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Nyovelt's Notes
  • Notes
  • CS 110: Computer Architecture I
  • EE 150: Signals and Systems
  • CS 131: Programming Languages and Compilers
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On this page
  • Basic concepts
  • Signals and Systems
  • Transformation
  • Even and Odd functions
  • Periodic
  • Eular's Formula
  • 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
  • Properties of System
  • Memory / Memoryless
  • Invertibility and Inverse System
  • Causality
  • Stability
  • Time-Invariance
  • Linearity
  • Convolution
  • Begin
  • Properties of Convolution
  • Properties of L.T.I System
  • Calculation
  • 逆变换
  • Eigen-function of L.T.I
  • Eigen-functions
  • e^{st} as eigenfunction of L.T.I
  • Orthonormal Basis
  • Fourier Analysis
  • CT & DT
  • Periodic signals & Fourier Series Expansion
  • Continuous-Time Fourier Transform (CTFT)
  • 变换与逆变换
  • Fourier Transform Pair
  • Dual Porperty
  • Properties
  • Normal CT Fourier Pairs
  • REF

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EE 150: Signals and Systems

ShanghaiTech University - Spring 2021

PreviousCS 110: Computer Architecture INextCS 131: Programming Languages and Compilers

Last updated 4 years ago

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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

Every signal function is x(t)=Even(x)+Odd(t)x(t) = Even(x) + Odd(t)x(t)=Even(x)+Odd(t) ,then

x(−t)=Even(−t)+Odd(−t)=Even(t)−Odd(t) x(-t) = Even(-t) + Odd(-t) = Even(t) - Odd(t) x(−t)=Even(−t)+Odd(−t)=Even(t)−Odd(t)

Even(t)=12(x(t)+x(−t)) Even(t) = \frac12 ( x(t) + x(-t) ) Even(t)=21​(x(t)+x(−t))

Odd(t)=12(x(t)−x(−t)) Odd(t) = \frac12 ( x(t) - x(-t) )Odd(t)=21​(x(t)−x(−t))

Preodic: x(t)=x(t+mT) x(t) = x(t+mT)x(t)=x(t+mT) or x[t]=x[t+mT] x[t] = x[t+mT]x[t]=x[t+mT]

Fundamental period TTT: Smallest positive TTT

ejω0t=cos⁡(ω0t)+j⋅sin⁡(ω0t) e^{j \omega0 t} = \cos ( \omega0 t ) + j \cdot \sin (\omega_0 t) ejω0t=cos(ω0t)+j⋅sin(ω0​t)

Fundamental period T0=2π/∣ω0∣T_0 = 2 \pi / \mid \omega _0 \midT0​=2π/∣ω0​∣

Acos⁡(ω0t+ϕ)=A2ejϕejω0t+A2e−jϕe−jω0tA \cos (\omega0 t + \phi) = \frac A2 e^{j \phi} e^{j \omega_0 t} + \frac A2 e^{-j \phi} e^{-j \omega_0 t}Acos(ω0t+ϕ)=2A​ejϕejω0​t+2A​e−jϕe−jω0​t

只要基础频率 ω0 \omega_0ω0​ 一样, 指数和三角函数可以互相转化

x(t)=Acos⁡(ω0t+ϕ) x(t) = A \cos (\omega _0 t + \phi )x(t)=Acos(ω0​t+ϕ)

unit: ω0\omega_0ω0​

radians: ϕ\phiϕ

phase: ω0t+ϕ \omega_0t+\phiω0​t+ϕ

δ[n] \delta[n]δ[n]

只有 n=0n=0n=0有正值

u[n] u[n]u[n]

只有 n≥0 n \geq 0 n≥0有正值

u[n] u[n] u[n]相当于 δ[n]\delta[n]δ[n]的积分

x[n]→y[n] x[n] \to y[n] x[n]→y[n] then x[n−n0]→y[n−n0] x[n-n_0] \to y[n-n_0] x[n−n0​]→y[n−n0​]

If x[n]→y[n] x[n] \to y[n] x[n]→y[n] then ax1[n]+bx2[n]→ay1[n]+by2[n] ax_1[n] + bx_2[n] \to ay_1[n] + b y_2 [n] ax1​[n]+bx2​[n]→ay1​[n]+by2​[n]

x[n]=0→y[n]=2x[n]=0x[ n]=0 \to y[n] = 2x[n] = 0 x[n]=0→y[n]=2x[n]=0

We can construct any signal by discrete function y[n]=x[k]δ[n−k] y[n] = x[k]\delta[n-k] y[n]=x[k]δ[n−k], so that y[n]y[n]y[n]can be valued only at kkk and get x[k]x[k] x[k]. By accumulation, the signal can be piled up. This is convolution, defined by x[n]=Σk=−∞∞x[k]δ[n−k]=x[n]∗h[n] x[n] =\Sigma _{k = - \infty} ^ {\infty} x[k]\delta[n-k] = x[n] * h[n] x[n]=Σk=−∞∞​x[k]δ[n−k]=x[n]∗h[n], which is called convolution sum.

Commutative: x(t)∗h(t)=h(t)∗x(t)x(t)∗h(t) =h(t)∗x(t)x(t)∗h(t)=h(t)∗x(t)

Bi-linear: (ax1(t)+bx2(t))∗h(t)=a(x1∗h)+b(x2∗h),x∗(ah1+bh2)=a(x∗h1)+b(x∗h2)(ax_1(t) +bx_2(t))∗h(t) =a(x_1∗h) +b(x_2∗h),x∗(ah_1+bh_2) =a(x∗h_1) +b(x∗h_2)(ax1​(t)+bx2​(t))∗h(t)=a(x1​∗h)+b(x2​∗h),x∗(ah1​+bh2​)=a(x∗h1​)+b(x∗h2​)

Shift: x(t−τ)∗h(t)=x(t)∗h(t−τ)x(t−τ)∗h(t) =x(t)∗h(t−τ)x(t−τ)∗h(t)=x(t)∗h(t−τ)

Identity: δ(t) is the identity signal, x∗δ=x=δ∗x x∗δ=x=δ∗x x∗δ=x=δ∗x

Identity is unique: i(t)=i(t)∗δ(t)=δ(t)i(t) =i(t)∗δ(t) =δ(t)i(t)=i(t)∗δ(t)=δ(t)

Associative: x1∗(x2∗x3)=(x1∗x2)∗x3x_1∗(x_2∗x_3) = (x_1∗x_2)∗x_3x1​∗(x2​∗x3​)=(x1​∗x2​)∗x3​

Smooth derivative: y(n)′=x(n)′∗h(n)=x(n)∗h(n)′ y(n)' = x(n)'*h(n) = x(n)*h(n)' y(n)′=x(n)′∗h(n)=x(n)∗h(n)′

Memoryless: x[n]≠0x[n] \neq 0x[n]=0 when n=0 n = 0 n=0

Causality: h[n]=0h[n] = 0 h[n]=0 when n<0 n < 0 n<0

y[n]=∑k=0∞h[k]×[n−k] y[n]=\sum_{k=0}^{\infty} h[k] \times[n-k] y[n]=∑k=0∞​h[k]×[n−k]

(∑k=−∞∞∣h[k]∣<∞) ( \sum_{k=-\infty}^{\infty}|h[k]|<\infty ) (∑k=−∞∞​∣h[k]∣<∞)

Convolving δ(t)δ(t)δ(t) with itself

(δ(t)∗δ(t))=δ(t) ( \delta(t) * \delta(t)) = \delta(t) (δ(t)∗δ(t))=δ(t)

Reverse the simpler one x(t)x(t) x(t)

Record reversed x(t)x(t)x(t) 's jumping points

Slide reversed x(t)x(t)x(t), for each g(t)g(t)g(t), is integral of multiplication

g(t)=∫ddtx∗h(t)dt=… g(t)=\int \frac{d}{d t} x * h(t) d t=\ldots g(t)=∫dtd​x∗h(t)dt=…

Consider the input to be x(t)=est x(t) = e^{st}x(t)=est, then the output is y(t)=∫h(τ)es(t−τ)dτ=est∫h(τ)e−sτdτ y(t)=\int h(\tau) e^{s(t-\tau)} d \tau=e^{s t} \int h(\tau) e^{-s \tau} d \tauy(t)=∫h(τ)es(t−τ)dτ=est∫h(τ)e−sτdτ

\begin{equation} H(s)=\int h(\tau) e^{-s \tau} d \tau \end{equation} is just a constant, i.e. eigenvalue for function est e^{st}est

When s ss purely imaginary \begin{equation} j k \omega_{0} \end{equation}, \begin{equation} e^{j k \omega_{0} t} \end{equation} is orthonormal and standard among different .

CT: ejωt e^{j\omega t } ejωt

DT: ejωn e^{j \omega n }ejωn

x(t)x(t)x(t) may be expressed as a Fourier series:

Where aka_kak​ can be obtained by

a0a_0a0​ controls a constant

x(t)=∑k=−∞∞akejkω0t=∑k=−∞∞(akcos⁡(kω0t)+jaksin⁡(kω0t))=a0+∑k>0((ak+a−k)cos⁡(kω0t)+j(ak−a−k)sin⁡(kω0t))\begin{aligned} x(t) &=\sum_{k=-\infty}^{\infty} a_{k} e^{j k \omega_{0} t} \\ &=\sum_{k=-\infty}^{\infty}\left(a_{k} \cos \left(k \omega_{0} t\right)+j a_{k} \sin \left(k \omega_{0} t\right)\right) \\ &=a_{0}+\sum_{k>0}\left(\left(a_{k}+a_{-k}\right) \cos \left(k \omega_{0} t\right)+j\left(a_{k}-a_{-k}\right) \sin \left(k \omega_{0} t\right)\right) \end{aligned}x(t)​=k=−∞∑∞​ak​ejkω0​t=k=−∞∑∞​(ak​cos(kω0​t)+jak​sin(kω0​t))=a0​+k>0∑​((ak​+a−k​)cos(kω0​t)+j(ak​−a−k​)sin(kω0​t))​
z(t)=αx(t)+βy(t)⟵S⟷⟶αak+βbkz(t)=\alpha x(t)+\beta y(t) \underset{\longleftrightarrow}{\longleftarrow S}{\longrightarrow} \alpha a_{k}+\beta b_{k}z(t)=αx(t)+βy(t)⟷⟵S​⟶αak​+βbk​
x(t−t0)⟷FSe−jkω0t0akx\left(t-t_{0}\right) \stackrel{F S}{\longleftrightarrow} e^{-j k \omega_{0} t_{0}} a_{k}x(t−t0​)⟷FS​e−jkω0​t0​ak​
x(−t)⟵FSa−kx(-t) \stackrel{F S}{\longleftarrow} a_{-k}x(−t)⟵FS​a−k​
x(αt)=∑k=−∞∞akejk(αω0)tx(\alpha t)=\sum_{k=-\infty}^{\infty} a_{k} e^{j k\left(\alpha \omega_{0}\right) t}x(αt)=k=−∞∑∞​ak​ejk(αω0​)t
x(t)y(t)⟷S⟷,hk=∑l=−∞∞albk−1x(t) y(t) \underset{\longleftrightarrow}{\longleftrightarrow S}, h_{k}=\sum_{l=-\infty}^{\infty} a_{l} b_{k-1}x(t)y(t)⟷⟷S​,hk​=l=−∞∑∞​al​bk−1​
dx(t)dt⟷FS,jkω0ak\frac{d x(t)}{d t} \longleftrightarrow F S, j k \omega_{0} a_{k}dtdx(t)​⟷FS,jkω0​ak​
∫−∞tx(τ)dτ FS ⟷,akjkω0\int_{-\infty}^{t} x(\tau) d \tau \underset{\longleftrightarrow}{\text { FS }}, \frac{a_{k}}{j k \omega_{0}}∫−∞t​x(τ)dτ⟷ FS ​,jkω0​ak​​
1T∫T∣x(t)∣2dt=∑k=−∞∞∣ak∣2\frac{1}{T} \int_{T}|x(t)|^{2} d t=\sum_{k=-\infty}^{\infty}\left|a_{k}\right|^{2}T1​∫T​∣x(t)∣2dt=k=−∞∑∞​∣ak​∣2
1T∫T∣x(t)∣2dt=1T∫T∑k1,k2ak1ak2∗ej(k1−k2)ω0tdt=∑k1,k2ak1ak2∗δ[k1−k2]=∑k∣ak∣2\begin{aligned} \frac{1}{T} \int_{T}|x(t)|^{2} d t &=\frac{1}{T} \int_{T} \sum_{k_{1}, k_{2}} a_{k_{1}} a_{k_{2}}^{*} e^{j\left(k_{1}-k_{2}\right) \omega_{0} t} d t \\ &=\sum_{k_{1}, k_{2}} a_{k_{1}} a_{k_{2}}^{*} \delta\left[k_{1}-k_{2}\right] \\ &=\sum_{k}\left|a_{k}\right|^{2} \end{aligned}T1​∫T​∣x(t)∣2dt​=T1​∫T​k1​,k2​∑​ak1​​ak2​∗​ej(k1​−k2​)ω0​tdt=k1​,k2​∑​ak1​​ak2​∗​δ[k1​−k2​]=k∑​∣ak​∣2​

Inverse fourier transform: x(t)=12π∫∞∞X(jω)ejωt dω x(t)=\frac{1}{2 \pi} \int_{\infty}^{\infty} X(j \omega) \mathrm{e}^{j \omega t} \mathrm{~d} \omega x(t)=2π1​∫∞∞​X(jω)ejωt dω

\begin{equation} X(j \omega) \end{equation}is called the "Spectrum" of x(t) x(t)x(t)

Means that when we put the \begin{equation} X(j \omega) \end{equation} in time domain, it will produce 2π 2\pi 2πtimes x(−t) x(-t) x(−t) waveform in freq domain

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Summary from UCB
A kind of signal
Fundamantal Frequency, 角频率越大, 振荡越大
Origin Signal
pile up
奇偶特性