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Unformatted text preview: 171 172 24681012141618200.511.522.533.544.5time (min)noisy dataalpha = 0.5alpha = 0.8analytical solutionXD Fig S17.1. Graphical comparison for noisy data, filtered data and analytical solution. As decreases, the filtered data give a smoother curve compared to the nofilter (=1) case, but this noise reduction is traded off with an increase in the deviation of the curve from the analytical solution. 17.2 The exponential filter output in Eq. 179 is ( )( )(1)(1)FmFykykyk= + (1) Replacing kby k1 in Eq. 1 gives (1)(1)(1)(2)FmFykykyk= + (2) Substituting for (1)Fykfrom (2) into (1) gives 2( )( )(1)(1)(1)(2)FmmFykykykyk= + + Successive substitution of (2)Fyk,(3)Fyk, gives the final form 1( )(1)()(1)(0)kikFmFiykykiy== + 173 17.3 Table S17.3 lists the unfiltered output and, from Eq. 179, the filtered data for sampling periods of 1.0 and 0.1. Notice that for sampling period of 0.1, the unfiltered and filtered outputs were obtained at 0.1 time increments, but they are reported only at intervals of 1.0 to preserve conciseness and facilitate comparison. The results show that for each value of t, the data become smoother as decreases, but at the expense of lagging behind the mean output y(t)=t. Moreover, lower sampling period improves filtering by giving smoother data and less lagg for the same value of . t=1 t=0.1 t =1 =0.8 =0.5 =0.2 =0.8 =0.5 =0.2 0 0 0 0 0 0 0 0 1 1.421 1.137 0.710 0.284 1.381 1.261 0.877 2 1.622 1.525 1.166 0.552 1.636 1.678 1.647 3 3.206 2.870 2.186 1.083 3.227 3.200 2.779 4 3.856 3.659 3.021 1.637 3.916 3.973 3.684 5 4.934 4.679 3.977 2.297 4.836 4.716 4.503 6 5.504 5.339 4.741 2.938 5.574 5.688 5.544 7 6.523 6.286 5.632 3.655 6.571 6.664 6.523 8 8.460 8.025 7.046 4.616 8.297 8.044 7.637 9 8.685 8.553 7.866 5.430 8.688 8.717 8.533 10 9.747 9.508 8.806 6.293 9.741 9.749 9.544 11 11.499 11.101 10.153 7.334 11.328 11.078 10.658 12 11.754 11.624 10.954 8.218 11.770 11.778 11.556 13 12.699 12.484 11.826 9.115 12.747 12.773 12.555 14 14.470 14.073 13.148 10.186 14.284 14.051 13.649 15 14.535 14.442 13.841 11.055 14.662 14.742 14.547 16 15.500 15.289 14.671 11.944 15.642 15.773 15.544 17 16.987 16.647 15.829 12.953 16.980 16.910 16.605 18 17.798 17.568 16.813 13.922 17.816 17.808 17.567 19 19.140 18.825 17.977 14.965 19.036 18.912 18.600 20 19.575 19.425 18.776 15.887 19.655 19.726 19.540 Table S17.3. Unfiltered and filtered output for sampling periods of 1.0 and 0.1 174 Graphical comparison: 24681012141618202468101214161820time, ty(t)=1=0.8=0.5=0.2Figure S17.3a. Graphical comparison for t = 1.0 24681012141618202468101214161820time, ty(t)=1=0.8=0.5=0.2Figure S17.3b. Graphical comparison for t = 0.1 175 17.4 Using Eq. 179 for = 0.2 and = 0.5, Eq. 1718 for N* = 4, and Eq. 1719 for y=0.5, the results are tabulated and plotted below....
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This homework help was uploaded on 04/10/2008 for the course CHE 242 taught by Professor Cummings during the Spring '08 term at Vanderbilt.
 Spring '08
 Cummings

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