Module 2 : Modeling Discrete Time Systems by Pulse Transfer Function

Lecture 1 : Motivation for using Z-transform

Exponential function is defined as:

$\displaystyle x(t)$
 $\displaystyle = e^{-at}, \;\;$   for$\displaystyle \;\; t \ge 0$    
  $\displaystyle =0, \;\;$   for$\displaystyle \;\; t < 0$    

We have $ x(kT)=e^{-akT}$ for $ k= 0,1,2 \cdots$. Thus,
$\displaystyle X(z)$
 $\displaystyle =$ $\displaystyle \frac{1}{1-e^{-aT}z^{-1}}$  
  $\displaystyle =$ $\displaystyle \frac {z}{z-e^{-aT}}$  


Similarly Z-transforms can be computed for sinusoidal and other compound functions. One should refer the Z-transform table provided in the appendix.

2.2 Properties of Z-transform

  1. Multiplication by a constant: $ Z[ax(t)]= aX(z)$, where $ X(z)= Z[x(t)]$.
  2. Linearity: If $ x(k)=\alpha f(k)\pm \beta g(k)$, then $ X(z)=\alpha F(z)\pm \beta G(z)$.
  3. Multiplication by $ a^k$: $ Z[a^kx(k)]=X(a^{-1}z)$

  5. Realshifting: $ Z[x(t-nT)]=z^{-n}X(z)$ and $ \displaystyle z[x(t+nT)]=z^{n} \left [X(z)-\sum_{k=0}^{n-1}x(kT)z^{-k} \right ]$

  6. Complex shifting:     $ Z[e^{\pm at}x(t)]=X(ze^{\mp aT})$
  7. Initial value theorem:

    $\displaystyle x(0)=\lim_{z\rightarrow\infty}X(z) $

  8. Final value theorem:

    $\displaystyle \lim_{k\rightarrow\infty}x(k)= \lim_{z\rightarrow 1}[(1-z^{-1})X(z)] $