Dual exponential Super Passband filter construction

Building the Super Passband in stages

Editorial framing: this note treats the Super Passband as a construction lab. Start with exponential smoothing, form a bandpass filter by subtracting two stages that sit at different critical frequencies, and treat the cancelled low-frequency remainder as the trend filter. The leftover residual can then be scaled with its own RMS amplitude.

The steps below restate the archive construction only. They do not add a trading rule.

Exponential smoothing as a low-pass filter

An exponential smoother can be written so the weight on the latest observation and the weight on the previous smoother value sum to one. That complementary pair is exponential smoothing: a recursive average that blends the latest observation with the previous average.

The same smoother acts as a low-pass filter. Low-frequency content is passed, while components above a critical frequency are increasingly attenuated.

From two smoothers to a bandpass filter

A bandpass filter is constructed by subtracting a second exponential-smoothing stage from the first. The second stage is assigned a lower critical frequency. The result is a mid-band response that rejects both the slowest drift and the fastest wiggles, and that passes the frequencies sitting between the two critical cutoffs.

At the lowest frequencies the two smoothers share the same gain, so their difference is zero by frequency cancellation. That trend-filter step removes the shared very-low-frequency gain so the residual no longer carries slow drift. Because the cancellation already does that work, the step does not add a separate high-pass delay.

After the slower smoother rolls off, the difference stays relatively flat until the faster smoother's critical frequency. That relatively flat mid-frequency region is the passband. Beyond it, the difference inherits the faster smoother's high-frequency attenuation.

Critical periods set the two coefficients

The two design inputs are the critical periods of the two smoothers. A critical period is the cycle length, the reciprocal of critical frequency, used to set each smoother's coefficient.

Each smoother coefficient is obtained from its period by the heuristic ratio of five divided by that period. Swapping the two periods inverts the output waveform.

A closed-form recurrence for the passband

A closed-form recurrence computes the passband value from the current close, the previous close, and the two most recent passband values. That single update mixes those four terms instead of computing two averages and subtracting them after the fact.

RMS amplitude as the scale reference

An RMS scale of the passband series is formed by averaging the squared output over a lookback illustrated as 50 bars, long enough to cover about one cycle, and taking the square root. That RMS amplitude is a root-mean-square scale of the passband residual, used as a reference envelope around the zero line.

Because the lowest frequencies are removed by cancellation, the passband output is described as having a guaranteed zero mean. Unity gain at the passband center is unnecessary when that RMS scale is the amplitude reference.