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Z Transform Basic Concepts

Introduction:

Z transform is a method to convert a discrete time domain signal to frequency domain. It is equivalent to the Laplace transform which converts the converts the continuous time domain signal to frequency domain. In digital signal processing, this mathematical tool is extremely useful as most of operations on digital signal are done in frequency domain. The two sided z-transform X(z) of a discrete signal x[n] is defined as

Z Transform
  • Where m is the position of the discrete time signal and z is the complex number.

  • Z Transform
    For eg:
  • The discrete time signal, x[n]={-3,2, 5,1,4}


  • Its z transform would be -3z^2 + 2z + 5 + z^-1 + 4z^-2

  • Now, as in the case of Laurent series it is necessary to define the region of convergence, in z transform also the region of convergence (ROC) is of great importance as it helps to know the whether the signal is causal or anti-causal. The three possible conditions of ROC are as follows:








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