1、u |svRandom Signal Analysis S I 0 S/vYY Email: LQQ: 914854024Typeset with LATEX and AMS-LATEX2008 M3I 11Copyright c 2008 by Jiming Lin, Rev01.01. Paper or electronic copies for noncommercial use maybe made freely without explicit permission from the authors. All other rights reserved.“ c1c q $11.1 q
2、 bW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 ql. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2Hq qd9 . . . . . . . . . . . . . . . . . . . . . . . . 31.2 M qs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 M . . . . . . . . . . .
3、 . . . . . . . . . . . . . . . . . . . . 51.3 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2Hs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3Hqs. . . . . . . . . . . . . . . . . .
4、. . . . . . . . . . . . . 81.3.4d9 (Statistical independent) . . . . . . . . . . . . . . . . . 81.4 M f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1B M f . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2= M f . . . . . . . . . . . . . . . . . . . . . . . . 111.5 M 3+. .
5、 . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6+f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6.1l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172c V212.1 V Q. . . .
6、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Vd9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 V 3V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 (ZZ. . . . . . . . . . . . . . .
7、. . . . . . . . . . . . . . 232.3.3M1f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 s |. . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.1B s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.2= s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8、- I -2.4.3 n s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.4 |. . . . . . . . . . . . . . . . . . . . . . . . . . . . 293c q 313.1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.1l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2M1f . .
9、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 q PX(!) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374c |YVL“d394.1L HM“d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.1
10、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.2 Hs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.3 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.1 X. 2. .
11、 . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2. 2YVLTI“d. . . . . . . . . . . . . . . . . . . . . . . . 424.2.3 K. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.4 K. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 435cN V455.1N V. . . . . . . . . . . . . . . . .
12、 . . . . . . . . . . . . 455.1.1d9+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.1.2 M q . . . . . . . . . . . . . . . . . . . . . . . 465.2?oFN V. . . . . . . . . . . . . . . . . . . . . . . . . . . 47- II -e1 p = el|(Signal) mb ha% MbT 0 (1.2) dBHq H L= MBA Yq i = 1;2; ;N(1.6)T
13、(1.6)$Bayes T(Bayes theroem)b P(Ai)(i = 1;2; ;N)5 q(Priori probability) k -XbP(AijB) 4BCHq/ YqAi? 3 q q(Posteriori prob-ability)bP(BjAj) YqAj kM YqB qM q(Transitionprobability)b dyTw5 -y YqAiT YqBby q5 qy y T q M q7 yQw y q qbyTwa. _5b1.2= m1.1 U ! 01?0 qq. p qp k p(1)a 01 q (2)a 1Hq/ 1 q# 0Hq/ 1 qb
14、X01ppY01m1.1= - 4 -5 I ny“ bWX = f0;1gX5 qP(X = 0) =qP(X) = 1 =) P(X = 1) = 1 qbp qp5?1 H l0 qP(Y = 0jX = 1) = p ?0 l1 qP(Y = 1jX = 0) = pbT“ bWY = f0;1gyNP(Y = 1jX = 1) = P(Y = 0jX = 0) = 1pb(1)a q TP(Y = 1) = P(Y = 1jX = 0)P(X = 0)+P(Y = 1jX = 1)P(X = 1)= pq +(1p)(1q) = 1(p+q)+2pq P(Y = 0) = P(Y =
15、 0jX = 0)P(X = 0)+P(Y = 0jX = 1)P(X = 1)= (1p)q +p(1q) = p+q 2pq(2)a pP(X = 1jY = 1)P(X = 1jY = 0)Bayes TP(X = 1jY = 1) = P(Y = 1jX = 1)P(X = 1)P(Y = 1jX = 1)P(X = 1)+P(Y = 1jX = 0)P(X = 0)= (1p)(1q)(1p)(1q)+pq = 1(p+q)+pq1(p+q)+2pq P(X = 1jY = 0) = P(Y = 0jX = 1)P(X = 1)P(Y = 0jX = 1)P(X = 1)+P(Y =
16、 0jX = 0)P(X = 0)= p(1q)p(1q)+(1p)q = ppqp+q 2pq1.2 M qs1.2.1 M k“ bWBil/“ZL) sb V|“ bWB L R 0“ L ) b“ L Yq L “=) M (Random Variable)M |S “ bWbBall ! q bWf;F;Pg B“i 2 (B L xi =X(i) 2 Rb “ 2 LBl Lf- 5 -X()X() M (Random variable r.v.)bi?v3 V U M Ml3 V U M V ? |be7 M L= “ bWB L R 0“ q bWb s + ? |K V Kb
17、pi = P(X = xi); i = 1;2; =)Xs p(s )(Distribution law) KK uW V “bA P(X = x) = 0 | i L q0=a qsf M 1 uWx1;x2 qPfx1 x1 HF(x2) F(x1) (2)a0 F(x) 1 Olimx!+1F(x) = 1limx!1F(x) = 0 (3)asf (Continuous from the right)limx!a+= F(a)b (Discrete)sf M Xsf F(x) =1Xi=1PfX = xigu(xxi) =1Xi=1piu(xxi) (1.8)u(x)f (Unit s
18、tep function)ba q f lB M X q f (Probability density function, pdf)l qsf :hy 2 (x1) 2 1i22(12) 229=;dy$f zB s TyN1ysz1yNfX(x) = 1p2 1exp(x1)221 fY (y) = 1p2 2exp(y 2)222Vn HssY XYB sb lHq q f(yjx) = fXY (x;y)fX(x)= 1p2 2p12 exp8:hy 2 (x1) 2 1i22(12) 229=; (2 +(x1) 2 1Z(12) 22B sb1.4 M f L=T p M f s5b
19、Ya|) 5|YV ) “d5 5b5 X M X qs OY = g(X)Z = g(X1;X2; ;Xn)p M YZ qsb5 I nB M f I n f b- 9 -1.4.1B M f BaMn5 I nf Y = g(X)iBQf X = h(Y) HXYBBbyNAKl uW = qMPfx0 0X M X (aZ 2 M p Y q by = cx2yNx = ryc;flflflfldxdyflflflfl = 12pcyY V ?yNy :1p2cy expy=c+22 2cosh2ryc; y 00; y :r2 2 exp r22 2; r 00; r :r2 exp
20、 r22 2; r 00; r :r2 2 exp r22 2 +racos( )2; r 00; r :S(!) =Z 11s(t)ej!tdts(t) =Z 11S(!)ej!td!M1 L= V|+f X(u) A q f f(x) fM(u| |Q)7f(x)X(u) f IM(x| |Q)f(x) F:T:! X(u);f(x) F:T:! X(u)b2 M +f 1+f b! M Y =nPi=1XiM yNY (u) = EejuY = E“ nYi=1ejuXi#=nYi=1EejuXi =nYi=1Xi(u) fM Q H yN M f f b3 3+ M Xni5 +f i
21、n i OEXk = (j)k dkX(u)dukflflflflu=0 (j)k(k)X (0) (1.34) TV +f H VZL b1.7 p M X N(; 2)+f b- 17 - M q fX(x) = 1p2 2 exp(x)22 27X0 = (X )= 5X0 N(0;1)blX0(u) =Z 111p2 expx22 +juxdx=Z 111p2 exp12 (xju)2 (ju)2dx= expu22Z 111p2 exp12 (xju)2dx= expu22X = X0 +yNX(u) = Eheju( X0+)i= ejuEeju X0 = ejuX0( u) =
22、exp 2u22 +ju1.8 p=s(Binomial distribution) (aZ+f bn M XiX =nPi=1Xi Xi(i = 1;2; ;n)s pP(Xi = 1) = p; P(Xi = 0) = 1 p = qb5 M XV(n;p)=s:X B(n;p)bn M 1 zCkQ qCknpk(1p)kyN=ss pPfX = kg = Pn(k) = Cknpkqkl9 (aZ9 N b I nV+f ?bXi+f Xi(u) = EejuXi = q +pejuX M yNX(u) =nYi=1Xi(u) = q +pejunp EX = j ddu q +pej
23、unflflflflu=0= npEX2 = (j)2 d2du2q +pejunflflflflu=0 = npq +n2p22X = EX2(EX)2 = npq- 18 -TBc51.81.161.18b- 19 - 20 -2c V L=Y“d|a HWa bW + M ( 2a;a)| HWf V Ux2(t);byN. 2K V ?oB. 2|BB HWofb k ( V q bWf;F;Pg bB M “ bWB L R 0“bn M ( _ )“ bW5n L bWRn0 bWb V V A M w M Q kTB q | L b B“i 2 (B L xi =X(i)b “i
24、 2 B 7 B M tMf xi(t) = X(t;i)N Ht“$“f (Sample Function)bB q bW V(Stochastic or Random Process)bY( HWtf yN |b +Y t ( HWbl 7 k q bWf;F;Pg “ bW B“i 2 9B f xi(t) = X(t;i); t 2 T“ 2 VBB1tf X(t;) VbB Bf V“f bVX(t;)e:X(t)“f e:x(t)b t L “0“b“f V Lf af _ f V9sY LVa V_ VbL= V Q VV Z 1a Q+ kTi5X(t;i)B HWf yN V
25、 V ABB“f “ 2a- 21 - + H YtiX(ti;)5 | %yNX(ti;) B M # V| V AG HWtBF M bN Vn -Bc M sTV M b ) V X(t;) f /cl(1)at; ( VM=)B HWf B= V (2)at VM%=)B HWf =B“f (3)at% VM=)B M (4)at%=)B = (State)b2.2 Vd9 d9 M qsf (CDF) q f (PDF) byN n5s V M “bVX(t) V A1 HWtBF M H Y VMY M byNn ! 1 H H Y M X(t1);X(t2); ;X(tn) Vb
26、yN V9iBCDFPDFbB qsf FX(x;t) = PfX(t) xg (2.1) HWt |x=f bB q f fX(x;t) = FX(x;t)x (2.2)VBd9+ Y M uY H HWtf boBp V H Yd9+ Q H Y W = “=)onp n qsf FX(x1;x2; ;xn;t1;t2; ;tn) = PfX(t1) x1;X(t2) x2; ;X(tn) xng(2.3)n i Vn q f fX(x1;x2; ;xn;t1;t2; ;tn) = nFX(x1;x2; ;xn;t1;t2; ;tn)x1x2xn (2.4)- 22 -A n “)Qo =
27、 “p s9Q V d9+bbd9 1 sY1 4) ! T Bd9? p5o 3Vp1 e Lb 3+ aZaM1f b9 M M7 7 HWf b2.3 V 3V2.3.1 mX(t) EfX(t)g =Z 11xfX(x;t)dx (2.5)V UV H Y M |s V Mb V 9 HWtf b d9 (Statistical Average)a“ (Assembly average) (Mean Value)bi HW ( uYb2.3.2 (ZZ(Z(Mean-square)= EfX2(t)g =Z 11x2fX(x;t)dx (2.6)Z(Variance)= DX(t) =
28、 2X(t) = EX(t)mX(t)2“= EX2(t)+m2X(t)2mX(t)X(t)“= EX2(t)“m2X(t)(2.7)2(t)Q V H Y M |M( () b (t)Sb il X(t)BB 5EfX2(t)gV U H qd9 ( 2(t) H qd9 (b- 23 -2.3.3M1f B( )V H Y(t1t2) M |MW1 G oM1p )1M1f (Autocorrelation function, ACF)M1(Cross-correlation function, CCF)bd9M19 V A B M MG 6B M b- HWWM1 5 - HWMl M1
29、 5 -MvbBa1M1f RX(t1;t2) = EfX(t1)X(t2)g =Z 11Z 11x1x2fX(x1;x2;t1;t2)dx1dx2 (2.8)f2(x1;x2;t1;t2)= q f t1t2 i H Yb t1 = t2 = tN H1M1f (Zb=axZf xZ(Covariance)f CX(t1;t2) = EfX(t1)mX(t1)X(t2)mX(t2)g=Z 11Z 11x1 mX(t1)x2 mX(t2)fX(x1;x2;t1;t2)dx1dx2(2.9)1M1“ X(t1;t2) = CX(t1;t2) X(t1) X(t2)(2.10)xZ1M1f i 1
30、“CX(t1;t2) =EfX(t1)mX(t1)X(t2)mX(t2)g=EfX(t1)X(t2)+mX(t1)mX(t2)mX(t1)X(t2)mX(t2)X(t1)g=EfX(t1)X(t2)g+mX(t1)mX(t2)mX(t1)EfX(t2)gmX(t2)EfX(t1)g=RX(t1;t2)EfX(t1)gEfX(t2)g = RX(t1;t2)mX(t1)mX(t2)t1 = t2 = t5CX(t1;t2) = 2X(t)b VnT VK+ L= M1f b- 24 -CX(t1;t2)RX(t1;t2)Vr Vd9M1+ l7X MB mX(t1)mX(t2)b M1xZl B
31、 VM1yN$1M1f 1xZf (Auto-covariance)b Q Vw VV fM1f xZf lbaM1f M1f RXY (t1;t2) = EfX(t1)Y(t2)g =Z 11Z 11xyfXY (x;y;t1;t2)dxdy (2.11)xZf CXY (t1;t2) = EfX(t1)mX(t1)Y(t2)mY (t2)g=Z 11Z 11xmX(t1)y mY (t2)fXY (x;y;t1;t2)dxdy(2.12)(Orthogonal)RXY (t1;t2) = 0 or CXY (t1;t2) = mX(t1)mY (t2) (2.13)M1RXY (t1;t2) = EfX(t1)gEfY(t2)g or CXY (t1;t2) = 0 (2.14)1a M1f axZf M1“ VM1xZM1f WB M
