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Practice Question, ECE438 Fall 2013, Prof. Boutin
On computing the inverse z-transform of a discrete-time signal.
Compute the inverse z-transform of
$ X(z) =\frac{1}{3-z}, \quad \text{ROC} \quad |z|>3 $.
(Write enough intermediate steps to fully justify your answer.)
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
$ X(z) =\frac{1}{\frac{3z}{z} - z} = \frac{-1}{z} \frac{1}{1-\frac{3}{z}} $
$ =\frac{-1}{z} \sum_{n=0}^{+\infty} (\frac{3}{z})^n = \sum_{n=0}^{+\infty} (-z^{-1}) (3z^{-1})^n $ NOTE: $ (3z^{-1})^n = (3^n) (z^{-n}) $ $ = \sum_{n=0}^{+\infty} -3^n z^{-n-1} = \sum_{n=-\infty}^{+\infty} -3^n u[n] z^{-n-1} $ NOTE: Let k=n+1
$ = \sum_{n=-\infty}^{+\infty} -3^{k-1} u[k-1] z^{-k} $ (compare with $ \sum_{n=-\infty}^{+\infty} x[n] z^{-k} $)
Therefore, $ x[n]= -3^{n-1} u[n-1] $
Answer 2
$ X(z) = \frac{1}{3-z} = -\frac{1}{z} \frac{1}{1-\frac{3}{z}} $
$ X(z) = \frac{1}{z} \sum_{n=0}^{+\infty} (\frac{3}{z})^{n} = \sum_{n=-\infty}^{+\infty} -u[n]3^{n}z^{-n-1} $
Let k = n+1, so n = k-1
$ X(z) = \sum_{n=-\infty}^{+\infty} -u[k-1]3^{k-1}z^{-k} $
By comparison with the z-transform equation
$ x[n] = -u[n-1] 3^{n-1} $
Answer 3
$ X(z) = \frac{1}{3-z} = -\frac{1}{z} \frac{1}{1-\frac{3}{z}} $
$ X(z) = \frac{1}{z} \sum_{n=0}^{+\infty} (\frac{3}{z})^{n} = \sum_{n=0}^{+\infty} (-3)^{n}({z})^{-n-1} = \sum_{n=-\infty}^{+\infty} -u[n]3^{n}z^{-(n+1)} $
Let k = n+1 then n = k-1
$ X(z) = \sum_{n=-\infty}^{+\infty} -u[k-1]3^{k-1}z^{-k} $
By comparison,
$ x[n] = -u[n-1] 3^{n-1} $
Answer 4
$ X(Z) = \frac{1}{3-Z} $
$ X(Z) = \frac{-1}{Z} \frac{1}{1-\frac{3}{Z}} $
Since,$ |3|<Z $
$ \frac{1}{1-\frac{3}{Z}} = \sum_{n=0}^{+\infty} (\frac{3}{Z})^{n} $
Thus,
$ X(Z) = \frac{-1}{Z} \sum_{n=0}^{+\infty} (\frac{3}{Z})^{n} $
$ X(Z) = -Z^{-1} \sum_{n=0}^{+\infty} (\frac{3}{Z})^{n} $
$ X(Z) = -\sum_{n=-\infty}^{+\infty} u[n] 3^{n}Z^{-n-1} $
Let k=n+1, then -k=-n-1,n=k-1
$ X(Z) = -\sum_{n=-\infty}^{+\infty} u[k-1] 3^{k-1}Z^{-k} $
Therefore, $ x(n) = -u[n-1] 3^{n-1} $
Answer 5
By Yixiang Liu
$ X(z) = \frac{1}{3-Z} $
$ X(z) = \frac{1}{-z}* \frac{1} {\frac{3}{-Z}+1} $
$ X(z) = \frac{1}{-z}* \frac{1} {1-\frac{3}{Z}} $
$ X(z) = \frac{1}{-Z} \sum_{n=0}^{+\infty} (\frac{3}{3})^n $
$ X(z) = \sum_{n=0}^{+\infty} 3^{n} Z^{-n-1} $
$ X(z) = \sum_{-\infty}^{+\infty} u[n] 3^{n} Z^{-n-1} $
Let -k = -n-1, so n = k-1
$ X(z) = \sum_{-\infty}^{+\infty} u[k-1] 3^{k-1} Z^{-k} $
By comparison with the x-transform formula
$ x[n] = 3^{n-1} u[n-1] $
Answer 6
$ X(z) = \frac{1}{3-z} $
we have |z| > 3,
$ X(z) = \frac{1}{3} \frac{1}{1-\frac{z}{3}} $
$ x[n] = (\frac{1}{3})^{-n+1} u[-n] $
Answer 7
Answer 6
$ X(z) = \frac{1}{3-z} $
$ X(z) = \frac{1}{\frac{3}{z} z -z} $
$ X(z) = \frac{1}{z} \frac{1}{\frac{3}{z} - 1} $
$ X(z) = \frac{-1}{z} \frac{1}{1-\frac{3}{z}} $
$ X(z) = \frac{-1}{z} \frac{1-\left( \frac{3}{z} \right) ^{n}}{1-\frac{3}{z}}, \quad \text{As n goes to} \quad \infty \text{ since ROC} \quad |z|>3 $
$ X(z) = \frac{-1}{z} \sum_{n=0}^{\infty} \left( \frac{3}{z} \right)^{n} $
$ X(z) = \frac{-1}{z} \sum_{n=0}^{\infty} 3^n z^{-n} $
$ X(z) = -\sum_{n=0}^{\infty} 3^n z^{-\left(n+1 \right)}, \quad \text{Replace n+1 with k} $
$ X(z) = -\sum_{k=1}^{\infty} 3^{k-1} z^{-k} $
$ X(z) = -\sum_{-\infty}^{\infty} 3^{k-1} u[k-1] z^{-k}, \quad \text{By comparing with DTFT equation we get} $
$ x[n] = -3^{n-1} u[n-1] $
