A bandpass filter bank for dominant cycle construction

Two stages, one measured period

Adaptive indicator construction begins by measuring the cycle periods present in the series. Once a dominant cycle is known, computation windows can be reset from that period, including setting an RSI lookback to half the dominant cycle.

Editorial: treat that adaptation as a two-stage build. First assemble a controllable bank of overlapping bandpass channels that locates where energy sits. Then retune a single bandpass to that measured period so every later lookback has an explicit spectral reason.

Estimate the spectrum with designed filters

A market spectrum can be estimated with bandpass filters rather than only a discrete Fourier transform, because filter selectivity and transient response can be set by design.

In this usage, a bandpass filter is a two-pole filter that passes a chosen relative-frequency neighborhood and rejects both slower trend and faster jitter when it is tuned to a steady cycle.

A simple two-pole bandpass is described as producing no output lag on a steady-state input at its tuned frequency, while more complicated filters generally add more lag.

When equal-amplitude components at all relative frequencies enter the filter, most transmitted energy lies in the relative-frequency interval from -0.5 to +0.5. Rejecting lower frequencies detrends and rejecting higher frequencies smooths, but only if the filter is tuned to the steady-state dominant cycle.

Assemble the overlapping channels

Spectral analysis here is a bar-by-bar map of energy versus cycle length obtained by comparing output amplitudes across a contiguous bank of overlapping bandpass channels.

A filter bank is that set of overlapping bandpass channels, with centers stepped in one-bar increments, used to read which period is active in the series. A spectrum is constructed by placing the channels side by side and comparing output amplitudes. The channel whose passband contains the input carries the larger amplitude.

The filter-bank construction processes each bar of midpoint or close price through a 40-bar high-pass detrend and a six-tap FIR smoother, then through channels centered from 8 to 50 bars, with relative half-bandwidth starting at 0.5 and later held at a floor of 0.15.

Relative half-bandwidth is the half-width of each channel relative to its center period. A wider value lets transients decay faster and a narrower value increases selectivity and ringing.

Read each channel as an inphase-quadrature pair

Each narrow channel output is treated as a sine of slowly varying phase. Multiplying its rate of change by period divided by two pi forms a cosine of the same amplitude, and the sum of the two squared waveforms is the channel amplitude.

That pairing is an inphase-quadrature pair: a sine-like filtered series and a cosine-like rate-of-change series that share the same channel filters so the sum of their squares is instantaneous amplitude.

Blend the strongest channels into a dominant cycle

Dominant cycle is the period that currently carries most of the isolated cyclic energy, recovered as a gravity-weighted blend of the strongest bandpass channels rather than the single loudest bin.

In this workflow the dominant cycle is computed as the amplitude-weighted center of gravity of channels no more than 3 dB below the peak, then taken as a 10-bar median of that gravity period, which is smoother than selecting only the single loudest channel.

Retune a single bandpass and reset later windows

After the period is measured, the same bandpass coefficients are retuned to that dominant cycle so the filter emits a sine that tracks a smoothed version of price and a cosine that leads the sine.

Editorial: later windows, including an RSI lookback set to half the dominant cycle, then rest on that measured period so the lookback has an explicit spectral reason.

Keep the passband wide enough to settle

Making the passband arbitrarily narrow is not useful for market series because a highly selective filter rings after a transient. A wider passband lets a sudden change decay faster, and gap openings on intraday data can still distort the filter response.