A low-pass filter lets slow changes through and blocks fast ones — so it smooths a jagged, noisy signal into its underlying trend. The simplest is a moving average: replace each point with the average of its neighbours. Fast wiggles cancel; slow trends survive. Every ‘smooth’ slider, every noise-reduction, every trend line is a low-pass filter. Slide the cutoff and watch noise melt away.
A low-pass filter attenuates frequencies above a cutoff while passing those below. A moving-average FIR filter y[n]=(1/M)Σ x[n−k] is the simplest: its frequency response is a sinc, with unity gain at DC (0 Hz) and nulls that suppress rapid oscillation — so broadband noise shrinks while the slow trend survives. Widening the window lowers the cutoff (more smoothing, more lag). Filtering is [[the-discrete-convolution|convolution]] with the filter’s impulse response; recursive (IIR) designs like Butterworth get sharper roll-off cheaply. Used for de-noising, anti-aliasing, and trend extraction. A fail-loud self-check throws unless DC passes at gain 1 and an alternating high-frequency signal is attenuated. ◆ real signal math, node-verified.
A moving average is a gentle filter (slow roll-off, ripple); sharper cutoffs need higher-order FIR/IIR designs. It also LAGS the signal. The unity DC gain and high-frequency attenuation are exact.