Definition of Z Transform
Z Transform is a very useful mathematical tool, mainly used to solve difference equations. It converts the difference equations into algebraic form, which are then easy to be solved and then the solution is inversely converted, using Inverse Z Transform, to the form it should initially be. It is the discrete equivalent of the Laplace Transform and a generalization of the Discrete Time Fourier Transform.
Z Transform of a discrete function or sequence x(n) is given by the equation:
Note that before the transform, the function had as independent variable the variable n (discrete time). After Z Transform, the independent variable of the function changed to z. In other words, Z Transform changes the domain of a function, from the domain of discrete time to the domain of z. This “z” is a dummy domain (variable) used only for the calculations and conclusions, it has no meaning for the real world. The result must be converted back to its original domain to have a meaningful result.
Z Transform is noted by a capital Z. For the resulting function we use the same letter of the initial function, but in capital. For example to note the Z Transform of the function x(n), we write it as follows:
Note again that before the transform we have a function of discrete time and after the transform we have a function of z.
Let see how we can calculate the Z Transform of the unit step function u(n):
For more examples about calculating Z Transform from the definition, have a look at the appropriate menu selection.
To solve a difference equation, we don’t have to transform it from the definition. We can use the transformation table to look up the answer. We do this for every factor of the difference equation, we solve the resulting algebraic equation and then, using the transformation tables again, we inversely transform the result to bring it back from the “z” world to the “real” discrete time world. For example:
For more examples about solving difference equations, have a look at the appropriate menu selection.
Where does Z Transform come from? Why this integral with this exponent is used?
All begin from the power series. Power series is the infinite summation of a product of a coefficient by a variable raised to a power. If this infinite summation leads to a number, then we say that it converges and in some cases it can lead to a certain function. For example the following summation is equal to the Euler number raised to the variable:
In a more compact notation we would write it as:
More generally we can have the summation of the product of a coefficient, which is a function of n, by a power of x:
In a more compact notation we would write it as:
We will make two changes to make it easier for calculations. The first change is to write C(n) as h(n), since this notation is common for discrete functions. The second change has to do with the x to the n power. This is not practical for further calculations, so we will convert it to something with e as its base.
In order to make the summation to converge it should be:
This assumption leads to an exponent that its value becomes smaller and smaller by the time, so the product of the function by this exponent becomes also smaller by the time making the integration to converge. Otherwise the integration becomes infinite and it is impossible to calculate it.
To make the notation simpler, we define:
Since x is generally a complex variable, then z is also a complex variable:
And we have
The exponent z raised to the power k, can also be imagined as a “flag”, indicating the position of the value of h(k), at the discrete time k, on the time line. So the factors:
have the meaning that the first factor is on the first place (i.e. first sample) on the time axis, the second on the forth place (i.e. forth sample) and the third on the eighth place (i.e. eighth sample) on the time axis.
is one discrete time (sampling time) delay.
The range of values of the complex z for which the Z Transform from the definition converges is called Region of Convergence. In exponential form z can be written as:
The range of values can be plotted on the Cartesian system, which in this case is called the “z plane”.
Let’s calculate the Region of Convergence of the following function:
In this example the integration converges only when:
This is the whole z plane except a disc of radius a, from the beginning of the axis.
The Region of Convergence for this example is:
Figure 1: Example of RoC
The Region of Convergence is the entire z plane except a disc of radius a.
Consider the following proper rational function:
The roots of the numerator are called zeros and the roots of the denominator are called poles. Obviously when z is equal to a zero, the value of the function becomes zero and when z is equal to a pole the value of the function becomes infinite. The values of z that are zeros or poles are important because they define the Region of Convergence. Zeros are noted by a “o” and poles are noted by a ”x”
For the above function plotting poles and zeros on the z plane would look like:
Figure 2: Example of Poles and Zeros