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Practice Problem on Z-transform computation
Compute the z-transform (including the ROC) of the following DT signal:
$ x[n]=3^n u[-n+3] \ $
Then use your answer to obtain the Fourier transform of the signal. (Write enough intermediate steps to fully justify your answer.)
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Answer 1
x[n] = 3nu[-n + 3]
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
$ X(z) = \sum_{n=-\infty}^{+\infty} 3^n u[-n+3] z^{-n} $
Let k = -n+3, n = -k+3
$ X(z) = \sum_{k=0}^{+\infty} (\frac{3}{z})^{-k+3} $
$ X(z) = (\frac{3}{z})^{3} \sum_{k=0}^{+\infty} (\frac{z}{3})^{k} $
$ X(z) = (\frac{27}{z^3}) \sum_{k=0}^{+\infty} (\frac{z}{3})^{k} $
By geometric series formula,
$ X(z) = (\frac{27}{z^3}) (\frac{1}{1-(\frac{z}{3})}) $ ,for |z| < 3
X(z) = diverges, else
So,
$ X(z) = (\frac{3}{3-z}) $ with ROC, |z| < 3
Answer 2
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} = \sum_{n=-\infty}^{+\infty} 3^n u[-n+3] z^{-n} $
Let k=-n+3, n=3-k, then
$ X(z) = \sum_{k=-\infty}^{+\infty} (3)^{n-k}u[k](z)^{-3+k} $
$ X(z) = (\frac{3}{z})^{3}\sum_{k=0}^{+\infty} (\frac{z}{3})^{k} $
$ X(z) = \left\{ \begin{array}{l l} (\frac{3}{z})^3 \frac{1}{1-\frac{z}{3}} &, if \quad |z| < 3\\ \text{diverges} &, \quad \text{otherwise} \end{array} \right. $
$ F(x[n]r^{-n}) = X(3^{e^{jw}}) = X(w) = (\frac{(\frac{3}{3e^{jw}})}{1-e^{jw}}) $
Answer 3
Write it here.
Answer 4
Write it here.