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


  • 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) ,then

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) )

Odd(t)=12(x(t)x(t)) Odd(t) = \frac12 ( x(t) - x(-t) )


  • 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

    • 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)+jsin(ω0t) e^{j \omega0 t} = \cos ( \omega0 t ) + j \cdot \sin (\omega_0 t)

  • Fundamental period T0=2π/ω0T_0 = 2 \pi / \mid \omega _0 \mid

  • Acos(ω0t+ϕ)=A2ejϕejω0t+A2ejϕejω0tA \cos (\omega0 t + \phi) = \frac A2 e^{j \phi} e^{j \omega_0 t} + \frac A2 e^{-j \phi} e^{-j \omega_0 t}

  • 只要基础频率 ω0 \omega_0 一样, 指数和三角函数可以互相转化

  • \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}


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

  • unit: ω0\omega_0

  • radians: ϕ\phi

  • phase: ω0t+ϕ \omega_0t+\phi

Discrete Time Unit Step and Unit Impulse Sequence

  • δ[n] \delta[n]

    • 只有 n=0n=0有正值

  • u[n] u[n]

    • 只有 n0 n \geq 0 有正值

u[n] u[n] 相当于 δ[n]\delta[n]的积分



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


All memoryless are causal

Output only depends on input at the same time or before


Bounded input gives Bounded output


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

x[n]y[n] x[n] \to y[n] then x[nn0]y[nn0] x[n-n_0] \to y[n-n_0]



Additivity and Scaling

If x[n]y[n] x[n] \to 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]

If linear, zero input gives zero output

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



We can construct any signal by discrete function y[n]=x[k]δ[nk] 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]=Σk=x[k]δ[nk]=x[n]h[n] x[n] =\Sigma _{k = - \infty} ^ {\infty} x[k]\delta[n-k] = x[n] * h[n] , which is called convolution sum.


Properties of Convolution

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

    • 交换律

  • Bi-linear: (ax1(t)+bx2(t))h(t)=a(x1h)+b(x2h),x(ah1+bh2)=a(xh1)+b(xh2)(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)

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

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

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

  • Associative: x1(x2x3)=(x1x2)x3x_1∗(x_2∗x_3) = (x_1∗x_2)∗x_3

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

Properties of L.T.I System

  • Memoryless: x[n]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]=k=0h[k]×[nk] y[n]=\sum_{k=0}^{\infty} h[k] \times[n-k]

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

  • Stability:

    • (k=h[k]<) ( \sum_{k=-\infty}^{\infty}|h[k]|<\infty )

  • Convolving δ(t)δ(t) with itself

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


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)=ddtxh(t)dt= g(t)=\int \frac{d}{d t} x * h(t) d t=\ldots

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


Eigen-function of L.T.I


  • 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(τ)es(tτ)dτ=esth(τ)esτdτ y(t)=\int h(\tau) e^{s(t-\tau)} d \tau=e^{s t} \int h(\tau) e^{-s \tau} d \tau

  2. \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}

Orthonormal Basis

When s s 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 .

  • 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: ejωt e^{j\omega t }

    • \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

  • DT: ejωn e^{j \omega n }

    • \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

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

\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”.

Where aka_k can be obtained by

\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!

  • a0a_0 controls a constant


x(t)=k=akejkω0t=k=(akcos(kω0t)+jaksin(kω0t))=a0+k>0((ak+ak)cos(kω0t)+j(akak)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}

Odd / Even



z(t)=αx(t)+βy(t)Sαak+βbkz(t)=\alpha x(t)+\beta y(t) \underset{\longleftrightarrow}{\longleftarrow S}{\longrightarrow} \alpha a_{k}+\beta b_{k}


x(tt0)FSejkω0t0akx\left(t-t_{0}\right) \stackrel{F S}{\longleftrightarrow} e^{-j k \omega_{0} t_{0}} a_{k}


x(t)FSakx(-t) \stackrel{F S}{\longleftarrow} 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(t)y(t)S,hk=l=albk1x(t) y(t) \underset{\longleftrightarrow}{\longleftrightarrow S}, h_{k}=\sum_{l=-\infty}^{\infty} a_{l} b_{k-1}

which is Convolution

conjugation & conjugate symmetry


dx(t)dtFS,jkω0ak\frac{d x(t)}{d t} \longleftrightarrow F S, j k \omega_{0} a_{k}

tx(τ)dτ FS ,akjkω0\int_{-\infty}^{t} x(\tau) d \tau \underset{\longleftrightarrow}{\text { FS }}, \frac{a_{k}}{j k \omega_{0}}

Parseval's identity

1TTx(t)2dt=k=ak2\frac{1}{T} \int_{T}|x(t)|^{2} d t=\sum_{k=-\infty}^{\infty}\left|a_{k}\right|^{2}


1TTx(t)2dt=1TTk1,k2ak1ak2ej(k1k2)ω0tdt=k1,k2ak1ak2δ[k1k2]=kak2\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}

Continuous-Time Fourier Transform (CTFT)


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

Inverse fourier transform: x(t)=12πX(jω)ejωt dω 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: \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

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

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}

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


Normal CT Fourier Pairs


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