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.

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