An odd-length smoother from a cycle cutoff

How the filter is built

This archive note follows a finite-impulse-response-filter from a cycle cutoff to a two-sided weighted average. The average sits on a finite odd-length-window centered on each observation, with the same number of observations on each side.

A cutoff becomes an odd-length window

Squaring spectral amplitude into a power-spectrum is presented as a way to make important cycle lengths easier to separate from unimportant ones. In the worked construction, the power-spectrum reading treats periods shorter than 20 days as less material for the longer move. That 20-day reading is converted into a 41-day filter that uses a hanning-weight by rebuilding length as twice the half-span plus one.

Swap the weights, keep the convolution

Three weight families are selectable. A moving-average assigns uniform linear weights. A hanning-weight is a cosine taper that down-weights observations farther from the window center. A triangular-weight is a linear taper that declines from the center toward each edge of the window.

The weight-assignment step is isolated so other moving-average weighting schemes can be added without rewriting the rest of the convolution. Chosen length is rebuilt as an odd integer, the weights are accumulated and scaled to unit sum, and the valid output range is shortened by the half-span at each end.

The routine writes a column, not a chart

The supplied routine is a BASIC computational block only. The user must load an ordered series into a fixed matrix, and the filter writes its study into a dedicated output column without printing or charting.

The smooth or the residual as a trend-filter

After smoothing, the study can be time-shifted by a user-specified number of days, then optionally replaced by raw minus smooth to emit a residual high-pass series. The trend-filter is either the smoothed series itself or that optional residual.