Best solutions to - "Determine the z-transform for each of the following..."

pozicijombx

Answered question

2022-01-22

Determine the z-transform for each of the following sequences and indicate the corresponding region of convergence for each of them.

x [n] = {(- \frac{1}{3})}^{n} u [- n - 2]

portafilses

Beginner2022-01-23Added 13 answers

We have

x [n] = {(- \frac{1}{3})}^{n} u [- n - 2]

Apply Z-transform on given sequence, we get

X (z) = \sum_{n = - \infty}^{\infty} {(- \frac{1}{3})}^{n} u [- n - 2] z^{- n}

= \sum_{- \infty}^{- 2} {(- \frac{1}{3})}^{n} z^{- n}

= \sum_{n = 2}^{\infty} {(- \frac{1}{3})}^{- n} z^{n}

put

n_{1} = n - 2

X (z) = \sum_{n_{1} = 0}^{\infty} {(- \frac{1}{3})}^{- n - 2} z^{n_{1} + 2}

= 9 z^{2} \sum_{n_{1} = 0}^{\infty} {(- \frac{1}{3})}^{- n_{1}} z^{n_{1}}

= 9 z^{2} \sum_{n_{1} = 0}^{\infty} {({(- \frac{1}{3})}^{- 1} z)}^{n_{1}}

= 9 z^{2} (\frac{1}{1 - {(- \frac{1}{3})}^{- 1} z})

= \frac{3 z}{1 + \frac{1}{3} z^{- 1}}

Hence, the sequence converges if

| z | < \frac{1}{3}

so that

R O C = | z | < \frac{1}{3}

. ROC is not including the unit

| z | = 1

circle, so that it doesn't have Fourier transform.

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