Use z-transform pair 3 in Table 8-1 to establish z- transform pairs 4 and 5 ( Hi

Question

Use z-transform pair 3 in Table 8-1 to establish z- transform pairs 4 and 5 ( Hint : First write * and differentiate both sides respect to /alpha

* This is pair 3-1)

Pair 4 is t -> (T*z^-1)/((1-z^-1)^2)

Pair 5 os t*e^(-/alpha*t) -> (T*e^(-/alpha*T)z^-1)/(1-e^(-/alpha*T)z^-1)^2

Explanation / Answer

pair 3 states : (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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