Signals and Systems Chapter 1

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发布时间：September 8, 2022
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<button class="btn m-b-xs btn-light btn-roundedbtn-addon" onclick='window.open("https://cdn.dabigu.xyz/100782181_p0.png","_blank")'><i class="fa fa-camera"></i>$\quad$ **ナヒーダ by 枸杞(産地直送)** </button>
# Chapter 1 An Overview

## Continuous-Time and Discrete-Time Signals

Signals describe a wide range of physical phenomena; **Mathematically**, signals are represented as functions of one or more variables. For example:

* Sound: represents acoustic pressure as a function of time $f(t)$
* Picture: represents brightness as a function of two spatial variables $f(x,y)$
* Video: consists of a sequence of images, called frames, and is a function of 3 variables: 2 spatial coordinate and time.

For signals, independent variables can be one or more; Generally we focus on signals which involve a **single** independent variable **time**. (although it may not represent time in specific applications)

**Continuous-Time signals**: the independent variable is continuous, and signals are defined for a **continuum** of values.

Represented as $x(t)$ where $t$ denotes the independent variable.

**Discrete-Time signals**: defines only at discrete times, and the independent variable only takes a discrete set of values. (e.g. closing price of a certain stock in one week)

Represented as $x[n]$, where $n$ denotes the independent variable.

For $x[n]$, it might be discrete or nature, or could be **sampling** of continuous time signal; It is defined only for **integer values** of $n$.

### Signal Energy and Power

Introduction: Let $v(t)$ and $i(t)$ be voltage and current across a resistor $R$, the instantaneous power is

$$
p(t)=v(t)i(t)=\frac{1}{R}v^2(t)
$$

So that the total energy over the time interval $t_1\leq t\leq t_2$ is

$$
E_{R}=\int_{t_1}^{t_2}p(t)\mathrm{d}t=\int_{t_1}^{t_2}\frac{1}{R}v^2(t)\mathrm{d}t
$$

And the average power over the interval $t_1\leq t\leq t_2$ is

$$
P_R=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}p(t)\mathrm{d}t=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}\frac{1}{R}v^2(t)\mathrm{d}t
$$

Similarly, for any signal $x(t)$ or $x[n]$, the **total energy** is defined as

$$
\begin{gather}
E=\int_{t_1}^{t_2}\left|x(t)\right|^2\mathrm{d}t,\quad t\in\left[t_1,t_2\right]\\
E=\sum_{n=n_1}^{n_2}\left|x[n]\right|^2,\quad t\in\left[t_1,t_2\right].
\end{gather}
$$

The **average power** is defined as

$\displaystyle P=\frac{E}{t_2-t_1}$ for continuous-time signal and $\displaystyle P=\frac{E}{n_2-n_1+1}$ for discrete-time signal.

Over **infinite** time interval $-\infty\leq t\leq \infty$ or $-\infty \leq n\leq \infty$, we have

$$
\left\{
\begin{array}{}
\displaystyle E_{\infty}\triangleq\lim_{T\rightarrow \infty}\int_{-T}^{T}\left|x(t)\right|^2\mathrm{d}t=\int_{-\infty}^{\infty}\left|x(t)\right|^2\mathrm{d}t\quad\text{Continuous}
\\
\displaystyle E_{\infty}\triangleq\lim_{N\rightarrow\infty}\sum_{n=-N}^{N}\left|x[n]\right|^2=\sum_{n=-\infty}^{\infty}\left|x[n]\right|^2\quad\text{Discrete}
\end{array}
\right.
$$

and

$$
\left\{\begin{array}{}
\displaystyle P_{\infty}\triangleq\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}\left|x(t)\right|^2\mathrm{d}t\quad\text{Continuous}\\
\displaystyle P_{\infty}\triangleq\lim_{N\rightarrow\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\left|x[n]\right|^2\quad\text{Discrete}
\end{array}\right.
$$

For any finite-energy signal $E_{\infty}<\infty$, its average power

$$
P_{\infty}=\lim_{T\rightarrow\infty}\frac{E_{\infty}}{2T}=0\quad\text{and}\quad P_{\infty}=\lim_{N\rightarrow\infty}\frac{E_{\infty}}{2N+1}=0.
$$

Finite-power signal(but not zero): $P_{\infty}<\infty, E_{\infty}=\infty$

Infinite energy & power signal: $P_{\infty}\rightarrow\infty,E_{\infty}\rightarrow\infty$.

## Transformations of the Independent Variable

### Time Shift

$$
x(t)\rightarrow y(t)=x(t-t_0)
$$

Relationship:

$$
y(t)\bigg|_{t=t_1}=x(t-t_0)\bigg|_{t=t_1}=x(t_1-t_0)=x(t)\bigg|_{t=t_1-t_0}.
$$

For discrete-time signal

$$
x[n]\rightarrow x[n-n_0].
$$

### Time Reversal

Flip along the axis where the independent variable $=0$.

$$
x(t)\rightarrow x(-t);\ x[n]\rightarrow x[-n]
$$

### Time Scaling

$$
\text{Compressed as }x(t)\rightarrow x(2t);\quad\text{Stretched as }x(t)\rightarrow x(t/2).
$$

### General Time Transformation

Combine all of the above as

$$
x(t)\rightarrow x(\alpha t+\beta)
$$

$\left|\alpha\right|>1$, compressed; $\left|\alpha\right|<1$, stretched; $\alpha<0$, reversed; $\beta\neq 0$, shifted.

### Periodic Signals

A continuous-time signal $x(t)$ is a periodic signal $\mathrm{iff.}$ there exists some $T$ $\mathrm{s.t.}$

$$
\forall t,\ x(t)=x(t+T).
$$

The smallest **positive** value of $T$ is called the **fundamental period** (基波周期) of the signal.

Similarly for discrete-time signals we have $x[n]=x[n+N]$ for all $n$.

### Even and Odd Signals

**Even Signal**:

$$
x(t)=x(-t)\quad x[n]=x[-n]
$$

**Odd Signal**:

$$
x(t)=-x(-t)\quad x[n]=-x[-n]
$$

**Any** Signal can be broken into a sum of two signals, one even and one odd, in the following method:

$$
x(t)=x_e(t)+x_o(t),\text{ where }\begin{cases}
x_e(t)=E_v\{x(t)\}=\displaystyle\frac{1}{2}[x(t)+x(-t)]\\
x_o(t)=O_d\{x(t)\}=\displaystyle\frac{1}{2}[x(t)-x(-t)]
\end{cases}.
$$

For discrete-time signals, it is similar that

$$
x[n]=x_e[n]+x_o[n],\text{ where }\begin{cases}
x_e[n]=\displaystyle\frac{x[n]+x[-n]}{2}\\
x_o[n]=\displaystyle\frac{x[n]-x[-n]}{2}
\end{cases}.
$$

## Exponential and Sinusoidal Signals

### Continuous-Time Complex Expo. and Sin. Signals

In the general case we have

$$
x(t)=Ce^{at}
$$

where $C$ and $a$ are complex number;

When $C$ and $a$ are real, it is a real exponential signal where

$$
\begin{cases}
a>0\Leftrightarrow x(t)\uparrow\text{ as }t\uparrow\\
a<0\Leftrightarrow x(t)\downarrow\text{ as }t\uparrow\\
a=0\Leftrightarrow x(t)\text{ is constant.}
\end{cases}
$$

**Periodic exponential signals**

In such case, $c$ is real, specifically $c=1$; $a$ is purely imaginary as $a=\omega_0j$

$$
x(t)=e^{j\omega_0t}
$$

To find its Fundamental period $T_0$, we have

$$
\begin{align}
x(t)&=e^{j\omega_0t}=e^{j\omega_0(t+T)}=e^{j\omega_
0t}\cdot e^{j\omega_0T}\Rightarrow e^{j\omega_0T}=1
\\&\Rightarrow \omega_0T=2k\pi,k\in\mathbb{Z}\setminus\{0\}
\\&\Rightarrow T=\frac{2k\pi}{\omega_0}
\\&\Rightarrow T_0=\frac{2\pi}{\left|\omega_0\right|}
\end{align}
$$

Notice: $T_0$ is undefined for $\omega_0=0$ as $x(t)$ is constant and any $T$ is its period.

**Sinusoidal Signals**

$$
x(t)=A\cos(\omega_0t+\phi)
$$

Which is closely related to complex exponential signals (Euler's Formula)

$$
\begin{align}
&\quad e^{j(\omega_0t+\phi)}=\cos(\omega_0t+\phi)+j\sin(\omega_0+\phi)
\\&\Leftrightarrow\begin{cases}
A\cos(\omega_0t+\phi)=A\cdot Re\{e^{j(\omega_0+\phi)}\}
\\A\sin(\omega_0t+\phi)=A\cdot Im\{e^{j(\omega_0+\phi)}\}
\end{cases}
\end{align}
$$

Its **fundamental frequency** (基波频率) is $\omega_0$.

Both $e^{j\omega_0}t$ and $A\cos(\omega_0t+\phi)$ has infinite total energy but finite average power:

$$
\begin{align}
\text{For }e^{j\omega_0t}&:
\begin{cases}
\displaystyle E_{period}=\int_0^{T_0}\left|e^{j\omega_0t}\right|^2\mathrm{d}t=\int_0^{T_0}1\mathrm{d}t=T_0
\\ P_{period}=\displaystyle\frac{1}{T_0}E_{period}=1
\end{cases}\\
\end{align}
$$

$$
\begin{align}
\text{For }A\cos(\omega_0t+\phi):E_{period}&=\int_0^{T_0}\left|A\cos(\omega_0+\phi)\right|^2\mathrm{d}t
\\&=A^2\int_0^{T_0}\cos^2(\omega_0t+\phi)\mathrm{d}t
\\{*}&=\frac{A^2}{2}\int_0^{T_0}1+\cos(2\omega_0t+2\phi)\mathrm{d}t
\\&=\frac{A^2}{2}T_0+\frac{A^2}{2}\int_0^{T_0}\cos(2\omega_0t+2\phi)\mathrm{d}t
\\&=\frac{A^2}{2}T_0+\frac{A^2}{2}\cdot\frac{1}{2\omega_0}\cdot\sin(2\omega_0t+2\phi)\bigg|_{0}^{T_0}
\\&=\frac{A^2}{2}T_0;
\\P_{period}&=\frac{1}{T_0}E_{period}=\frac{A^2}{2}.
\end{align}
$$

**Harmonically related (谐波) complex exponentials**

Sets of periodic exponentials (**with different frequencies**), all of which are periodic with a common period $T_0$

To have the common period $T_0$, they must satisfy that

$$
e^{j\omega t}=e^{j\omega(t+T_0)}=e^{j\omega t}\cdot e^{j\omega T_0}\Rightarrow \omega T_0=2k\pi\Rightarrow\omega=\frac{2k\pi}{T_0}=k\omega_0\Rightarrow \omega_0=\frac{2\pi}{T_0}
$$

We call

$$
\phi_k(t)=e^{jk\omega_0t},k\in\mathbb{Z}\setminus\{0\}
$$

a **harmonically related set**.

For any $k\neq 0$, they have fundamental frequency $\left|k\right|\omega_0$ and fundamental period $\displaystyle\frac{2\pi}{\left|k\right|\omega_0}=\frac{T_0}{\left|k\right|}$.

**Back to the general case**:

$$
x(t)=Ce^{at}
$$

Since that complex numbers $C,a$ can be written in the form

$$
C=\left|C\right|e^{j\theta},\ a=r+j\omega_0
$$

We have

$$
\begin{align}
Ce^{at}&=\left|C\right|e^{j\theta}e^{(r+j\omega_0t)}=\left|C\right|e^{rt}e^{j(\omega_0t+\theta)}
\\&=\left|C\right|e^{rt}\cos(\omega_0t+\theta)+j\left|C\right|e^{rt}\sin(\omega_0t+\theta)
\end{align}
$$

### Discrete-Time Complex Expo. and Sin. Signals

In the general case

$$
x[n]=C\alpha^n
$$

where $C$ and $\alpha$ are complex numbers;

> $x[n]$ can be expressed in exponential form as $x[n]=Ce^{\beta n}$ if we let $\alpha=e^{\beta}$ using Euler's Formula, but for discrete-time signals it is easier to handle in the former format.

**Sinusoidal signals**

In such case $C$ is real, specifically $C=1$; $\beta$ is purely imaginary:

$$
x[n]=e^{j\omega_0n}
$$

Which is closely related to $A\cos(\omega_0n+\phi)$ since that

$$
\begin{align}
e^{j\omega_0n}&=\cos\omega_0n+j\sin\omega_0n\\
A\cos(\omega_0n+\phi)&=\frac{A}{2}\cdot e^{j\theta}e^{j\omega_0n}+\frac{A}{2}\cdot e^{-j\theta}e^{-j\omega_0n}
\end{align}
$$

Similarly, it has infinite total energy but finite average power since

$$
\left|e^{j\omega_0}n\right|^2=1.
$$

**General Signals**

$$
\begin{align}
x[n]&=C\alpha^n\text{ where }C=\left|C\right|e^{j\theta},\alpha=\left|\alpha\right|e^{j\omega_0}
\\\Rightarrow x[n]&=\left|C\right|\left|\alpha\right|^n\cos(\omega_0n+\theta)+j\left|C\right|\left|\alpha\right|^n\sin(\omega_0n+\theta)
\end{align}
$$

**Periodicity properties**

Focusing on $\omega_0$ of $x[n]=e^{j\omega_0n}$:

It is obvious that $e^{j\omega_0n}$ has the same value at $\omega_0$ and $\omega_0+2k\pi$;

Only consider interval $0\leq\omega_0\leq 2\pi$:

> Notice that here we are considering different periods $\omega_0$ rather than one period of a specific signal.

From $0$ to $\pi$: when $\omega_0$ increases, the **oscillation rate** (振荡率) of $e^{j\omega_0n}$ **increases**;

From $\pi$ to $2\pi$: when $\omega_0$ increases, the oscillation rate of $e^{j\omega_0n}$ **decreases**.

The maximum oscillation rate is taken at $\omega_0=\pi$ where

$$
e^{j\pi n}=(e^{j\pi})^n=(-1)^n
$$

Then, we focus on $n$ of $x[n]=e^{j\omega_0n}$:

In order for $e^{j\omega_0n}$ to be periodic with $N>0$, it must satisfy that

$$
e^{j\omega_0(n+N)}=e^{j\omega_0N}\cdot e^{j\omega_0n}=e^{j\omega_0n}\Rightarrow\omega_0N=2m\pi\Rightarrow\frac{\omega_0}{2\pi}=\frac{m}{N}
$$

So that $\displaystyle\frac{\omega_0}{2\pi}$ must be a rational number; In such case,

Fundamental frequency: $\displaystyle\frac{2\pi}{N}=\frac{\omega_0}{m}$; Fundamental period: $N=\displaystyle\frac{2m\pi}{\omega_0}$.

## The Unit Impulse and Unit Step Functions

### Discrete-time Unit Impulse and Unit Step Sequences

**Unit Impulse** (unit sample, 单位冲击函数) is defined as

$$
\delta[n]=\begin{cases}0&n\neq0\\1&n=0\end{cases}
$$

**Unit Step** (单位阶跃函数) is defined as

$$
u[n]=\begin{cases}0&n<0\\1&n\geq0\end{cases}
$$

The impulse is the first difference (差分) of the step:

$$
\delta[n]=u[n]-u[n-1]
$$

Conversely, the step is the *running* sum of the step:

$$
u[n]=\sum_{m=-\infty}^n\delta[m]\rightarrow\begin{cases}n<0&\sum=0\\n\geq0&\sum=1\end{cases}
$$

Or let $m=n-k$, we have

$$
u[n]=\sum_{k=\infty}^0\delta[n-k]\quad\text{or}\quad\sum_{k=0}^{\infty}\delta[n-k].
$$

**Sampling Property**

$$
\begin{align}
&x[n]\delta[n]=x[0]\delta[n],\ \text{since }\delta[n]\text{ has value only when }n=0;\\
&\text{More Generally, }x[n]\delta[n-n_0]=x[n_0]\delta[n-n_0].
\end{align}
$$

### Continuous-time Unit Impulse and Unit Step Functions

Unit step is defined as

$$
u(t)=\begin{cases}0&t<0\\1&t>0\end{cases}
$$

> As we can see, according to the definition, the function is **discontinuous** at $t=0$.

Still, the continuous unit step $u(t)$ is the **running integral** of unit impulse $\delta(t)$:

$$
u(t)=\int_{-\infty}^t\delta(\tau)\mathrm{d}\tau
$$

And $\delta(t)$ is the first derivative of $u(t)$:

$$
\delta(t)=\frac{\mathrm{d}u(t)}{\mathrm{d}t}
$$

* Since $u(t)$ is discontinuous at $t=0$, How can we get $\delta(t)$?
  
  We can consider $\delta(t)$ as a **approximation** of some function $u_{\Delta}(t)$, whose value changes from $0$ to $1$ in a very short time $\Delta$.
  
  $$
  u(t)=\lim_{\Delta\rightarrow0}u_{\Delta}(t)\Rightarrow\delta_{\Delta}(t)=\frac{\mathrm{d}u_{\Delta}(t)}{\mathrm{d}t}\Rightarrow\delta(t)=\lim_{\Delta\rightarrow0}\delta_{\Delta}(t)
  $$
  
  Also shown in the figure below:
  
  <img src="https://cdn.dabigu.xyz/QQ截图20220914234113.png" width="600px">

The area of the impulse maintains to be $1$ which is the change of value for $u_{\Delta}(t)$; When $\Delta\rightarrow0$, the area is concentrated at $t=0$. From this interpretation, $\delta(t)$ can actually be written as

$$
\delta(t)=\begin{cases}1&t=0\\0&t\neq0\end{cases}
$$

which is exactly the same as $\delta[t]$ for discrete-time signals.

Or let $\sigma=t-\tau$, and we have

$$
u(t)=\int_0^{\infty}\delta(t-\sigma)\mathrm{d}\sigma.
$$

**Sampling Property**

Similarly we have

$$
x_1(t)=x(t)\delta_{\Delta}(t)\approx x(0)\delta_{\Delta}(t)\\
$$

The approximation is made since that $\Delta$ is so small here that we can consider $x(t)$ as a constant $x(0)$; So that we have the following result

$$
\begin{gather}
x(t)\delta(t)=\lim_{\Delta\rightarrow0}x(t)\delta_{\Delta}(t)=x(0)\lim_{\Delta\rightarrow0}\delta_{\Delta}(t)=x(0)\delta(t)
\\
\text{More Generally,}\ x(t)\delta(t-t_0)=x(t_0)\delta(t-t_0).
\end{gather}
$$

> Here the property is actually *very similar* to that of discrete-time signals; However the approach is different since $\delta(t)$ has a different definition here, $\mathrm{i.e.}$ $\delta(t)=\displaystyle\frac{\mathrm{d}u(t)}{\mathrm{d}t}$.

## Continuous-Time and Discrete Time Systems

For continuous-time systems, the Input & Output are continuous:

$$
x(t)\rightsquigarrow\text{System}\rightsquigarrow y(t)
$$

And for discrete-time systems, the Input & Output are discrete:

$$
x[t]\rightsquigarrow\text{System}\rightsquigarrow y[t]
$$

### **Example of Systems**

**RC Circuit**

For an RC circuit, we have

$$
\begin{cases}
i(t)=\displaystyle\frac{v_s(t)-v_c(t)}{R}&\text{current in circuit}\\
i(t)=C\cdot\displaystyle\frac{\mathrm{d}v_c(t)}{t}&\text{current through }C
\end{cases}\Rightarrow\frac{\mathrm{d}v_c(t)}{\mathrm{d}t}+\frac{1}{RC}v_c(t)=\frac{1}{RC}v_s(t).
$$

The latter equation is the relationship between input signal $v_s(t)$ and output signal $v_c(t)$.

**Balance in a bank account**

For balance in a bank account as an example of discrete-time system, we have

$$
y[n]=1.01y[n-1]+x[n]
$$

where $y[n]$ is the balance at the end of $n$th month, $x[n]$ is next deposit, and the interest rate is $1\%$.

The relationship between input $x[n]$ and output $y[n]$ is

$$
y[n]-1.01y[n-1]=x[n].
$$

**Interconnections of Systems**

### Basic System Properties

### System with and without Memory

**System without Memory **: Output is dependent **only** on the current input.

$$
\mathrm{e.g.}\ \begin{cases}y[n]=\big(2x[n]-x^2[n]\big)^2\\y(t)=Rx(t)\end{cases}\ ,\text{where the only variable is the current input }n/t.
$$

**System with Memory **: Output is dependent on the current **and** previous inputs.

$$
\mathrm{e.g.}\ \begin{cases}y[n]=\displaystyle\sum_{k=-\infty}^nx[k]\\y[n]=x[n-1]\rightarrow\text{延迟单元，存储输入并在下一时刻输出}\\y(t)=\displaystyle\frac{1}{C}\int_{-\infty}^tx(\tau)\mathrm{d}\tau\end{cases}
$$

**Memory** : retaining or storing information about input values at times

Specifically, for *physical systems*, memory is associated with the storage of energy.

### Invertibility and Inverse System

**Invertible** : Distinct inputs lead to distinct outputs. (单射, $f^{-1}$ exists)

For example, the Accumulator $\displaystyle y[n]=\sum_{k=-\infty}^n x[k]$, where the difference between two successive outputs is precisely the inputs $y[n]-y[n-1]=x[n]$.

**Noninvertible** : $f^{-1}$ does not exist, examples: $y[n]=0,\ y(t)=x^2(t).$

### Causality 系统的因果性

**Casual** : The output at any time depends only on the inputs at the **present time** and in the **past**. (But not in the future; In other words, the system has no ability for prediction.)

### Stability 系统的稳定性

**Informally**, small inputs lead to responses that do not diverge.

**Formally**, bounded input leads to bounded output.

Here a function $f(t)$ is bounded if such $B$ exist that $\left|f(t)\right|<B$.

Examples :

$$
\begin{align}
1.\quad y[n]&=\frac{1}{2M+1}\sum_{k=-M}^{+M}x[n-k]\text{ which is the Average on }n\in[-M,M]\\
&\Rightarrow\text{Bounded}\Rightarrow\text{Stable};
\\
2.\quad y[n]&=\sum_{k=-\infty}^nu[k]=(n+1)u[n]\rightarrow\infty\Rightarrow\text{Unstable}.
\end{align}
$$

## Time Invariance 时不变

**Time Invariant** : A time shift in the input signal results in an identical time shift in the output signal.

$$
\begin{align}
x[n]\rightsquigarrow y[n]&\Leftrightarrow x[n-n_0]\rightsquigarrow y[n-n_0];\\
x(t)\rightsquigarrow y(t)&\Leftrightarrow x(t-t_0)\rightsquigarrow y(t-t_0).
\end{align}
$$

Method for proving time invariance:

$$
\begin{cases}
x_2(t)=x_1(t-t_0)\\y_2(t)=f\{x_2(t)\}\\y_2'(t)=y_1(t-t_0)
\end{cases}\Rightarrow y_2(t)=y_2'(t)?
$$

Examples :

$$
\begin{align}
&1.\quad y(t)=\sin[x(t)]\\
&\text{If }x_2(t)=x_1(t-t_0),y_2(t)=\sin[x_1(t-t_0)]\\
&y_2'(t)=y_1(t-t_0)=\sin[x_1(t-t_0)]=y_2(t)\\
&\text{Time Invariant.}\\\\
&2.\quad y[n]=nx[n]\\
&\text{If }x_2[n]=x_1[n-n_0],y_2[n]=f\{x_2[n]\}=n\cdot x_1[n-n_0];\\
&y_2'[n]=y_1[n-n_0]=(n-n_0)x_1[n-n_0]\neq y_2[n].\\
&\text{Time Variant.}\\\\
&3.\quad y(t)=x(2t)\\
&\text{If }x_2(t)=x_1(t-t_0),y_2(t)=f\{x_2(t)\}=x_2(2t)=x_1(2t-t_0);\\
&y_2'(t)=y_1(t-t_0)=x_1(2t-2t_0)\neq y_2(t).\\
&\text{Time Variant.}\\
\end{align}
$$

> 也即，先进行变换$y(t)=f\{x(t)\}$再平移($x\rightarrow x-x_0$)与先平移再进行变换所得到的两种结果应该是完全一致的，否则不具有时不变性。

## Linearity

A system is linear if and only if it has superposition property, $\mathrm{i.e.}$ additivity(可加性) and homogeneity(齐次性).

$$
\begin{cases}
x_1(t)\rightsquigarrow y_1(t)\\
x_2(t)\rightsquigarrow y_2(t)
\end{cases}\Rightarrow ax_1(t)+bx_2(t)\rightsquigarrow ay_1(t)+by_2(t).
$$

Method for proving linearity:

$$
\begin{cases}
x_3(t)=ax_1(t)+bx_2(t)\\
y_3(t)=f\{x_3(t)\}\\
y_3'(t)=ay_1(t)+by_2(t)
\end{cases}\Rightarrow y_3(t)=y_3'(t)?
$$

Examples :

$$
\begin{align}
&1.\quad y(t)=tx(t)\\
&\text{If }x_3(t)=ax_1(t)+bx_2(t),y_3(t)=f\{x_3(t)\}=t[ax_1(t)+bx_2(t)];\\
&y_3'(t)=ay_1(t)+by_2(t)=atx_1(t)+btx_2(t)=y_3(t).\\
&\text{Linear.}\\\\
&2.\quad y(t)=x^2(t)\\
&\text{If }x_3(t)=ax_1(t)+bx_2(t),y_3(t)=f\{x_3(t)\}=[ax_1(t)+bx_2(t)]^2;\\
&y_3'(t)=ay_1(t)+by_2(t)=ax_1^2(t)+bx_2^2(t)\neq y_3(t).\\
&\text{Not Linear.}\\\\
&3.\quad y[n]=\mathrm{Re}\{x[n]\}\\
&\text{If }x_3[n]=ax_1[n]+bx_2[n],y_3[n]=f\{x_3[n]\}=\mathrm{Re}\{ax_1[n]+bx_2[n]\};\\
&y_3'[n]=ay_1[n]+by_2[n]=a\cdot\mathrm{Re}\{x_1[n]\}+b\cdot\mathrm{Re}\{x_2[n]\}\\
&y_3[n]\neq y_3'[n]\text{ if }a\text{ and }b\text{ are Complex numbers.}\\
&\text{Not Linear.}
\end{align}
$$

Last modification：September 15, 2022

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Signals and Systems Chapter 1

Bi_Gu • 2022 年 09 月 08 日

## Continuous-Time and Discrete-Time Signals

Signals describe a wide range of physical phenomena; **Mathematically**, signals are represented as functions of one or more variables. For example:

**Continuous-Time signals**: the independent variable is continuous, and signals are defined for a **continuum** of values.

Represented as $x(t)$ where $t$ denotes the independent variable.

**Discrete-Time signals**: defines only at discrete times, and the independent variable only takes a discrete set of values. (e.g. closing price of a certain stock in one week)

Represented as $x[n]$, where $n$ denotes the independent variable.

For $x[n]$, it might be discrete or nature, or could be **sampling** of continuous time signal; It is defined only for **integer values** of $n$.

### Signal Energy and Power

Introduction: Let $v(t)$ and $i(t)$ be voltage and current across a resistor $R$, the instantaneous power is

$$
p(t)=v(t)i(t)=\frac{1}{R}v^2(t)
$$

So that the total energy over the time interval $t_1\leq t\leq t_2$ is

$$
E_{R}=\int_{t_1}^{t_2}p(t)\mathrm{d}t=\int_{t_1}^{t_2}\frac{1}{R}v^2(t)\mathrm{d}t
$$

And the average power over the interval $t_1\leq t\leq t_2$ is

$$
P_R=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}p(t)\mathrm{d}t=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}\frac{1}{R}v^2(t)\mathrm{d}t
$$

Similarly, for any signal $x(t)$ or $x[n]$, the **total energy** is defined as

$$
\begin{gather}
E=\int_{t_1}^{t_2}\left|x(t)\right|^2\mathrm{d}t,\quad t\in\left[t_1,t_2\right]\\
E=\sum_{n=n_1}^{n_2}\left|x[n]\right|^2,\quad t\in\left[t_1,t_2\right].
\end{gather}
$$

The **average power** is defined as

$\displaystyle P=\frac{E}{t_2-t_1}$ for continuous-time signal and $\displaystyle P=\frac{E}{n_2-n_1+1}$ for discrete-time signal.

Over **infinite** time interval $-\infty\leq t\leq \infty$ or $-\infty \leq n\leq \infty$, we have

and

For any finite-energy signal $E_{\infty}<\infty$, its average power

$$
P_{\infty}=\lim_{T\rightarrow\infty}\frac{E_{\infty}}{2T}=0\quad\text{and}\quad P_{\infty}=\lim_{N\rightarrow\infty}\frac{E_{\infty}}{2N+1}=0.
$$

Finite-power signal(but not zero): $P_{\infty}<\infty, E_{\infty}=\infty$

Infinite energy & power signal: $P_{\infty}\rightarrow\infty,E_{\infty}\rightarrow\infty$.

## Transformations of the Independent Variable

### Time Shift

$$
x(t)\rightarrow y(t)=x(t-t_0)
$$

Relationship:

$$
y(t)\bigg|_{t=t_1}=x(t-t_0)\bigg|_{t=t_1}=x(t_1-t_0)=x(t)\bigg|_{t=t_1-t_0}.
$$

For discrete-time signal

$$
x[n]\rightarrow x[n-n_0].
$$

### Time Reversal

Flip along the axis where the independent variable $=0$.

$$
x(t)\rightarrow x(-t);\ x[n]\rightarrow x[-n]
$$

### Time Scaling

$$
\text{Compressed as }x(t)\rightarrow x(2t);\quad\text{Stretched as }x(t)\rightarrow x(t/2).
$$

### General Time Transformation

Combine all of the above as

$$
x(t)\rightarrow x(\alpha t+\beta)
$$

$\left|\alpha\right|>1$, compressed; $\left|\alpha\right|<1$, stretched; $\alpha<0$, reversed; $\beta\neq 0$, shifted.

### Periodic Signals

A continuous-time signal $x(t)$ is a periodic signal $\mathrm{iff.}$ there exists some $T$ $\mathrm{s.t.}$

$$
\forall t,\ x(t)=x(t+T).
$$

The smallest **positive** value of $T$ is called the **fundamental period** (基波周期) of the signal.

Similarly for discrete-time signals we have $x[n]=x[n+N]$ for all $n$.

### Even and Odd Signals

**Even Signal**:

$$
x(t)=x(-t)\quad x[n]=x[-n]
$$

**Odd Signal**:

$$
x(t)=-x(-t)\quad x[n]=-x[-n]
$$

**Any** Signal can be broken into a sum of two signals, one even and one odd, in the following method:

$$
x(t)=x_e(t)+x_o(t),\text{ where }\begin{cases}
x_e(t)=E_v\{x(t)\}=\displaystyle\frac{1}{2}[x(t)+x(-t)]\\
x_o(t)=O_d\{x(t)\}=\displaystyle\frac{1}{2}[x(t)-x(-t)]
\end{cases}.
$$

For discrete-time signals, it is similar that

$$
x[n]=x_e[n]+x_o[n],\text{ where }\begin{cases}
x_e[n]=\displaystyle\frac{x[n]+x[-n]}{2}\\
x_o[n]=\displaystyle\frac{x[n]-x[-n]}{2}
\end{cases}.
$$

## Exponential and Sinusoidal Signals

### Continuous-Time Complex Expo. and Sin. Signals

In the general case we have

$$
x(t)=Ce^{at}
$$

where $C$ and $a$ are complex number;

When $C$ and $a$ are real, it is a real exponential signal where

$$
\begin{cases}
a>0\Leftrightarrow x(t)\uparrow\text{ as }t\uparrow\\
a<0\Leftrightarrow x(t)\downarrow\text{ as }t\uparrow\\
a=0\Leftrightarrow x(t)\text{ is constant.}
\end{cases}
$$

**Periodic exponential signals**

In such case, $c$ is real, specifically $c=1$; $a$ is purely imaginary as $a=\omega_0j$

$$
x(t)=e^{j\omega_0t}
$$

To find its Fundamental period $T_0$, we have

Notice: $T_0$ is undefined for $\omega_0=0$ as $x(t)$ is constant and any $T$ is its period.

**Sinusoidal Signals**

$$
x(t)=A\cos(\omega_0t+\phi)
$$

Which is closely related to complex exponential signals (Euler's Formula)

Its **fundamental frequency** (基波频率) is $\omega_0$.

Both $e^{j\omega_0}t$ and $A\cos(\omega_0t+\phi)$ has infinite total energy but finite average power:

**Harmonically related (谐波) complex exponentials**

Sets of periodic exponentials (**with different frequencies**), all of which are periodic with a common period $T_0$

To have the common period $T_0$, they must satisfy that

$$
e^{j\omega t}=e^{j\omega(t+T_0)}=e^{j\omega t}\cdot e^{j\omega T_0}\Rightarrow \omega T_0=2k\pi\Rightarrow\omega=\frac{2k\pi}{T_0}=k\omega_0\Rightarrow \omega_0=\frac{2\pi}{T_0}
$$

We call

$$
\phi_k(t)=e^{jk\omega_0t},k\in\mathbb{Z}\setminus\{0\}
$$

a **harmonically related set**.

For any $k\neq 0$, they have fundamental frequency $\left|k\right|\omega_0$ and fundamental period $\displaystyle\frac{2\pi}{\left|k\right|\omega_0}=\frac{T_0}{\left|k\right|}$.

**Back to the general case**:

$$
x(t)=Ce^{at}
$$

Since that complex numbers $C,a$ can be written in the form

$$
C=\left|C\right|e^{j\theta},\ a=r+j\omega_0
$$

We have

### Discrete-Time Complex Expo. and Sin. Signals

In the general case

$$
x[n]=C\alpha^n
$$

where $C$ and $\alpha$ are complex numbers;

> $x[n]$ can be expressed in exponential form as $x[n]=Ce^{\beta n}$ if we let $\alpha=e^{\beta}$ using Euler's Formula, but for discrete-time signals it is easier to handle in the former format.

**Sinusoidal signals**

In such case $C$ is real, specifically $C=1$; $\beta$ is purely imaginary:

$$
x[n]=e^{j\omega_0n}
$$

Which is closely related to $A\cos(\omega_0n+\phi)$ since that

$$
\begin{align}
e^{j\omega_0n}&=\cos\omega_0n+j\sin\omega_0n\\
A\cos(\omega_0n+\phi)&=\frac{A}{2}\cdot e^{j\theta}e^{j\omega_0n}+\frac{A}{2}\cdot e^{-j\theta}e^{-j\omega_0n}
\end{align}
$$

Similarly, it has infinite total energy but finite average power since

$$
\left|e^{j\omega_0}n\right|^2=1.
$$

**General Signals**

**Periodicity properties**

Focusing on $\omega_0$ of $x[n]=e^{j\omega_0n}$:

It is obvious that $e^{j\omega_0n}$ has the same value at $\omega_0$ and $\omega_0+2k\pi$;

Only consider interval $0\leq\omega_0\leq 2\pi$:

> Notice that here we are considering different periods $\omega_0$ rather than one period of a specific signal.

From $0$ to $\pi$: when $\omega_0$ increases, the **oscillation rate** (振荡率) of $e^{j\omega_0n}$ **increases**;

From $\pi$ to $2\pi$: when $\omega_0$ increases, the oscillation rate of $e^{j\omega_0n}$ **decreases**.

The maximum oscillation rate is taken at $\omega_0=\pi$ where

$$
e^{j\pi n}=(e^{j\pi})^n=(-1)^n
$$

Then, we focus on $n$ of $x[n]=e^{j\omega_0n}$:

In order for $e^{j\omega_0n}$ to be periodic with $N>0$, it must satisfy that

$$
e^{j\omega_0(n+N)}=e^{j\omega_0N}\cdot e^{j\omega_0n}=e^{j\omega_0n}\Rightarrow\omega_0N=2m\pi\Rightarrow\frac{\omega_0}{2\pi}=\frac{m}{N}
$$

So that $\displaystyle\frac{\omega_0}{2\pi}$ must be a rational number; In such case,

Fundamental frequency: $\displaystyle\frac{2\pi}{N}=\frac{\omega_0}{m}$; Fundamental period: $N=\displaystyle\frac{2m\pi}{\omega_0}$.

## The Unit Impulse and Unit Step Functions

### Discrete-time Unit Impulse and Unit Step Sequences

**Unit Impulse** (unit sample, 单位冲击函数) is defined as

$$
\delta[n]=\begin{cases}0&n\neq0\\1&n=0\end{cases}
$$

**Unit Step** (单位阶跃函数) is defined as

$$
u[n]=\begin{cases}0&n<0\\1&n\geq0\end{cases}
$$

The impulse is the first difference (差分) of the step:

$$
\delta[n]=u[n]-u[n-1]
$$

Conversely, the step is the *running* sum of the step:

$$
u[n]=\sum_{m=-\infty}^n\delta[m]\rightarrow\begin{cases}n<0&\sum=0\\n\geq0&\sum=1\end{cases}
$$

Or let $m=n-k$, we have

$$
u[n]=\sum_{k=\infty}^0\delta[n-k]\quad\text{or}\quad\sum_{k=0}^{\infty}\delta[n-k].
$$

**Sampling Property**

$$
\begin{align}
&x[n]\delta[n]=x[0]\delta[n],\ \text{since }\delta[n]\text{ has value only when }n=0;\\
&\text{More Generally, }x[n]\delta[n-n_0]=x[n_0]\delta[n-n_0].
\end{align}
$$

### Continuous-time Unit Impulse and Unit Step Functions

Unit step is defined as

$$
u(t)=\begin{cases}0&t<0\\1&t>0\end{cases}
$$

> As we can see, according to the definition, the function is **discontinuous** at $t=0$.

Still, the continuous unit step $u(t)$ is the **running integral** of unit impulse $\delta(t)$:

$$
u(t)=\int_{-\infty}^t\delta(\tau)\mathrm{d}\tau
$$

And $\delta(t)$ is the first derivative of $u(t)$:

$$
\delta(t)=\frac{\mathrm{d}u(t)}{\mathrm{d}t}
$$

$$
\delta(t)=\begin{cases}1&t=0\\0&t\neq0\end{cases}
$$

which is exactly the same as $\delta[t]$ for discrete-time signals.

Or let $\sigma=t-\tau$, and we have

$$
u(t)=\int_0^{\infty}\delta(t-\sigma)\mathrm{d}\sigma.
$$

**Sampling Property**

Similarly we have

$$
x_1(t)=x(t)\delta_{\Delta}(t)\approx x(0)\delta_{\Delta}(t)\\
$$

The approximation is made since that $\Delta$ is so small here that we can consider $x(t)$ as a constant $x(0)$; So that we have the following result

## Continuous-Time and Discrete Time Systems

For continuous-time systems, the Input & Output are continuous:

$$
x(t)\rightsquigarrow\text{System}\rightsquigarrow y(t)
$$

And for discrete-time systems, the Input & Output are discrete:

$$
x[t]\rightsquigarrow\text{System}\rightsquigarrow y[t]
$$

### **Example of Systems**

**RC Circuit**

For an RC circuit, we have

The latter equation is the relationship between input signal $v_s(t)$ and output signal $v_c(t)$.

**Balance in a bank account**

For balance in a bank account as an example of discrete-time system, we have

$$
y[n]=1.01y[n-1]+x[n]
$$

where $y[n]$ is the balance at the end of $n$th month, $x[n]$ is next deposit, and the interest rate is $1\%$.

The relationship between input $x[n]$ and output $y[n]$ is

$$
y[n]-1.01y[n-1]=x[n].
$$

**Interconnections of Systems**

### Basic System Properties

### System with and without Memory

**System without Memory **: Output is dependent **only** on the current input.

$$
\mathrm{e.g.}\ \begin{cases}y[n]=\big(2x[n]-x^2[n]\big)^2\\y(t)=Rx(t)\end{cases}\ ,\text{where the only variable is the current input }n/t.
$$

**System with Memory **: Output is dependent on the current **and** previous inputs.

**Memory** : retaining or storing information about input values at times

Specifically, for *physical systems*, memory is associated with the storage of energy.

### Invertibility and Inverse System

**Invertible** : Distinct inputs lead to distinct outputs. (单射, $f^{-1}$ exists)

For example, the Accumulator $\displaystyle y[n]=\sum_{k=-\infty}^n x[k]$, where the difference between two successive outputs is precisely the inputs $y[n]-y[n-1]=x[n]$.

**Noninvertible** : $f^{-1}$ does not exist, examples: $y[n]=0,\ y(t)=x^2(t).$

### Causality 系统的因果性

**Casual** : The output at any time depends only on the inputs at the **present time** and in the **past**. (But not in the future; In other words, the system has no ability for prediction.)

### Stability 系统的稳定性

**Informally**, small inputs lead to responses that do not diverge.

**Formally**, bounded input leads to bounded output.

Here a function $f(t)$ is bounded if such $B$ exist that $\left|f(t)\right|<B$.

Examples :

## Time Invariance 时不变

**Time Invariant** : A time shift in the input signal results in an identical time shift in the output signal.

$$
\begin{align}
x[n]\rightsquigarrow y[n]&\Leftrightarrow x[n-n_0]\rightsquigarrow y[n-n_0];\\
x(t)\rightsquigarrow y(t)&\Leftrightarrow x(t-t_0)\rightsquigarrow y(t-t_0).
\end{align}
$$

Method for proving time invariance:

$$
\begin{cases}
x_2(t)=x_1(t-t_0)\\y_2(t)=f\{x_2(t)\}\\y_2'(t)=y_1(t-t_0)
\end{cases}\Rightarrow y_2(t)=y_2'(t)?
$$

Examples :

> 也即，先进行变换$y(t)=f\{x(t)\}$再平移($x\rightarrow x-x_0$)与先平移再进行变换所得到的两种结果应该是完全一致的，否则不具有时不变性。

## Linearity

A system is linear if and only if it has superposition property, $\mathrm{i.e.}$ additivity(可加性) and homogeneity(齐次性).

$$
\begin{cases}
x_1(t)\rightsquigarrow y_1(t)\\
x_2(t)\rightsquigarrow y_2(t)
\end{cases}\Rightarrow ax_1(t)+bx_2(t)\rightsquigarrow ay_1(t)+by_2(t).
$$

Method for proving linearity:

$$
\begin{cases}
x_3(t)=ax_1(t)+bx_2(t)\\
y_3(t)=f\{x_3(t)\}\\
y_3'(t)=ay_1(t)+by_2(t)
\end{cases}\Rightarrow y_3(t)=y_3'(t)?
$$

Examples :

Signals and Systems Chapter 1

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