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Signals & Systems
Second Edition
电子工程学院孔斌
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电子电路类电路分析基础 模拟电路基础 数字逻辑设计及应用电子技术实验基础集成电路应用实验现代电子技术综合实验电子信息工程综合实践电磁场与微波类 电磁场与波 微波技术基础计算机类 软件技术基础微机原理与应用信号与信息类信息论导论 信号与系统数字信号处理 通信技术与系统制图与控制类 现代工程设计制图 自动控制原理专业基础虚拟仪器基础实验单片机原理及接口
MATLAB工具软件
EDA技术
DSP技术高级语言随机信号分析 PCA 技术物理电子类 量子力学与物理电子学 固体电子学 半导体物理
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课程说明教学计划学时,80 学分,5
教学内容
1,课堂理论教学 (68学时 )
2,上机实验 (8学时 )
3,课堂习题课 (4学时 )
考核方式半期 20%+上机实验 10%+期末 70%
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课程说明参考书目,
Exploration in Signals and Systems Using MATLAB
John R.Buck,Michael M,Daniel
刘树棠 译，西安交通大学出版社
,信号与系统分析,
吕幼新 张明友 电子工业出版社
,信号与系统复习考研例题详解,
吕幼新 张明友 电子工业出版社
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课程说明
1,课件下载（密码,2005）
校园网 教师社区 电子工程 学院 孔斌 课件发布
3,作业要求
① 用中、英文书写 ;
② 每周五交作业 ;
校园网 精品课程 信号与系统
2,网络教学资源
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4,答疑安排课程说明答疑时间 星期一19:30-21:30 星期三19:30-21:30
答疑地点 电子通信大楼 136
答疑教师邹林王洪张花国贾聪智谢小强李万春王忠荣
2005年 9月 12日起执行
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Chapter 1
Signals and Systems
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Chapter 1 Signals and Systems
(1) A simple RC circuit
Source voltage Vs and
Capacitor voltage Vc
(2) An automobile
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(3) A Speech Signal
Chapter 1 Signals and Systems
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(4) A Picture
Chapter 1 Signals and Systems
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(5) Vertical Wind Profile
Chapter 1 Signals and Systems
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信号的描述 频率特性信号可表示成一个或多个自变量的函数 ;
tzyxf,,,
电压电流
tv
ti
信号与系统分析的两个共同的基本点：
2,系统：对给定的信号作出响应，并产生新的信号
Chapter 1 Signals and Systems
1,信号（一个或多个自变量）
时间特性
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§ 1.1 Continuous-Time and Discrete-Time Signals
连续时间信号和离散时间信号
§ 1.1.1 Examples and Mathematical Representation
1,Continuous-Time Signals
—— The independent variable is continuous
0 t
tf
0 t
tf
Chapter 1 Signals and Systems
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Chapter 1 Signals and Systems
2,Discrete-Time Signals
—— The independent variable is discrete

nx
n
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1
2 31 4
5
6
4
1
7
10 8
11
n is integer number
Continuous-time signals Discrete-time signals
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Chapter 1 Signals and Systems
3,Time-averaged power（平均功率）
Energy signal
0 t
tx
0 t
tx
Power signal
0 t
tx
1,Instantaneous power（瞬时功率）
2,Total energy
§ 1.1.2 Signal Energy and Power
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Chapter 1 Signals and Systems
§ 1.2 Transformations of the independent variable
§ 1.2.1 Examples of transformations of the independent variable
1,Time shift( 时移 )
0ttt
0nnn
0ttxtx
0nnxnx
Example 1
tx
1/tt 10 tt
0 otherwise
Please indicate
0ttx?
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Chapter 1 Signals and Systems
0ttxtx与 波形相同
00?t 相当于 左移（超前） totx
2,Time reversal( 时域反折 )
tt nn
x(-t) is a reflection of x(t) about t=0
00?ttx相当于 右移（延迟） to
时移对信号的影响
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Classification of signals （信号的分类）
1.Continuous Time（ CT） and discrete time（ DT） signals
连续时间信号和离散时间信号
2,Periodic and Aperiodic Signals
周期信号和非周期信号
3,Energy and Power Signals
能量信号和功率信号 (能量有限信号和能量无限信号 )
4,Deterministic and Stochastic (nondeterministic) Signals
确定性信号随机信号 (非确定性信号 )
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3,Time- scaling( 尺度变换 )
Chapter 1 Signals and Systems
atxtxatt
a>1 信号压缩 a倍
0<a<1 信号扩展 1/ a 倍
① Continuous- time signals
② Discrete- Time signals x[ an] n,an ∈ N
n1? 0 1 2
3
1
2
x[n]
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Chapter 1 Signals and Systems
0 1 2 t
tx1
Solution
Example 1.1
Given the signal 3
12xt
-2/3 0 2/3 t
1
3 1
2xt


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§ 1.2.2 Periodic signals
Chapter 1 Signals and Systems
Synthesize the signal's periodic
tx1 T1tx2 T2
当 T1/T2为有理数时,为周期的tbxtax
21?
2211 TnTnT
其中 n1,n2 互质其周期
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Chapter 1 Signals and Systems
§ 1.2.3 Even and Odd Signals
txOdtxEvtx
—— Even parttxtxtxEv
2
1
—— Odd parttxtxtxOd
2
1
0 1 2 t
tx
1
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Chapter 1 Signals and Systems
§ 1.3 Exponential and Sinusoidal Signals
指数信号和正弦信号
§ 1.3.1 Continuous-Time Complex Exponential
and Sinusoidal Signals
stCetx t
1,Real Exponential Signals
atCetx?
a is real
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Chapter 1 Signals and Systems
2,Periodic Complex Exponential and Sinusoidal Signals
① Period
② Euler’ s relation
③ Average Power
④ Harmonic relation
3,General Complex Exponential Signals
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Chapter 1 Signals and Systems
§ 1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals
nCnx n
nCenx where e?
1,Real Exponential Signals
a real
nanx?
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Chapter 1 Signals and Systems
2,Complex Exponential Signals and Sinusoidal Signals
① Average Power
② Euler’ s relation
④ Periodicity Properties
⑤ Harmonic relation
③ Frequency Properties
3,General Complex Exponential Signals
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Chapter 1 Signals and Systems
( a) ω0=0 N=1
( b) ω0= π /8 N=16
( c) ω0= π /4 N=8
( d) ω0 = π /2 N=4
( e) ω0 = π N=2
低频高频
( f) ω0 =3 π/2 N=4
( g) ω0 =7 π/4 N=8
( h) ω0 =15 π/8 N=16
( i) ω0 =2 π N=1
nnx 0co sFigure 1.27 2,00?
ω0=2 kπ 时,信号频率低
ω0=(2 k+1)π 时,信号频率高
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ω0不同,信号不同,ω0相差 2 kπ,信号相同,
ω0越大,频率越高,ω0 =2 k π时,频率低 ;
ω0 =(2 k+1)π时,频率高,
对任意的 ω0,
信号均为周期的,
为有理数时,
信号为周期的,
Chapter 1 Signals and Systems
tje 0? nje 0?
2,00?
0
0
2
T
mN
0
2

2/0
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Chapter 1 Signals and Systems
§ 1.4 The Unit Impulse and Unit Step Functions
单位冲激与单位阶跃函数
§ 1.4.1 The Discrete-Time Unit Impulse and Unit Step Sequences
Unit Impulsen?
1 n=0
0 n ≠ 0
n0
1

n?
Unit Step
nu
1 n ≥ 0
0 n<0
n1? 0 1 2 3
1
nu
n
m
u n m?


1n u n u n
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Chapter 1 Signals and Systems
2,Sifting property
筛选特性
1,Sampling property
取样特性
The properties of n?
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Chapter 1 Signals and Systems
§ 1.4.2 The Continuous- Time Unit Step and Unit Impulse Functions
1,Unit Step Function
tu
1 t > 0
0 t < 0 0 t
1
tu
0 △ t
1
tu?
tutu 0lim
2,Unit Impulse Function
1 C
+
-tvc
tict=0
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Chapter 1 Signals and Systems
dt tdut①
②t?
0 t ≠ 0
1 dtt?
100 dtt?
If 1
dttslimkt ks kt
③
dtu t
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Chapter 1 Signals and Systems
§ 1.4.3 The Properties of Unit Impulse Functions
1,Sampling and Sifting properties
①tfttf 0?
If f(t) is continuous at the point of t=0
②
0fdtttf
Sampling property
Sifting property
2,Scaling property
taat 1?If a is real,a ≠ 0
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Chapter 1 Signals and Systems
Example2
dttt 2/22/s i n
4
2
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Chapter 1 Signals and Systems
§ 1.4.4 信号的计算
1,信号的加、减、乘
2,信号的基本表示
-τ 0 τ t
tP?2
0 1 t
1
xt
-1 0 1 t
1
xt
3,信号的微分、积分运算
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Chapter 1 Signals and Systems
Example 1.7
x(t) is depicted in Figure 1.40(a),determine
the derivative of x(t).
2
1
0 1 2 3 4 t-1
x(t)
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Chapter 1 Signals and Systems
§ 1.5 Continuous-time and discrete-time systems
System
Be constituted by some units
Contact with some rule
The system's function
System analysis （系统分析）
System synthesizes （系统综合）Research system
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Chapter 1 Signals and Systems
§ 1.5.1 Simple Examples of Systems
Example 1.9
tvtf m
tv?
Linear constant-coefficient differential equation
Example 1.8
tvs
R
ti
Ctvc
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Chapter 1 Signals and Systems
Continuous-Time
System
k
kM
k
kk
kN
k
k dt
txdb
dt
tyda

00
Discrete-Time
Systemknxbknya M
k
k
N
k
k
00
N-order Linear Constant-coefficient
Differential Equation
N-order Linear Constant-coefficient
Difference Equation
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Chapter 1 Signals and Systems
§ 1.5.2 Interconnection of Systems
Series interconnection
(级联 )
Parallel interconnection
(并联 )
Feedback interconnection
(反馈 )
System 1 System 2
input output
System 1
System 2
input output
System 1
System 2
input output
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Chapter 1 Signals and Systems
§ 1.6 Basic System Properties
§ 1.6.1 Systems with and without Memory
有记忆、无记忆系统无记忆系统,在任意时刻 (t)的输出仅仅与同时刻 (t)的输入有关。
—— memoryless（无记忆）
—— identity system,memoryless
22 2 nxnxny①
txty?②
③ summer
kxny
n
k

④ delay1 nxny
⑤ integrate dxty t?

Systems with memory
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Chapter 1 Signals and Systems
§ 1.6.2 Invertibility and Inverse Systems
可逆系统与可逆性可逆系统,不同的输入导致不同的输出（一一对应）。
System
nx
txty
ny
Inverse System
nxnw?
txtw?
tx
txty 2?
tytxtw?
tytw 21?
kxny n
k

nxny
1 nynynw
nxnw?
0?nytxty 2? —— noninvertible systems不可逆系统
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Chapter 1 Signals and Systems
§ 1.6.3 Causality （因果性）
因果系统,在任意时刻 (t)的输出只取决于同时刻 (t)或以前 (<t)
的输入。 (与该时刻以后的输入无关）
③ Systems without memory
①1 txty
② dxty t?
Causal systems
因果系统不可预测系统物理上可实现
④txtty 1c o s
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Chapter 1 Signals and Systems
非因果系统,适用于非时间自变量信号的处理,
knxMny M
Mk

12
1
nxny
txty 2?
Not Causal
§ 1.6.4 Stability （稳定性）
Stable System Bty Mtx
①ttxty?
②txety?
—— not stable
—— stable
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Chapter 1 Signals and Systems
§ 1.6.5 Time Invariance （时不变性）
时不变系统,系统参数不随时间改变 (恒参系统 ),系统的输出波形仅仅取决于输入波形,而与输入作用的时刻无关,
txLty?
If00 ttyttxL
Time invariant
时不变
Example 1.14txty s i n?
Example 1.15nnxny?
Example 1.16txty 2?
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Chapter 1 Signals and Systems
§ 1.6.6 Linearity （线性）
tvtx m
0
1,Initial State （初始状态）
2,Linearity
① Additivity
② Scaling
3,Linear System
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Chapter 1 Signals and Systems
Example 1.17ttxty?
Example 1.18txty 2?
Example 1.19nxny Re?
Example 1.20 32 nxny
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Chapter 1 Signals and Systems
ty
nynx
tx 线性系统
tyxnyx
tyf
nyf
线性系统的三个特性tytf?
① 微分特性
dt
tdy
dt
tdf?
② 积分特性 dydf tt
③ 频率保持性：
信号通过线性系统不会产生新的频率分量
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Example 1 an LTI system
0 2 t
tf1
1
211 tftf
0 1 2 t
ty1
1L
211 tyty
0 2 4 t
1
-1
0 2 4 t
tf2
1 L
ty2
1?
Chapter 1 Signals and Systems
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Chapter 1 Signals and Systems
作业：
1.14 1.15 1.16 1.17
1.21 (d) (e) (f)
1.22 (d) (g) 1.23
1.24 (a) (b) 1.26 (a) (b)
1.27 1.31
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Classification of signals （信号的分类）
1.Continuous Time（ CT） and discrete time（ DT） signals
连续时间信号和离散时间信号
2,Periodic and Aperiodic Signals
周期信号和非周期信号
3,Energy and Power Signals
能量信号和功率信号 (能量有限信号和能量无限信号 )
4,Deterministic and Stochastic (nondeterministic) Signals
确定性信号随机信号 (非确定性信号 )
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Chapter 1 Signals and Systems
nxnnx 0?
knkxnx
k



mnu n
m


1 nunun?
n?
1 n=0
0 n ≠ 0
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①tfttf 0?
②
0fdtttf
1 a t ta?
③
The Properties of Unit Impulse Functions