Module 2 : Modeling Discrete Time Systems by Pulse Transfer Function

Lecture 1 : Motivation for using Z-transform

Complex convolution:

$\displaystyle Z[f_{1}(t)f_{2}(t)]=\frac{1}{2\pi j} \oint_{\Gamma} \frac {F_{1}(\xi) F_{2}(z\xi^{-1})}{\xi} d\xi$


$ \Gamma :$ circle / closed path in z-plane which lie in the region $ \sigma < \vert\xi \vert < \dfrac {\vert z\vert}{\sigma_{2}}$
$ \sigma_{1}$: radius of convergence of $ F_{1}(\xi)$
$ \sigma_{1}$: radius of convergence of $ F_{2}(\xi)$

2.5 Limitation of Z-transform method

  1. Ideal sampler assumption
    $ \Rightarrow$ z-transform represents the function only at sampling instants.


  2. Non uniqueness of z-transform.


  3. Accuracy depends on the magnitude of the sampling frequency $ w_{s}$ relative to the highest frequency component contained in the function f(t).


  4. A good approximation of f(t) can only be interpolated from f(kT), the inverse z-transform of F(z), by connecting f(kT) with a smooth curve.

2.6 Application of Z-transform in solving Difference Equation

One of the most important applications of Z-transform is in the solution of linear difference equations. Let us consider that a discrete time system is described by the following difference equation.

$\displaystyle y(k+2)+0.5y(k+1)+0.06y(k)=-(0.5)^{k+1}$