The Z-transform and advanced Z-transform were introduced (under the Z-transform name) by E. I. Jury in 1958 in Sampled-Data Control Systems (John Wiley & Sons). The idea contained within the Z-transform was previously known as the "generating function method".
The (unilateral) Z-transform is to discrete-time signals what the one-sided Laplace transform is to continuous-time signals.
Definition
The Z-transform, like many other integral transforms, can be defined as either a one-sided or two-sided transform.
Bilateral Z-transform
The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z) defined as
where n is an integer and z is, in general, a complex number:
An important example of the unilateral Z-transform is the probability-generating function, where the component is the probability that a discrete random variable takes the value , and the function is usually written as , in terms of . The properties of Z-transforms (below) have useful interpretations in the context of probability theory.
Inverse Z-transform
The inverse Z-transform is
where is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). The contour or path, , must encircle all of the poles of .
The Z-transform with a finite range of n and a finite number of uniformly-spaced z values can be computed efficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform (DTFT) (not to confuse with the discrete Fourier transform (DFT)) is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.
Region of convergence
The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges.
Example 1 (No ROC)
Let . Expanding on the interval it becomes
Looking at the sum
There are no such values of that satisfy this condition.
The last equality arises from the infinite geometric series and the equality only holds if which can be rewritten in terms of as . Thus, the ROC is . In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".
Using the infinite geometric series, again, the equality only holds if which can be rewritten in terms of as .
Thus, the ROC is . In this case the ROC is a disc centered at the origin and of radius 0.5.
What differentiates this example from the previous example is only the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
Examples conclusion
Examples 2 & 3 clearly show that the Z-transform of is unique when and only when specifying the ROC. Creating the pole-zero plot for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will never contain poles.
In example 2, the causal system yields an ROC that includes while the anticausal system in example 3 yields an ROC that includes .
ROC shown as a blue ring
In systems with multiple poles it is possible to have an ROC that includes neither nor . The ROC creates a circular band. For example, has poles at 0.5 and 0.75. The ROC will be , which includes neither the origin nor infinity. Such a system is called a mixed-causality system as it contains a causal term and an anticausal term .
The stability of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., ) then the system is stable. In the above systems the causal system (Example 2) is stable because contains the unit circle.
If you are provided a Z-transform of a system without an ROC (i.e., an ambiguous ) you can determine a unique provided you desire the following:
Stability
Causality
If you need stability then the ROC must contain the unit circle.
If you need a causal system then the ROC must contain infinity.
If you need an anticausal system then the ROC must contain the origin.
where is the continuous-time function being sampled, the nth sample, is the sampling period, and with the substitution: .
Likewise the unilateral Z-transform is simply the one-sided Laplace transform of the ideal sampled function. Both assume that the sampled function is zero for all negative time indices.
The Bilinear transform is a useful approximation for converting continuous time filters (represented in Laplace space) into discrete time filters (represented in z space), and vice versa. To do this, you can use the following substitutions in or :
from Laplace to z (Tustin transformation);
from z to Laplace.
Relationship to Fourier
The Z-transform is a generalization of the discrete-time Fourier transform (DTFT). The DTFT can be found by evaluating the Z-transform at or, in other words, evaluated on the unit circle. In order to determine the frequency response of the system the Z-transform must be evaluated on the unit circle, meaning that the system's region of convergence must contain the unit circle. Otherwise, the DTFT of the system does not exist.
Linear constant-coefficient difference equation
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the autoregressive moving-average equation.
Both sides of the above equation can be divided by , if it is not zero, normalizing and the LCCD equation can be written
This form of the LCCD equation is favorable to make it more explicit that the "current" output is a function of past outputs , current input , and previous inputs .
Transfer function
Taking the Z-transform of the equation (using linearity and time-shifting laws) yields
Where is the zero and is the pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the pole-zero plot.
In simple words, zeros are the solutions to the equation obtained by setting the numerator equal to zero, while poles are the solutions to the equation obtained by setting the denominator equal to zero.
In addition, there may also exist zeros and poles at and . If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.
By factoring the denominator, partial fraction decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the impulse response and the linear constant coefficient difference equation of the system.
Output response
If such a system is driven by a signal then the output is . By performing partial fraction decomposition on and then taking the inverse Z-transform the output can be found. In practice, it is often useful to fractionally decompose before multiplying that quantity by to generate a form of which has terms with easily computable inverse Z-transforms.