EE 150: Signals and Systems

ShanghaiTech University - Spring 2021

Basic concepts

Signals and Systems

  • Signal: a function of one or more independent variables (e.g., time and spatial variables); typically contains information about the behavior or nature of some physical phenomena

    • 信号是一系列独立变量

  • System: responds to a particular signal input by producing another signal(output)

    • 系统是其一个信号进, 一个信号出

Transformation

  • Time reflection: x(t) ←→ x(−t), x[n] ←→ x[−n]

  • Time scaling: x(t) ←→ x(ct)

  • Time shift: x(t) ←→ x(t − t0), x[n] ←→ x[n − n0]

  • Usually do shift then scaling to avoid complex mathematics

Even and Odd functions

Periodic

    • 如果是合成的 signal, 其 Fundamental Period 是最小公倍数

  • Aperiodic: Non-preodic

Eular's Formula

  • \begin{equation} \mathrm{e}^{-\mathrm{j} \omega t}=\cos (\omega t)-\mathrm{j} \sin (\omega t) \end{equation}

  • \begin{equation} \cos (\omega t)=\frac{1}{2}\left(\mathrm{e}^{\mathrm{j} \omega t}+\mathrm{e}^{-\mathrm{j} \omega t}\right) \end{equation}

  • \begin{equation} \sin (\omega t)=\frac{1}{2 \mathrm{j}}\left(\mathrm{e}^{\mathrm{j} \omega t}-\mathrm{e}^{-\mathrm{j} \omega t}\right) \end{equation}

Sin

Discrete Time Unit Step and Unit Impulse Sequence

常见信号与周期判断

功率与能量

Other concepts

  • Energy and Power of Periodic Signals: 积分与积分后的处理

  • 谐振: 同一个角频率的集合

Properties of System

Memory / Memoryless

Output only depends on input at the same time

Invertibility and Inverse System

Distinct input leads to distinct output

Causality

All memoryless are causal

Output only depends on input at the same time or before

Stability

Bounded input gives Bounded output

Time-Invariance

A time-shift in the input causes a same time-shift in the output

Example:

Linearity

Additivity and Scaling

If linear, zero input gives zero output

Convolution

Begin

Example:

Properties of Convolution

    • 交换律

Properties of L.T.I System

  • Invertibility: A system is invertible only if an inverse system exists.

    • \begin{equation} y[n]=\sum_{k=-\infty}^{n} x[k] h[n-k] \end{equation}

  • Stability:

Calculation

Sliding window:

\begin{equation} g(t)=\frac{d}{d t} \int x * h(t) d t=\ldots \end{equation}

逆变换

Eigen-function of L.T.I

Eigen-functions

  • A signal for which the system's output is just a constant (possibly complex) times the input.

  • Eigen Basis: If an input signal can be decomposed to a weighted sum of eigenfunctions (eigen basis), then the output can be easily found.

So goal: Getting an Eigen basis of an L.T.I system.

e^{st} as eigenfunction of L.T.I

Orthonormal Basis

  • Definition of inner-product of perioidic functions: \begin{equation} <x_{1}(t), x_{2}(t)>=\frac{1}{T_{0}} \int_{T_{0}} x_{1}(t) x_{2}^{*}(t) d t \end{equation}

Fourier Analysis

CT & DT

    • \begin{equation} \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \longrightarrow \mathrm{H}(\mathrm{j} \omega) \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \end{equation}

    • 纯虚数

    • \begin{equation} \mathrm{e}^{j \omega n} \rightarrow H\left(\mathrm{e}^{j \omega}\right) \mathrm{e}^{j \omega n} \end{equation}

Periodic signals & Fourier Series Expansion

\begin{equation} x(t)=\sum_{k=-\infty}^{\infty} a_{k} \cdot e^{j k \omega_{0} t} \end{equation}, \begin{equation} x(t) \leftarrow{ }^{F . S .} \rightarrow a_{k} \end{equation}

sum of sinusoids whose frequencies are multiple of ω0, the “fundamental frequency”.

\begin{equation} a_{k}=\frac{1}{T_{0}} \int_{T_{0}} x(\tau) e^{-j k \omega_{0} \tau} d \tau \end{equation}

  • Case 0 is often special!

And,

Odd / Even

Approximation

Linearity

Time-shift

Time-reverse

Time-scaling

Multiplication

which is Convolution

conjugation & conjugate symmetry

7

Parseval's identity

Proof:

Continuous-Time Fourier Transform (CTFT)

变换与逆变换

Fourier series: \begin{equation} x(t)=\sum_{k=-\infty}^{\infty} a_{k} \mathrm{e}^{j k \omega_{0} t} \end{equation}

Fourier Transform Pair

Fourier Transform: \begin{equation} \mathcal{F}: X(j \omega)=\int_{-\infty}^{\infty} x(t) e^{-j \omega t} d t \end{equation}

  • For periodic signals, \begin{equation} X(j \omega)=2 \pi \sum_{-\infty}^{\infty} a_{k} \delta\left(\omega-k \omega_{0}\right) \end{equation}, i.e. the Fourier Series

Inverse F.T. \begin{equation} \mathcal{F}^{-1}: x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega) e^{j \omega t} d \omega \end{equation}

Dual Porperty

\begin{equation} \mathcal{F}(\mathcal{F}(x(t)))=2 \pi \cdot x(-t) \end{equation}

Properties

Normal CT Fourier Pairs

REF

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