Constructing simple and exponential moving averages as specified filters

A moving average as a low-pass filter

A moving average is constructed as a low-pass filter that lets slower variation through and attenuates high-frequency variation. The construction maps ordered price observations over a defined lookback into one filtered value at each sampling interval.

Finite memory in the simple average

A simple moving average has a finite-impulse-response. It is nonzero only while an impulse remains inside a fixed observation window, and it can be updated by adding the newest sample and dropping the oldest. The filter reaction lasts only while that single impulse remains inside the window.

Cycle response and phase delay

On a pure sine-wave input, a simple moving average is most sensitive when the observation window equals one half-cycle, and the averaged series lags the input by a quarter cycle. That lag is the phase delay of the filtered series relative to the generating cycle.

A simple moving average cancels to zero when an even number of complete cycles fills the window and leaves a residual fluctuation at odd cycle multiples. The frequency response follows a sin(x)/x shape, where x is pi times the frequency-ratio, meaning how many input cycles fit inside one observation window.

Recursive memory in exponential smoothing

An exponential moving average is formed recursively as a mix of the latest observation and the previous average, so older samples never fully leave the calculation. That recursive reaction is an infinite-impulse-response: after an impulse it theoretically never reaches exact zero.

The smoothing factor is the constant that sets how much of each new exponential average comes from the latest observation. An exponential smoothing factor can be matched to a window length by setting the factor approximately equal to two divided by that length.

A same-lookback comparison

Simple-average phase delay increases linearly with frequency, whereas exponential-average phase delay is nonlinear and stays nearly constant at high frequencies. Simple and exponential averages have similar overall attenuation except for periodic peaks in the simple average, and both constructions lag the generating series because they combine historical observations.