One second-order transfer function, a family of trend filters

One equation and many coefficient tables

The Swiss Army Knife indicator is a single second-order rational transfer function whose coefficient table implements several common technical filters. The general indicator uses that rational second-order transfer function plus an optional delayed input term, so one programmed equation can host many coefficient tables.

A trend filter is a discrete linear operator that maps ordered price samples into a filtered series used as a quantitative baseline. Band-stop and related coefficient branches are selected by a type flag, and the same second-order recurrence is evaluated only after the lookback bar count is exceeded.

Transfer functions in Z-transform form

A discrete linear filter is specified as a transfer function: the ratio of filter output to filter input expressed as polynomials in the delay operator. In Z-transform notation, Z to the minus one represents one sample period of delay, matching daily or intraday bar sampling.

Exponential smoothing as a first-order coefficient set

Exponential smoothing is a recursive smoother whose current output is a weighted blend of the newest input and the previous output, recovered here as one coefficient set of the general transfer function. The first-order coefficient choice a0=1, b0=alpha, b1=0, a1=-(1-alpha) reduces the general difference equation to the familiar exponential moving average.

Alpha is the exponential-smoothing weight that also sets the attenuation corner relative to a chosen cycle period. Alpha can be set from an equivalent simple-average length L by the relation alpha = 2/(L+1), or from a cycle period at which attenuation is intended to begin. A check value given in the source is that attenuating cycle components shorter than 20 bars yields alpha of 0.2735, roughly a six-bar simple average.

When the input and feedback coefficients of that exponential smoother sum to one, the zero-frequency gain is unity. DC gain is that zero-frequency response; unity DC gain means a long constant input leaves the output nearly equal to the input.

A sliding simple average as a finite window

A finite-impulse-response filter is a filter whose output depends on a finite window of input samples and then drops the oldest term, as in the sliding simple average coefficient set. A simple moving average of length N is recovered as a finite-window coefficient set that adds the newest sample and drops the oldest sample divided by N.

That sliding-average implementation needs an initialization guard, because starting from zero on a nonzero price series delays recovery until the window fills.