2006 Fall Signals and Systems Lecture 2 Complex Exponentials Unit Impulse and Unit Step Signal...

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


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


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


(This decomposition of the sinusoid can be traced to Euler's relation)

ω0 :Fundamental Frequency

Φ :Phase

A :Amplitude

2006 Fall


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


§ 1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals

nCnx n

nCenx where e

1. Real Exponential Signals

a real

nanx

2006 Fall


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


( 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 .


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


§ 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


2. Sifting property

筛选特性

1. Sampling property

取样特性

The properties of n

2006 Fall


nxnnx 0

knkxnxk

mnun

m

1 nunun

n1 n=0

0 n ≠ 0

2006 Fall

][][][][ 000 nnnxnnnx

][][][ 00 nxnnnxn

ntegersareabandaabnban i][][


2006 Fall


§ 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


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


2006 Fall


§ 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


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


§ 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


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)

2006 Fall Signals and Systems Lecture 2 Complex Exponentials Unit Impulse and Unit Step Signal...

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