Shared coefficient construction for recursive price filters

One routine, several responses

One recursive routine can produce simple averaging, exponential averaging, and momentum-style measurements by switching a type argument and the coefficients that type selects. That shared object is a unified recursive filter: a single second-order update whose coefficients are switched to produce different responses from one ordered price series.

After more than N bars, the shared update is a linear mix of the current price, the two prior prices, the two prior outputs, and the price at lag N, each scaled by type-specific coefficients.

Exponential smoothing as the one-pole case

Exponential-smoothing mode is the one-pole case of the shared update. It sets the current-price weight to an alpha computed from the sine and cosine of 360 divided by Period, and sets the prior-output weight to one minus that alpha. The angle 360 divided by Period is the period angle used for those sine and cosine gains.

Editorial: this lesson treats the Kalman filter as the conceptual parent of carrying the prior output as state, a predict-and-revise estimator that retains the last output until the new price revises it.

Simple averaging with a lag-N zero

Simple-average mode requires a nonzero lookback N, sets both the current-price weight and the lag-N subtractive weight to 1/N, and holds the first-order feedback coefficient at 1. The recursion also allows an optional extra zero at lag N, which is that lag-N subtractive term.

Two-pole gains and the alpha-beta step

Two-pole Gaussian and Butterworth modes share a beta equal to 2.415 times one minus the cosine of 360/Period, then derive alpha from that beta. They differ mainly in feedforward gain and extra input taps.

Editorial: those period-derived alpha and beta terms are used here as the two-gain pattern of an alpha-beta filter. They set how strongly new prices revise the retained state.

Warmup bars and a zero period

High-pass and two-pole high-pass modes seed the first N bars at zero, while averaging and Gaussian modes seed those bars with the input price. The first N observations are the warmup bars: they receive a type-specific seed instead of the full recursion. Most period-using modes abort if Period is zero.

Band-pass, band-stop, and the chirped test wave

Band-pass and band-stop modes introduce a width parameter through a cosine term at twice the period angle, then solve for alpha from that term. The two modes differ in the signs and placement of the feedforward taps. Band-pass keeps variation near the chosen period and rejects slower and faster variation. Band-stop attenuates variation near the chosen period.

The same band-pass constructor can be run at two nearby periods and subtracted to form a two-line difference. A chirped test wave, a synthetic sine whose frequency declines across the sample, is used as a known periodic test input for inspecting those responses.

Rational transfer and a lasting seed offset

The recursion matches a rational transfer function with at most two poles and an optional extra zero at lag N. Simple-average reconstruction is sensitive to the opening seed, and a constant initialization error can persist as a permanent level offset.

Editorial: specify the taps, the period angle, and the warmup seed completely on the chirped test wave before the same recursion is applied to a market series.