??????(Discrete Random Variables)

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??????(Discrete Random Variables)

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Title: ??????(Discrete Random Variables)

1
??????? ??????????

2
??

5-1 ????
??????(Discrete Random Variables)
??????(Continuous Random Variables)
5-2 ????
????(Random Processes)????
???????????
????????(Wide Sense Stationary Random Processes)
????????(Power Spectral Density)
5-3 ?????????????(Transmission of a Random
Processes through Linear Time-Invariant Filter)
5-4 ??
??????(Gaussian Random Noise)
????
????(Narrowband Noise)
?????????(Sinusoidal Wave Plus Narrowband Noise)

3
??

5-1 ????
??????(Discrete Random Variables)
??????(Continuous Random Variables)
5-2 ????
????(Random Processes)????
???????????
????????(Wide Sense Stationary Random Processes)
????????(Power Spectral Density)
5-3 ?????????????(Transmission of a Random
Processes through Linear Time-Invariant Filter)
5-4 ??
??????(Gaussian Random Noise)
????
????(Narrowband Noise)
?????????(Sinusoidal Wave Plus Narrowband Noise)

4
????

?????????????????????????????????????????????(S,F,
P),???????
????(sample space) S
????S???????????????????????????(sample point or
outcome)???????
????F
?????????
? ? ?
????,?P???

5
????

??????????
P(S)1,??P(???)0?
?????F??A, ?
?????????A?B,
?
?? 5-1-1
???????,????????(S,F,P)
S
F
P(???)0,
, , P(S)1

6
????

?????????????(Axioms)
??3??

???
7
????

???????A?B?????S?????,?P(B)gt0,
????A???????,??B????????????
??????? ?
?A?B??,??? ,?
?
?A?B????,?
?
?
??A?B????(??????)?

8
????

?????(Total Probability Theorem)
?????S??????E1 , E2 , , En ,?????A??,??
????(Bayes Theorem)
?E1 , E2 , , En ?????S??????,?????A,P(A)gt0,??

9
????

????(Random Variables)
??????????????,?????????????
???????????? ,??
?????,???X,??????X S ?R,??R????

?5-1-1
10
????

????S?????????,?????R ,???????? (range)
SX???????????????????????
???????????????????,?????????????
??????????????????????????????????,???????????????
??

?5-1-2
11
??

??(event)??????????,??????????????????????????????
???????
??A?????O????,A? O,?A????????????
?????????X????B,???
???A???,??B??????,??A?B???????,????????????,PAP
B?

?5-1-3
12
????

?? 5-1-2
??????????????,????????X?????????
????SX0,1,2,3?
??Fx2???,???,????
P(x2)P(???,???,???)3/8?

???????????X
13
??????

??????(Discrete Random Variables)
????????X???SX?????,???????????,??????????
????????,??????(Probability mess function,
PMF)???????????????????
????,??????
?

14
??????

????????X,????????Px(x)????SX
?????????????x, ????????Px(x) ? 0
?????????????????????1
????????B?SX, ???B ?????PB?

15
??????

??????(cumulative distribution function, CDF)
?????X????????PX(x),????????????
???????????

16
??????

?? 5-1-3
?????????,????X?????????,?????????????????
??

????X 0 1 2 3
PX(x) 1/8 3/8 3/8 1/8
?5-1-4
17
??????

????,???????????

?5-1-5
18
??????

???????????

?????????1
?5-1-6
John G. Proakis, Communication System
Engineering Chapter 4, 2th Edition, Prentice
Hall,2002
????????????????
19
???

???(???)
????????X????????PX(x) ,?g(x)????X???????,?g(X)???
??
????????X1, X2,, Xn,??????????,?g????X1, X2,,
Xn???????,?

20
???

???????????????
?X??????????PX(x)?????,?
??

21
???

???(variance)
???????X????????
???(Standard deviation)
???????X????????
????,????????????,?????? ,??????????X????????

22
??????

23
??????

?? 5-1-4
???????R,????????,????3/2,?????R?????
??
???(5-1-1)??,VarXEX2-(EX)2

24
??????

?????
???????X????????
????????????????,??????????????????????,???????xi
?????,????????PX(xi)?PX(xi)dx????!
n???(nth moment)???????X?n???????

25
??????

??????(Moment Generating Function, MGF)
??
??

26
????

????(Random Vector)??????????????????????
???????????X?Y,???????????
(joint PMF)?
?? ?
??????????????????(joint CDF)?

27
????????

????????????
??

????????? ??????
???? ???fXY(x,y)
?5-1-7
?5-1-8
??????? ??????
??????? ????????
28
????

??????????????
????
????????X??????
??????X?????????
?????????n?????,????????,???????
???????????????????,????????????????

29
????

?????
??
??
?5-1-9
?5-1-10
??????? ????????
??????? ????????
???? http//wikimediafoundation.org/wiki/Home
30
?????

?????
????????X??????
??????X??????,??XB(1, p)?
????????
??????????,??????
?????????????
?????????????????
?????,????????????????????????????

31
?????

?? 5-1-5
?????XB(1, p),??(1)????? (2)???
(3)???
??
????????
???(5-1-2),
(2) MX(t)?t0 ??????

(???)
32
?????

??
??
(3)???(5-1-1)

33
????

????
????????X??????
??????X?????,??XB(n, p)?
??????????,?????????p,??????1-p,??n?????,????????
??X,????0n??,?X????????????

34
????

?? 5-1-6
?????XB(n,p),??X????????????????
??
?????
???(5-1-4),???
????

35
????

?? 5-1-7
?????X??n????????????????,???X?????
??
????????,??????X??B(n, 1/6)???
???(5-1-1),

36
?????

?????
???????X??????
??X??????,????
?????????????????,????
???????
?????????

X????
37
?????

?? 5-1-8
?????X??????,?????????????????
??
?
???(5-1-3)???
???

??
38
?????

?? 5-1-9
???????,?????bit?????0.001??????????1000
bit???????bit???bit????????
??
??????X??1000 bit????bit??

????? ????? ???????
39
?????
???
??
??
?5-1-12
?5-1-11
???????? ????????
???????? ????????
???? http//wikimediafoundation.org/wiki/Home
40
?????

?????
???????X??????
??X??????,????
??????X???????????????,???r?????,????????,X??????
???????

41
?????

?? 5-1-10
?????X??????,?????????????????
??
???(5-1-2)

??
(???)
42
?????

?????????????
???(5-1-4)??

43
?????
? ?
r??????
????
?5-1-13
???????????? ?????????????
???? http//wikimediafoundation.org/wiki/Home
44
????

????
???????X??????
??X?????,????
??????X???????????????,???1?????,????????,X??????
??????
???????????????????r??????

45
????

?? 5-1-11
?10????,??????,????,?????????????????????,???????X
??????????????,??X?????????????
??
???????????,??????,????????

?????????
?x??????
(???)
46
????

????????

47
?????

?????
???????X??????
??X??????,????
???N????,???m???????????n?????,?????X????????????
?,?X???????????

48
?????

?? 5-1-12
??????200?????50?????,????????????????????????????
X?????????,??????????????,?X????????
??
???????????,?????????

49
?????

?? 5-1-13
??????300?,??5?????????????5??,???5??????????????
??
??
????????? ??
????X????5????????,?XHG(300,5,15)

50
??????

??????(Continuous Random Variables)
????????X???SX??????,??????????
??????(Probability Density Function, PDF)
??????????fx(x)??????
?fx(x)??????X????????
?????????

?????? ???????!
51
??????

??????(Cumulative Distribution Function, CDF)
?????X????????fX(x),??????
??????
???????????
???????????(Right-continuous)?

52
??????

?? 5-1-14
?????X?????(0,3)???????,???????X????????
??
?X???????fX(x)k,??????????2,
??

53
??????

?????????

?5-1-14
????????????????
John G. Proakis, Communication System
Engineering Chapter 4, 2th Edition, Prentice
Hall,2002
54
???

???
????????X????????fX(x),?g(x)????X???????,?g(X)????
?
????????X1, X2,, Xn,??????????,?g????X1, X2,,
Xn???????,?

55
???

???
?????X????????fX(x)????????FX(x),????
?Var(x)??,??x??????????
?????????????X???x?????????

56
??????

??????
??????(??c?????)

???f(x)??????, ?????????????
57
??????

?? 5-1-15
?????X? ???????FX(x)1-e-x2 , xgt0?
???
??X?????
??
???(5-1-5)
???(5-1-6)

58
??????

?????
???????X????????
????????????????,??????????????????????,???????xi
?????,????????fX(xi)?fX(xi)dx????!
n???(nth moment)???????X?n???????

59
??????

??????(Moment Generating Function, MGF)
??
??
??2??
????

60
????

????(Characteristic Functions)
????X????????fX(x),?X?????????
?????????fX(x)??????,???X????????????????
?? 5-1-16
?????????,???m,??? ,??????

61
????

?????????yg(x)??,?????X ??Y g(X)??????????fY
(y)?
?? 5-1-17
?????X?????, ,??????YaXb???????
??

62
????

????(Random Vector)????????????????????????
?S??????????(X,Y)???,E?S????????,????fX,Y(x,y)??
??fX,Y(x,y)?(X,Y)??????????
??????????

63
????????

??X?Y?????????(joint CDF)?
?????(Leibnitzs)???????????????????????????
???Xx????????????

64
????????

??????????????X?Y,???
?g(x,y)????X,Y???????,?????g(x,y)?????
????g(x,y)?XY,?EXY??????X?Y?????

65
??-?????

??-?????(Cauchy-Schwartz Inequality)
??
? ?????,
????
??

66
???

???(Covariance)
????X,Y??????
??????
?????X?Y????,?Cov(X,Y)gt0
?????X?Y???????,?Cov(X,Y)lt0
?????X?Y????,?Cov(X,Y)0

(??) Cov(X,Y)EXY-EXEY
EXEY-EXEY0
67
????

????(Correlation Coefficient)
????X,Y???????
???????
???1?-1???
? ,??X?Y?????(uncorrelated)?
?X?Y????,?X?Y?????,?????

68
????

???????1??
?????U?V
?????????????X?Y?????????????,????????1?-1,?????
????????????1?-1,??X?Y??????

69
????

??????????????
????
????????X????????
??????X????????,??U(a,b)?
????????????????,???????????????,????????????????
????????????

70
????

?? 5-1-18
?????X?????,??????X?????????????????
??
????(5-1-7),
????????,???(5-1-5)?(5-1-6)

71
????
??????
?5-1-15
??????? ????????
72
????

????(????)
????????X????????
??????X?????,?? ?
? ?,??X????????
????????????,
??????,??f(x)?x????????
??????,??f(x)??????

73
????
?? ????????
??
?? ???????
????????????
?5-1-16
???? http//wikimediafoundation.org/wiki/Home
74
????
??
?5-1-17
?????????????
???? http//wikimediafoundation.org/wiki/Home
75
????
??????? ?????????
??
????????????
?5-1-18
???? http//wikimediafoundation.org/wiki/Home
76
????

?? 5-1-19
?????X?????,
,??????X?????????????????
??
????(5-1-7)

(???)
77
????

????????
???????

???(5-1-8)
78
????

?? 5-1-20
?????X?????,
,??????X2?????????
??
?? ,?? EX0
?
,??
???(5-1-9)?X????????
,???(5-1-8)?
?

79
????

????,?
,??X????????
????????????????????
????

?5-1-19
80
????

?????????
????????,??

?5-1-20
81
Q?????

??????Q?????,????Q????????

??Q????
82
Q?????

????????????Q????

83
Q?????

?????????,????Q???????

84
??????

??????(????????)
????????(X,Y )??????????
??
??????(X,Y )???????,
??

85
??????

??????????
?????????X,Y????,?????????
? ,??X?Y???????
??? ?,
?? 5-1-21
?????X?Y?????????????

86
??????

??
??
??????
????????????????
??
??

X?Y????
87
??????

?????????????????,???????
????????X?Y??????Z g (X, Y), W h (X,
Y)??Z?W??????????????

88
??????

??5-1-22
?????X?Y???????? ,??????
?????????????????
??
??????X?Y?????????

89
??????

????
??
???????,??
??

?????(Rayleigh)?? ???????
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
90
??????

??????
????????????(independent and identically
distributed)?????X1? X2 ? ?Xn ,??????Xj?????
,???
?????Zn?
?
????????????(????????)????????? ??????????

??????????????
91
??

5-1 ????
??????(Discrete Random Variables)
??????(Continuous Random Variables)
5-2 ????
????(Random Processes)????
???????????
????????(Wide Sense Stationary Random Processes)
????????(Power Spectral Density)
5-3 ?????????????(Transmission of a Random
Processes through Linear Time-Invariant Filter)
5-4 ??
??????(Gaussian Random Noise)
????
????(Narrowband Noise)
?????????(Sinusoidal Wave Plus Narrowband Noise)

92
????????

?????????????????????
??????????????
???????(Deterministic signal model)
????????????????
??????????
???????(Random or Stochastic signal model)
??????????????
???????????????
????????????????

93
????????

????????????,?????????????
????????????
????
???????,?????????????
???????????????
????
???????,?????????????????
????????????????,??????????
???????
???????,??????????????

94
????????

????
????S??????????????????????????????
???? (Sample function)
???????? ????????x(t si)?
???? (Random processes)
??????X(t , s)??????????,????
???? ????????? x(t ,
si),?????
????si??????PrsiPrx(t,si)????
?? (Ensemble)
????????????????????????????

95
????????
X(tk)? ????
ttk
????S
x(tk ,si)???
????
?5-2-1

???????,????s?????X(t)?????????
???????tk ,?????????????????
? t tk,s si ????,x(tk ,si) ???????

96
????????

?? 5-2-1
???????T 0, 1, 2, , ?????????,?????????????NT
????(1 ? NT ? 6),????????????????X(t),???T ? t ?
T1?, X(t) NT?
????,????????????????,????????????????????????????
??
???????????
????????????

97
????????

?? 5-2-1???????????????

????
??s1? ??????
??s2? ??????
?5-2-2
98
????????

????????????
?????????tk?????????
? ?????????
?????????????? ???????
????

???? X(tk)
ttk
?5-2-3
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
99
????????

?? 5-2-2
????????????X(t)Rcos2?ft?????????R,??R?????????
???(exponential distributed)????? ?????????
?????? ???????
??
??

100
?????????

X(t)???????
??????????????
?? 5-2-3
??5-2-2?,?Xn X(nT),?Xn???????(PMF)???
??

101
?????????

???????????????
????????X1??,???????????
??????X1????????
???????????X1?X2,?????????
????????(joint PDF)
?
????????X(t)??????,????k??????k???t1, t2, ,
tk????????????X(t1), , X(tk),??????????????????
???

102
??????????

??????????????
(Discrete-value and Continuous-value R.P.s)
??????X(t),????????t???????(outcomes)?????,???????
?????,??X(t) ??????????????????????????,??X(t)
????????????
??????????????
(Discrete-time and Continuous-time R.P.s)
?????X(t)?????tnnT???? (X(nT)??????,T??????n???),
??X(t)????????????X(t)?????t????,??X(t)??????????
??

103
??????????

????????????????
?????????(x??)??????
???????????????????????????(???y??)?????

?5-2-4
?5-2-5
????,????
????,????
104
??????????

????????????????
????????Xdc(t)?????????????Xcc(t)?T0.1???????????
?????
?????????(y??)??????

?5-2-6
?5-2-7
????,????
????,????
105
????????

????????X(t),?????t1????X(t1)???????,??X(t1)??????
?
?????????t???,???????????????,?

???? X(t1)
John G. Proakis, Communication System
Engineering Chapter 4, 2th Edition, Prentice
Hall,2002
?5-2-8
??????????
106
????????

?????X(t)??????????(deterministicn function)?
??X(t1)??????? fX(t1) ??,?X(t1)????
?? 5-2-4
??????5-2-2?????X(t)Rcos2?ft????
??

????
?????
107
?????

????? (Autocorrelation function)
??????????????????????????? t1? t2 ?????????
?????

??????????? t1 ? t2 ?????????
??????????? t1 ? t2 ?????????
?5-2-9
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
????????????????
108
?????

????? (Autocovariance function)
??????????????????????????? t1 ? t2 ??????????
?????
??????????????
????X(t)????(variance)?

109
?????

??????(Cross-correlation function )
???????? X(t) ? Y(t),????????????RX(t1, t2) ??
RY(t1, t2),??X(t)?Y(t)??????????????
??????t1 ?t2 ,???
,?X(t)?Y(t)???(orthogonal)???
??????????????

110
?????

?????
?????(Cross-covariance function )
??X(t)?Y(t)?????????????
??????t1 ?t2 ,???
?X(t)?Y(t)???(uncorrelated)???

111
?????

?? 5-2-5
????X(t)?????????XI(t) ? XQ(t)????
?fc??????,????????0, 2p??????
????X(t)??????,???????
??
???,
(???)

????
112
?????

??(??)
?XI(t1) ? XQ(t2)????,???

?????????????????
??????? t1 t2?,RXI XQ (t1,t2) EXI(t1)XQ(t2)
0,???????????????????XI(t) ? XQ(t)
,?????????,???XI(t)?XQ(t)????????
113
??????????

??????????????
(Auto-correlation function for
complex-valued random process )

114
??????????

??????????????
(Cross-correlation function for
complex-valued random process )
???1/2???????????????????

115
??????

??????(Stationary random processes )???
????????X(t)??Xn ,??????m m1,2,??????(t1,,
tm),????????????? ????????????
??????????????
??????????m ?M???,??X(t)??Xn ????M????????

116
??????

????????m????????m?????(random vector)X(t1),X(t2)
,,X(tm)??
?m?????????????????????
?????????????(t1, t2 ,, tm)???????????
??????(Stationary random processes
)????????????,?????????????????
??????????????????????????????????????
???????????????????????????????

117
??????

?????????????
?X(t)???????,?X(t)???????????????,????????????????
?????????????
???????
??????

???????????????
118
??????

?????
?????

?????????? ?????????
?????????? ?????????
119
??????

?? 5-2-6
??????X(t)???????????fc???????????,????0,2??????
???,???????
?????????
??????X(t)????????????????

????????
120
??????

??
X(t)???????
??????????

??????????? ?????,?????0
121
??????

?????0
?????????????t1?t2? ????t1-t2???t1?t2??????
122
????????

????????????????????
??????????????????

?5-2-9
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
?5-2-10
123
????????

????????????????
????X(t)???????????????
????
?X(t)????????(wide-sense stationary,
WSS)????
(????5-2-6) ?
??????????????????????,???????????????????????????
?????

????????
????????t1?t2??? ??t1-t2???t1?t2??????
124
????????

?????????????????(t1-t2 )???t1???t2???????
?X(t)???????????,??tt1-t2 ,??????
??????????????,??????????????????????????????,??
????????????

125
??????????

???????? X(t) ? Y(t),???????????
X(t) ? Y(t)?????????(WSS)?
????????????????(t1-t2 )??
??
?X(t)?Y(t)??????????(jointly
WSS)????(????5-2-5) ?

126
????????

?X(t)?????????,?X(t)??????RX(?)
?????????
?????????f(x) f(- x)
,??Y?,
?? ?????(???)

?? 0,RX(0)???????????
RX(?)??????
???????????? 0
?5-2-11
???? http//wikimediafoundation.org/wiki/Home
127
????????

????
???
??(???????)
??????? (Cauchy Inequality)
128
????????

?? 5-2-7
????????(???????????????????)
?????????????????1?0??????????X(t)?????x(t)???????
???
?? 1?0???A ?-A???????,???T??
?????????,?????????????td????0?T?????????
?? td?????????????Td????,??????????

129
????????

????? (n?1)Tlt t?td ltnT ?(n???),??????????????????0
??1?????,?????????1????,?????????0?
??0?1???????????,?????????????0?1????????????

?5-2-12
130
????????

X(t)?????
X(t)???????
? ?
? ?
?
??
??

131
????????

??X(t) ?td?????????
??????Td????????
? ??Td???????

??????????? ti?tk?????ti-tk??
132
????????

X(t)?????????,? ,??????
??????

?5-2-14
133
????????

????????X(t)??????
?? 5-2-8
????????W,???????fW(w),?????? ????(??,?)?????,
?W????????X(t)?????,
?X(t) ????????????
??
????X(t)?????0,???????
?tt1-t2???,???X(t)??????????

134
????????

????(ensemble average)?????(time average)
?????????????????(??????)??????,??????????????????
????????X(t),?????t0???????????X(t0),??????g(X),??
?????
??????????????????x(t ,si)???????????????g(x),????
???

135
????

???????????x(t ,si)??,???????????????
????????
????????X(t),??????????????????????????
????(Ergodic)??? ????????X(t)
???????g(x)????????x(t ,si),????
?X(t)??????
????X(t)?????????????????????????

136
????

?? 5-2-9
??X(t)????????
?????t?, X(t)??????1,2,3,4,5,6
??????
??X(t)???0,T,2T,,10T?????????,??????1, 2, 3, 4,
3, 2, 5, 6, 4, 1 ,?????????

???,??????
???????!
137
????

??????????
????X(t)????????????
?
????X(t)????????????????
?

138
????

??????????????????????????????? ?
X(t)???????,?????????????????????5-2-9,??????!
??????X(t)?????,?X(t)??????????
???,??????X(t)???,?X(t)?????????
????????????????????????????????????????

139
????????

????
???????????????
?????
??????????????????????????
??????(Power Spectrum Density, PSD)
???????????????????????????
???????????????????????????(Fourier
transform)???(Einstein-Wiener-Khintchine
relation)

??????SX(f)????watt/Hz?
140
????????
??????SX(f)???????????????????????????
?????????????????????
141
??????????

??????????
?????????????
????
??????????
???????????????????
??? ,??????????????
??????????

????????????????!
??????????????????????!
142
????????

??3 ???
?????X(t)???????,?????????
???????

143
????????

????Y(t)??????
??????fc,???

?5.3???
???4??????
144
????????

??4?????????????
??5????????????
?? ?????,?????????????
,??

???z????z????z????
???4??????????????
145
????????

?? 5-2-10
?????????
(????5.6) ,???
? -?,? )?????,?????????
??
?(5-3-1),

146
????????

??????????????
?????????

?5-2-15
Haykin / Communication Systems,4th Ed., 2000,John
Wiley Sons, Inc.
147
????????

?? 5-2-11
??????(Random Binary Wave)(????5-2-7)
??0?1????????A?-A?????T?????,?????????
???(5-2-7)??????

148
????????

?????????(5-3-1)

149
????????
???????????????
?5-2-16
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
150
????????(??)
??(bandlimited)?? ?????????
?5-2-17
??(bandpass)?? ?????????
?5-2-18
John G. Proakis, Communication System
Engineering Chapter 4, 2th Edition, Prentice
Hall,2002
151
????????

????????(cross spectral density)
????????X(t)??Y(t),???????????????
??
?X(t)?Y(t)?????????(jointly WSS)????,?

152
????????

??????????X(t)?Y(t)????????????
??
???

153
????????

?? 5-2-12
?X(t)?Y(t)??????????,??????????Z(t) X(t) Y(t)
,?Z(t)????????????
??

(???)
154
????????

??X(t)?Y(t)??????????,?
?????????
??X(t)?Y(t)?????,?X(t)??Y(t)??????,
?

??????????????????? ?????????????????!
155
??

5-1 ????
??????(Discrete Random Variables)
??????(Continuous Random Variables)
5-2 ????
????(Random Processes)????
???????????
????????(Wide Sense Stationary Random Processes)
????????(Power Spectral Density)
5-3 ?????????????(Transmission of a Random
Processes through Linear Time-Invariant System)
5-4 ??
??????(Gaussian Random Noise)
????
????(Narrowband Noise)
?????????(Sinusoidal Wave Plus Narrowband Noise)

156
?????????????

?????,?????????????????????????X(t)??????Y(t)?????
?????
??X(t)???????????
X(t)?Y(t)???????????

??Y(t)?????????????
??????(Convolution integration)
157
?????????????

????Y(t)????
????X(t)?????Y(t)????????

??X(t)?????
1. ????Y(t)??????(???t??) 2. ???????????????????
H(0)
158
?????????????

???
??
X(t)?Y(t)???????????????tt1-t2
??,??????Y(t)?????????,??X(t)?Y(t)?????????(jointl
y WSS)?????

??X(t)?????
?uv-t2
?tt1-t2
159
?????????????

????Y(t)??????
??
??Y(t)??????,?????????t1?t2????tt1-t2
??,??Y(t)??????????

??X(t)?Y(t)???????
?uv-t2
?tt1-t2
160
?????????????

????????????X(t)?????Y(t)????????
????????????????????,??????????????????
????????????
??

??RX(t)????
??SX(f)?????
161
?????????????

????????Y(t)?????????
??????????Y(t)?????(?????),?????????????Y(t)?????
????
??????????X(t)?????????(???),????????????????????
????(?????)??Y(t)?????????

?????????????????????
162
?????????????

?? ??
?
??
???????????

??h(t)???,?h(u) h(u)
???????????????? ???????????????
??zajb????za-jb????? z2a2b2
163
?????????????

??????????????????????
?? 5-3-1
?? ???,? ????

2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
?5-3-2
164
?????????????

?
??????? ??? SX ? fc ??????????f
???f ???,SX ( fc ) ??????? fc ??????,?????????????
?????,? SX ( f ) ???????????

??f ???
165
?????????????

?? 5-3-2
??????????(??),????X(t)??????, X(t)???????????
???????????????

?? ???
?5-3-3
166
?????????????

?????
???????????
?
??
???

167
????????

?? 5-3-3
?????????????(?????????h1(t)? h2(t)),?????????????
(jointly WSS)????X(t)?Y(t),????????????V(t)?Z(t)??
????????
??
?

2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
(???)
168
????????

??X(t)?Y(t)???????
?tt1-t2
V(t)?Y(t)??????????
???V(t)?Z(t)??????????
169
????????

?????????????V(t)?Z(t)?????

170
??

5-1 ????
??????(Discrete Random Variables)
??????(Continuous Random Variables)
5-2 ????
????(Random Processes)????
???????????
????????(Wide Sense Stationary Random Processes)
????????(Power Spectral Density)
5-3 ?????????????(Transmission of a Random
Processes through Linear Time-Invariant Filter)
5-4 ??
??????(Gaussian Random Noise)
????
????(Narrowband Noise)
?????????(Sinusoidal Wave Plus Narrowband Noise)

171
???????

???????
??,??????????????,??????????????
???????????????????????????????
????????????????
??????????????????
???????????????????????
?,????????????????

172
????

????????????????

???? ??????
???? ?????? (salt and pepper noise)
????
Fromvision.cse.psu.edu
173
????

???????
??????????
??????????
????
2002, Prentice Hall Gonzalez and Woods Digital
Image Processing 2nd Edition(DIP/2e)
????????????
??????????
174
??????

??????
??????X(t)????k??????????????t1 ?t2 ? ?
tk????????????X(t1), , X(tk)???????????????,??X
(t)???????????

?????? ???????
John G. Proakis, Communication System
Engineering Chapter 4, 2th Edition, Prentice
Hall,2002
?5-4-1
175
??????

??????????
???????????????????,???????????????(??????????????
??????????)
???????????k???????????????X(t1), ,
X(tk)?????????????
????????????????????(WSS),??????????????????(SSS)
?
????3??
??XX(t1), , X(tk) ??????
??????(covariance
matrix) C?

176
??????

??????T???
????
?????????? ? ???X(t) ???????
1. ??????????
2. Cij????????i??j????,????
????
?

?????????????????????????????????, ???????????????
?,??X(t)???????
177
??????

??????(Gaussian Random Noise)???????????????? ?
?????????,??????????????

??????????????
?5-4-2
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
178
??????

?????????????????

????????????????
????
2002, Prentice Hall, Gonzalez and Woods Digital
Image Processing 2nd Edition(DIP/2e)
179
????

????(Shot noise)
???????????????????
?????????,?????????????,???????????????????
?k???????? ??,?????????????
?? ? ??????????

180
????

???(0,t???????????N(t),
?
N(t)????????????(Poisson Counting Process)
,???????

?5-4-3
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
181
????

????,N(0)0?
??????0ltt1ltt2,N(t1)- N(t2)????????(t2 - t1)
????????????
?t1?t2???????????k??????????????
????X(t)???????,??????????????,?????????????????,?
??

????????????????
182
????

?? 5-4-1
??p(t)??????A????T????
???(5-5-1)?(5-5-2)???????X(t)??????????

???
?5-4-4
183
???

???(Thermal Noise)
???????????????????
????????????(0oK)?????
?????f Hz?,????R(??)?????????VTN?
VTN??????,????
,???(??2)?
??
???????(Boltzmanns constant) ,
T??????(oK) ?

184
???

????????????
???(Thevenin)?????????R ?????? ?????
??(Norton)????
,???(??2)
???????????,???????????????VTN?????,????0?

2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
?5-4-5
185
????

????
??????????????????????,????????????????,?????????
????
?????????? ?

???????????
??????????
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
?5-4-6
?5-4-7
186
????

?????????????,???????????(??????)????????(uncorrel
ated),???
????????????????????,
,??????????
?????????????????????????,?????????????????????,?
????????????,??????????????????????
?? 5-4-2
?????????????? (Ideal Low-Pass Filtered
White Noise)
????????0?????,???????????????????????????????B,??
??????1?

187
????

?????????,????0?????????????
?????
?????,??????2B????????????,????????????(uncorrelat
ed)?

?5-4-8
?5-4-9
??????
?????
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
188
????

?? 5-4-3
?????????????????????????

????(local carrier)
???(correlator)
?5-4-10
189
????

????????????????????? ?????????
?????

190
????
?w(t)???????,?N??????, ???????????????????
191
????

????
??????????,??????????????????????????????????
??,???????????????????,
??????????

????????
??????????
?5-4-11
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
?5-4-12
192
????

??????????????(In-Phase and Quadrature
Components)????????
????n(t)????nI(t)??nQ(t)??????,??????
????????

???????? ???????????? (???)
?5-4-13
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
193
????

??,??????????????????????
????,????
nI(t)??nQ(t)????
nI(t)??nQ(t)????0?
?n(t)???,?nI(t)??nQ(t)?????(joint Gaussian)

?????????????? ??????? (???)
?5-4-14
2000,John Wiley Sons, Inc. Haykin /
Communication Systems,4th Ed
194
????

nI(t)??nQ(t)????
3. ?n(t)???,?nI(t)??nQ(t)??????
4. nI(t)??nQ(t)???????(????)??
SN(f)????
? ?
5. nI(t)??nQ(t)?????n(t)??,
?

195
????

nI(t)??nQ(t)????
6. nI(t)??nQ(t)???????????
7. n(t)????SN(f)???fc,?nI(t)?nQ(t)????

SN(f)???????
?5-4-15
???????? ???????
John G. Proakis, Communication System
Engineering Chapter 4, 2th Edition, Prentice
Hall,2002
?5-4-16
196
????