第4章连续时间信号采样

第4章连续时间信号的采样4.04.14.24.34.44.54.64.7引言周期采样采样的频域表示由样本重构带限系统连续时间信号的离散时间处理离散时间信号的连续时间处理抗混叠滤波器小结y(t)

=

x(t)

*h(t)信号系统时域x(t)

y(t)y(t)

=

x(t)h(t

-t)dt频域X

(jW

=

x

(t

e-

jW

t

dtY

(jW

=

X

(jW

H

(jWS域X

(S

=

x

(t

e-st

dtY

(S

=

X

(S H

(S0104050600.60.40.20-0.2-0.4-0.60.75

Cos(t)20

30时时/s幅幅01050601.510.50-0.5-1-1.5Cos(t1)+

0.75Cos(t2)20

30

40时时/s幅幅h(t)0104050600.60.40.20-0.2-0.4-0.60.75

Cos(t)20

30时时/s幅幅01050601.510.50-0.5-1-1.5Cos(t1)+

0.75Cos(t2)20

30

40时时/s幅幅h

ny

n

=

x

n

*h

n信号系统时域x

n

y

ny

n

=

x

m

h

n

-

mm频域X

(e

jw

=

x n

e-

jw

nnY

(e

jw

=

X

(e

jw

H

(e

jwz域X

(z

=

x

n

z-nnY

(z

=

X

(z H

(z0104050600.60.40.20-0.2-0.4-0.60.75

Cos(t)20

30时时/s幅幅01050601.510.50-0.5-1-1.5Cos(t1)+

0.75Cos(t2)20

30

40时时/s幅幅h

ny

n

=

x

n

*h

n信号系统时域x

n

y

ny

n

=

x

m

h

n

-

mm频域-

j

k

nX

[k

]=

x[n]e

2pnY

n

=

X

n

H

nz域X

(z

=

x

n

z-nnY

(z

=

X

(z H

(z4.0引言如何将连续时间信号离散时间信号进行数字处理如何将离散时间信号连续时间信号4.1周期采样连续时间信号：xc

(t)x[n]

=

xc

(nT

),-¥

<

n

<

¥sTf

=

1sT=

2p离散化其中：采样周期：T采样率：弧度/秒（rad/s）表示采样频率：W理想连续（ContinuousTime）时间到离散时间（Discrete

Time）转换器C/Dxc

(t)x[n]

=

xc

(nT

)T的转换将CD分为两步在数学表示上比较方便。冲击串调制器

+

序列转换器C/D转换器冲击串到离散时间序列xc

(t)x[n]

=

xc(nT

)2T

3T-

-3T

2T

-T

0

Txc

(t)s(t)-3T-2T-T

0

T

2T3Txc

(t)s(t)x[n]-3

-2

-1

0

1

2

3-3

-2

-1

0

1

2

3xs(t)和x[n]的本质差别在于：在某种意义上xs(t)

还是一个连续时间信号（即一个脉冲串），除了T的整倍数时刻外其他时刻值为零；另外，采样引入了时间归一化，即x[n]中不包括采样率信息。xc(t)的样本在x[n]中使用有限数值表示的，而不是xs(t)中以单位冲击的面积。¥n=-¥4.1采样的频域表示周期冲击串调制s(t)

=

s

(t

-

nT

)¥n=-¥¥xs

(t

)=

xc

(t

)s(t)

=

xc

(t

)

s

(t

-

nT

)xs

(t

)=

xc

(nT

)s

(t

-

nT

)n=-¥则go¥¥s(t)

=

s

(t

-

nT

)n=-¥周期函数的傅利叶变换等于其傅利叶级数的系数。（即谐波分量）S

(

jW

)

=

2p

aks

(W

-

kW

s

)k

=-¥2kTaTT-=

1T

12

s

(t)e-

jw0kt

dt

=s2p¥S

(

jW

)

=Ts

(W

-

kW

)k

=-¥其系数为：傅立叶变换1sscscscscsTTT1T2p2p

1

2pk

=-¥¥k

=-¥¥¥s

W

-

WX

(

jW

)

=

1

X

(

jW

)*

S

(

jW

)X

(

jW

)

=X

(

jW

)*s

(W

-

kW

)=X

(

jW

)*s

(W

-

kW

)=X

(

jW

-

jkW

)S

(

jW

)

=

2p

(

k

)k

=-¥¥k

=-¥WW

N-

W

NXc

(

jW

)1WW

s2W

s0-

W

s-

2W

sS

(

jW

)2pTWW

s2W

s0-

W

s-

2W

s-

W

N

W

NW

s

-

W

NXs

(

jW

)1TWW

s2W

s0-

2W

s

-

W

sW

N-

W

NW

s

-

W

NXs

(

jW

)1T奈奎斯特采样定理xc(t)为一个带限信号,即

W

>

W

N

Xc

(

jW

)

=

0

时那么xc(t)能唯一由他的样本x[n]=xc(nT),n＝0，+-1，+-2….所确定。要求W

N2W

NNsTW

=

2p

>

2W奈奎斯特频率奈奎斯特率X

r

(

jW

)

=

Hr

(

jW

)

X

s

(

jW

)W

N

<

W

c

<

(W

s

-

W

N

)X

r

(

jW

)

=

X

c

(

jW

)xc

(t)信号恢复s(t)xs

(t)xr

(t)Hr

(

jW

)WW

N-

W

NXc

(

jW

)1WW

s2W

s0-

W

s-

2W

s-

W

N

W

NW

s

-

W

NXs

(

jW

)1TW

s

>

2W

NWW

C-

W

Cc

<

(W

s

-

W

N

)THr

(

jW

)W

N

<

WWW

N-

W

NX

r

(

jW

)1WW

00-

W

0Xc

(

jW

)pW0X

c

(

jW

)pTW

0-

W

0W

s-

W

sW

s

>

2W

NT2W

sW0X

c

(

jW

)W

0-

W

0W

s-

W

sW

s

<

2W

NTpT2W

sWW

00-

W

0X

r

(

jW

)pWX

r

(

jW

)p-(W

s

-

W

s

)0

(W

s

-

W

s

)cxc

(nT

)x

(nT

)e¥x(t)e-

jW

t

dt-¥¥¥-

jW

t-¥

n=-ˆ¥¥-

jW

t-¥-

jW

nt=s

(t

-

nT

)e

dt=n=-¥¥n=-¥X（jW

）=X（s

jW

）=

x(t)s

(t

-

nT

)e

dt因为sx[n]e-

jW

nTn=-¥X

(

jW

)

=x[n]

=

xc

(nT

),¥-¥

<

n

<

¥¥X

(e

jw

)

=

x[n]e-

jw

nn=-¥jwXs

(

jW

)

=

X

(e

)w

=W

T=

X

(e

jW

T

)带入上式和则所以可表示为：c1TTT

T¥k

=-¥k

=-¥Xs

(

jW

)

=Xc

(

jW

-

jkW

s

)X

(

j

w

-

jk

2p

)X

(e

jw

)

=

1

¥进行了频率尺度归一化w

=

W

T例4.1用采样周期对连续时间信号采样，得到其中判断T

=

16000xc

(t

=

cos

(4000ptx[n]

=

xc

(nT

=

cos

(4000pnT

=

cos

(w0

n03w

=

4000pT

=

2

psTW

=

2p

=12000pW

0

=

4000pW

s

>

2W

0满足奈奎斯特采样定理。利用归一化频率判断准则w

0

<

pWXs

(

jW

)Hr

(

jW

)TpTpTpTpTpTpT16000p4000p

6000p

8000p-

8000p

-

6000p

-

4000p

0-16000pw(a)X

(e

jw

)

=

X

(

jw

T

)s3-

2p0(b)pppppp3-

4p3-

8p2p

34p

38p

3......图4.6W

0

=4000p

和采样周期T=1/6000的已采样余弦信号的连续时间(a)和离散时间(b)的傅立叶变换WXs

(

jW

)Hr

(

jW

)TpTpTpTpTpTpT4000p1000p

1500p

2000p-

2000p

-1500p

-1000p

0-

4000pw(a)X

(e

jw

)

=

X

(

jw

T

)s3-

2p0(b)pppppp3-

4p3-

8p2p

34p

38p

3......图4.7W

0

=

4000p和采样周期T=1/1500的已采样余弦信号的连续时间(a)和离散时

的傅立叶变换4.3由样本重构带限系统¥¥用调制脉冲串和低通滤波器可以使得输出信号的傅立叶变换与原来输入信号的福利也变化相同。以上讨论的是信号的时域表示，下面讨论上述频域特性在时域的形式。由前面的公式可得xs

(t

)=

xc

(nT

)s

(t

-

nT

)n=-¥xr

(t

)=

xc

(nT

)hr

(t

-

nT

)n=-¥这种选择对任何截至频率一般选取重构滤波器的响应序列到冲击串的转换理想重构系统xr

(t)xs

(t)x[n]理想重构滤波器W

N

（

W

s

>2W

N）都是适合的W

sTc=

p=

W

sW2D/Cxr

(t)x[n]TWpTHr

(

jW

)TT-

ptThr

(t)1-T3T-

3Trrx

[n]pt

/

T傅立叶反变换h

(t)

=

sin

pt

/

Tsin

[p

(t

-

nT

)

/

T

]x

(t)

=p

(t

-

nT

)

/

T¥n=-¥(a)(b)n

=

–1,

–2,注意hr

(0)

=1hr

(nT

)

=

0

,可得即重构信号在各采样时刻与原连续信号相同，且与采样周期无关。xr

(mT

)

=

xc

(mT

)t下图为采样和恢复的时域示意图xc

(t)xs

(t)-3T

-2T

-T

0

T

2T

3T-3T

-2T

-T

0

T

2T

3Txr

(t)如果不存在混叠，低通滤波器就内插出样本之间的准确值。x[n]sin

p

(t

-

nT

)

/

T

]p

(t

-

nT

)

/

T频率上重新标定了。输入带限信号输出也为带限信号。低通滤波器的截至频率一般取采样频率的一半由rr¥-

jW

TnX

(

jW

)

=x[n]H

(

jW

)en=-¥¥n=-¥xr

(t

)=

xc

(nT

)hr

(t

-

nT

)W

T

«

wjW

TXr

(

jW

)

=

Hr

(

jW

)

X

(e

)则得4.4连续时间信号的离散时间处理离散时间系统的主要应用场合时连续时间处理C/Dxc

(t)x[n]TD/Cyr

(t)y[n]T离散时间系统C/D产生离散时间信号x[n]

=

xc

(nT

)k

=-¥cT

TTX

(

j

w

-

j

2pk

)X

(e

jw

)

=

1

¥¥n=-¥ry

(t)

=y[n]

sin[

p

(t

-

nT

)

/

T

]p

(t

-

nT

)

/

T其傅立叶变换为：D/C产生离散时间信号rY

(

jW

)

=

H

(

jW

)Y

(e

jW

T

)rTY

(e

jW

T

),

W

<

p

/

T=

0,其他具上节结论X

(

jW

)

=

H

(

jW

)

X

(e

jW

T

)r

r则以上是连续系统进行离散处理的基本部分，接着是离散系统处理。一个简单例子（恒等系统）4.4.1线性时不变离散时间系统k

=-¥cT

TTX

(

j

w

-

j

2pk

)X

(e

jw

)

=

1

¥系统的频域表示Y(e

jw

)

=

H

(e

jw

)

X

(e

jw

)带入连续时间系统Y

(

jW

)

=

H

(

jW

)H

(e

jW

T

)

X

(e

jW

T

)r

r其中则¥k

=-¥cjW

TT1TYr

(

jW

)

=

Hr

(

jW

)H

(e

)X

(

jW

-

j

2pk

)whenW

‡

p

TXc

(

jW

)

=

0对于重构滤波器则cjW

TrH

(e

)

X

(

jW

),Y

(

jW

)

=

0,

W

‡

p

/

TW

<

p

/

THeff

(

jW

)

=

0,

W

>=

p

/

TH

(e

jW

T

),

W

<

p

/

T即：连续时间系统等效于一个线性时不变系统jW

T，其有效的频率响应如上

Heff

(e

)是所示因此，如果输入带限，并且满足奈奎斯特采样 率，则Y

(

jW

)

=

H

(e

jW

T

)

X

(

jW

)r

eff

c（4.38）系统为线性时不变，依赖一下两点：1.离散时间系统必须是线性时不变的。2.输入信号必须是带限的，且采样率要足够高，以使得任何混叠分量都被该离散时间系统说抵消。以书上的单位冲击响应为例，说明2混叠:满足奈奎斯特定理，或者离散时间系统的系统函数将混叠部分滤掉。例4.4频率响应如下：其频率响应如图所示Cjw0,w

C

<

w

£

p1,

w

<

wH

(e

)

=w

C-w

CH

(e

jw

)1-

2p2p其：Ceff0,

W

T1,

W

TH

(

jW

)

=c>

wc

W

>

wc

/

T<

w

W

<

w

/

TWTw

CHeff

(

jW

)1T-

w

CWW

N-

W

NXc

(

jW

)1W0-

W

N

W

NNT2p

-

WjwX

s

(

jW

)

=

X

(e

)1T2pTT-

2pW1T2p

-

W

NTX

(e

jw

)2p-

2pH

(e

jw

)-

W

NT

0

W

NT-w

cWY

(e

jw

)1T2p-

2p-w

c

0

w

cWY

(

jW

)TT1TH

(

jW

)eff-

2p

-

p

-

w

c

0

w

c

p

2pTWT

T

T

T1Yr

(

jW)T

T-

w

c

0

w

c只要满足：(2p

-W

NT

)>w

c则输出中不存在混叠奈奎斯特定理要求(2p

-

W

NT

)

>

W

NT例4.5

带限微分器系统输入输出关系：dtd[xc

(t)]yc

(t)

=Hc

(

jW

)

=

jW

,H

eff

(

jW

)

=

0,

W

‡

p

/

T

jW

,

W

<

p

/

TWT-

ppTHeff

(

jW)WT-

ppTpTHeff

(

jW)p

2p2-（a）即由于是带限系统，所以有效的频率响应：离散系统的频域表示H

(e

jw

)

=

jw

/

T0Tpn2h[n]

=

pn

cospn

-sin

pnn

=

0=

cospnn

„

0

TnpTp

2H(jW)H

(e

jw

)2p--2pp-p-pw2p

w（b）4.4.2冲击响应不变(4.49),

w

<

pH

(e

jw

)

=

H

(

jw

/T

)c如何实现连续时间系统由于

Hc

(

jW

)

是带限的，(4.38)给出了选择

H

(e

jw的)

方法，特别是HEFF

(

jW

)

=

Hc

(

jW

)yr

(t)

=

yc

(t)C/Dxc

(t)x[n]TD/Cy

(t)ry[n]T离散时间系统xc

(t)yc

(t)连续时间LTI系统接下来需要选择合适的T，满足如下条件：(4.50)W

‡

p

/

T(4.51)即，离散时间系统的单位脉冲响应就是一个在幅度上受到加权的hc(t)的采样序列。这样的离散时间系统叫做连续时间系统的一个脉冲响应不变型式。Hc

(

jW

)

=

0

,则h

n

=

Thc

nT将系统响应h（t）看作一个x（t），可用以前的公式进行证明(4.52)(4.53)cTT

Tk

=-¥h

n

=

hc

(nT

)H

(

j

w

-

j2pk

)H

(e

jw

)

=

1

¥（4.50）成立的条件：1(4.54)cTTjwwH

(e

)

=

H

(

j)

,

w

<

p幅度问题(4.55)(4.56)cwTh

n

=

Thc

(nT

)

,jwH

(e

)

=

H

(

j)

,

w

£

p例4.7 求一离散时间低通滤波器可以先设计连续系统低通滤波器，然后用冲击响应不变法求离散系统。连续时间低通滤波器：cc1,

W

<

W

cH

(

jW

)

=0,

W

‡

Wccptsin(W

t)h

(t)

=单位冲击响应：则离散时间的单位冲击响应：[

]cccsin(w

n)pntpnsin(W

nT

)h

n

=

Th

(nT

)

=

T=cw

<

wc1,H

(e

jw

)

=0,

w

<

w

£

p其傅立叶变换：例4.8具有有理系统函数的连续系统大部分连续时间系统可以表示为：0s

tcu(t)h

(t)

=

Ae0cAH

(s)

=s

-

s0s

Tncu

nh

n

=

Th

(nT

)

=

Ae拉普拉斯变换：离散时间系统：z变换ATH

(z)

=1-

es0T

z-1ATH

(e

jw

)

=1-

es0T

e-

jw频率响应4.5离散时间信号的连续时间处理一般不会用到，可以对有些离散时间信号做出有用的解释D/Cxc

(t)x[n]TC/Dyc

(t)y[n]T连续时间系统hc

(t)Hc

(

jW

)h(n),

H

(e

jw

)由以前的证明可知：(4.57)(4.58)cx

[n]y

[n]n=-¥sin

p

(t

-

nT

)

/

Tx

(t)

=p

(t

-

nT

)

/

Tsin

[p

(t

-

nT

)

/

T

]y(t)

=x

[n]=

xc

(nT

)p

(t

-

nT

)

/

Ty

[n]=

yc

(nT

)¥n=-¥¥式中频域关系如下1c(4.59a)(4.59b)(4.59c)Tjww

/

T

)

,W

<

p

/

TjW

TXC

(

jW

)

=

TX

(e

)

,Yc

(

jW

)

=

H

(

jW

)

Xc

(

jW

)

,W

<

p

/

TY

(e

)

=

Y

(

jw

<

p则系统响应：(4.60)H

(e

jw

)

=

H

(

jw

/

T

)

,

w

<

pc(4.61)jW

THc

(

jW

)

=

H

(e

)

,W

<

p

/

T即,如果连续时间系统的频域响应为：则上述系统的总响应为：H

(e

jw

)例4.9离散系统的频率响应为利用连续时间系统则(4.62)(4.63)(4.64)(4.65)jW

T

-

jWD

THc

(

jW

)

=

H

(e

)

=

eyc

(t)

=

xc

(t

-

DT

)H

(e

jw

)

=

e-

jwD

,

w

<

p当

D为整数时y

n]=

x

n

-

D](4.66)==¥k

=-¥t

=nT¥k

=-¥x

kx

ky

n]=

yc

(nT

)

=

xc

(nT

-

DT

)p

(n

-

k

-

D)[

]sin

p

(n

-

k

-

D)p

(t

-

DT

-

kT)

/

T[

]sin

[p

(t

-

DT

-

kT)

/

T

]按照卷积的定义，可知系统的响应为：不为整数时，h[n]有无限长。当当为整数时h[n]=

sinp(n

-D)

,-¥

<

n

<

¥p(n

-D)DD

=

n0h

n

=

s

[n

-

n0

]（a）中的离散时间序列的连续时间处理可以产生

一个“半样本间隔”延迟的新序列，如图（b）所示xc

(t)x[n]t0

T

2T(a)2c

cy

(t)

=

x

(t

-

T

)y[n]0

T

2Tt(b)例4.10具有非整数延迟的滑动平均系统[

]1(4.67)11)

e2)

e2sin(w

/

2)(4.68)-

j

0.25pnsin

w

(M

+1)

/

2

e-

jw

M

/2

,H

(e

jw

)

=w

<

p(M

+1)y

[n]=

w

[n

-

M

/

2]y

n

=

H

(e

j

0.25pj

0.25p

n

+

H

(e-

j

0.25p=

1

sin

[3(0.25p

)]e-

j

(0.25p

)5

/

2e

j

0.25p

n2

6

sin(0.125p

)+

1

sin

[3(-0.25p

)]e

j

(0.25p

)5

/

2e-

j

0.25pn2

6

sin(-0.125p

)=

0.308

cos[0.25p

(n

-

2.5)]M

sin(w

/

2)1

sin(w

(M

+1)

/

2)e-

jw

M

/2x[n]y

[

n

]w[n]H

(e

jw

)输入信号x[n]=cos(0.25pn)6点滑动平均滤波器的对应输出10.50-0.5-1-50510152010.50-0.5-1-5

0

510

1520n（a）n(b)4.6利用离散时间处理改变采样率x[n]

=

xc

(nT

)x

'[n]

=

xc

(nT

')改变采样率可以通过离散信号连续处理（模拟）的方法，也可以利用其离散域的性质在离散域得到。根据采样率的增减变化可分为：采样率压缩器（压缩器），即减采样；采样率扩展器（扩展器），即增采样。也可分为：采样率按整数变换；采样率按非整数变换。4.6.1采样率按整数因子减少xd

[n]

=

x[nM

]

=

xc

(nMT

)减采样时需要考虑信号混叠问题。fl

Mx[n]dx

[n]

=

x[nM

]采样周期T采样周期T

'

=

MT图4.20离散时间采样器或压缩器的表示cdcTT

TX

(e

1

MTMT

MTjwk

=-¥¥r

=-¥X

(

j

w

-

jk

2p

)x[n]fl

Mxd

[n]

=

x[nM

])

=X

(

j

w

-

jr

2p

)X

(e

jw

)

=

1

¥r是k的M倍1cX

(

jMTMTTMTjwwr

=

i

+

Mki

=

0,

2,...M

-1M

-1

1

¥2p

2pi

Xd

(e

)

=-

jk

-

j

)

i=0

k

=-¥dcdcX

(eX

(eMT

TjwjwM

-1¥k

=-¥M

-1¥k

=-¥1

1M

1

M)

=X

(

j

w

-

jk

2p

-

j

2pi

)TMT

T

MT

i=0

)

=

1

X

(

j

w

-

2pi

-

jk

2p

)Ti=0

cTMTT¥X

(e

j(w

-2pi)

M

)1

X

(

j

w

-

2pi=-

jk

2p

)k

=-¥(

))dX

(eX

(e

1

MjwM

-1j

w

-2pi

Mi=0)

=图4.21

减采样的频域说明(a)(c)(d)(e)1

Xc

(jw

)-W

NW

N-W

N

(b)

W

NWWsX

(jW

)

=

X

(e

jW

T

)2pTT-

2p1Tw

=W

TX

(e

jw

)2p-2pw

N

=W

NT-w

N1MT1)]2djw

jw

/2j

(w

-2p

)

/

2X

(e

)

=

[

X

(e

)+

X

(e(M

=

2)(M

=

2)dX

(e

jW

T

)T

'-

4pT

'-

2p4pT

'2pT

'TwW=

'-2p2ppp1T

'w

=W

T（a）~(c)有混叠的减采样(a)(c)1

Xc

(jW

)-W

NW

NWw

=W

TX

(e

jw

)2p-2p1MTw

=W

T

'X

(e

jw

)2p-2pw

N

=W

NT-w

N1T-w

N2(b)

w

N=

p(M

=

3)(d)~(f)既有为避免混叠的预滤波的减采样(d)-2p2pppw

=W

T(e)-2p2pppw

=W

T(f)1MTw

=W

T

'X

d

(e

jw

)2p-2pw

N

=W

NT-w

NjwHd

(e

)1pMcMw=

p1T3-

pM

3p

=

pX

(e

jw

)

=

H

(e

jw

)

X

(e

jw

)d抽取器采样周期T采样周期Tfl

M低通滤波器增益=1截止频率=p

/Mx[n]x[n]x

d

[n]

=

x[nM

]采样周期T

'=MT4.6.2采样率按整数因子增加和DC转换有点类似。xi

[n]

=

x[n

L]

=

xc

(nT

L)

n

=

0,–L,–2L,...0

其他x

[n]

=

e¥ex

[n]=

x[n

/

L]s

[n

-

kL]k

=-¥采样周期T

采样周期Tx[n

L],

n

=

0,–L,–2L,...fl

M低通滤波器增益=1截止频率=p

/Mx[n]x[n]x

d

[n]

=

x[

nM

]采样周期T

'=MT频域解释1T

'

T1=

L(a)(b)(c)1

Xc

(jW

)-W

NW

NWw

=W

TX

(e

jw

)2p-2p1Tw

=W

T

'eX

(e

jw

)

=

X

(e

jw

L)L-

2ppLL-

p1T(L=

2)L-

4p2pLL4p

=

2p(d)iH

(e

jw

)-2p2pp-pw

=W

T

'(e)-2p2pppw

=W

T'LL-

ppLT

'T

1

=

LL-

ppLiX

(e

jw

)ejw-

jwn¥¥

]n=-¥

k

=-¥-

kL

eX

(e

)=x[n

/

L]s

[n¥xe

[n]

=

x[n

/

L]s

[n

-

kL]k

=-¥¥=

x[k

]e-

jwkL

=

X

(e

jwL

)k

=-¥T

'=

T

L内插器pn

/

Lh

[n]

=

sin

pn

/

L)¥k

=-¥ix

[n]

=[p

(n

-

kL)/

L]x[k

]sin

p

(n

-

kL)/

L]ihi

[0]

=1hi

[n]

=

0,

n

=

–L,–2L,...14/53/52/51/50hlin

[n]L=5线性内插可实现，lin1-

n

/

L,

n

£

Lh

[n]

=

0,其他¥

¥xlin

[n]

=

xe

[k

]hlin

[n

-

k

]

=

x[k

]hlin

[n

-

kL]k

=-¥

k

=-¥图4.26

线性内插的单位脉冲响应误差-

pp5-

2p5

5-

4p2p

54p

5-ppwL

=

5LjwHi

(e

)jwHlin

(e

)（a）5（b）图4.27

（a）线性内插的说明；（b）线性内插滤波器与理想低通内插滤波器频率响应的比较如果原来的信号以奈奎斯特采样得到，则恢复的信号不太好，因外有许多代外能量。原信号的采样越高，回复的越好。hlin[0]

=1hlin[n]

=

0,

n

=

–L,–2L,...xlin[n]

=

x[n

/

L]0,

n

=

–L,–2L,...Hlin

e(

)=

L

wL

/

21

sin

(wL

/

2)2jw4.6.3采样率按非整数因子变换x[n]ex

[n]ix

[n]x

i

[n]x

d

[n]低通滤波器增益=1截止频率=p

/L低通滤波器增益=1截止频率=p

/M›

Lfl

M内插器抽取器采样周期TTLTLTMLfl

M›

Lx[n]ex

[n]xi

[n]xd

[n]低通滤波器增益=L截止频率min(p

/

L,p

/

M

)采样周期TTLT

LTM

LTL（a）(b)T

'

=

MT

L截至频率min

(p

L

,p

MM>L减采样M<L增采样例4.11先内插L＝2抽取M＝3T

'

=

3

2T-W

NW

NWXc

(

jW

)1X

(e

jw

)1T-2p-pp2pw

=W

Tw

=W

T

/

LjwXe

(e

)1T(L=2)L-

pL-

2pL-

4p4pLp

2pL

Lw

=W

T

/

L2p-2p-ppM-

pcMw

=

pw

=W

T

/

L2p-2p-pp3-

pLTLH

(e

jw

)dX

i

(e

jw

)

=

H

(e

jw

)

X

(e

jw

)d

ew

=W

TM

/

L2p-2p-ppLMTp

=

pM

3X

d

(e

jw

)图4.29

按非整数因子变换采样率的说明4.7多采样率信号处理用途：降低计算量。过采样噪声整形ADC,DAC。实现滤波器组。4.7.1滤波和减采样/增采样的互换bX

e

jw

)=

H

e

jw

M

)X

e

jw

)

1M

-1i=0

M M

beXY

(e

)

=j

w

-

ji

2p

jw

Mfl

Mfl

MH

(zM

)x[n]x[n]y[n]y[n]H

(zM

)xa

[n](a)xb

[n](b)图4.30

基于减采样恒等的两个等效系统jwjwbjw

jweXM=

H

(e

)X

a

(e

M M

M

-1i=0j

w

-

ji

2p

1Y

(e

)

=

H

(e

)H

e

j

(w

-2pi

))=

H

e

jw

1M

-1i=0

M M

beXXa

(e

)

=j

w

-

ji

2p

jw

M1M

-1i=0

M M

bH

(e

)eXY

(e

)

=j

(w

-2pi

)

j

w

-

ji

2p

jw

Mfl

Mfl

Mfl

Mfl

M›

M›

M›

M›

Mz-1z-(

M

-1)+0e

[n]1e

[n]2e

[n]e[n]M

-10h

[n]1h

[n]2h

[n]h

[n]M

-1h[n]h[n]zh[n

+1]h[n]z2h[n

+

2]..zM

-1h[n

+

M

-1]....z-2..图4.32

利用ek

[n]分量的滤波器h[n]的多相分解图4.33

利用ek

[n]分量和延迟链的滤波器h[n]的多相分解fl

Mfl

Mfl

Mfl

M›

M›

M›

M›

Mzzzz-1z-1z-1+++h[n]h[n

+1]h[n

+

2]h[n

+

M

-1]e0[n]1e

[n]2e

[n]eM

-1[n]h0[n]h1[n]2h

[n]hM

-1[n]h[n]h[n]»»......+0E

(zM

)1E

(zM

)2E

(zM

)E

(zM

)M

-1y[n]x[n]z-1z-1z-1»...图4.34

基于

h[n]

多项解的实现结构fl

MH

(z)x[n]y[n]w[n]

=

y[nM

]图4.35

抽取系统+0E

(zM

)1E

(zM

)2E

(zM

)MEM

-1

(z

)x[n]w[n]z-1z-1z-1»...图4.36

利用多相分解的滤波器的实现fl

Mfl

Mfl

Mfl

M...+x[n]w[n]z-1z-1z-1»...图4.37

将减采样恒等关系用于多相分解的抽取滤波器的实现..fl

Mfl

Mfl

Mfl

ME0

(z)E1

(z)E2

(z).EM

-1

(z)H

(z)›

Lx[n]y[n]y[n]图4.38

内插系统x[n]z-1z-1z-1»...图4.39利用多相分解的内插滤波器的实现..›

L›

L›

L›

L0E

(zL

)1E

(z

L

)E2

(z

).LE

(z

)M

-1+++y[n]x[n]z-1z-1z-1»..图4.40将增采样恒等关系用于多相分解的内插滤波器的实现..›

L›

L›

L.›

L0E

(zL

)1E

(z

L

)E2

(z

).LE

(z

)M-1+++y[n]图4.41

（a）连续时间信号的离散时间过滤；（b）模拟信号的数字处理C/D离散时间系统D/Ccx

(t)x[n]y[n]ry

(t)TT(a)抗混叠滤波器采样保持A/D转换器离散时间系统D/A转换器补偿重构滤波器T

TTxc

(t)xa

(t)x0

(t)x[n]y[n]yDA

(t)H

r

(

jW)Haa

(

jW

)y

(t)r(b)4.8.1消除混叠的预滤波实际：所采样的信号＝有用信号+宽带噪声。需要抗混叠滤波器要求滤波器是锐截至的，单其实现起来比较困难，且成本昂贵。有了数字信号处理技术，可以用简单的滤波器，便可以实现。C/D离散时间系统D/Ccx

(t)x[n]y[n]ry

(t)TT图4.42

为消除混叠的预滤波的应用抗混叠滤波器ax

(t)2p

T

-W

c

>

W

NT

'

=

MTNW

=

p

T

'Xc

(

jW

)1-W

c-W

NW

NW

cWW

c

WW

Nc-W-W

NXa

(

jW

)11TNT

=

p

/(M

W

)N

NMw

=W

T

=

p-2p2pNww

=W

T-2p

-p

p

2p1T

'T

'

=

MTdX

(e

jw

)X

(e

jw

)简单抗混叠滤波器高频噪声信号信号经过滤的噪声锐截止抽取滤波器混叠噪声w

=W

T图4.44

在C/D转换器采用过采样再紧跟抽取的说明4.8.2

ADC采样保持A/D转换器Tax

(t)x0

(t)xB

[n]T图4.45

模数转换的实际结构ADC量化及二进制表示+xa

(t)sx

(t)¥s(t)

=

d(t

-

nT

)n=-¥零阶保持h0

(t)0x

(t)采样保持（a）x0

(t)xa

(t)T

2T-3T

-2T

-T

0

3T

t(b)图4.46

（a）理想采样保持的表示（b）采样保持典型的输入输出信号C/D量化器xa

(t)T图4.47

图4.45系统的概念性表示编码器x[n]x[n]x

B

[n]-D2D3D2D23D5D

7D

9D2

2

22-

D

D2

2

229D

-

7D

-

5D

-

3D-x

=

Q(x)补偿二进制码111补码011010001000110101100x

111011110010101001100000-2D-3D-4D2

Xm图4.48

用于A/D转换的典型量化器原始信号量化样本未量化样本理想采样保持的的输出D/A转换器输出幅度0T2T3T4T5T000

100

t

110

011图4.49用3位量化器的采样,量化，编码和D/A转换011-D-2D-3D-4D02DD3Dx

B0[n1]1:4.8.3量化误差分析量化器Q(.)x[n]x[n]

=

Q(x[n])x[n]x[n]

=

x[n]

+

e[n]+e[n]图4.50

量化器的加性噪声模型均匀分布-D

2

<

e[n]£

D

22212ee[n]deD

2-D

2s

=1

=

DD12e2-2

B

X

2s

2

=

m

2BXD

=

m

1010

x

mX

2s

2s

2

s

2

12

·

22

Bs

2

SNR

=10

log=10

log

x

e

X

2

=

6.02B

+10.8

-

20

log10

m

x

2-

DD2mD

=

2-B

Xpen

(e)1

D图4.52

舍入量化器（图4.48）量化误差的概率密度函数4.8.4

DAC图4.53

（a）D/A转换器方框图（b）利用零阶保持的表示D/A转换器x[n]DAx

(t)（a）加权转换为冲激零阶保持x[n]x

B

[n]xDA

(t)（b）Hr

(

jW

)