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The Z-Transform

The z-transform is an extension of the DTFT (The Discrete-Time Fourier Transform) to overcome two shortcomings of the DTFT approach: First, there are many important signals for which the DTFT does not exist. Take the case of the step u[n]. Second, the transient response of a system due to initial conditions or due to changing inputs cannot be computed using the DTFT approach.

In consequence, the bilateral version of the z-transform provides another domain in which a larger class of sequences and systems can be analyzed. Meanwhile, the unilateral version of the z-transform can be used to obtain the response of systems with initial conditions or changing inputs.  

THE BILATERAL Z-TRANSFORM

The z-transform of an arbitrary sequence x[n] is given by:

where z is a complex variable called the complex frequency:

The set  of values for which X[z] exists is called the region of convergence (ROC) and is given by:

For some non-negative numbers Rx- and Rx+. Since the ROC defined in terms of the module of z, the shape of the ROC is an open ring, as show in Figure 1:

Figure 1. A general region of convergence.

Another way of defining the ROC is:

null

When:

Hence, the DTFT X[e] may be viewed as a special case of the z-transform X[z].

The inverse z-transform of a complex function X[z] is given by:

where C is a counterclockwise contour encircling the origin and lying in the ROC.  

Example 1. 

Let x1[n] a positive-time sequence:

Then:

Note: in example 1:

That is, has a zero at the origin (z=0) and a pole in z=a.

Summarizing:

Example 1 is s special case of a right-side sequences, defined as a sequence x[n]  that is zero for some n<n0. The ROC of a right-side sequence is always outside of a circle of radius Rx-. In the case of example 1, Rx-=a. If n0 0, then the right-side sequence is also called a casual sequence. Note that if a=1, example 1 is the z-transform of the unit step. That is to say:

Example 2

Let x2[n] a negative-time sequence:

Then:

Example 2 is s special case of a left-side sequences, defined as a sequence x[n]  that is zero for some n>n0. The ROC of a left-side sequence is always inside of a circle of radius Rx+. In the case of example 2, Rx+=b. If n0<=0, then the right-side sequence is also called an anticasual sequence.

Example 3

Let x3[n] a two-side sequence:

Then:

Example 3 is s special case of a two-side sequences. The ROC of a two-side sequence is always an open ring Rx+ <IzI<Rx+ if it exist.

Another considerations about ROC are as follows:

  • The sequences that are zero for n<n1 and n>n2 are called finite-duration sequences. The ROC of such sequences is the entire z-plane. If n1<0, then z=+∞ is not in the ROC. If n2>0, then z=0 is not in the ROC.
  • The ROC cannot include a pole since X(z) converges uniformly in there.
  • There is at least one pole on the boundary of a ROC of a rational X(z).
  • The ROC is one contiguous region; that is, the ROC does not come in pieces.

In digital signal processing, signals are assumed to be casual since almost every digital data is acquired in real time. Therefore, the only ROC of interest is those of the same type of example 1.

Previous: The Discrete-Time Fourier TransformThe Frequency Response of an LTI system

Source:

  • Digital Signal Processing Using Matlab, 3erd ed
  • Fundamentos_de_Señales_y_Sistemas_usando la Web y Matlab
  • Oppenheim – Señales y Sistemas
  • Análisis de Sistemas Lineales Asistido con Scilab – Un Enfoque desde la Ingeniería Eléctrica.

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