102_1_W08-EE102-Week4-Recap - 14 NHAN LEVAN WEEK 4 RECAP(i...

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14 NHAN LEVAN WEEK 4 RECAP (i) Shift in s : L s { f ( t ) } := F ( s ) L s { e β t f ( t ) } = 0 e - st e β t f ( t ) dt = 0 e - ( s ± β ) t f ( t ) dt = F ( s ± β ) This property allows you to find L s of such function as e - 102 t t 142 U ( t ), etc... . (ii) Shift in t : L s { f ( t - t d ) U ( t - t d ) } = 0 e - st f ( t - t d ) U ( t - t d ) dt = t d e - st f ( t - t d ) dt = e - st d F ( s ) How so? Remark. Consider the system S defined by: x ( t ) [ S ] y ( t ) := x ( t - t d ) , t R where t d > 0. S is often called a“delayed” (by t d unit) system, or a right “shift-by- t d ” system, or a right “translation-by- t d ” system. Why so? Let t 0, then t - t d is a time before t , i.e., t - t d < t i.e., t - t d is “earlier” than the time t . Thus the value y ( t ) at t is equal to the value x ( t - t d ) at the earlier time t - t d . But x ( t - t d ) can be considered as x ( t ) being delayed by the amount t d . Then since y ( t ) := x ( t - t d ) y ( t ) is simply x ( t ) delayed by t d . Hence S is called a delayed-by- t d system. Next, if you want to get the value y ( t ) at time t you will have to bring the value x ( t - t d ) to the time t . To do this you simply shift x ( t - t d ) , to the right, to the time t . Hence the name right shift-by- t d . Finally Shifting and Translating are synonymous. 12. Easy Steps To Find Inverse Laplace Transforms L - 1 s {·} Let F ( s ) be the Laplace Transform of f ( t ) then f ( t ) is called the Inverse Laplace Transform of F ( s ) and we write L - 1 s { F ( s ) } := f ( t ) . Here we are only interested in Rational Functions of s , that is, F ( s ) is of the form: F ( s ) = P ( s ) Q ( s ) , where P ( s ) and Q ( s ) are polynomials in s . The ROT (Rule Of Thumb) is:
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15 Step 1: Look at degree m of P ( s ) and degree n of Q ( s ). If degree P ( s ) degree Q ( s ) Devide P ( s ) by Q ( s ) until degree of the top < degree of the bottom Step 2: Do Partial Fraction Expansion (PFE) Case 1.
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