# Overview

Overview of the Unilateral Z-transform properties, pairs and initial/final theorem. Includes links to the the proofs.

The tables below introduce commonly used properties, common input functions and initial/final value theorems that I collected over time.

The time-domain function is usually given in terms of a discrete index $$n$$, rather than time. Since $$t=nT$$, we may replace $$f[n]$$ with $$f(nT)$$ where $$T$$ is the sampling period.

## Unilateral Z-Transform properties

Unilateral Z-Transform properties
Time domain $$z$$-domain
Z-transform $$f[n]$$ $$\def\lfz#1{\overset{\Large#1}{\,\circ\kern-6mu-\kern-7mu-\kern-7mu-\kern-6mu\bullet\,}} \def\lfzraised#1{\raise{10mu}{#1}} \def\ztransform{\lfz{\mathcal{Z}}} \lfzraised\ztransform$$ $$X(z)=\sum_{n=0}^{\infty}z^{-n}f[n]$$ proof
Linearity $$a\,f[n]+b\,g[n]$$ $$\lfzraised\ztransform$$ $$a\,F[n]+b\,G[n]$$ proof
Time delay $$f[n-a]\,\color{grey}{\gamma[n}-a\color{grey}{]}$$ $$\lfzraised\ztransform$$ $$z^{-a}F(z)$$ proof
Time delay #2 $$f[n-a]\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$z^{-a}\left(F(z)+\sum_{m=1}^{a} z^m\ f[-m]\right)$$ proof
Time advance $$f[n+a]\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$z^a\left(F(z)-\sum_{m=0}^{a-1} z^{-m}\ f[m]\right)$$ proof
Time advance ($$a=1$$) $$f[n+1]\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$z\left(F(z)-f[0]\right)$$ proof
Time multiply $$n\,f[n]\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$-z\dfrac{\text{d}F(z)}{\text{d}z}$$ proof
Modulation $$a^n\,f[n]\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$F(a^{-1}z)$$ proof
Convolution $$(f\ast g)[n]\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$F(z)\ G(z)$$ proof
Conjugation $$f^\star[n]$$ $$\lfzraised\ztransform$$ $$F^{\star}(z^{\star})$$ proof
First difference $$f[n]-f[n-1]$$ $$\lfzraised\ztransform$$ $$\left(1-z^{-1}\right)F(z)$$ proof
Accumulation $$\sum_{i=0}^{n}x[i]$$ $$\lfzraised\ztransform$$ $$F(z)\,\dfrac{z}{z-1}$$ proof
Double poles $$(n+1)p^n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{1}{(1-pz^{-1})^2}$$ proof
Real part $$\Re{f[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{X(z)+X^{\star}(z^{\star})}{2}$$
Imaginary part $$\Im{f[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{X(z)-X^{\star}(z^{\star})}{2j}$$

## Unilateral Z-transform pairs

Unilateral Z-transform pairs
Time domain $$z$$-domain ($$z$$) $$z$$-domain ($$z^{-1}$$) ROC
Impulse $$\small{\delta[n]\triangleq\begin{cases}1,&n=0\\0,&n\neq0\end{cases}}$$ $$\lfzraised\ztransform$$ $$1$$ $$1$$ all $$z$$ proof
Delayed impulse $$\delta[n-a]$$ $$\lfzraised\ztransform$$ $$\small{\begin{cases}z^{-a},&a\geq0\\0,&a\lt0\end{cases}}$$ $$\small{\begin{cases}z^{-a},&a\geq0\\0,&a\lt0\end{cases}}$$ $$z\neq0$$ proof
Unit step $$\small{\gamma[n]\triangleq\begin{cases} 0,&n\lt0\\1,&n\geq 0\end{cases}}$$ $$\lfzraised\ztransform$$ $$\dfrac{z}{z-1}$$ $$\dfrac{1}{1-z^{-1}}$$ $$|z|\gt1$$ proof
Scaled $$a^n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{z}{z-a}$$ $$\dfrac{1}{1-az^{-1}}$$ $$|z|\gt |a|$$ proof
Delayed scaled $$a^{n-1}\,\gamma[n-1]$$ $$\lfzraised\ztransform$$ $$\dfrac{1}{z-a}$$ $$\dfrac{z^{-1}}{a-z^{-1}}$$ $$|z|\gt1$$ proof
n-scaled $$n\,a^n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{az}{(z-a)^2}$$ $$\dfrac{az^{-1}}{(1-az^{-1})^2}$$ $$|z|\gt |a|$$ proof
Ramp $$n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{z}{(z-1)^2}$$ $$\dfrac{z^{-1}}{(1-z^{-1})^2}$$ $$|z|\gt1$$ proof
Binomial scaled, $$|z|\gt |a|$$ $$n^2a^n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{az(z-1)}{(z-a)^3}$$ $$\dfrac{az^{-1}\left(1+az^{-1}\right)}{(1-az^{-1})^3}$$ $$|z|\gt |a|$$ proof
Binomial scaled, $$|z|\gt |a|$$ $$\tfrac{1}{2}n(n-1)a^n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{a^2z}{(z-a)^3}$$ $$\dfrac{a^2z^{-2}}{(1-az^{-1})^3}$$ $$|z|\gt |a|$$ proof
Binomial scaled, $$|z|\lt |a|$$ $$\small{\left(\begin{array}{c}n+m-1\\m-1\end{array}\right)}\,a^n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{z^m}{(z-a)^m}$$ $$\dfrac{1}{(1-az^{-1})^m}$$ $$|z|\gt |a|$$ proof
Binomial scaled, $$|z|\lt |a|$$ $$\small{(-1)^m\left(\begin{array}{c}-n-1\\m-1\end{array}\right)}\,a^n\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{z^m}{(z-a)^m}$$ $$\dfrac{1}{(1-az^{-1})^m}$$ $$|z|\lt |a|$$ proof
Exponential $$\mathrm{e}^{-anT}\ \gamma[n]$$ $$\lfzraised\ztransform$$ $$\frac{z}{z-\mathrm{e}^{-aT}}$$ $${|\mathrm{e}^{-aT}|\lt |z|}$$ proof
Sine $$\sin(\omega n)\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{z\sin(\omega)}{z^2-2z\cos(\omega)+1}$$ $$\dfrac{z^{-1}\sin(\omega)}{1-2z^{-1}\cos(\omega)+z^{-2}}$$ $$|z|\gt1$$ proof
Cosine $$\cos(\omega n)\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{z^2-z\cos(\omega)}{z^2-2z\cos(\omega)+1}$$ $$\dfrac{1-z^{-1}\cos(\omega)}{1-2z^{-1}\cos(\omega)+z^{-2}}$$ $$|z|\gt1$$ proof
Decaying Sine $$a^n\sin(\omega n)\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{az\sin(\omega)}{z^2-2az\cos(\omega)+a^2}$$ $$\dfrac{az^{-1}\sin(\omega)}{1-2az^{-1}\cos(\omega)+a^2z^{-2}}$$ $$|z|\gt|a|$$ proof
Decaying Cosine $$a^n\cos(\omega n)\,\color{grey}{\gamma[n]}$$ $$\lfzraised\ztransform$$ $$\dfrac{1-az\cos(\omega)}{z^2-2az\cos(\omega)+a^2}$$ $$\dfrac{z^{-1}\left(z^{-1}-a\cos(\omega)\right)} {1-2az^{-1}\cos(\omega)+a^2z^{-2}}$$ $$|z|\gt|a|$$ proof

The binomial coefficient, used in the table above, is defined as

$$\left(\begin{array}{c}a\\ b\end{array}\right)\triangleq\frac{a!}{b!(a-b)!} \nonumber$$

## Initial and final value theorem

Initial and final value theorem
Time domain $$z$$-domain
Initial Value $$f(0^+)$$ $$f[0]=\lim_{z\to\infty}F(z)$$ proof
Final Value $$f(\infty)$$ $$\lim_{n\to\infty}f[n]=\lim_{z\to1}(z-1)F(z)$$ proof

The proof for these transforms can be found in the post Z-Transforms Proofs.

I recommend reading through the proofs for these Z-transforms. If you want to skip ahead, I suggest Discrete Transfer Functions for follow-up reading.