第02章离散时间信号和系统

Chapter

2

Discrete-time

signalsandsystems1Discrete-time

signals:sequencesDiscrete-time

systemFrequency-domain

representation

ofdiscrete-time

signal

and

system2.1

Discrete-time

signals:sequences2DefinitionClassification of

sequenceBasic

sequencesPeriod

of

sequenceSymmetry

of

sequenceEnergy

of

sequenceThe

basic

operations

of

sequences2.1.1

Definition-

¥

<

n

<

+¥x

=

{x[n]}EXAMPLE枚举法表示序列x[n]

=

{1,2,1.2,0,-1,-2,-2.5},-1

£

n

£

5x[n]

=

0.9n

cos(0.2pn

+

p

/

2),0

£

n

<10函数法表示序列3-20246-3-2-10120510-1-0.500.54图形表示序列MATLAB产生并画序列n=-1:5x=[1,2,1.2,0,-1,-2,-2.5]stem(n,x,

'.')n=0:9y=0.9.^n.*cos(0.2*pi*n+pi/2)stem(n,y,'.')5Figure

2.26EXAMPLE对连续时间信号的采样x[n]

=

xa

(t)

|t

=nT

=

xa

(nT

)7演示ULTRAEDIT显示的WAV声音文件内容整个波形局部放大演示COOLEDIT显示的WAV声音文件波形82.1.2

Classification of

sequence9Right-sideN1

£

n

<

+¥x[n]

„

0,

forx[n]

„

0,

for-

¥

<

n

£

N2-

¥

<

n

<

+¥x[n]

„

0,

forN1

£

n

£

N2x[n]

„

0,

forx[n]

=

0,

for n

<

0x[n]

„

0,

for n

<

0Left-sideTwo-sideFinite-lengthNoncausalCausal1.

Unit

sample

sequence2.1.3 Basic

sequences2．Theunit

stepsequence01n

=

0n

„

0d[n]

=

0n

‡

0n

<

0u[n]

=

11u[n]0n1δ[n]0n103．Therectangular

sequence1R[n]0N-1n0111

0

£

n

£

N

-1otherRN

[n]

=

4. Exponential

sequencex[n]

=

an12135.

Sinusoidal

sequencex[n]

=

A

cos(w

n

+

F

)数字角频率体现序列变化的快慢14=

cos(1.8pn)2.1.4 Period

of

sequencex[n]

=

x[n

+

N

],-¥

<

n

<

¥x(t)

=

Asin(

W

t

+

F

)

=

Asin(W

t

+

F

+

2p

)=

Asin(W

(t

+

2p

/

W

）+

F

)

T

=

2p

/

Wx[n]

=

Asin(nw

+

F

)

=

Asin(nw

+

F

+

2p

)=

Asin((n

+

2p

/w

)w

+

F

)=

x[n+

2p

/w

]151序列周期性的三种情况2p

/w

=

N2p

/w

=

P

/

Q，N

=

P6172.1.5

Symmetry

of

sequencesequencesequencex[n]

=

x[-n],evenx[n]

=

-x[-n],oddx[n]

=

x*[-n]

Conjugate-symmetricsequencex[n]

=

-x*[-n]

Conjugate-antisymmetricsequence18x[n]

=

xe

[n]

+

xo

[n]2exe

[n]

=

xe

[-n]x

[n]

=

x[n]

+

x[-n]2oxo

[n]

=

-xo

[-n]x

[n]

=

x[n]

-

x[-n]x[n]

=

xe

[n]

+

xo

[n]ex

[n]

=

x

*[-n]o2ex[n]

+

x*[-n]xe

[n]

=2xo[n]

=

-x

*[-n]x[n]

-

x*[-n]xo

[n]

=19x[n]

=

(n

+1)R6[n]n=[-5:5];x=[0,0,0,0,0,1,2,3,4,5,6];xe=(x+fliplr(x))/2;xo=(x-fliplr(x))/2;subplot(3,1,1)stem(n,x)subplot(3,1,2)stem(n,xe)subplot(3,1,3)stem(n,xo)EXAMPLE实序列分解成对称序列x[n]

=

(1

+

j)n

R6[n]).^[0:5]20n=[-5:5];x=zeros(1,11);x((n>=0)&(n<=5))=(1+jxe=(x+conj(fliplr(x)))/2;xo=(x-conj(fliplr(x)))/2subplot(3,2,1);stem(n,real(x))subplot(3,2,2);stem(n,imag(x))subplot(3,2,3);stem(n,real(xe))subplot(3,2,4);stem(n,imag(xe))subplot(3,2,5);stem(n,real(xo))subplot(3,2,6);stem(n,imag(xo))-505-4-220-505-4-220-505-202-505-202-505-202-505-210-1EXAMPLE复序列分解成对称序列2.1.6 Energy

of

sequence21¥

¥E

=

|

x[n]

|2

=

x[n]x*[n]n=-¥

n=-¥2.1.7

The

basic

operations

of

sequences221.y[n]

=

x[n

-

n0

]2.y[n]

=

x[-n]3.y[n]

=

a

+

x[n]4.y[n]

=

x[n]

+

w[n]5.y[n]

=

a

x[n]6.y[n]

=

x[n]

w[n]图示序列的基本运算23原始音乐序列原始语音序列矢加后的序列标乘后的序列矢乘后的序列回声24y[n]

=

3x[n

+

2]

+

x[n

-

4]-50510-20-1002040EXAMPLEx[n]

=[1,-2,4,6,-5,8,10]序列相加的MATLAB编程25x=[1,-2,4,6,-5,8,10]

;x1=x;n=[-4:2]

;%x1[n]=x[n+2]n1=n-2;%x2[n]=x[n-4]n2=n+4;x2=x;%y[n]m=[min(min(n1),min(n2)):

max(max(n1),max(n2))]

;y1=zeros(1,length(m))

;

y2=y1;y1((m>=min(n1))&(m<=max(n1)))=x1;y2((m>=min(n2))&(m<=max(n2)))=x2;

y=3*y1+y2;

stem(m,y)Output:y

=3 -6

12 18

-15

24

31 -2

4

6 -5

8

102627¥

¥y[n]

=

x[n]*h[n]

=

x[k

]h[n

-

k

]

=

x[n

-

k

]h[k

]k

=-¥

k

=-¥¥¥rxy

[n]

=

x[k

]y[k

+

n]

=

x[-n]*

y[n]

=

ryx

[-n]k

=-¥or,rxy

[n]

=

x[k

]y[k

-

n]k

=-¥8.crosscorrelation:7.convolution

sum:x[n]

=

R4

[n],h[n]

=

d[n

-1]computation

convolutionsum28EXAMPLEnx=0:10x=0.5.^nxnh=-1:4h=ones(1,length(nh))y=conv(x,h)stem([min(nx)+min(nh):max(nx)+max(nh)],y)附MATLAB程序29图示自相关和互相关rx1

x1rx2

x2rx1

x2rx2

x1x1x2302.1

序列小结31•2.1

定义•2.2

分类•2.3

基本序列•2.4

周期性•2.5

对称性•2.6

能量•2.7

基本运算要求： 判断序列的周期图解法和解析法求卷积32重点： 卷积运算2.2 Discrete-time

system33Definition：input-output

description

of

systemsClassification

of

discrete-time

systemLinear

time-invariant

system（LTI）Linear

constant-coefficient

difference

equation2.2.5.

Direct

implementation

of

discrete-time

systemx[n]T[

]y[n]2.2.1

definition：input-output

description

of

systems34y[n]

=

T[x[n]]0accumulator

:

y[n]

=

x[n

-

k

]k

=¥ideal delay

:

y[n]

=

x[n

-

5]3511

2M

2k

=-M1x[n

-

k

]moving

average

:

y[n]

=M

+

M

+1backward

difference:

y[n]

=

x[n]-

x[n

-1]forward

difference:

y[n]

=

x[n

+1]

-

x[n]EXAMPLEecho

system

:

y[n]

=

x[n]

+

ax[n

-

nd

]y[n]

=

2x[n]y[n]

=

max{x[n

-1],

x[n],

x[n

+1]}362.2.2 classification

of

discrete-time

systemMemoryless(static)

systemLinear

systemT[ax1[n]

+

bx2[n]]

=

aT[x1[n]]

+

bT[x2[n]]Time-invariant

system：if

T[x[n]]

=

y[n],then

T[x[n

-

n0

]]

=

y[n

-

n0

]Causalsystem：Stable

system：if

|

x[n]

|<

¥

,

then

|

T[x[n]]

|<

¥2.2.3

linear

time-invariant

system（LTI）37characterized

by

h[n]¥

¥y[n]

=

x[n]*

h[n]

=

x[k

]h[n

-

k

]

=

x[n

-

k

]h[k

]k

=-¥

k

=-¥0k

=¥h[n]

=

d[n

-

k

]

=

u[n]h[0]

=1,

h[50]

=

0.5(1)

y[n]

=

2x[n],

h[n]

=

2d[n](2)

y[n]

=

x[n]

+

0.5x[n

-

50]h[n]

=

d[n]

+

0.5d[n

-

50](3)

y[n]

=

x[n

-

k

]k

=¥01M

2M1

+

M

2

+1

k

=-M(4)

y[n]

=

1

x[n

-

k

]£

n

£

M0,

other,-M+

M

+11M

+

M

+11h[n]

=212121M

2k

=-M1d[n

-

k

]

=

MEXAMPLELTI的单位取样响应38Figure

2.12LTI的性质h[n]h[n]x[n]h1[n]h2[n]h2[n]h1[n]h1[n]

*h2[n]392.2.4

linear

constant-coefficient

difference

equation40N

M

ak

y[n

-

k

]

=

bk

x[n

-

k

]k

=0

k

=02FIRh[n]

=

d[n]

-

d[n

-1]

+

2d[n

-

2]y[n]

=

x[n]*

h[n]

=

h[k

]x[n

-

k

]

=

x[n]

-

x[n

-1]

+

2x[n

-

2]k

=0x[n]z-1z-1y[n]++-122.2.5.

Direct

implementation

of

discrete-time

system41EXAMPLE+¥IIRh[n]

=

u[n]y[n]

=

x[n]*

h[n]

=

x[n

-

k

]k

=0y[n]

=

x[n]

+

y[n

-1]x[n]z-1y[n]+EXAMPLE42B=1;y=filter(B,A,x);A=[1,-1]n=[0:100];stem(n,y);x=[n>=0];axis([0,20,0,20])LTI的直接实现的

MATLAB编程43定义：系统的输入输出描述分类（根据输入输出描述）线性时不变系统的h[n]描述和分类系统的常系数线性差分方程描述离散时间系统的直接实现442.2

离散时间系统小结要求：判断系统类型LTI的输入输出关系和卷积式间的关系常系数线性差分方程的递推解法IIR系统实现必须有反馈，FIR则可以无重点和难点：LTI卷积的实质就是将输入信号加权组合成输出信号，

h[n]就是权重；FIR与IIR的区别452.3

frequency-domain

representation

ofdiscrete-time

signal

and

system46definition

of

fourier

transformfrequency

response

of

systemproperties

of

fourier

transformx(t)

=

cos(2p100t)

+

0.5

cos(2p

200t),

f1

=100Hz,

f

2

=

200Hz-10120

0.01

0.02

0

100

200

300

400

5000510EXAMPLE信号的频域表示的直观意义47y(t)

=

T{x(t)}

=

cos(2p100t),

f

=

100

Hz-0.5-10.5010

0.01

0.02

0

100

200

300

400

5000510150Hz

f频响系统的频域表示的直观意义48EXAMPLE低通和高通滤波对图象信号所起的作用49带阻去噪过程的频域分析50傅立叶变换定义的引出w

3x[n]

=

|

X

(w

)

|

cos(wn

+

f(w

))X

(w

)

=|

X

'(w

)

|

e

jf(w

)w

3=

X

(w

)e

jwnw

=-w

3=w

=-w

3|

X

'(w

)

|

e

j(wn+f(w

))251w

=w

0w

3w

=w

0w

3

j(wn+f(w

))

-

j(wn+f(w

))=

|

X

(w

)

|

e

+

e2.3.1

definition

of

fourier

transform52p-pX

(e

jw

)e

jwndw

1

2px[n]

=+¥X

(e

jw

)=

x[n]e-

jwnn=-¥53title('实部')title('虚部')title('幅度')title('相位')1

-

0.2e-

jw1x[n]

=

0.2nu[n],

X

(e

jw

)

=-5050.51-505-0.50subplot(2,2,1);

fplot('real(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);subplot(2,2,2);

fplot('imag(1/(1-0.2*exp(-1*j*w)))',[-2*pi

,2*pi]);subplot(2,2,3);

fplot('abs(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);subplot(2,2,4);

fplot('angle(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]);实

实 虚

实1.5

0.511.5幅幅050.5 -0.5-5

0

5 -500.5相相EXAMPLE画信号频谱的MATLAB程序+¥X

(ejw

)=

2pd

(w

+

2pr

)r

=-¥,-¥

<

n

<

¥x[n]

=pnsin(w

cn)cjw0,|

w

|>

wc1,|

w

|£

wX

(e

)

=[=+¥

1

2p

1

2pp-pp-pdw

=1[d(w

)]e2pd(w

+

2pr

)]e

dwjwnjwnr

=-¥jw

IFT

[

X

(e

)]

=EXAMPLEx[n]

=

1EXAMPLEccpnwjwnsin(w

n)2p1e

dw

=IFT

[

X

(e

jw

]

=-w

c非绝对可和信号的傅立叶变换542.3.2

frequency

response

of

system01|

w

|<

w

cw

c

<|

w

|£

pjwlpH

(e

)

=clpcpnwjwnsin(w

n)

1

2pe

dw

=h

[n]

=-wc-π

-ωcωcπωH(ejω)EXAMPLE理想低通滤波器的频域和时域55echo

system:

y[n]

=

x[n]

+

0.5x[n

-10]h=[1,0,0,0,0,0,0,0,0,0.5]freqz(h,1)EXAMPLEMATLAB编程画系统频响5657Figure

2.20h[0]h[N-1]h[0]FIR系统作用于因果信号IIR系统作用于因果信号58Sin(0.1*pi*n)h[n]=[1,1,1,1,1,1,1,1,1,1]/4B=[1,0,1,0,1];A=[1,0.81,0.81,0.81]592.3.3

properties

of

fourier

transform1.linearity:

ax[n]

+

by[n]‹

F

fi

aX

(e

jw

)

+

bY

(e

jw

)F

fi

e-

jwn0

X

(e

jw

)2.time

shifting

:

x[n

-

n0

]‹F

fi

X

(e

j

(w

-w

0

)

)3.

frequency

shifting

:

e

jw

0n

x[n]‹60x1[n]

=

x[n

-

5]x2[n]

=

x[n]e

j0.5pnX,X1X2XX1X2EXAMPLE频移信号幅度原信号和时移信号幅度时移信号相位原信号相位频移信号相位4.x[-n]‹for

realF

fi

X

(e-

jw

),sequence:

x[-n]‹

F

fi

X

*

(e

jw

)dw61FdX

(e

jw

)5.nx[n]‹

fi

j6.x[n]

*

y[n]‹

F

fi

X

(e

jw

)Y

(e

jw

)62p-pjwjwF

1

2pX

(e

)*Y

(e

)

=

1

2p7.x[n]y[n]‹

fiX

(e

jq

)Y

(e

j

(w

-q

))dqp-p

1

2px[n]y

[n]

=¥8.

parseval

:

n=-¥X

(e

jw

)Y

*

(e

jw

)dw*p-p2|

X

(e