6.3: Weighted Averages
Weighted, tapered, or windowed averages are straightforward generalizations of simple averages. They take the form
\[x=\sum_{n=1}^{N} w_{n} u_{n} \nonumber \]
with the constraint that the “weights in the window,” \(w_n\), sum to 1:
\[\sum_{n=1}^{N} w_{n}=1 \nonumber \]
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
\[w_{n}=\frac{2}{N+1}(1-\frac{2}{N+1}\Bigg |\frac{N+1}{2}-n|) \nonumber \]
This window, illustrated in Figure 1, weights the input \(u_{(N+1) / 2}\) by \(\dfrac{2}{N+1}\) and the inputs \(u_1\) and \(u_N\) by \(\left(\dfrac{2}{N+1}\right)^{2}\). The most general triangular window takes the form
\[w_{n}=\alpha(1-\beta|\frac{N+1}{2}-n|) ; \alpha, \beta>0, \quad N \text { odd. } \nonumber \]
Determine the constraints on \(\alpha\) and \(\beta\) to make the general triangular window a valid window (i.e., \(\sum_{n=1}^{N} w_{n}=1\)). Show that \(\alpha=\frac{2}{N+1}=\beta\) is a valid solution. Propose another solution that you like.
You are taking three 3-credit courses, one 5-credit course, and one 2-credit course. Write down the weighted average for computing your GPA in a system that awards 4.0 points for an A, 3.0 points for a B,..., and (horrors!) 0 points for an F.