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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 pre processing 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 average with the two adjacent values: $$\frac{1}{3} y_{k+1}+\frac{1}{3} y_{k}+\frac{1}{3} y_{k-1}=z_{k} \quad \text { for } k=1,2, \ldots$$Suppose a signal $y_{k},$ for $k=0, \ldots, 14,$ is$$9,5,7,3,2,4,6,5,7,6,8,10,9,5,7$$Use the filter to compute $z_{1}, \ldots, z_{13}$ . Make a broken-line graph that superimposes the original signal and the smoothed signal.

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Calculus 3

Chapter 4

Vector Spaces

Section 8

Applications to Difference Equations

Vectors

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University of Michigan - Ann Arbor

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In mathematics, a vector (…

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Correlated bit noise. Let …

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A The $R C$ high-pass filt…

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In Exercises $7-12$ , assu…

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by putting together the transformation for the signal. Why in green and the signal w in red, we followed the mattress is A and B which have 16 rose and the A s three columns and be as only one. So this is the first step. And now the coefficients then we want so a zero I wanted to. They described The transform are given by the least square solution off the system a x equal Be so First of all, we observed that and he's on exercise for you. The columns of a I don't forget. So you just compute the inner product between every column of a and you see that there are four go. That is good because it allows us to simplify the computation by far. So since we know the columns that are a foregone all, we know that we can just compute the projection onto the columns of a off the vector B in an easy way. We just need to compute the inner products between each column and then a product off, be against each column. And once we know these data, we know that the entries off our list square solution X hat is going to be given by simply the entries. Off be times, eh? I divided by I times a I So once we have these numbers, we know immediately what the least square solution is. So it is just an issue of computation. So you can check that a wine daughter one is equal to eight to dot es tu, which is equal to a $3.3 and they're all equal to 7.92. Then we compete in the problem, be against the columns of a so be against a one turns out to be 2.8 be against a two is 3.96 and me dot es three is to burn eight. And now we have all the ingredients and we need because the exact is going to be well be against a column divided by the square norm of the column. So the first century we have 2.8 divided by 7.82 which is roughly 0.35 Then the second entry is be against a two divided by the square length away, too. She's 0.5, which is just, you know, three point in the six divided by 7 92 And finally, for the third row again, we have to 18 Development 7 92 which is 0.35 again. So these are the numbers that describe the transform that we want.

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