# 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
$x(t) = Even(x) + Odd(t)$
,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

### Sin

$x(t) = A \cos (\omega _0 t + \phi )$
• unit:
$\omega_0$
$\phi$
• phase:
$\omega_0t+\phi$ Fundamantal Frequency， 角频率越大， 振荡越大

### 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 $u[n]$

$\delta[n]$

### 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

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: Origin Signal pile up

### 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
$n < 0$
• $y[n]=\sum_{k=0}^{\infty} h[k] \times[n-k]$
• Stability:
• $( \sum_{k=-\infty}^{\infty}|h[k]|<\infty )$
• Convolving
$δ(t)$
with itself
• $( \delta(t) * \delta(t)) = \delta(t)$

### Calculation

Sliding window:
1. 1.
Reverse the simpler one
$x(t)$
2. 2.
Record reversed
$x(t)$
's jumping points
3. 3.
Slide reversed
$x(t)$
, for each
$g(t)$
, is integral of multiplication
$g(t)=\int \frac{d}{d t} x * h(t) d t=\ldots$

### 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. 1.
Consider the input to be
$x(t) = e^{st}$
, then the output is
$y(t)=\int h(\tau) e^{s(t-\tau)} d \tau=e^{s t} \int h(\tau) e^{-s \tau} d \tau$
2. 2.
is just a constant, i.e. eigenvalue for function
$e^{st}$

### Orthonormal Basis

When
$s$
purely imaginary
,
is orthonormal and standard among different .
• Definition of inner-product of perioidic functions:

## Fourier Analysis

### CT & DT

• CT:
$e^{j\omega t }$
• 纯虚数
• DT:
$e^{j \omega n }$

### Periodic signals & Fourier Series Expansion

$x(t)$
may be expressed as a Fourier series:
,
sum of sinusoids whose frequencies are multiple of ω0, the “fundamental frequency”.
Where
$a_k$
can be obtained by
• 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:
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:
• For periodic signals,
, i.e. the Fourier Series
• is called the "Spectrum" of
$x(t)$
Inverse F.T.

### Dual Porperty

Means that when we put the
in time domain, it will produce
$2\pi$
times
$x(-t)$
waveform in freq domain

### Properties  ### Normal CT Fourier Pairs  