Question

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 \({y_k}\) by its with the two adjacent values:

\(\frac{{\bf{1}}}{{\bf{3}}}{y_{k + {\bf{1}}}} + \frac{{\bf{1}}}{{\bf{3}}}{y_k} + \frac{{\bf{1}}}{{\bf{3}}}{y_{k - {\bf{1}}}} = {z_k}\), for \(k = {\bf{1}},{\bf{2}},....\)

Suppose a signal \({y_k}\), for \(k = {\bf{0}},.....,{\bf{14}}\), is

9, 5, 7, 3, 2, 4, 6, 5, 7, 6, 8, 10, 9, 5, 7

Use the filter to compute \({z_{\bf{1}}}\),…..\({z_{{\bf{13}}}}\), Make a broken line graph that superimpose the original signal and the smoothed signal.

Short answer

\(k\)	\({y_k}\)	\({z_k} = \frac{1}{3}{y_{k + 1}} + \frac{1}{3}{y_k} + \frac{1}{3}{y_{k - 1}}\)
0	9
1	5	7
2	7	5
3	3	4
4	2	3
5	4	4
6	6	5
7	5	6
8	7	6
9	6	7
10	8	8
11	10	9
12	9	8
13	5	7
14	7

Step-by-step solution

Find the values of \({z_k}\)

\(k\)	\({y_k}\)	\({z_k} = \frac{1}{3}{y_{k + 1}} + \frac{1}{3}{y_k} + \frac{1}{3}{y_{k - 1}}\)
0	9
1	5	7
2	7	5
3	3	4
4	2	3
5	4	4
6	6	5
7	5	6
8	7	6
9	6	7
10	8	8
11	10	9
12	9	8
13	5	7
14	7

Plot \({y_k}\) and \({z_k}\)

The figure below represents the plot of \({y_k}\) and \({z_k}\).