2006 Fall
Signals and SystemsSignals and Systems Lecture 2Lecture 2
Complex ExponentialsUnit Impulse and Unit Step SignalSingular Functions
2006 Fall
Chapter 1 Signals and Systems
§ 1.3 Exponential and Sinusoidal Signals复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
Decaying Exponential, when α<0Growing Exponential, when α>0 Figure 1.19
2006 Fall
Chapter 1 Signals and Systems
2. Periodic Complex Exponential and Sinusoidal Signals
① Period
② Euler’ s relation( 欧拉关系 )
2cos
tjtj eet
j
eet
tjtj
2sin
tjte tj sincos
00
2
T
2006 Fall
Sinusoids and Complex Exponential Probably the most important elemental signal that we
will deal with is the real-valued sinusoid. In its continuous-time form, we write the general form as
)cos()( 0 tAtx0
0
2
T
Chapter 1 Signals and Systems
Maybe as important as the general sinusoid, the complex exponential function will become a critical part of our study of signals and systems. Its general form is also written as
tCetx )(
a is complex
2006 Fall
Sinusoids and Complex Exponential Decomposition: The complex exponential signal can
thus be written in terms of its real and imaginary parts }Re{)cos( )(
00 tjeAtA
}Im{)sin( )(0
0 tjeAtA
Chapter 1 Signals and Systems
(This decomposition of the sinusoid can be traced to Euler's relation)
ω0 :Fundamental Frequency
Φ :Phase
A :Amplitude
2006 Fall
Chapter 1 Signals and Systems
③ Average Power
④ Harmonic relation
3. General Complex Exponential Signals
00
200 TdteE
T tjperiod 1
1
0
periodperiod ET
P
,2,1,0,)( 0 ket tjkk
)sin()cos( 00
)( 0
teCjteC
eeCCertrt
tjrjt
2006 Fall
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
2006 Fall
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
0
2
mN
)sin()cos( 00 nCjnCCnnn
2006 Fall
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
( 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 Low Frequency
High Frequency
nnx 0cosFigure 1.27 2,00
ω0=2 kπ, low frequency
ω0=(2 k+1)π, high frequency
Frequency Properties
2006 Fall
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
00
2
T mN0
2
2/0
Table 1.1 Comparison of the andtje 0 nje 0
2006 Fall
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 Impulse n1 n=0
0 n ≠ 0 0 n
1
n
Unit Step nu 1 n ≥ 0
0 n<0n1 0 1 2 3
1
nu
n
m
u n m
1n u n u n
2006 Fall
Chapter 1 Signals and Systems
2. Sifting property
筛选特性
1. Sampling property
取样特性
The properties of n
2006 Fall
Chapter 1 Signals and Systems
nxnnx 0
knkxnxk
mnun
m
1 nunun
n1 n=0
0 n ≠ 0
2006 Fall
][][][][ 000 nnnxnnnx
][][][ 00 nxnnnxn
ntegersareabandaabnban i][][
Chapter 1 Signals and Systems
2006 Fall
Chapter 1 Signals and Systems
§ 1.4.2 The Continuous- Time Unit Step and Unit Impulse Functions
1. Unit Step Function
tu1 t > 0
0 t < 00 t
1 tu
0 △ t
1 tu
tutu
0lim
2. Unit Impulse Function
1 C+
- tvc
tict=0
2006 Fall
Chapter 1 Signals and Systems
dt
tdut ①
② t0 t ≠ 0
1
dtt
10
0
dtt
If 1
dtts lim
kt ks kt
③ dtut
2006 Fall
Engineering Model forEngineering Model for )(t
otherwise
ttf0
22
1)(
otherwise
tt
tt
tf
0
0)(1
0)(1
)(2
2
Properties of Engineering Model – The value at t=0 is very large– The duration is very short– The area is one
Model 1
Demo
Model 2
Demo
Chapter 1 Signals and Systems
2006 Fall
Chapter 1 Signals and Systems
§ 1.4.3 The Properties of Unit Impulse Functions
1. Sampling and Sifting properties
① tfttf 0
If f(t) is continuous at the point of t=0
② 0fdtttf
Sampling property
Sifting property
2. Scaling property
ta
at 1If a is real, a ≠ 0 )(
1)(
a
bt
abat
2006 Fall
Chapter 1 Signals and Systems
Example2 dttt 2/22/sin
4
2
2006 Fall
CT Unit Impulse FunctionCT Unit Impulse Function The Dirac delta function is defined by
where f(t) is any function that is continuous at t=0
)(t
)0()()( fdtttf
Mathematical Properties :
)0(
0,0)( tfort
evenist)(
1)()(0
0
dttdtt
2006 Fall
Chapter 1 Signals and Systems
§ 1.4.4 信号的计算
1. 信号的加、减、乘、除
2. 信号的基本表示
-τ 0 τ t
tP 2
0 1 t
1
x t
-1 0 1 t
1
x t
3. 信号的微分、积分运算
2006 Fall
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)
2006 Fall
Models for Models for )(t
Derivation
2006 Fall
Properties of Properties of
)()( tt
)(t
)()()()()()( 00000 tttftttftttf
0,)(1
0,)(1
)(
2
2
aa
bt
a
aa
bt
abat
otherwiseatuat
tandatatdpap
t
,)()(
0,)()(
)(11
)( taa
at
2006 Fall
Higher Derivation of Higher Derivation of
)(11
)( )()( taa
at kk
k
)(t
)0()1()()( )()( nnn fttf
)()1()( )()( tt kkk
2006 Fall
SummarySummaryWhat we have learned?
– Complex Exponentials– Unit Impulse and Unit Step Signal– Singular Functions
What was the most important point in the lecture?What was the muddiest point?What would you like to hear more about?
2006 Fall
Reading ListReading List
Signals and Systems : – 1.5,1.6,
Question :– Classification of Systems
2006 Fall
Problem SetProblem Set
1.21(e),(f)1.22(e),(f)1.25(c),(d)1.26(a),(e)