Whitened lowpass filters for trend and countertrend

Treat price as pink noise, not as a waveform plus noise

The construction treats market prices as pink noise with memory rather than as a deterministic waveform plus additive white noise that can be stripped away by smoothing. Pink noise is a spectrum whose power is stronger at lower frequencies, so longer swings dominate unless the series is first whitened.

With the memory exponent taken as one, pink-noise amplitude doubles when frequency is halved. That change is described as a 6 dB per octave rise in spectral power.

An exponential moving average applied to a white-noise source has a 6 dB per octave rolloff. That average is used to synthesize a pink-noise price series with lightly filtered high and low perturbations.

Whiten first so later cycles are comparable

That 6 dB per octave rise produces spectral dilation. Spectral dilation is the low-frequency amplitude bias that appears when unwhitened prices are treated as if every cycle period had equal power. Cycle amplitudes are unequal across periods unless the series is whitened before later frequency-domain work.

A practical whitening step is a two-bar average of successive one-bar differences, equal to half the two-bar price change. That step avoids amplifying noise at the two-sample Nyquist period. Whitening is a short differencing step that inverts the pink-noise memory so residual power is more uniform across frequencies before the lowpass is applied.

Pass the residual through a two-pole lowpass

The whitened residual is passed through a two-pole lowpass with a 12 dB per octave rolloff and a transmission zero at the two-bar period. The only required input is a band-edge lookback. The lowpass is a two-pole smoother whose band-edge period removes short-cycle content from a whitened residual and is the only required lookback in the oscillator.

The default band-edge is 20 bars on daily data. A smaller setting keeps more high-frequency content and a more irregular oscillator. A larger setting smooths the waveform and adds lag. Band-edge is the lowpass period that attenuates components shorter than the chosen lookback, trading irregularity against lag.

Scale the output without another smoother

A peak tracker that decays the reference by 0.991 each bar, then resets on a new absolute peak, scales the lowpass output into a roughly minus-one to plus-one oscillator without an extra smoothing stage. Automatic-gain-control is that peak-tracking scaler. It holds the filter output near a fixed range by decaying the reference until a new peak resets it.

Compare the zero line with a moving-average trend

The moving average appears twice. An exponential average turns white noise into a pink-noise simulator. A simple average is the baseline trend rule the oscillator is compared against.

At a 50-bar band-edge, oscillator zero-line crossings are described as closely associated with price crossings of a 50-bar simple moving average. The same zero-line rule can therefore stand in for that moving-average trend definition.

Trend-following is a testable long-horizon use of the same oscillator, in which staying above or below zero stands in for a price-versus-average trend definition. Band-edge settings in the 150-to-200-bar range are presented as a practical window for longer trend identification. Two-sided long-and-short use is limited by an upward market bias.