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Determine the z transform & the ROC of the discrete time signal. $X[n] = \{ 2, 10, 1, 2, 5, 7, 2\}$

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 05

Year : MAY 2014

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We have $x(-3) = 2, x(-2) = 10, x(-1) =1, x(0) = 2, x(1) = 5, x(2) = 7, x(3) =2$

By using z-transform,

$X (z) = ∑\limits_{n=-∞}^∞ x [n] z^{-n} \\ ∴X (z)= ∑\limits_{n=-3}^3 \space x[n]z^{-n} \\ = x(-3)z^{-3} + x(-2) z^{-2} +x(-1)z^{-1} +x(0)z^0+x(1)z^1+ x(2)z^2+x(3)z^3 \\ =2z^{-3} + 10z^{-2}+1z^{-1}+2z^0+5z^1+ 7z^2+2z^3 \\ X(z) = 2z^0+ 5z^1+7z^2+2z^3 + \dfrac 1z+\dfrac {10}{z^2} + \dfrac 2{z^3 }$

Put $z=0$ we get $x (z) =∞$

Similarly, put $z=∞$ we get $x (z) =∞$

ROC is the entire z-plane except $z=0$ and $∞$