Determine the z transform of x1[n] = u[n] − u[n − 10].
Added by Jeffery B.
Step 1
To determine the z-transform of the signal \( x_1[n] = u[n] - u[n - 10] \), we will follow these steps: Show more…
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Determine the z-transform for the unit impulse $\left\{\delta_{k}\right\}=\{1,0,0,0, \ldots\}$ The z-transform of $\left\{\delta_{k}\right\}$ is given by: $$ Z\left\{\delta_{k}\right\}=F(z)=\sum_{k=0}^{\infty} \frac{\delta_{k}}{z^{k}}=\frac{1}{z^{0}}+\frac{0}{z^{1}}+\frac{0}{z^{2}}+\ldots=1 $$ i.e. $\quad \mathbf{Z}\left\{\boldsymbol{\delta}_{k}\right\}=\mathbf{1}$ valid for all values of $\mathrm{z}$ )
Problem 2. Determine the z-transform for the unit step sequence $\left\{u_{k}\right\}=\{1,1,1,1, \ldots\}=\{1\}$ The z-transform of $\left\{u_{k}\right\}$ is given by: $$ \begin{aligned} Z\left\{u_{k}\right\} &=F(z)=\sum_{k=0}^{\infty} \frac{u_{k}}{z^{k}}=\sum_{k=0}^{\infty} \frac{1}{z^{k}} \\ &=\frac{1}{z^{0}}+\frac{1}{z^{1}}+\frac{1}{z^{2}}+\frac{1}{z^{3}}+\frac{1}{z^{4}}+\ldots \end{aligned} $$ i.e. $\quad \mathbf{Z}\left\{u_{k}\right\}=1+\frac{1}{z}+\frac{1}{z^{2}}+\frac{1}{z^{3}}+\frac{1}{z^{4}}+\ldots$ Using the binomial theorem for $(1+x)^{n}$, the series expansion of $\frac{1}{1-x}$ may be determined: $$ \begin{aligned} \frac{1}{1-x}=&(1-x)^{-1} \\ =& 1+(-1)(-x)+\frac{(-1)(-2)}{2 !}(-x)^{2} \\ &+\frac{(-1)(-2)(-3)}{3 !}(-x)^{3}+\ldots \\ &=1+x+x^{2}+x^{3}+\ldots \text { valid for }|x|<1 \end{aligned} $$ Comparing equations (1) and (2) gives: $\mathrm{F}(\mathrm{z})=\frac{1}{1-\frac{1}{z}}$ provided $\left|\frac{1}{z}\right|<1$ $\frac{1}{1-\frac{1}{z}}=\frac{1}{\frac{z-1}{z}}=\frac{z}{z-1}$ hence, $z\left\{u_{k}\right\}=\frac{z}{z-1}$ provided $|z|>1$
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