8-13 through 8-15 Use z-transform pair 3 in Table 8-1 to establish z-transform p
ID: 2999945 • Letter: 8
Question
8-13 through 8-15
Use z-transform pair 3 in Table 8-1 to establish z-transform pairs 4 and 5. (Hint: First write and differentiate both sides with respect to alpha) Use z-transform pair 3 in Table 8-1 to establish z-transform pairs 6 and 7. (Hint: Again first write Let alpha=jb and equate real and imaginary parts.) Use z-transform pair 3 in Table 8-1 to establish z-transform pairs 8 and 9. (Hint: The procedure is the same as in Problem 8-14. What should you let alpha be now?) Show that the z-transform of a*x(nT) is X(z/a). Use this result to show that entry 3 in Table B-1 follows from entry 2.Explanation / Answer
Multiple questions in a single post is not allowed, Please split up the questions I'm solving the first one , post the others separately 8-13) (summation) e^(-anT)z^-n = 1/{1 - e^-aT*z^-1} differentiating both sides w.r.t a => (summation) -nT*e^(-anT)z^-n = -{Te^(-aT)*z^-1}/{1 - e^-aT*z^-1} ----------- (1) put a = 0 in (1) => (summation) -nT*z^-n = -(T*z^-1)/{1-z^-1} => (summation) nT*z^-n = (T*z^-1)/{1-z^-1} => Z{nT} = (T*z^-1)/{1 - z^-1} => Z{t} = (T*z^-1)/{1 - z^-1} Hence 4 is proved Now,from (1) (summation) -nT*e^(-anT)z^-n = -{Te^(-aT)*z^-1}/{1 - e^-aT*z^-1} => (summation) nT*e^(-anT)z^-n = {Te^(-aT)*z^-1}/{1 - e^-aT*z^-1} => Z{nTe^(-anT)} = {Te^(-aT)*z^-1}/{1 - e^-aT*z^-1} => Z{te^(-at)} = {Te^(-aT)*z^-1}/{1 - e^-aT*z^-1} Hence 5 is proved
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