when-a-signal-is-produced-from-a-sequence-of-measurements-made-on-a-process-a-chemical-reaction-a-fl

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 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:  
$$
\frac{1}{3} y_{k+1}+\frac{1}{3} y_{k}+\frac{1}{3} y_{k-1}=z_{k} \quad 	ext { for } k=1,2, \ldots
$$
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.

Angelo Rendina · Accepted Answer

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&#x27;s on exercise for you. The columns of a I don&#x27;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&#x27;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&#x27;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.

Nikolaos Alexandrakis · Answer

So, with the signal at the discreet times, the equals 01 of them. And we apply this with them, which can be found on example three on Bates to 48. And we need to evaluate Zakia. Is this good times? Eso We proceed by start of calculations and we find that for J equals zero, their signals at zero equals, huh? 1.4 for Syria calls one. We have zero for K equals two will have minus one over floor. Let&#x27;s what they get in here, Okay? Of course. Two week of minds one over four. As we&#x27;ve just said for Galia, Quinn&#x27;s three will have my most 1.9 for Gaia Quinn&#x27;s four. We have minus one point for. Are you going? Blues five? We have zero. Forget. Vehicle six will have 1.4. Um, we proceed for K vehicle seven. We&#x27;re having one with nine Ah time. Eight week of, uh, one on one blowing for time. Nine without zero. And at times, then we have Is it off them? Equals minus one. Going for now, we know that we still have been in the city with period. It wa ce eight as it happens, exhibiting an example of three in the signal, in example, for you on Bates to 48. And, uh, the signal bits to 48 on example. Three is just this. They&#x27;re off the signal that Cassandra is off by the over four. Here. We don&#x27;t have traditional therapies. We have co signers. Three bitey over four and plus a general this term exams fighting over four with mixed wise. The signal. Bless something that it&#x27;s a lot of difference in there in the results, but that was it.

Zulfiqar Ali · Answer

well is the filter consists off like a pastor. Ca Bester Blades charge slowly making the outward will teach league and out of phase with input and beautiful or fees changes. Andi. So option. Option D is right. So the reason is capacitor plates, uh, charge slowly, making the output ball teach, making the output voltage lake and out of phase with the input wall Teach. So there is a feeds change, so Option D&#x27;s right.

Zulfiqar Ali · Answer

12 option is the correct option. So option Hey, is ah the correct option. Right? It law frequency. Like it? Uh, no. Frequency capacitor reactant works crossing the frequency signals and, uh, it high frequency faster. Reactant schtrops by short circuiting on their for low boss filter.

Zulfiqar Ali · Answer

Well, let me draw Ah, the circuit. So these aren&#x27;t inputs. And then here we have resistor r right. And here is the pastor, see? And from here we take on the outward. So these are the output right here we have in purple teas and then we have output voltage, an electrical. And I is in this direction This is our no anger frequencies Omega and input Will TT&#x27;s here weaken And the, uh, lake xsi is equal to our so capacity reactions is equal to resistance. Right now, impotence off the circuit is given binds. It is equal to square root off our square bliss x c square. And since both are equal therefore we have that is equal to boot, uh, to our right route to times are and further, um, output will teach is equal to will teach across stick bastions We out eyes equal to like Oh, I don&#x27;t act See I don&#x27;t exceeds the walt it across the capacitance Right now we out is equal to we in and divided by said times x c So we are people&#x27;s we out music will dio we in divided by route two times are multiply by are. And so we out is equal to we in divided by I really do and so options. He&#x27;s right, Option C is correct.

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Let $\left\{y_{k}\right\}$ be the sequence produc…

Problem 21 Easy Difficulty

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graph cannot copy

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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graph cannot copy

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications