EE 150: Signals and Systems

ShanghaiTech University - Spring 2021
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Summary from UCB
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
A kind of signal
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
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)
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​Even(t)=12(x(t)+x(−t)) Even(t) = \frac12 ( x(t) + x(-t) ) Even(t)=21​(x(t)+x(−t))
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​Odd(t)=12(x(t)−x(−t)) Odd(t) = \frac12 ( x(t) - x(-t) )Odd(t)=21​(x(t)−x(−t))
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Periodic
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
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如果是合成的 signal, 其 Fundamental Period 是最小公倍数
Aperiodic: Non-preodic
Eular's Formula
​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)
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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
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只要基础频率 ω0 \omega_0ω0​
 一样， 指数和三角函数可以互相转化
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Sin
​x(t)=Acos⁡(ω0t+ϕ) x(t) = A \cos (\omega _0 t + \phi )x(t)=Acos(ω0​t+ϕ)
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unit: ω0\omega_0ω0​
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radians: ϕ\phiϕ
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phase: ω0t+ϕ \omega_0t+\phiω0​t+ϕ
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Fundamantal Frequency， 角频率越大， 振荡越大
Discrete Time Unit Step and Unit Impulse Sequence
​δ[n] \delta[n]δ[n]
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只有 n=0n=0n=0
有正值
​u[n] u[n]u[n]
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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
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
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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
​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​]
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Example:
Linearity
Additivity and Scaling
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]
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If linear, zero input gives zero output 
​x[n]=0→y[n]=2x[n]=0x[ n]=0 \to y[n] = 2x[n] = 0 x[n]=0→y[n]=2x[n]=0
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Convolution
Begin
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.
Example:
Origin Signal
pile up
Properties of Convolution
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)
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交换律
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​)
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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)
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Identity: δ(t) is the identity signal,  x∗δ=x=δ∗x x∗δ=x=δ∗x x∗δ=x=δ∗x
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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)
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Associative: x1∗(x2∗x3)=(x1∗x2)∗x3x_1∗(x_2∗x_3) = (x_1∗x_2)∗x_3x1​∗(x2​∗x3​)=(x1​∗x2​)∗x3​
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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)′
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Properties of L.T.I System
Memoryless: x[n]≠0x[n] \neq 0x[n]=0
 when n=0 n = 0 n=0
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Invertibility: A system is invertible only if an inverse system exists.
Causality: h[n]=0h[n] = 0 h[n]=0
 when  n<0 n < 0 n<0
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​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]
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Stability: 
 (∑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)
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Calculation
Sliding window: 
1.Reverse the simpler one x(t)x(t) x(t)
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2.Record reversed x(t)x(t)x(t)
 's jumping points
3.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=…
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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
1.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
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2.​
 is just a constant, i.e. eigenvalue for function est e^{st}est
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Orthonormal Basis
When s ss
 purely imaginary 
, 
 is orthonormal and standard among different .
Definition of inner-product of perioidic functions: 
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Fourier Analysis
CT & DT
CT: ejωt e^{j\omega t } ejωt
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纯虚数
DT: ejωn e^{j \omega n }ejωn
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Periodic signals & Fourier Series Expansion
​x(t)x(t)x(t)
 may be expressed as a Fourier series:
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, 
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sum of sinusoids whose frequencies are multiple of ω0, the “fundamental frequency”.
Where aka_kak​
 can be obtained by
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Case 0 is often special!
​a0a_0a0​
 controls a constant 
And, 
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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))​
Odd / Even
奇偶特性
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Approximation
Linearity
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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​
Time-shift
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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​
Time-reverse
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x(−t)⟵FSa−kx(-t) \stackrel{F S}{\longleftarrow} a_{-k}x(−t)⟵FS​a−k​
Time-scaling
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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
Multiplication
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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​
which is Convolution
conjugation & conjugate symmetry
7
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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​
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∫−∞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​​
Parseval's identity
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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
Proof:
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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​
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Continuous-Time Fourier Transform (CTFT)
变换与逆变换
Fourier series: 
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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
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Fourier Transform Pair
Fourier Transform: 
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For periodic signals, 
, i.e. the Fourier Series
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is called the "Spectrum" of x(t) x(t)x(t)
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Inverse F.T.  
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Dual Porperty
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Means that when we put the 
 in time domain, it will produce 2π 2\pi 2π
times x(−t) x(-t) x(−t)
 waveform in freq domain
Properties
Normal CT Fourier Pairs
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REF
​https://www.josehu.com/assets/file/signals-systems.pdf​
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