Constructing a coincident moving average as a lowpass filter

A rectangular window is a lowpass filter

A simple moving average is built by giving every observation inside a finite window equal weight and every observation outside that window zero weight. That construction is a lowpass filter whose time-domain shape is a rectangular pulse.

The resulting average can be plotted on the current bar and scored for amplitude loss and delay. Window length is the number of sampling intervals given nonzero weight.

Attenuation of a named cycle

The frequency response of that rectangular window is a sine-over-argument curve whose argument, in radians, equals pi times window length divided by the cycle period under study.

When the cycle period is twice the averaging window, the simple-moving-average attenuation equals 0.637. When the cycle period is four times the averaging window, the simple-moving-average attenuation equals 0.9, with one-fourth of that cycle inside the window at any chosen phase.

Attenuation is the reduction in cycle amplitude caused by averaging, stated as the ratio of filter output to the input cycle.

Phase lag grows with window length

Phase lag of a simple moving average scales linearly with window length. A half-cycle window lags by a quarter cycle, or 90 degrees, and a quarter-cycle window lags by 45 degrees.

A full-cycle simple moving average would imply a half-cycle lag, but its phase delay is indeterminate because the output amplitude is exactly zero at the first sign change of the sine-over-argument response.

First-order averages and a softer cutoff

The attenuation and lag relations apply to simple moving averages. Exponential smoothing retains more high-frequency variation because its cutoff is less sharp, and it introduces less lag.

For first-order averages, the simple-moving-average attenuation and lag figures are treated as reasonable approximations. Exponential smoothing is a first-order average that weights recent observations more heavily than an equal-weight window of similar reach.

Offset from extremes and travel around the mean

The vertical offset of an advanced, coincident-plotted average from price extremes is the sum of attenuation, which reduces that offset, and a scale effect, which increases it. As the window lengthens, the scale effect dominates. That net price-point offset is the vertical displacement after amplitude loss and the competing scale effect are combined.

Lookback should also reflect how far price travels around a centered average. A narrow range can use a shorter window, while a wide range with large gaps needs a longer one. Those centered-mean excursions are the typical travel of price around an average aligned to the middle of its window.