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

    • δΏ‘ε·ζ˜―δΈ€η³»εˆ—η‹¬η«‹ε˜ι‡

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

    • η³»η»Ÿζ˜―ε…ΆδΈ€δΈͺδΏ‘ε·θΏ›οΌŒ δΈ€δΈͺ俑号出

A kind of signal

Transformation

  • Time reflection: x(t) ←→ x(βˆ’t), x[n] ←→ x[βˆ’n]

  • Time scaling: x(t) ←→ x(ct)

  • Time shift: x(t) ←→ x(t βˆ’ t0), x[n] ←→ x[n βˆ’ 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) ,then

​x(βˆ’t)=Even(βˆ’t)+Odd(βˆ’t)=Even(t)βˆ’Odd(t) x(-t) = Even(-t) + Odd(-t) = Even(t) - Odd(t) ​

​Even(t)=12(x(t)+x(βˆ’t)) Even(t) = \frac12 ( x(t) + x(-t) ) ​

​Odd(t)=12(x(t)βˆ’x(βˆ’t)) Odd(t) = \frac12 ( x(t) - x(-t) )​

Periodic

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

    • Fundamental period TT: Smallest positive TT​

    • ε¦‚ζžœζ˜―εˆζˆηš„ signal, ε…Ά Fundamental Period ζ˜―ζœ€ε°ε…¬ε€ζ•°

  • Aperiodic: Non-preodic

Eular's Formula

​ejΟ‰0t=cos⁑(Ο‰0t)+jβ‹…sin⁑(Ο‰0t) e^{j \omega0 t} = \cos ( \omega0 t ) + j \cdot \sin (\omega_0 t) ​

  • Fundamental period T0=2Ο€/βˆ£Ο‰0∣T_0 = 2 \pi / \mid \omega _0 \mid

  • ​Acos⁑(Ο‰0t+Ο•)=A2ejΟ•ejΟ‰0t+A2eβˆ’jΟ•eβˆ’jΟ‰0tA \cos (\omega0 t + \phi) = \frac A2 e^{j \phi} e^{j \omega_0 t} + \frac A2 e^{-j \phi} e^{-j \omega_0 t}​

  • εͺθ¦εŸΊη‘€ι’‘ηŽ‡ Ο‰0 \omega_0 δΈ€ζ ·οΌŒ ζŒ‡ζ•°ε’ŒδΈ‰θ§’ε‡½ζ•°ε―δ»₯δΊ’η›Έθ½¬εŒ–

  • ​​

  • ​​

  • ​​

Sin

​x(t)=Acos⁑(Ο‰0t+Ο•) x(t) = A \cos (\omega _0 t + \phi )​

  • unit: Ο‰0\omega_0​

  • radians: Ο•\phi​

  • phase: Ο‰0t+Ο• \omega_0t+\phi​

Fundamantal Frequency, θ§’ι’‘ηŽ‡θΆŠε€§οΌŒ 振荑袊倧

Discrete Time Unit Step and Unit Impulse Sequence

  • ​δ[n] \delta[n]​

    • εͺζœ‰ n=0n=0ζœ‰ζ­£ε€Ό

  • ​u[n] u[n]​

    • εͺζœ‰ nβ‰₯0 n \geq 0 ζœ‰ζ­£ε€Ό

​u[n] u[n] η›Έε½“δΊŽ Ξ΄[n]\delta[n]ηš„η§―εˆ†

εΈΈθ§δΏ‘ε·δΈŽε‘¨ζœŸεˆ€ζ–­

εŠŸηŽ‡δΈŽθƒ½ι‡

Other concepts

  • Energy and Power of Periodic Signals: η§―εˆ†δΈŽη§―εˆ†εŽηš„ε€„η†

  • 谐振: εŒδΈ€δΈͺθ§’ι’‘ηŽ‡ηš„ι›†εˆ

  • ​

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]β†’y[n] x[n] \to y[n] then x[nβˆ’n0]β†’y[nβˆ’n0] x[n-n_0] \to y[n-n_0] ​

Example:

Linearity

Additivity and Scaling

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

If linear, zero input gives zero output

​x[n]=0β†’y[n]=2x[n]=0x[ n]=0 \to y[n] = 2x[n] = 0 ​

Convolution

Begin

We can construct any signal by discrete function y[n]=x[k]Ξ΄[nβˆ’k] y[n] = x[k]\delta[n-k] , so that y[n]y[n]can be valued only at kk and get x[k]x[k] . By accumulation, the signal can be piled up. This is convolution, defined by x[n]=Ξ£k=βˆ’βˆžβˆžx[k]Ξ΄[nβˆ’k]=x[n]βˆ—h[n] x[n] =\Sigma _{k = - \infty} ^ {\infty} x[k]\delta[n-k] = x[n] * h[n] , which is called convolution sum.

Example:

Origin Signal
pile up

Properties of Convolution

  • Commutative: x(t)βˆ—h(t)=h(t)βˆ—x(t)x(t)βˆ—h(t) =h(t)βˆ—x(t)​

    • 亀捒律

  • Bi-linear: (ax1(t)+bx2(t))βˆ—h(t)=a(x1βˆ—h)+b(x2βˆ—h),xβˆ—(ah1+bh2)=a(xβˆ—h1)+b(xβˆ—h2)(ax_1(t) +bx_2(t))βˆ—h(t) =a(x_1βˆ—h) +b(x_2βˆ—h),xβˆ—(ah_1+bh_2) =a(xβˆ—h_1) +b(xβˆ—h_2)​

  • Shift: x(tβˆ’Ο„)βˆ—h(t)=x(t)βˆ—h(tβˆ’Ο„)x(tβˆ’Ο„)βˆ—h(t) =x(t)βˆ—h(tβˆ’Ο„)​

  • Identity: Ξ΄(t) is the identity signal, xβˆ—Ξ΄=x=Ξ΄βˆ—x xβˆ—Ξ΄=x=Ξ΄βˆ—x ​

    • Identity is unique: i(t)=i(t)βˆ—Ξ΄(t)=Ξ΄(t)i(t) =i(t)βˆ—Ξ΄(t) =Ξ΄(t)​

  • Associative: x1βˆ—(x2βˆ—x3)=(x1βˆ—x2)βˆ—x3x_1βˆ—(x_2βˆ—x_3) = (x_1βˆ—x_2)βˆ—x_3​

  • Smooth derivative: y(n)β€²=x(n)β€²βˆ—h(n)=x(n)βˆ—h(n)β€² y(n)' = x(n)'*h(n) = x(n)*h(n)' ​

Properties of L.T.I System

  • Memoryless: x[n]β‰ 0x[n] \neq 0 when n=0 n = 0 ​

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

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

    • ​y[n]=βˆ‘k=0∞h[k]Γ—[nβˆ’k] y[n]=\sum_{k=0}^{\infty} h[k] \times[n-k] ​

    • ​​

  • Stability:

    • (βˆ‘k=βˆ’βˆžβˆžβˆ£h[k]∣<∞) ( \sum_{k=-\infty}^{\infty}|h[k]|<\infty )

  • Convolving Ξ΄(t)Ξ΄(t) with itself

    • ​(Ξ΄(t)βˆ—Ξ΄(t))=Ξ΄(t) ( \delta(t) * \delta(t)) = \delta(t) ​

Calculation

Sliding window:

  1. Reverse the simpler one x(t)x(t) ​

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

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

​g(t)=∫ddtxβˆ—h(t)dt=… g(t)=\int \frac{d}{d t} x * h(t) d t=\ldots ​

​​

ι€†ε˜ζ’

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}, then the output is y(t)=∫h(Ο„)es(tβˆ’Ο„)dΟ„=est∫h(Ο„)eβˆ’sΟ„dΟ„ y(t)=\int h(\tau) e^{s(t-\tau)} d \tau=e^{s t} \int h(\tau) e^{-s \tau} d \tau​

  2. ​ is just a constant, i.e. eigenvalue for function est e^{st}​

Orthonormal Basis

When s s purely imaginary , is orthonormal and standard among different .

  • Definition of inner-product of perioidic functions: ​

Fourier Analysis

CT & DT

  • CT: ejΟ‰t e^{j\omega t } ​

    • ​​

    • ηΊ―θ™šζ•°

  • DT: ejΟ‰n e^{j \omega n }​

    • ​​

Periodic signals & Fourier Series Expansion

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

​, ​

sum of sinusoids whose frequencies are multiple of Ο‰0, the β€œfundamental frequency”.

Where aka_k can be obtained by

​​

  • Case 0 is often special!

  • ​a0a_0 controls a constant

And,

​

x(t)=βˆ‘k=βˆ’βˆžβˆžakejkΟ‰0t=βˆ‘k=βˆ’βˆžβˆž(akcos⁑(kΟ‰0t)+jaksin⁑(kΟ‰0t))=a0+βˆ‘k>0((ak+aβˆ’k)cos⁑(kΟ‰0t)+j(akβˆ’aβˆ’k)sin⁑(kΟ‰0t))\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}

Odd / Even

ε₯‡εΆη‰Ήζ€§

​

Approximation

Linearity

​

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}

Time-shift

​

x(tβˆ’t0)⟷FSeβˆ’jkΟ‰0t0akx\left(t-t_{0}\right) \stackrel{F S}{\longleftrightarrow} e^{-j k \omega_{0} t_{0}} a_{k}

Time-reverse

​

x(βˆ’t)⟡FSaβˆ’kx(-t) \stackrel{F S}{\longleftarrow} a_{-k}

Time-scaling

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x(Ξ±t)=βˆ‘k=βˆ’βˆžβˆžakejk(Ξ±Ο‰0)tx(\alpha t)=\sum_{k=-\infty}^{\infty} a_{k} e^{j k\left(\alpha \omega_{0}\right) t}

Multiplication

​

x(t)y(t)⟷S⟷,hk=βˆ‘l=βˆ’βˆžβˆžalbkβˆ’1x(t) y(t) \underset{\longleftrightarrow}{\longleftrightarrow S}, h_{k}=\sum_{l=-\infty}^{\infty} a_{l} b_{k-1}

which is Convolution

conjugation & conjugate symmetry

7

​

dx(t)dt⟷FS,jkΟ‰0ak\frac{d x(t)}{d t} \longleftrightarrow F S, j k \omega_{0} a_{k}

​

βˆ«βˆ’βˆžtx(Ο„)dτ FS ⟷,akjkΟ‰0\int_{-\infty}^{t} x(\tau) d \tau \underset{\longleftrightarrow}{\text { FS }}, \frac{a_{k}}{j k \omega_{0}}

Parseval's identity

​

1T∫T∣x(t)∣2dt=βˆ‘k=βˆ’βˆžβˆžβˆ£ak∣2\frac{1}{T} \int_{T}|x(t)|^{2} d t=\sum_{k=-\infty}^{\infty}\left|a_{k}\right|^{2}

Proof:

​

1T∫T∣x(t)∣2dt=1T∫Tβˆ‘k1,k2ak1ak2βˆ—ej(k1βˆ’k2)Ο‰0tdt=βˆ‘k1,k2ak1ak2βˆ—Ξ΄[k1βˆ’k2]=βˆ‘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}

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Continuous-Time Fourier Transform (CTFT)

ε˜ζ’δΈŽι€†ε˜ζ’

Fourier series: ​

Inverse fourier transform: x(t)=12Ο€βˆ«βˆžβˆžX(jΟ‰)ejΟ‰tΒ dΟ‰ x(t)=\frac{1}{2 \pi} \int_{\infty}^{\infty} X(j \omega) \mathrm{e}^{j \omega t} \mathrm{~d} \omega ​

Fourier Transform Pair

Fourier Transform: ​

  • For periodic signals, , i.e. the Fourier Series

  • ​is called the "Spectrum" of x(t) x(t)​

Inverse F.T. ​

Dual Porperty

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Means that when we put the in time domain, it will produce 2Ο€ 2\pi times x(βˆ’t) x(-t) waveform in freq domain

Properties

Normal CT Fourier Pairs

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REF

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