In mathematics and signal processing, the Ztransform converts a discretetime signal, which is a sequence of real or complex numbers, into a complex frequencydomain (zdomain or zplane) representation.^{[1]}^{[2]}
It can be considered as a discretetime equivalent of the Laplace transform (sdomain).^{[3]} This similarity is explored in the theory of timescale calculus.
Whereas the continuoustime Fourier transform is evaluated on the Laplace sdomain's imaginary line, the discretetime Fourier transform is evaluated over the unit circle of the zdomain. What is roughly the sdomain's left halfplane, is now the inside of the complex unit circle; what is the zdomain's outside of the unit circle, roughly corresponds to the right halfplane of the sdomain.
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 basic idea now known as the Ztransform was known to Laplace, and it was reintroduced in 1947 by W. Hurewicz^{[4]}^{[5]} and others as a way to treat sampleddata control systems used with radar. It gives a tractable way to solve linear, constantcoefficient difference equations. It was later dubbed "the ztransform" by Ragazzini and Zadeh in the sampleddata control group at Columbia University in 1952.^{[6]}^{[7]}
The modified or advanced Ztransform was later developed and popularized by E. I. Jury.^{[8]}^{[9]}
The idea contained within the Ztransform is also known in mathematical literature as the method of generating functions which can be traced back as early as 1730 when it was introduced by de Moivre in conjunction with probability theory.^{[10]} From a mathematical view the Ztransform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.
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:
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 (below) have useful interpretations in the context of probability theory.
The inverse Ztransform is

(Eq.3) 
where C 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 C must encircle all of the poles of .
A special case of this contour integral occurs when C 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 n and a finite number of uniformly spaced z 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 z to lie on the unit circle.
The region of convergence (ROC) is the set of points in the complex plane for which the Ztransform summation converges.
Let . Expanding x[n] on the interval (−∞, ∞) it becomes
Looking at the sum
Therefore, there are no values of z that satisfy this condition.
Let (where u is the Heaviside step function). Expanding x[n] on the interval (−∞, ∞) it becomes
Looking at the sum
The last equality arises from the infinite geometric series and the equality only holds if 0.5z^{−1} < 1, which can be rewritten in terms of z as z > 0.5. Thus, the ROC is z > 0.5. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".
Let (where u is the Heaviside step function). Expanding x[n] on the interval (−∞, ∞) it becomes
Looking at the sum
Using the infinite geometric series, again, the equality only holds if 0.5^{−1}z < 1 which can be rewritten in terms of z as z < 0.5. Thus, the ROC is z < 0.5. 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 X(z) of x[n] 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 z = ∞ while the anticausal system in example 3 yields an ROC that includes z = 0.
In systems with multiple poles it is possible to have a ROC that includes neither z = ∞ nor z = 0. 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 (0.5)^{n}u[n] and an anticausal term −(0.75)^{n}u[−n−1].
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 x[n]). We can determine a unique x[n] 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 x[n] can then be found.
Time domain  Zdomain  Proof  ROC  

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

Decimation  ohiostate.edu or ee.ic.ac.uk  
Time delay 
with and 
ROC, except z = 0 if k > 0 and z = ∞ if k < 0  
Time advance 
with 
Bilateral Ztransform:
Unilateral Ztransform:^{[12]}


First difference backward 
with x[n] = 0 for n < 0 
Contains the intersection of ROC of X_{1}(z) and z ≠ 0  
First difference forward  
Time reversal  
Scaling in the zdomain  
Complex conjugation  
Real part  
Imaginary part  
Differentiation  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 x[n] is causal, then
Final value theorem: If the poles of (z − 1)X(z) 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, ω, 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 2πperiodic 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, T, equal the x[n] sequence. Then the DTFT of the x[n] sequence can be written as follows.


(Eq.5) 
When T 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 ω = 2π corresponds to . And now, with the substitution Eq.4 can be expressed in terms of the Fourier transform, X(•):


(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 x(nT) 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, X(z), of a timesampled function, the corresponding starred transform produces a Laplace transform and restores the dependence on sampling parameter, T:
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 α_{0}, if it is not zero, normalizing α_{0} = 1 and the LCCD equation can be written
This form of the LCCD equation is favorable to make it more explicit that the "current" output y[n] is a function of past outputs y[n − p], current input x[n], and previous inputs x[n − q].
Taking the Ztransform of the above equation (using linearity and timeshifting laws) yields
and rearranging results in
From the fundamental theorem of algebra the numerator has M roots (corresponding to zeros of H) and the denominator has N roots (corresponding to poles). Rewriting the transfer function in terms of zeros and poles
where q_{k} is the kth zero and p_{k} is the kth 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 z = 0 and z = ∞. 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 H(z) is driven by a signal X(z) then the output is Y(z) = H(z)X(z). By performing partial fraction decomposition on Y(z) and then taking the inverse Ztransform the output y[n] can be found. In practice, it is often useful to fractionally decompose before multiplying that quantity by z to generate a form of Y(z) which has terms with easily computable inverse Ztransforms.