7.2 Filtering: weighted averages

Weighted, tapered, or windowed averages are straightforward generalizations of simple averages. They take the form

with the constraint that the “weights in the window,” $w_n$ , sum to 1:

When $w_n=\frac{1}{N}$ then $x$ is the simple average studied in the section on "Simple Averages" .

There are many windows that are commonly used in engineering practice. For $N$ odd, the standard triangular window is

This window, illustrated in Figure 1 , weights the input $u_{(N+1)/2}$ by $\frac{2}{N+1}$ and the inputs $u_1$ and $u_N$ by ${\left(\frac{2}{N+1}\right)}^2$ The most general triangular window takes the form