MovingAverageFilter

Movingaveragefilter is a discrete-time linear filter used to smooth data by replacing each sample with the average of neighboring samples within a fixed window. It is commonly implemented as a finite impulse response (FIR) filter and is widely used in digital signal processing to reduce high-frequency noise while preserving slower trends.

A simple moving-average (SMA) with window length M computes, for each sample n, y[n] = (1/M) sum_{k=0}^{M-1} x[n-k], assuming sufficient past samples. In noncausal or centered forms, the window is symmetric around the current sample, which introduces a delay of (M-1)/2 samples but yields linear phase. Edge handling strategies include zero-padding, reflection, or truncating the window at the start.

Variants include cumulative moving average (CMA), which uses all past samples up to n, and weighted moving averages (WMA), where different weights are assigned to samples within the window. Exponential moving average (EMA) is a related smoothing filter with a recursive form y[n] = α x[n] + (1-α) y[n-1], producing exponentially decaying weights and an infinite impulse response (IIR) behavior.

Frequency response and characteristics: the SMA has a low-pass frequency response that attenuates high-frequency components. The magnitude response is H(e^{jω}) = (1/M) |sum_{k=0}^{M-1} e^{-jωk}|, and its phase is linear for centered filters. The main trade-off is smoothness versus lag; larger M provides smoother output but greater delay.

Implementation considerations: a straightforward implementation is O(nM) per data series, but the SMA can be computed in O(n) using a sliding window or by maintaining a running sum. Extensions to two dimensions yield 2D moving-average filters used in image smoothing.