Z-transform

Not to be confused with Fisher z-transformation.

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.

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.

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]

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 ${\displaystyle x[n]}$ is the formal power series ${\displaystyle X(z)}$ defined as:

${\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=-\infty }^{\infty }x[n]z^{-n}}$

where ${\displaystyle n}$ is an integer and ${\displaystyle z}$ is, in general, a complex number. In polar form, ${\displaystyle z}$ may be written as:

${\displaystyle z=Ae^{j\phi }=A\cdot (\cos {\phi }+j\sin {\phi })}$

where ${\displaystyle A}$ is the magnitude of ${\displaystyle z}$, ${\displaystyle j}$ is the imaginary unit, and ${\displaystyle \phi }$ is the complex argument (also referred to as angle or phase) in radians.

Unilateral Z-transform

Alternatively, in cases where ${\displaystyle x[n]}$ is defined only for ${\displaystyle n\geq 0}$, the single-sided or unilateral Z-transform is defined as:

${\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=0}^{\infty }x[n]z^{-n}.}$

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 ${\displaystyle x[n]}$ is the probability that a discrete random variable takes the value. The properties of Z-transforms (listed in § Properties) have useful interpretations in the context of probability theory.

The inverse Z-transform is:

${\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}={\frac {1}{2\pi j}}\oint _{C}X(z)z^{n-1}dz}$

where ${\displaystyle C}$ 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 ${\displaystyle C}$ must encircle all of the poles of ${\displaystyle X(z)}$.

A special case of this contour integral occurs when ${\displaystyle C}$ is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when ${\displaystyle X(z)}$ 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:

${\displaystyle x[n]={\frac {1}{2\pi }}\int _{-\pi }^{+\pi }X(e^{j\omega })e^{j\omega n}d\omega .}$

The Z-transform with a finite range of ${\displaystyle n}$ and a finite number of uniformly spaced ${\displaystyle z}$ 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 ${\displaystyle z}$ to lie on the unit circle.

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):

${\displaystyle \mathrm {ROC} =\left\{z:\left|\sum _{n=-\infty }^{\infty }x[n]z^{-n}\right|<\infty \right\}}$

Example 1 (no ROC)

Let ${\displaystyle x[n]=(.5)^{n}\ .}$ Expanding ${\displaystyle x[n]}$ on the interval ${\displaystyle (-\infty ,\infty )}$ it becomes

${\displaystyle x[n]=\left\{\dots ,(.5)^{-3},(.5)^{-2},(.5)^{-1},1,(.5),(.5)^{2},(.5)^{3},\dots \right\}=\left\{\dots ,2^{3},2^{2},2,1,(.5),(.5)^{2},(.5)^{3},\dots \right\}.}$

Looking at the sum

${\displaystyle \sum _{n=-\infty }^{\infty }x[n]z^{-n}\to \infty .}$

Therefore, there are no values of ${\displaystyle z}$ that satisfy this condition.

Example 2 (causal ROC)

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

Let ${\displaystyle x[n]=(.5)^{n}\,u[n]}$ (where ${\displaystyle u}$ is the Heaviside step function). Expanding ${\displaystyle x[n]}$ on the interval ${\displaystyle (-\infty ,\infty )}$ it becomes

${\displaystyle x[n]=\left\{\dots ,0,0,0,1,(.5),(.5)^{2},(.5)^{3},\dots \right\}.}$

Looking at the sum

${\displaystyle \sum _{n=-\infty }^{\infty }x[n]z^{-n}=\sum _{n=0}^{\infty }(.5)^{n}z^{-n}=\sum _{n=0}^{\infty }\left({\frac {.5}{z}}\right)^{n}={\frac {1}{1-(.5)z^{-1}}}.}$

The last equality arises from the infinite geometric series and the equality only holds if ${\displaystyle |(.5)z^{-1}|<1,}$ which can be rewritten in terms of ${\displaystyle z}$ as ${\displaystyle |z|>(.5).}$ Thus, the ROC is ${\displaystyle |z|>(.5).}$ 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 ${\displaystyle x[n]=-(.5)^{n}\,u[-n-1]}$ (where ${\displaystyle u}$ is the Heaviside step function). Expanding ${\displaystyle x[n]}$ on the interval ${\displaystyle (-\infty ,\infty )}$ it becomes

${\displaystyle x[n]=\left\{\dots ,-(.5)^{-3},-(.5)^{-2},-(.5)^{-1},0,0,0,0,\dots \right\}.}$

Looking at the sum

${\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }x[n]\,z^{-n}&=-\sum _{n=-\infty }^{-1}(.5)^{n}\,z^{-n}\\&=-\sum _{m=1}^{\infty }\left({\frac {z}{.5}}\right)^{m}\\&=-{\frac {(.5)^{-1}z}{1-(.5)^{-1}z}}\\&=-{\frac {1}{(.5)z^{-1}-1}}\\&={\frac {1}{1-(.5)z^{-1}}}\\\end{aligned}}}$

and using the infinite geometric series again, the equality only holds if ${\displaystyle |(.5)^{-1}z|<1}$ which can be rewritten in terms of ${\displaystyle z}$ as ${\displaystyle |z|<(.5).}$ Thus, the ROC is ${\displaystyle |z|<(.5).}$ 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 conclusion

Examples 2 & 3 clearly show that the Z-transform ${\displaystyle X(z)}$ of ${\displaystyle x[n]}$ 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 a ROC that includes ${\displaystyle |z|=\infty }$ while the anticausal system in example 3 yields an ROC that includes ${\displaystyle |z|=0.}$

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 ${\displaystyle |z|=\infty }$ nor ${\displaystyle |z|=0.}$ The ROC creates a circular band. For example,

${\displaystyle x[n]=(.5)^{n}\,u[n]-(.75)^{n}\,u[-n-1]}$

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 ${\displaystyle (.5)^{n}\,u[n]}$ and an anticausal term ${\displaystyle -(.75)^{n}\,u[-n-1].}$

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 Z-transform of a system without a ROC (i.e., an ambiguous ${\displaystyle x[n]}$). We can determine a unique ${\displaystyle x[n]}$ provided we desire the following:

• Stability
• Causality

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 ${\displaystyle x[n]}$ can then be found.

Properties of the z-transform

Property	Time domain	Z-domain	Proof	ROC
Definition of Z-transform	${\displaystyle x[n]}$	${\displaystyle X(z)}$	${\displaystyle X(z)={\mathcal {Z}}\{x[n]\}}$ (definition of the z-transform) ${\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}}$ (definition of the inverse z-transform)	${\displaystyle r_{2}<|z|<r_{1}}$
Linearity	${\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]}$	${\displaystyle a_{1}X_{1}(z)+a_{2}X_{2}(z)}$	${\displaystyle {\begin{aligned}X(z)&=\sum _{n=-\infty }^{\infty }(a_{1}x_{1}[n]+a_{2}x_{2}[n])z^{-n}\\&=a_{1}\sum _{n=-\infty }^{\infty }x_{1}[n]\,z^{-n}+a_{2}\sum _{n=-\infty }^{\infty }x_{2}[n]\,z^{-n}\\&=a_{1}X_{1}(z)+a_{2}X_{2}(z)\end{aligned}}}$	Contains ROC1 ∩ ROC2
Time expansion	${\displaystyle x_{K}[n]={\begin{cases}x[r],&n=Kr\\0,&n\notin K\mathbb {Z} \end{cases}}}$ with ${\displaystyle K\mathbb {Z}$ :=\{Kr:r\in \mathbb {Z} \}}	${\displaystyle X(z^{K})}$	${\displaystyle {\begin{aligned}X_{K}(z)&=\sum _{n=-\infty }^{\infty }x_{K}[n]z^{-n}\\&=\sum _{r=-\infty }^{\infty }x[r]z^{-rK}\\&=\sum _{r=-\infty }^{\infty }x[r](z^{K})^{-r}\\&=X(z^{K})\end{aligned}}}$	${\displaystyle R^{\frac {1}{K}}}$
Decimation	${\displaystyle x[Kn]}$	${\displaystyle {\frac {1}{K}}\sum _{p=0}^{K-1}X\left(z^{\tfrac {1}{K}}\cdot e^{-i{\tfrac {2\pi }{K}}p}\right)}$	ohio-state.edu or ee.ic.ac.uk
Time delay	${\displaystyle x[n-k]}$ with ${\displaystyle k>0}$ and ${\displaystyle x:x[n]=0\ \forall \,n<0}$	${\displaystyle z^{-k}X(z)}$	${\displaystyle {\begin{aligned}{\mathcal {Z}}\{x[n-k]\}&=\sum _{n=0}^{\infty }x[n-k]z^{-n}\\&=\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}&&j=n-k\\&=\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}\\&=z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}\\&=z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}&&x[\beta ]=0,\beta <0\\&=z^{-k}X(z)\end{aligned}}}$	ROC, except ${\displaystyle z{=}0}$ if ${\displaystyle k>0}$ and ${\displaystyle z{=}\infty }$ if ${\displaystyle k<0}$
Time advance	${\displaystyle x[n+k]}$ with ${\displaystyle k>0}$	Bilateral Z-transform: $${\displaystyle z^{k}X(z)}$$ Unilateral Z-transform:[12] $${\displaystyle z^{k}\,X(z)-z^{k}\sum _{n=0}^{k-1}x[n]\,z^{-n}}$$
First difference backward	${\displaystyle x[n]-x[n-1]}$ with ${\displaystyle x[n]{=}0}$ for ${\displaystyle n<0}$	${\displaystyle (1-z^{-1})\,X(z)}$		Contains the intersection of ROC of ${\displaystyle X_{1}(z)}$ and ${\displaystyle z\neq 0}$
First difference forward	${\displaystyle x[n+1]-x[n]}$	${\displaystyle (z-1)\,X(z)-z\,x[0]}$
Time reversal	${\displaystyle x[-n]}$	${\displaystyle X(z^{-1})}$	${\displaystyle {\begin{aligned}{\mathcal {Z}}\{x(-n)\}&=\sum _{n=-\infty }^{\infty }x[-n]z^{-n}\\&=\sum _{m=-\infty }^{\infty }x[m]z^{m}\\&=\sum _{m=-\infty }^{\infty }x[m]{(z^{-1})}^{-m}\\&=X(z^{-1})\\\end{aligned}}}$	${\displaystyle {\tfrac {1}{r_{1}}}<|z|<{\tfrac {1}{r_{2}}}}$
Scaling in the z-domain	${\displaystyle a^{n}x[n]}$	${\displaystyle X(a^{-1}z)}$	${\displaystyle {\begin{aligned}{\mathcal {Z}}\left\{a^{n}x[n]\right\}&=\sum _{n=-\infty }^{\infty }a^{n}x[n]z^{-n}\\&=\sum _{n=-\infty }^{\infty }x[n](a^{-1}z)^{-n}\\&=X(a^{-1}z)\end{aligned}}}$	${\displaystyle |a|r_{2}<|z|<|a|r_{1}}$
Complex conjugation	${\displaystyle x^{*}[n]}$	${\displaystyle X^{*}(z^{*})}$	${\displaystyle {\begin{aligned}{\mathcal {Z}}\{x^{*}(n)\}&=\sum _{n=-\infty }^{\infty }x^{*}[n]z^{-n}\\&=\sum _{n=-\infty }^{\infty }\left[x[n](z^{*})^{-n}\right]^{*}\\&=\left[\sum _{n=-\infty }^{\infty }x[n](z^{*})^{-n}\right]^{*}\\&=X^{*}(z^{*})\end{aligned}}}$
Real part	${\displaystyle \operatorname {Re} \{x[n]\}}$	${\displaystyle {\tfrac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]}$
Imaginary part	${\displaystyle \operatorname {Im} \{x[n]\}}$	${\displaystyle {\tfrac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]}$
Differentiation in the z-domain	${\displaystyle n\,x[n]}$	${\displaystyle -z{\frac {dX(z)}{dz}}}$	${\displaystyle {\begin{aligned}{\mathcal {Z}}\{n\,x(n)\}&=\sum _{n=-\infty }^{\infty }n\,x[n]z^{-n}\\&=z\sum _{n=-\infty }^{\infty }n\,x[n]z^{-n-1}\\&=-z\sum _{n=-\infty }^{\infty }x[n](-n\,z^{-n-1})\\&=-z\sum _{n=-\infty }^{\infty }x[n]{\frac {d}{dz}}(z^{-n})\\&=-z{\frac {dX(z)}{dz}}\end{aligned}}}$	ROC, if ${\displaystyle X(z)}$ is rational; ROC possibly excluding the boundary, if ${\displaystyle X(z)}$ is irrational[13]
Convolution	${\displaystyle x_{1}[n]*x_{2}[n]}$	${\displaystyle X_{1}(z)\,X_{2}(z)}$	${\displaystyle {\begin{aligned}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}&={\mathcal {Z}}\left\{\sum _{l=-\infty }^{\infty }x_{1}[l]x_{2}[n-l]\right\}\\&=\sum _{n=-\infty }^{\infty }\left[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}[n-l]\right]z^{-n}\\&=\sum _{l=-\infty }^{\infty }x_{1}[l]\left[\sum _{n=-\infty }^{\infty }x_{2}[n-l]z^{-n}\right]\\&=\left[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}\right]\!\!\left[\sum _{n=-\infty }^{\infty }x_{2}[n]z^{-n}\right]\\&=X_{1}(z)X_{2}(z)\end{aligned}}}$	Contains ROC1 ∩ ROC2
Cross-correlation	${\displaystyle r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]}$	${\displaystyle R_{x_1,x_{}}}$