# 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 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: $\omega_0$

• radians: $\phi$

• 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

$u[n]$相当于 $\delta[n]$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

## 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: 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. Reverse the simpler one $x(t)$

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

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

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

# 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 $x(t) = e^{st}$, 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. 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 奇偶特性

### 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

### 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