= Finite impulse response
{title2=$y[n]=\sum_{j=0}^{M-1}h[j]x[n-j]$}
{wiki}
= FIR filter
{c}
{synonym}
A <linear time-invariant system> has finite impulse response when its <impulse response> is nonzero at only finitely many indices. For a <causal filter> supported at indices $0,\ldots,M-1$,
$$
y[n]=\sum_{j=0}^{M-1}h[j]x[n-j].
$$
This is a finite <discrete convolution>. The coefficients $h[j]$ specify the complete operation; they need not be positive or equal. Such a filter has a simple bounded-input estimate, $|y[n]|\leq\|x\|_\infty\sum_j|h[j]|$, so bounded inputs remain bounded. The <moving average> chooses $h[j]=1/M$.
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