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Filtering: Weighted Averages

Module by: Louis Scharf. E-mail the author

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This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

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

x=∑n=1Nwnunx=∑n=1Nwnun
(1)

with the constraint that the “weights in the window,” w n w n , sum to 1:

∑n=1Nwn=1.∑n=1Nwn=1.
(2)

When wn=1Nwn=1N then x x is the simple average studied in the section on "Simple Averages".

Example 1

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

wn=2N+1(1-2N+1|N+12-n|).wn=2N+1(1-2N+1|N+12-n|).
(3)

This window, illustrated in Figure 1, weights the input u(N+1)/2u(N+1)/2 by 2N+12N+1 and the inputs u 1 u 1 and u N u N by (2N+1)2(2N+1)2 The most general triangular window takes the form

wn=α(1-β|N+12-n|);α,β>0,N odd . wn=α(1-β|N+12-n|);α,β>0,Nodd.
(4)
Figure 1: Triangular Window
This graph consist of a x axis dashed at equal distances numbered 1-11 where the space that would be labeled 6 is actually a y- axis like line. Starting above dash 1 there is a line of points proceeding with positive slope with a point directly above each dashed mark. This line ends at the line from where dash 6 should be. This point is labeled W_n. A similar line of dots descends at a negative slope from point 6-11. Point 11 is labeled (2/n-1)^2. The x axis is labeled n

Exercise 1

Determine the constraints on α α and β β to make the general triangular window a valid window (i.e., ∑n=1Nwn=1∑n=1Nwn=1). Show that α=2N+1=βα=2N+1=β is a valid solution. Propose another solution that you like.

Exercise 2

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,3.0A,3.0 points for a B,...B,..., and (horrors!) 0 points for an F F.

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