Constructing adaptive channel breakouts from a measured cycle period

A shared period estimate

The dominant cycle is the slowly varying market cycle length estimated from recent price observations and used to size subsequent calculations. In this construction the finished period is not a separate overlay. It is reused to compensate the next bar's Hilbert amplitude distortion, to drive adaptive averages or sinewave indicators, and to feed a simple channel-breakout signal.

A channel-breakout construction can size its entry and exit lookbacks as multiples of a Hilbert-estimated period when fixed lookback inputs are left at zero. Those windows are adaptive lookbacks: a channel or average length expressed as a multiple of the measured period instead of a fixed bar count.

Smoothing and detrending before the transform

The period function first applies a four-bar weighted moving average, whose lag is described as one bar, then detrends the result. The detrend step is required because the Hilbert transform needs a detrended input.

In-phase and quadrature components

Fourier analysis, in this setting, is the decomposition of an ordered price series into in-phase and quadrature components so that the one-bar phase change can be read as cycle period. After a three-bar-lag quadrature calculation, the in-phase component is taken as the detrended series lagged three bars.

The pair is then phase-advanced and smoothed with an exponential average of alpha 0.15 before multiplication. Noise is reduced first, and only then is the complex product formed.

Reading period with a homodyne discriminator

A homodyne discriminator estimates period by multiplying the complex signal by its one-bar lag and reading the one-bar phase change as the cycle length. The multiplication produces a zero-frequency beat whose phase rate is that cycle length. The estimator is presented as preferable in a low signal-to-noise setting.

Compensating Hilbert amplitude error

A two-dimensional Hilbert numbering scheme that reduces computation also introduces amplitude error in the quadrature component. That error is offset with a straight-line compensation term taken from the prior bar's frequency. Amplitude compensation is this correction: it is applied so that a computationally cheap Hilbert transform does not distort the period estimate.

The prior-bar term is justified by the construction assumption that cycle period cannot change drastically from one bar to the next. The finished period is then written back into the next bar's compensation step.

Limits on the finished period

The finished period is change-limited, clamped between 6 and 50 bars, and exponentially smoothed. After those constraints it is reused to compensate the next bar's Hilbert amplitude distortion and to drive adaptive averages or sinewave indicators.

Sizing the channel from the same estimate

The matching signal is constructed as a simple channel breakout whose buy and exit orders consume the HilbertPeriod function output. When the fixed lookback inputs are left at zero, those orders inherit entry and exit lookbacks as multiples of the same Hilbert-estimated period.

The plotted Hilbert channel indicator is specified to use the same inputs and maximum reference-bar setting as the corresponding signal after those inputs have been optimized. The plot and the orders are therefore built from one period function rather than from a second, independent window.