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

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

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

$Even(t) = \frac12 ( x(t) + x(-t) )$

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

Periodic

Preodic: $x(t) = x(t+mT)$ or $x[t] = x[t+mT]$
- Fundamental period $T$ : Smallest positive $T$
- 如果是合成的 signal, 其 Fundamental Period 是最小公倍数
Aperiodic: Non-preodic

Eular's Formula

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

Fundamental period $T_0 = 2 \pi / \mid \omega _0 \mid$
$A \cos (\omega0 t + \phi) = \frac A2 e^{j \phi} e^{j \omega_0 t} + \frac A2 e^{-j \phi} e^{-j \omega_0 t}$
只要基础频率 $\omega_0$ 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}

Sin

$x(t) = A \cos (\omega _0 t + \phi )$

unit: $\omega_0$
radians: $\phi$
phase: $\omega_0t+\phi$

Discrete Time Unit Step and Unit Impulse Sequence

$\delta[n]$
- 只有 $n=0$ 有正值

$u[n]$
- 只有 $n \geq 0$ 有正值

$u[n]$ 相当于 $\delta[n]$ 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

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

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

Example:

Linearity

Additivity and Scaling

If $x[n] \to y[n]$ then $ax_1[n] + bx_2[n] \to ay_1[n] + b y_2 [n]$

If linear, zero input gives zero output

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

Convolution

Begin

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

Example:

Properties of Convolution

Commutative: $x(t)∗h(t) =h(t)∗x(t)$
- 交换律
Bi-linear: $(ax_1(t) +bx_2(t))∗h(t) =a(x_1∗h) +b(x_2∗h),x∗(ah_1+bh_2) =a(x∗h_1) +b(x∗h_2)$
Shift: $x(t−τ)∗h(t) =x(t)∗h(t−τ)$
Identity: δ(t) is the identity signal, $x∗δ=x=δ∗x$
- Identity is unique: $i(t) =i(t)∗δ(t) =δ(t)$
Associative: $x_1∗(x_2∗x_3) = (x_1∗x_2)∗x_3$
Smooth derivative: $y(n)' = x(n)'*h(n) = x(n)*h(n)'$

Properties of L.T.I System

Memoryless: $x[n] \neq 0$ when $n = 0$
Invertibility: A system is invertible only if an inverse system exists.
Causality: $h[n] = 0$ when $n < 0$
- $y[n]=\sum_{k=0}^{\infty} h[k] \times[n-k]$
- \begin{equation} y[n]=\sum_{k=-\infty}^{n} x[k] h[n-k] \end{equation}
Stability:
- $( \sum_{k=-\infty}^{\infty}|h[k]|<\infty )$
Convolving $δ(t)$ with itself
- $( \delta(t) * \delta(t)) = \delta(t)$

Calculation

Sliding window:

Reverse the simpler one $x(t)$
Record reversed $x(t)$ 's jumping points
Slide reversed $x(t)$ , for each $g(t)$ , is integral of multiplication

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

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

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

Consider the input to be $x(t) = e^{st}$ , then the output is $y(t)=\int h(\tau) e^{s(t-\tau)} d \tau=e^{s t} \int h(\tau) e^{-s \tau} d \tau$
\begin{equation} H(s)=\int h(\tau) e^{-s \tau} d \tau \end{equation} is just a constant, i.e. eigenvalue for function $e^{st}$

Orthonormal Basis

When $s$ 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 & DT

CT: $e^{j\omega t }$
- \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: $e^{j \omega n }$
- \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

$x(t)$ 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 $a_k$ 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!
$a_0$ controls a constant

And,

\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

Approximation

Linearity

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

Time-shift

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

Time-reverse

x(-t) \stackrel{F S}{\longleftarrow} a_{-k}

Time-scaling

x(\alpha t)=\sum_{k=-\infty}^{\infty} a_{k} e^{j k\left(\alpha \omega_{0}\right) t}

Multiplication

x(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

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

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

Parseval's identity

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

Proof:

\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)

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}

Inverse fourier transform: $x(t)=\frac{1}{2 \pi} \int_{\infty}^{\infty} X(j \omega) \mathrm{e}^{j \omega t} \mathrm{~d} \omega$

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 $x(t)$

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 $2\pi$ times $x(-t)$ waveform in freq domain