In mathematics and signal processing, the Ztransform converts a discretetime signal, which is a sequence of real or complex numbers, into a complex valued frequencydomain (the zdomain or zplane) representation.^{[1]}^{[2]}
It can be considered a discretetime equivalent of the Laplace transform (the sdomain or splane).^{[3]} This similarity is explored in the theory of timescale calculus.
While the continuoustime Fourier transform is evaluated on the sdomain's vertical axis (the imaginary axis), the discretetime Fourier transform is evaluated along the zdomain's unit circle. The sdomain's left halfplane maps to the area inside the zdomain's unit circle, while the sdomain's right halfplane maps to the area outside of the zdomain'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 sdomain to the zdomain, 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.
The foundational concept now recognized as the Ztransform, which is a cornerstone in the analysis and design of digital control systems, was not entirely novel when it emerged in the mid20th century. Its embryonic principles can be traced back to the work of the French mathematician PierreSimon 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 Ztransform were significantly advanced in 1947 by Witold Hurewicz and colleagues. Their work was motivated by the challenges presented by sampleddata control systems, which were becoming increasingly relevant in the context of radar technology during that period. The Ztransform provided a systematic and effective method for solving linear difference equations with constant coefficients, which are ubiquitous in the analysis of discretetime signals and systems.^{[4]}^{[5]}
The method was further refined and gained its official nomenclature, "the Ztransform," in 1952, thanks to the efforts of John R. Ragazzini and Lotfi A. Zadeh, who were part of the sampleddata control group at Columbia University. Their work not only solidified the mathematical framework of the Ztransform but also expanded its application scope, particularly in the field of electrical engineering and control systems.^{[6]}^{[7]}
The development of the Ztransform did not halt with Ragazzini and Zadeh. A notable extension, known as the modified or advanced Ztransform, was later introduced by Eliahu I. Jury. Jury's work extended the applicability and robustness of the Ztransform, 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 discretetime control systems, contributing significantly to the field of digital signal processing.^{[8]}^{[9]}
Interestingly, the conceptual underpinnings of the Ztransform 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 Ztransform. From a mathematical perspective, the Ztransform 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 Ztransform but also illustrates its versatility and broad applicability across different branches of mathematics and engineering.^{[10]}
The Ztransform can be defined as either a onesided or twosided transform. (Just like we have the onesided Laplace transform and the twosided Laplace transform.)^{[11]}
The bilateral or twosided Ztransform of a discretetime 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.
Alternatively, in cases where is defined only for , the singlesided or unilateral Ztransform is defined as:

(Eq.2)

In signal processing, this definition can be used to evaluate the Ztransform of the unit impulse response of a discretetime causal system.
An important example of the unilateral Ztransform is the probabilitygenerating 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 Ztransforms (listed in § Properties) have useful interpretations in the context of probability theory.
The inverse Ztransform 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 Ztransform simplifies to the inverse discretetime Fourier transform, or Fourier series, of the periodic values of the Ztransform around the unit circle:

(Eq.4)

The Ztransform with a finite range of and a finite number of uniformly spaced values can be computed efficiently via Bluestein's FFT algorithm. The discretetime Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Ztransform obtained by restricting to lie on the unit circle.
See also: Pole–zero_plot § Discretetime systems 
The region of convergence (ROC) is the set of points in the complex plane for which the Ztransform summation converges (i.e. doesn't blow up in magnitude to infinity):
Let Expanding on the interval it becomes
Looking at the sum
Therefore, there are no values of that satisfy this condition.
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".
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 2 & 3 clearly show that the Ztransform 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
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 mixedcausality 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 Ztransform 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 rightsided sequence. If we need an anticausal system then the ROC must contain the origin and the system function will be a leftsided 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.
Property 
Time domain  Zdomain  Proof  ROC 

Definition of Ztransform  (definition of the ztransform)
(definition of the inverse ztransform) 

Linearity  Contains ROC_{1} ∩ ROC_{2}  
Time expansion 
with 

Decimation  ohiostate.edu or ee.ic.ac.uk  
Time delay 
with and 
ROC, except if and if  
Time advance 
with 
Bilateral Ztransform:
Unilateral Ztransform:^{[12]}


First difference backward 
with for 
Contains the intersection of ROC of and  
First difference forward  
Time reversal  
Scaling in the zdomain  
Complex conjugation  
Real part  
Imaginary part  
Differentiation in the zdomain  ROC, if is rational;
ROC possibly excluding the boundary, if is irrational^{[13]}  
Convolution  Contains ROC_{1} ∩ ROC_{2}  
Crosscorrelation  Contains the intersection of ROC of and  
Accumulation  
Multiplication   
Initial value theorem: If is causal, then
Final value theorem: If the poles of are inside the unit circle, then
Here:
is the unit (or Heaviside) step function and
is the discretetime unit impulse function (cf Dirac delta function which is a continuoustime version). The two functions are chosen together so that the unit step function is the accumulation (running total) of the unit impulse function.
Signal,  Ztransform,  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 
Further information: Discretetime Fourier transform § Relationship to the Ztransform 
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 bilateral transform reduces to a Fourier series:

(Eq.4)

which is also known as the discretetime 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 faxis. 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 amplitudevariant 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 Discretetime Fourier transform § Periodic data.)
Further information: Laplace transform § Ztransform 
Main article: Bilinear transform 
The bilinear transform can be used to convert continuoustime filters (represented in the Laplace domain) into discretetime filters (represented in the Zdomain), and vice versa. The following substitution is used:
to convert some function in the Laplace domain to a function in the Zdomain (Tustin transformation), or
from the Zdomain to the Laplace domain. Through the bilinear transformation, the complex splane (of the Laplace transform) is mapped to the complex zplane (of the ztransform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire axis of the splane onto the unit circle in the zplane. As such, the Fourier transform (which is the Laplace transform evaluated on the axis) becomes the discretetime Fourier transform. This assumes that the Fourier transform exists; i.e., that the axis is in the region of convergence of the Laplace transform.
Main article: Starred transform 
Given a onesided Ztransform of a timesampled 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 impulsesampled function.
The linear constantcoefficient difference (LCCD) equation is a representation for a linear system based on the autoregressive movingaverage 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
Taking the Ztransform of the above equation (using linearity and timeshifting laws) yields:
where and are the ztransform of and respectively. (Notation conventions typically use capitalized letters to refer to the ztransform 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:
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 (zplane) 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 multipleorder 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.
If such a system is driven by a signal then the output is By performing partial fraction decomposition on and then taking the inverse Ztransform 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 Ztransforms.