• 离散时间
• 离散振幅

### 离散时间信号

#### 定义

• 一维（目前为止）
• 记号：$x [n]$
• $x: \mathbb{Z} \rightarrow \mathbb{C}$
• $n$是时间维度
• 分析：定期测量
• 合成：生成样本

#### 四种信号类型

• 有限长
• 无限长
• 周期
• 紧支集
##### 有限长
• $x[n], \quad n=0,1, \ldots, N-1$
• 向量形式：$\mathbf{x}=\left[x_{0} x_{1} \ldots x_{N-1}\right]^{T}$
##### 无限长
• $x[n], \quad n \in \mathbb{Z}$
##### 周期信号
• $\tilde{x}[n]=\tilde{x}[n+k N], \quad n, k, N \in \mathbb{Z}$
• 和长度为$N$的信号信息相同

#### 数字和物理频率

• 采样的时间间隔为$T_s$

• $M$个样本的周期$\longrightarrow$ $MT_ {s}$秒的周期

• 实际频率

#### Karplus-Strong

• 建立延迟为$M$的递归循环
• 选择一个仅在$0 \leq n <M$为非零的信号$\bar {x} [n]$
• 选择一个衰减因子$\alpha$
• 向系统输入$\bar {x} [n]$
• 播放输出

### 习题

#### 1

(Difficulty: $\star$) What are the advantages of using digital signals over analog ones? Choose the correct answer(s).

• Digital signals are more robust to noise.
• Digital signals can be easily stored.
• Physical quantities are naturally represented by digital signals.
• Digital signals contains more information than analog ones.

1,2

#### 2

(Difficulty: $\star$) Amongst the signals listed below, select those that are in digital format.

• Music recorded on a vinyl record.
• JPEG image on a website.
• Music recorded on a CD.
• A handwritten book manuscript.

2,3

#### 3

Consider the following finite support signal:

Consider also its periodic repetition

Compute the energy of $x[n]$

#### 4

Consider the same signals as the previous question. Compute the power of $x[n]$.

#### 5

Consider the same signals as the previous question. Compute the energy of $y[n]$.

#### 6

Consider the same signals as the previous question. Compute the power of $y[n]$.

#### 7

Consider the signal

Consider now its moving average, i.e. the signal

Select the correct expressions from the options below.

• $y[n]=\delta[n]+2 \delta[n-1]+5 \delta[n-2]+3 \delta[n-3]$
• The output for $n \geq 4$ is always zero.

#### 8

A music song recorded in a studio is stored as a digital sequence on a CD. The analog signal representing the music is 2 minutes long and is sampled at a frequency $f_{s}=44100 s^{-1}$. How many samples should be stored on the CD? Write the number of samples without commas or dots. Assume that the audio file is mono, or in other words, single channel.

#### 9

Given the following filter

What is the input-output relationship?

• $y[n]=b(a x[n]+x[n-1])-(c+1) x[n-3]$
• $y[n]=(b a x[n]+x[n-1])+(c x[n-3]+x[n-4])$
• $y[n]=a b x[n]+b x[n-1]+c x[n-3]-x[n-4]$
• $y[n]=b(a x[n]+x[n-1])-(c x[n-3]+x[n-4])$

#### 10

What is the minimum period $P$ (in samples) of the signal $e^{j(M / N) 2 \pi n}$ for $M=1, N=3$

#### 11

What is the minimum period $P$ (in samples) of the signal $e^{j(M / N) 2 \pi n}$ for $M=5, N=7$

#### 12

What is the minimum period $P$ (in samples) of the signal $e^{j(M / N) 2 \pi n}$ for $M=35, N=15$

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