7.3 Filtering: moving averages

Moving averages are generalizations of weighted averages. They are designed to “run along an input sequence, computing weighted averages as they go.” A typical moving average over N inputs takes the form

The most current input, $u_n$ , is weighted by $w_0$ ; the next most current input, $u_{n-1}$ , is weighted by $w_1$ ; and so on. This weighting is illustrated in Figure 1 . The sequence of weights, $w_0$ through $w_{N-1}$ , is called a “window,” a “weighting sequence,” or a “filter.” In the example illustrated in Figure 6.8, the currentvalue $u_n$ is weighted more heavily than the least current value. This is typical (but not essential) because we usually want $x_n$ to reflect more of the recent past than the distant past.

When the weights $w_0,w_1,...,w_{N-1}$ are all equal to $\frac{1}{N}$ , then the moving average $x_n$ is a “simple moving average”:

This is the same as the simple average , but now the simple average moves along the sequence of inputs, averaging the $N$ most current values.

When the weights $w_n$ equal $w_0{a}^n$ for $n=0,1,...,N-1$ , then the moving average $x_n$ takes the form

$x_n=w_0\sum k=0N-1a^k{u}_{n-k}$ .

When $a<1$ , then u _n is weighted more heavily than $u_{n-(N-1);}$ when $a>1$ , $u_{n-(N-1)}$ is weighted more heavily than u _n ; when $a=1,u_n$ is weighted the same as $u_{n-(N-1)}.$