Constructing a cycle-aligned bandpass from paired lowpass filters

An oscillator as a passband

An oscillator can be restated in the frequency domain as a filter that removes long-period trend and short-period noise so that only a chosen band of cycle frequencies remains. The kept band is the passband, and the period at the middle of that band is the passband center.

Subtracting two three-pole lowpass filters

The bandpass output is the difference of two three-pole recursive lowpass filters that share the same input series but use unequal cutoff periods. Each three-pole lowpass is a recursive smoothing filter whose output depends on three prior outputs as well as the current and recent inputs, and it is used here as each edge of the bandpass.

A three-pole design is presented as a compromise between extra smoothing and extra lag. Butterworth-type lowpass lag is given as NP/π², where N is the pole count and P is the cutoff period.

The shorter-cutoff filter attenuates high-frequency content. The two filters pass long-period content at similar amplitude so that content cancels in the subtraction. The remaining passband is what does not cancel.

Phase at the passband center

Lag is described as approximately zero at the passband center because the indicator is a difference of two lagging functions. Components longer than that center lead in phase, and shorter components lag. The passband center is the reference for that phase description and for the lead-line scale factor.

A companion lead line from the same output

A leading signal is a same-amplitude companion series formed from the bandpass output plus a normalized one-step change, producing a smaller phase lead than a raw difference. In this workflow the companion lead line is formed by scaling the change in the bandpass output by Pcenter/(2π), adding that term to the bandpass output, and dividing by 1.414, which yields about 45 degrees of lead at matched amplitude.

How the band edges are placed

Band edges may be set as a single fixed pair, as independent edges chosen from recent historical profitability, or from a center period taken from the observed cycle with the edges placed as a fractional offset around that center.

A fixed daily example uses cutoffs of 6 and 30, implying a center period of 13.4, but that universal pair is treated as analogous to a one-size oscillator length and is not recommended as the default.

When the center follows a dominant cycle, an octave bandwidth is suggested. That band-edge rule sets the longer cutoff at twice the shorter cutoff once the center has been taken from an observed cycle, and the passband is placed at half the measured cycle period. That period can be counted between successive highs or lows, or doubled from a high-to-low span. The dominant cycle is the prevailing cyclic period in an ordered price series, obtained by bar counts between swings or by a spectral estimate, and used to place the passband.

Tracking a measured dominant cycle

The same bandpass can be retuned bar by bar from a maximum-entropy spectrum estimate of the dominant cycle so the passband tracks the measured cycle mode. The maximum-entropy spectrum is a spectral estimate of the current cycle period used to retune the bandpass center as that period changes.