Direct Z Transform

Sources

1. DirectZTransform (Lecture Slides)
2. Lecture Notes

Example 1

Problem 1.a.

$$x_1[n]=\{\underset{\uparrow }{1},2,5,7,0,1\}$$

$$\begin{array}{rl}{x}_1[n]& =\delta [n]+2\delta [n-1]+5\delta [n-2]+7\delta [n-3]+\delta [n-5]\\ & \\ X_1(z)& =\sum _{n=0}^{5}x(n)z^{-n}\\ & =x[0]z^{-0}+x[1]z^{-1}+x[2]z^{-2}+x[3]z^{-3}+x[4]z^{-4}+x[5]z^{-5}\end{array}$$ $$\boxed{X_1(z)=1+2z^{-1}+5z^{-2}+7z^{-3}+z^{-5}}$$

Region of Convergence (ROC): entire z-plane

Problem 1.b.

$$x_2[n]=\{1,2,\underset{\uparrow }{5},7,0,1\}$$

$$\boxed{X_2(z)=z^2+2z+5+7z^{-1}+z^{-3}}$$

Problem 1.c.

$$x_3[n]=\{\underset{\uparrow }{0},0,1,2,5,7,0,1\}$$

$$\boxed{X_3(z)=z^{-2}+2z^{-3}+5z^{-4}+7z^{-5}+z^{-7}}$$

ROC: entire z-plane, except $z=0$

Problem 1.d.

$$\begin{array}{r}{x}_4[n]=\delta [n]\end{array}$$

$$\begin{array}{rl}{X}_4(z)& =\sum _{n=-\infty }^{\infty }\delta [n]z^{-n}\\ & =\delta [0]z^{-0}\\ & =1(1)\end{array}$$ $$\boxed{X_4(z)=1}$$

ROC: entire z-plane

Problem 1.e.

$$\begin{array}{rlr}{x}_5[n]=\delta [n-k],& & k>0\end{array}$$

$$\begin{array}{rl}{X}_5(z)& =\sum _{-\infty }^{\infty }\delta [n-k]z^{-n}\\ & =\delta [0]z^{-k}\end{array}$$ $$\boxed{X_5(z)=z^{-k},k>0}$$

Problem 1.f.

$$\begin{array}{rlr}{x}_6[n]=\delta [n+k],& & k>0\end{array}$$

$$\boxed{X_6(z)=z^k,k>0}$$

Example 2

Problem 2.a.

$$\begin{array}{r}x[n]={\alpha }^{n}u[n]=\{\begin{array}{ll}{a}^n& n\ge 0\\ 0& n<0\end{array}\end{array}$$

$$\begin{array}{rlr}X(z)& =\sum _{n=0}^{\infty }{\alpha }^n{z}^{-n}& \\ & =\sum _{n=0}^{\infty }(\alpha z^{-1}{)}^n& \\ & =\frac{1}{1-\alpha z^{-1}}& |\alpha z^{-1}|<1\\ & =\frac{1}{1-\alpha z^{-1}}& |z|>|\alpha |\end{array}$$

Tip 2.a.1. Summation of Exponentials $(0\le n<\infty )$

$$\begin{array}{rlr}\sum _{n=0}^{\infty }{A}^n& =1+A+A^2+A^3+\dots & \\ & =\frac{1}{1-A}& |A|<1\end{array}$$

Tip 2.a.2. $|z|>|\alpha |$

$$\begin{array}{rlr}|\alpha z^{-1}|& <1& \text{negative exponent means inverse/reciprocate}\\ |\alpha \left(\frac{1}{z}\right)|& <1& \text{multiply both sides by }z\\ |\alpha |& <|z|& \\ |z|& >|a|& \end{array}$$

Transform pairs:

$$\boxed{a^{n}u[n]\stackrel{\text{z}}{\leftrightarrow }\frac{1}{1-a{z}^{-1}}\text{, }|z|>|\alpha |}$$

let $\alpha =1$:

$$\boxed{u[n]\stackrel{\text{z}}{\leftrightarrow }\frac{1}{1-z^{-1}}\text{, }|z|>1}$$

Problem 2.b.

$$\begin{array}{r}x[n]=-{\alpha }^{n}u[-n-1]=\{\begin{array}{ll}0& n\ge 0\\ -{\alpha }^n& n\le -1\end{array}\end{array}$$

Tip 2.b.1. What to do with $u[-n-1]$?

$$\begin{array}{rlr}u[n]& \to 0\le n<\infty & \text{fold and shift}\\ u[-n-1]& \to -\infty <n\le -1& \end{array}$$

$$\begin{array}{rlr}X(z)& =\sum _{n=-\infty }^{-1}-{\alpha }^n{z}^{-n}& \\ & =-\sum _{n=-\infty }^{-1}{\alpha }^n{z}^{-n}& \\ & =-\sum _{n=1}^{\infty }{\alpha }^{-n}{z}^n& \\ & =-\frac{{\alpha }^{-1}z}{1-{\alpha }^{-1}z}& |a^{-1}z|<1\\ & =-\frac{1}{{\alpha }^{-1}z-1}& |z|<|\alpha |\\ & =\frac{1}{1-{\alpha }^{-1}z}& |z|<|\alpha |\end{array}$$

Tip 2.b.2. Summation of Exponentials $(1\le n<\infty )$

$$\begin{array}{rlr}\sum _{n=1}^{\infty }{A}^n& =A+A^2+A^3+\dots & \\ & & \\ & =A(1+A+A^2+\dots )& \\ & =A(\frac{1}{1-A})& |A|<1\end{array}$$

Transform pairs

$$\boxed{-a^{n}u[-n-1]\stackrel{\text{z}}{\leftrightarrow }\frac{1}{1-a{z}^{-1}}\text{, }|z|<|\alpha |}$$

Problem 2.c.

$$x[n]={\alpha }^{n}u[n]+b^{n}u[-n-1]$$