Z Transforms

Z transforms of unit step function is

• $\frac{\text{z}}{\text{z-a}}$

• $\frac{\text{z}}{\text{z-1}}$

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Which of the following is damping rule?

• $Z\left\{n\left(f\left(n\right)\right)\right\}=-z\frac{\text{d}}{\text{d}}F\left(\text{}z\right)$

• $Z\left\{f\left(n-k\right)\right\}=z^{-n}F\left(z\right)$

• $Z\left\{f\left(n+1\right)\right\}=zF\left(z\right)-zf\left(0\right)$

• $Z\left\{a^nf\left(n\right)\right\}=F\left(\frac{z}{a}\right)$

 $If\ Z\left\{f\left(n\right)\right\}=F\left(z\right)\ then\ f\left(0\right)=\lim_{z\rightarrow\infty}F\left(z\right)\ is\ called$  

•  $First\ shifting\ theorem$  

•  $Second\ shifting\ theorem$  

•  $Initial\ value\ theorem$  

•  $Final\ value\ theorem$  

The convolution theorem of Z- Transform is

• $Z^{-1}\left\{F\left(z\right)G\left(z\right)\right\}=f\left(n\right)\cdot g\left(n\right)$ $Z^{-1}\left\{F\left(z\right)G\left(z\right)\right\}=f\left(n\right)\cdot g\left(n\right)$

• $Z^{-1}\left\{F\left(z\right)+G\left(z\right)\right\}=f\left(n\right)+g\left(n\right)$

• $Z^{-1}\left\{F\left(z\right)-G\left(z\right)\right\}=f\left(n\right)-g\left(n\right)$

• $Z^{-1}\left\{\left(\frac{F\left(z\right)}{G\left(z\right)}\right)\right\}=\frac{f\left(n\right)}{g\left(n\right)}$

What are the types used to solve inverse Z Transform

 $Z\left[\frac{1}{n}\right]=$  

•  $\log\left(\frac{1}{z-1}\right)$  

•  $\log\left(\frac{z}{z-1}\right)$  

•  $\log\ \left(\frac{1}{z+1}\right)$  

•  $\log\left(z-1\right)$  

 $Z\left\{\delta\left(n-3\right)\right\}=$  

•  $\frac{1}{z}$   $\frac{1}{z}$ 

•  $\frac{1}{z^2}$  

•  $\frac{1}{z^3}$  

•  $\frac{1}{z^k}$  

 $Z\left\{\frac{1}{n!}\right\}=$  

•  $e^{-z}$  

•  $e^z$  

•  $e^{-\frac{1}{z}}$  

•  $e^{\frac{1}{z}}$  

 $Z\left[k\right]=$  

•  $\frac{1}{z-1}$  

•  $\frac{z}{z+1}$  

•  $k\frac{z}{z-1}$  

•  $\frac{z}{z-1}$  

 $Z\left\{nC_k\right\}=$  

•  $\left(1+z^{-1}\right)^n$  

•  $\left(1-z^{-1}\right)^n$  

•  $\left(1+z^{-1}\right)^k$  

•  $\left(1-z^{-1}\right)^k$