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 (the z-domain or z-plane) representation.[1][2]

It can be considered a discrete-time equivalent of the Laplace transform (the s-domain or s-plane).[3] 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 imaginary 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.

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.[4][5]

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.[6][7]

The development of the Z-transform did not halt with Ragazzini and Zadeh. 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.[8][9]

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 (Laurent) 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.[10]

Definition

The Z-transform can be defined as either a one-sided or two-sided transform. (Just like we have the one-sided Laplace transform and the two-sided Laplace transform.)[11]

Bilateral Z-transform

The bilateral or two-sided Z-transform of a discrete-time signal is the formal power series defined as:

 

 

 

 

(Eq.1)

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 (also referred to as angle or phase) in radians.

Unilateral Z-transform

Alternatively, in cases where is defined only for , the single-sided or unilateral Z-transform is defined as:

 

 

 

 

(Eq.2)

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 , and the function is usually written as in terms of . The properties of Z-transforms (listed in § Properties) have useful interpretations in the context of probability theory.

Inverse Z-transform

The inverse Z-transform is:

 

 

 

 

(Eq.3)

where is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2), 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:

 

 

 

 

(Eq.4)

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 (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting to lie on the unit circle.

Region of convergence

See also: Pole–zero_plot § Discrete-time systems

The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges (i.e. doesn't blow up in magnitude to infinity):

Example 1 (no ROC)

Let Expanding on the interval it becomes

Looking at the sum

Therefore, there are no values of that satisfy this condition.

Example 2 (causal ROC)

ROC (blue), |z| = .5 (dashed black circle), and the unit circle (dotted grey circle).

Let (where is the Heaviside step function). Expanding on the interval it becomes

Looking 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 (anti causal ROC)

ROC (blue), |z| = .5 (dashed black circle), and the unit circle (dotted grey circle).

Let (where is the Heaviside step function). Expanding on the interval it becomes

Looking 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 & 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 0.5 < |z| < 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 < |z| < 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 (i.e., |z| = 1) then the system is stable. In the above systems the causal system (Example 2) is stable because |z| > 0.5 contains the unit circle.

Let us assume we are provided a Z-transform of a system without a ROC (i.e., an ambiguous ). We can determine a unique provided we desire the following:

For stability the ROC must contain the unit circle. If we need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If we need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If we need both stability and causality, all the poles of the system function must be inside the unit circle.

The unique can then be found.

Properties

Properties of the z-transform

Property

Time domain Z-domain Proof ROC
Definition of Z-transform (definition of the z-transform)

(definition of the inverse z-transform)

Linearity Contains ROC1 ∩ ROC2
Time expansion

with

Decimation ohio-state.edu or ee.ic.ac.uk
Time delay

with and

ROC, except if and if
Time advance

with

Bilateral Z-transform:

Unilateral Z-transform:[12]

First difference backward

with for

Contains the intersection of ROC of and
First difference forward
Time reversal
Scaling in the z-domain
Complex conjugation
Real part
Imaginary part
Differentiation in the z-domain ROC, if is rational;

ROC possibly excluding the boundary, if is irrational[13]

Convolution Contains ROC1 ∩ ROC2
Cross-correlation Contains the intersection of ROC of and
Accumulation
Multiplication -

Parseval's theorem

Initial value theorem: If is causal, then

Final value theorem: If the poles of are inside the unit circle, then

Table of common Z-transform pairs

Here:

is the unit (or Heaviside) step function and

is the discrete-time unit impulse function (cf Dirac delta function which is a continuous-time version). The two functions are chosen together so that the unit step function is the accumulation (running total) of the unit impulse function.

Signal, Z-transform, ROC
1 1 all z
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17 , for positive integer [13]
18 , for positive integer [13]
19
20
21
22

Relationship to Fourier series and Fourier transform

Further information: Discrete-time Fourier transform § Relationship to the Z-transform

For values of in the region , known as the unit circle, we can express the transform as a function of a single real variable by defining And the bi-lateral transform reduces to a Fourier series:

 

 

 

 

(Eq.4)

which is also known as the discrete-time Fourier transform (DTFT) of the sequence. This -periodic function is the periodic summation of a Fourier transform, which makes it a widely used analysis tool. To understand this, let be the Fourier transform of any function, , whose samples at some interval equal the sequence. Then the DTFT of the sequence can be written as follows.

 

 

 

 

(Eq.5)

where has units of seconds, has units of hertz. Comparison of the two series reveals that is a normalized frequency with unit of radian per sample. The value corresponds to . And now, with the substitution Eq.4 can be expressed in terms of (a Fourier transform):

 

 

 

 

(Eq.6)

As parameter T changes, the individual terms of Eq.5 move farther apart or closer together along the f-axis. In Eq.6 however, the centers remain 2π apart, while their widths expand or contract. When sequence represents the impulse response of an LTI system, these functions are also known as its frequency response. When the sequence is periodic, its DTFT is divergent at one or more harmonic frequencies, and zero at all other frequencies. This is often represented by the use of amplitude-variant Dirac delta functions at the harmonic frequencies. Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler discrete Fourier transform (DFT). (See Discrete-time Fourier transform § Periodic data.)

Relationship to Laplace transform

Further information: Laplace transform § Z-transform

Bilinear transform

Main article: Bilinear transform

The bilinear transform can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z-domain), and vice versa. The following substitution is used:

to convert some function in the Laplace domain to a function in the Z-domain (Tustin transformation), or

from the Z-domain to the Laplace domain. Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire axis of the s-plane onto the unit circle in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the axis is in the region of convergence of the Laplace transform.

Starred transform

Main article: Starred transform

Given a one-sided Z-transform of a time-sampled function, the corresponding starred transform produces a Laplace transform and restores the dependence on (the sampling parameter):

The inverse Laplace transform is a mathematical abstraction known as an impulse-sampled function.

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. By normalizing with 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 above equation (using linearity and time-shifting laws) yields:

where and are the z-transform of and respectively. (Notation conventions typically use capitalized letters to refer to the z-transform of a signal denoted by a corresponding lower case letter, similar to the convention used for notating Laplace transforms.)

Rearranging results in the system's transfer function:

Zeros and poles

From the fundamental theorem of algebra the numerator has roots (corresponding to zeros of ) and the denominator has roots (corresponding to poles). Rewriting the transfer function in terms of zeros and poles

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 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.

See also

References

  1. ^ Mandal, Jyotsna Kumar (2020). "Z-Transform-Based Reversible Encoding". Reversible Steganography and Authentication via Transform Encoding. Studies in Computational Intelligence. Vol. 901. Singapore: Springer Singapore. pp. 157–195. doi:10.1007/978-981-15-4397-5_7. ISBN 978-981-15-4396-8. ISSN 1860-949X. S2CID 226413693. Z is a complex variable. Z-transform converts the discrete spatial domain signal into complex frequency domain representation. Z-transform is derived from the Laplace transform.
  2. ^ Lynn, Paul A. (1986). "The Laplace Transform and the z-transform". Electronic Signals and Systems. London: Macmillan Education UK. pp. 225–272. doi:10.1007/978-1-349-18461-3_6. ISBN 978-0-333-39164-8. Laplace Transform and the z-transform are closely related to the Fourier Transform. z-transform is especially suitable for dealing with discrete signals and systems. It offers a more compact and convenient notation than the discrete-time Fourier Transform.
  3. ^ Palani, S. (2021-08-26). "The z-Transform Analysis of Discrete Time Signals and Systems". Signals and Systems. Cham: Springer International Publishing. pp. 921–1055. doi:10.1007/978-3-030-75742-7_9. ISBN 978-3-030-75741-0. S2CID 238692483. z-transform is the discrete counterpart of Laplace transform. z-transform converts difference equations of discrete time systems to algebraic equations which simplifies the discrete time system analysis. Laplace transform and z-transform are common except that Laplace transform deals with continuous time signals and systems.
  4. ^ E. R. Kanasewich (1981). Time Sequence Analysis in Geophysics. University of Alberta. pp. 186, 249. ISBN 978-0-88864-074-1.
  5. ^ E. R. Kanasewich (1981). Time sequence analysis in geophysics (3rd ed.). University of Alberta. pp. 185–186. ISBN 978-0-88864-074-1.
  6. ^ Ragazzini, J. R.; Zadeh, L. A. (1952). "The analysis of sampled-data systems". Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry. 71 (5): 225–234. doi:10.1109/TAI.1952.6371274. S2CID 51674188.
  7. ^ Cornelius T. Leondes (1996). Digital control systems implementation and computational techniques. Academic Press. p. 123. ISBN 978-0-12-012779-5.
  8. ^ Eliahu Ibrahim Jury (1958). Sampled-Data Control Systems. John Wiley & Sons.
  9. ^ Eliahu Ibrahim Jury (1973). Theory and Application of the Z-Transform Method. Krieger Pub Co. ISBN 0-88275-122-0.
  10. ^ Eliahu Ibrahim Jury (1964). Theory and Application of the Z-Transform Method. John Wiley & Sons. p. 1.
  11. ^ Jackson, Leland B. (1996). "The z Transform". Digital Filters and Signal Processing. Boston, MA: Springer US. pp. 29–54. doi:10.1007/978-1-4757-2458-5_3. ISBN 978-1-4419-5153-3. z transform is to discrete-time systems what the Laplace transform is to continuous-time systems. z is a complex variable. This is sometimes referred to as the two-sided z transform, with the one-sided z transform being the same except for a summation from n = 0 to infinity. The primary use of the one sided transform ... is for causal sequences, in which case the two transforms are the same anyway. We will not, therefore, make this distinction and will refer to ... as simply the z transform of x(n).
  12. ^ Bolzern, Paolo; Scattolini, Riccardo; Schiavoni, Nicola (2015). Fondamenti di Controlli Automatici (in Italian). MC Graw Hill Education. ISBN 978-88-386-6882-1.
  13. ^ a b c A. R. Forouzan (2016). "Region of convergence of derivative of Z transform". Electronics Letters. 52 (8): 617–619. Bibcode:2016ElL....52..617F. doi:10.1049/el.2016.0189. S2CID 124802942.

Further reading