Z-transform properties

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.

	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	\(\Im{f[n]}\)	\(\lfzraised\ztransform\)	\(\dfrac{X(z)-X^{\star}(z^{\star})}{2j}\)

	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

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