#!/usr/bin/env python
# coding: utf-8

# # The $z$-Transform
# 
# *This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Comunications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*

# ## Summary of Properties, Theorems and Transforms
# 
# The [properties](properties.ipynb), [theorems](theorems.ipynb) and transforms of the two-sided $z$-transform as derived in the previous sections are summarized in the following. The corresponding tables serve as a reference for the application of the $z$-transform in the theory of signals and systems. Please refer to the respective sections for details.

# ### Definition
# 
# The $z$-transform and its inverse are defined as
# 
# \begin{align}
# X(z) &= \sum_{k = -\infty}^{\infty} x[k] \, z^{-k} \\
# x[k] &= \frac{1}{2 \pi j} \oint_{C} X(z) \, z^{k - 1} \; dz
# \end{align}
# 
# where $z \in \text{ROC} \{ x[k] \}$.

# ### Properties and Theorems
# 
# The properties and theorems of the two-sided $z$-transform are given as
# 
# |  | $x[k]$ | $X(z) = \mathcal{Z} \{ x[k] \}$ | ROC |
# |:---|:---:|:---:|:---|
# | [Linearity](properties.ipynb#Linearity) | $A \, x_1[k] + B \, x_2[k]$ | $A \, X_1(z) + B \, X_2(z)$ | $\supseteq \text{ROC}\{x_1[k]\} \cap \text{ROC}\{x_2[k]\}$ |
# | [Conjugation](properties.ipynb#Symmetry-for-Real-Valued-Signals) | $x^*[k]$ | $X^*(z^*)$ | $\text{ROC}\{ x[k] \}$ |
# | [Real-valued signal](properties.ipynb#Symmetry-for-Real-Valued-Signals) | $x[k] = x^*[k]$ | $X(z) = X^*(z^*)$ | |
# | [Linear Convolution](theorems.ipynb#Convolution-Theorem) | $x[k] * h[k]$ | $X(z) \cdot H(z)$ | $\supseteq \text{ROC}\{x[k]\} \cap \text{ROC}\{h[k]\}$ |
# | [Shift](theorems.ipynb#Shift-Theorem) | $x[k - \kappa]$ | $z^{- \kappa} \cdot X(z)$ | $\supseteq \text{ROC}\{x[k]\} \setminus \{0, \infty \}$ |
# | [Modulation](theorems.ipynb#Modulation-Theorem) | $z_0^k \cdot x[k]$ | $X\left( \frac{z}{z_0} \right)$ | $\{z: \frac{z}{z_0} \in \text{ROC} \{ x[k] \} \}$ |
# | [Reversal](theorems.ipynb#Reversal) | $x[-k]$ | $X \left( \frac{1}{z} \right)$ | $\{z: \frac{1}{z} \in \text{ROC} \{ x[k] \} \}$ |
# 
# where $A, B, z_0 \in \mathbb{C}$ and $\kappa \in \mathbb{Z}$

# ### Selected Transforms
# 
# Two-sided $z$-transforms which are frequently used are given as
# 
# | $x[k]$ | $X(z) = \mathcal{Z} \{ x[k] \}$ | ROC |
# |:---:|:---:|:---|
# | $\delta[k]$ | $1$ | $\mathbb{C}$ |
# | $\epsilon[k]$ | $\frac{z}{z-1}$ | $|z| > 1$ |
# | $k \epsilon[k]$ | $\frac{z}{(z-1)^2}$ | $|z| > 1$ |
# | $z_0^{k} \epsilon[k]$ | $\frac{z}{z - z_0}$ | $|z| > |z_0|$ |
# | $-z_0^{k} \epsilon[-k-1]$ | $\frac{z}{z - z_0}$ | $|z| < |z_0|$ |
# | $\sin(\Omega_0 k) \epsilon[k]$ | $\frac{z \sin(\Omega_0)}{z^2 - 2 z \cos(\Omega_0) + 1}$ | $|z| > 1$ |
# | $\cos(\Omega_0 k) \epsilon[k]$ | $\frac{z ( z - \cos(\Omega_0))}{z^2 - 2 z \cos(\Omega_0) + 1}$ | $|z| > 1$ |
# 
# where $z_0 \in \mathbb{C}$, $\Omega_0 \in \mathbb{R}$ and $n \in \mathbb{N}$. More one- and two-sided transforms may be found in the literature or [online](https://en.wikipedia.org/wiki/Z-transform#Table_of_common_Z-transform_pairs).

# **Copyright**
# 
# This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*.