Rational Z-Transform and Transfer Function

Sources

1. Rational Z-Transform and Transfer Function (Lecture Slides)
2. Lecture Notes

Example 1 - Poles and Zeros Part 1

Problem 1

Determine the pole-zero plot for the signal

$$x[n]=a^{n}u[n]$$

$$\begin{array}{rl}X(z)& =\frac{1}{1-a{z}^{-1}}\cdot \frac{z}{z}\end{array}$$ $$\boxed{\begin{array}{rl}X(z)& =\frac{z}{z-a}\\ \text{zero: }{z}_1& =0\\ \text{pole: }z-a& =0\\ z& =a\text{, }{P}_1=a\end{array}}$$

Example 2 - Poles and Zeros Part 2

Problem 2

Determine the pole-zero plot for the signal

$$x[n]=\{\begin{array}{ll}{a}^n& 0\le n\le M-1\\ 0& \text{elsewhere}\end{array}$$

$$\begin{array}{rl}X(z)& =\sum _{n=0}^{M-1}{a}^n{z}^{-n}\\ & =\sum _{n=1}^N(a{z}^{-1}{)}^n\\ & =\frac{1-(a{z}^{-1}{)}^M}{1-a{z}^{-1}}\end{array}$$

WARNING

I get the feeling that the reference is incomplete for this one, so I’m guessing that this is not the final answer yet

Tip 2. Summation of $A^n$

$$\sum _{n=1}^N{A}^n=\frac{1-A^{N+1}}{1-A}\text{, }|A|<1$$

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TO BE CONTINUED