Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain representation.
It can be considered a discrete-time counterpart of the Laplace transform. This similarity is explored in the theory of time-scale calculus.
While the continuous-time Fourier transform is evaluated on the s-domain's vertical axis, the discrete-time Fourier transform is evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle.
In signal processing, one of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical approximation. Such methods tend not to be accurate except in the vicinity of the complex unity, i.e. at low frequencies.
History
The foundational concept now recognized as the Z-transform, which is a cornerstone in the analysis and design of digital control systems, was not entirely novel when it emerged in the mid-20th century. Its embryonic principles can be traced back to the work of the French mathematician Pierre-Simon Laplace, who is better known for the Laplace transform, a closely related mathematical technique. However, the explicit formulation and application of what we now understand as the Z-transform were significantly advanced in 1947 by Witold Hurewicz and colleagues. Their work was motivated by the challenges presented by sampled-data control systems, which were becoming increasingly relevant in the context of radar technology during that period. The Z-transform provided a systematic and effective method for solving linear difference equations with constant coefficients, which are ubiquitous in the analysis of discrete-time signals and systems.The method was further refined and gained its official nomenclature, "the Z-transform", in 1952, thanks to the efforts of John R. Ragazzini and Lotfi A. Zadeh, who were part of the sampled-data control group at Columbia University. Their work not only solidified the mathematical framework of the Z-transform but also expanded its application scope, particularly in the field of electrical engineering and control systems.
A notable extension, known as the modified or advanced Z-transform, was later introduced by Eliahu I. Jury. Jury's work extended the applicability and robustness of the Z-transform, especially in handling initial conditions and providing a more comprehensive framework for the analysis of digital control systems. This advanced formulation has played a pivotal role in the design and stability analysis of discrete-time control systems, contributing significantly to the field of digital signal processing.
Interestingly, the conceptual underpinnings of the Z-transform intersect with a broader mathematical concept known as the method of generating functions, a powerful tool in combinatorics and probability theory. This connection was hinted at as early as 1730 by Abraham de Moivre, a pioneering figure in the development of probability theory. De Moivre utilized generating functions to solve problems in probability, laying the groundwork for what would eventually evolve into the Z-transform. From a mathematical perspective, the Z-transform can be viewed as a specific instance of a Laurent series, where the sequence of numbers under investigation is interpreted as the coefficients in the expansion of an analytic function. This perspective not only highlights the deep mathematical roots of the Z-transform but also illustrates its versatility and broad applicability across different branches of mathematics and engineering.
Definition
The Z-transform 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 is the formal power series defined as:where is an integer and is, in general, a complex number. In polar form, may be written as:
where is the magnitude of, is the imaginary unit, and is the complex argument in radians.
Unilateral Z-transform
Alternatively, in cases where is defined only for, the single-sided or unilateral Z-transform is defined as:In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system.
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. The properties of Z-transforms 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. In the case where the ROC is causal, this means the path must encircle all of the poles of.
A special case of this contour integral occurs when is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around the unit circle:
The Z-transform with a finite range of and a finite number of uniformly spaced values can be computed efficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform —not to be confused with the discrete Fourier transform —is a special case of such a Z-transform obtained by restricting to lie on the unit circle.
The following three methods are often used for the evaluation of the inverse -transform,
Direct evaluation by contour integration
This method involves applying the Cauchy Residue Theorem to evaluate the inverse Z-transform. By integrating around a closed contour in the complex plane, the residues at the poles of the Z-transform function inside the ROC are summed. This technique is particularly useful when working with functions expressed in terms of complex variables.Expansion into a series of terms in the variables ''z'' and ''z''−1
In this method, the Z-transform is expanded into a power series. This approach is useful when the Z-transform function is rational, allowing for the approximation of the inverse by expanding into a series and determining the signal coefficients term by term.Partial-fraction expansion and table lookup
This technique decomposes the Z-transform into a sum of simpler fractions, each corresponding to known Z-transform pairs. The inverse Z-transform is then determined by looking up each term in a standard table of Z-transform pairs. This method is widely used for its efficiency and simplicity, especially when the original function can be easily broken down into recognizable components.Example
A) Determine the inverse Z-transform of the following by series expansion method,Solution:
Case 1:
ROC:
Since the ROC is the exterior of a circle, is causal.
thus,
.
Note that in each step of long division process we eliminate lowest power term of.
Case 2:
ROC:
Since the ROC is the interior of a circle, is anticausal.
By performing long division we get
Note that in each step of long division process we eliminate lowest power term of.
Note:
- When the signal is causal, we get positive powers of and when the signal is anticausal, we get negative powers of.
- indicates term at and indicates term at.
Eliminating negative powers if and dividing by,
By partial fraction expansion,
Case 1:
ROC:
Both the terms are causal, hence is causal.
Case 2:
ROC:
Both the terms are anticausal, hence is anticausal.
Case 3:
ROC:
One of the terms is causal and other is anticausal, hence is both sided.
Region of convergence
The region of convergence is the set of points in the complex plane for which the Z-transform summation absolutely converges:Example 1 (no ROC)
Let Expanding on the interval it becomesLooking at the sum
Therefore, there are no values of that satisfy this condition.
[|Example 2] (causal ROC)
Let . Expanding on the interval it becomesLooking at the sum
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".
Example 3 (anticausal ROC)
Let . Expanding on the interval it becomesLooking at the sum
and 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 and 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 a ROC that includes while the anticausal system in example 3 yields an ROC that includes
Image:Region of convergence 0.5 0.75 mixed-causal.svg|thumb|250px|ROC shown as a blue ring 0.5 < < 0.75
In systems with multiple poles it is possible to have a 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 0.5 < < 0.75, 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 then the system is stable. In the above systems the causal system is stable because > 0.5 contains the unit circle.
Let us assume we are provided a Z-transform of a system without a ROC. We can determine a unique provided we desire the following:
- Stability
- Causality
The unique can then be found.