When a signal is produced from a sequence of measurements made on a process (a chemical reaction, a flow of heat through a tube, a moving robot arm, etc.), the signal usually contains random noise produced by measurement errors. A standard method of preprocessing the data to reduce the noise is to smooth or filter the data. One simple filter is a moving average that replaces each yk by its average with the two adjacent values:

Use the filter to compute z₁,.....,z₁₃. Make a broken-line graph that superimposes the original signal and the smoothed signal.

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Yk+1 + k + YK-1 =Zk for k = 1,2,... Suppose a signal yk, for k = 0,..., 14, is 9, 5, 7, 3, 2, 4, 6, 5, 7, 6, 8, 10, 9, 5, 7