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

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

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

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.