first angle is the arctangent of the imaginary component divided by the real component. A quadrant correction and a sine-wave reference then place the phase angle, which is wrapped to one full turn and expressed in degrees.
  • If the absolute real component is not greater than 0.001, the arctangent step is skipped and the angle is set to a signed quarter-turn from the sign of the imaginary component.
  • The same one-cycle sums yield 180 degrees on a linear rising slope and 0 degrees on a linear falling slope, so constructed phase can mark a trend as well as a cycle.
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Current trackFourier analysis41 readingsDominant cycle detection180 readings

One-cycle sums form the phasor

Phase construction sums price against a cosine over one dominant cycle to form a real component and against a sine over the same window to form an imaginary component. The real component is the horizontal part of that phasor. The imaginary component is the vertical part.

The dominant cycle is the bar length that sets both the summing window and the period of the sine and cosine weights. Without that length there is no pair to assemble and no phase angle to compute.

Correcting the first angle estimate

The first angle estimate is the arctangent of the imaginary component divided by the real component. That estimate places the result in the first or fourth quadrant.

If the real component is negative, a quadrant correction adds half a turn so the angle is moved out of the wrong half-plane. A sine-wave reference then adds a quarter-turn so the cycle origin is referenced to a sine wave. After those corrections the angle is wrapped into one full turn and, when computed in radians, converted to degrees. The wrapped reading is the phase angle.

If the absolute real component is not greater than 0.001, the arctangent step is skipped. The angle is set to a signed quarter-turn from the sign of the imaginary component.

A required dominant-cycle length

Dominant-cycle length is a required input. It may be a fixed default of 15 bars, counted between notable highs or lows, or remeasured on every bar. The same length governs the cosine weights, the sine weights, and the summing window.

The same sums on a linear slope

The same one-cycle sums yield 180 degrees on a linear rising slope and 0 degrees on a linear falling slope. Constructed phase can mark a trend as well as a cycle because those identities are properties of the assembled pair.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
15 of 26 in the Fourier analysis track
19991-8 pp.Next on Fourier analysisSinusoidal pseudo-securities for indicator evaluationA sinusoidal pseudo-security is a constructed price series built from one or more sine waves and imported as a controlled series whose future path is already known.
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 year1996121 readings
All 41 readings tagged Fourier analysis
Also on Fourier analysis5 readings
1984Building moving-window spectral cycle forecasts1984Constructing maximum-entropy spectra for dominant-cycle forecasts1986Why Fourier windows limit dominant-cycle resolution1988Fourier cycle models break in major swings1988Fourier cycles versus calendar seasonal regimes