Prereqisting Discussion About Z Transform

For analysis of continuous time LTI system Laplace transform is used. And for analysis of discrete time LTI system z transform is used. Z transform is mathematical tool used for conversion of time domain into frequency domain (z domain) and is a function of the complex valued variable Z. The z transform of a discrete time signal x(n) denoted by X(z) and given as

Z transform is an infinite power series because summation index varies from -∞ to ∞. But it is useful for values of z for which sum is finite. The values of z for which f (z) is finite and lie within the region called as ―region of convergence (ROC).

ADVANTAGES OF Z TRANSFORM

1. The DFT can be determined by evaluating z transform.

2. Z transform is widely used for analysis and synthesis of digital filter.

3. Z transform is used for linear filtering. z transform is also used for finding Linear convolution, cross-correlation and auto-correlations of sequences.

4. In z transform user can characterize LTI system (stable/unstable, causal/anti- causal) and its response to various signals by placements of pole and zero plot.

ADVANTAGES OF ROC(REGION OF CONVERGENCE)

1. ROC is going to decide whether system is stable or unstable.

2. ROC decides the type of sequences causal or anti-causal.

3. ROC also decides finite or infinite duration sequences. 

Z Transform Plot

Fig show the plot of z transforms. The z transform has real and imaginary parts. Thus a plot of imaginary part versus real part is called complex z-plane. The radius of circle is 1 called as unit circle.