Signals and Systems Chapter 0

Author： Bi_Gu
发布时间：September 6, 2022
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# Signals and Systems Chapter 0

## General Introduction

**Signals**: functions containing information about the behavior or nature of some phenomenon.

**Systems**: respond to particular signals by producing other signals or some desired behavior.

**Application**: Communication systems, Medical imaging, Geophysics, Signal processing, Optical computing

## Pre-knowledge

### Complex Numbers

Cartesian notation: $z=Re\{z\}+j\cdot Im\{z\}$

Polar notation: $z=\left|z\right|e^{j\theta}$, where $\displaystyle\theta=\arctan\frac{Im\{z\}}{Re\{z\}}$.

Complex conjugation (共轭): $z^{*}=Re\{z\}-j\cdot Im\{z\}$

Euler's Formula:

$$
e^{j\theta}=\cos\theta+j\sin\theta\ ;\quad\begin{cases}\cos\theta=\displaystyle\frac{e^{j\theta}+e^{-j\theta}}{2}\\\sin\theta=\displaystyle\frac{e^{j\theta}-e^{-j\theta}}{2}\end{cases}\ .
$$

### Important geometric series

For some (possibly complex) number $z_0$, we have

$$
\sum_{n=0}^{\infty}z_0^n=\frac{1}{1-z_0}\quad\mathrm{iff.}\quad \left|z_0\right|<1.
$$

### Zeros of a complex equation

For some (complex) number $a$, find zeros of $z^N-a=0$

We have that

$$
z^N=a=\left|a\right|e^{j\cdot(2k\pi+\theta)}\Rightarrow z_k=\left|a\right|^{\frac{1}{N}}\cdot e^{j\cdot\frac{2k\pi+\theta}{N}}\ \text{for }k=0,1,\cdots,N-1.

Last modification：September 8, 2022

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