Các tính chất phổ - Dương Hoài Nghĩa

Spectral analysis 1

Chng 1 : GII THIU

1.1 TN HIU NNG LNG Nng lng ca tn hiu x(t)

Ex = +

-|x(t)|2 dt =

+

-x(t)x*(t) dt =

+

- +

-X(f)ej2pift df x*(t) dt

= +

-X(f)

+

-x*(t)ej2pift dt df =

+

-X(f)X*(f) df =

+

-|X(f)|2 df

Ex = +

-|x(t)|2 dt =

+

-|X(f)|2 df (ng thc Parseval) (1.1)

Sx(f) = |X(f)|2 (mt ph nng lng) (1.2)

Hm tng quan cho gia cc tn hiu x(t) v y(t)

rxy() = +

-x(t)y*(t-) dt (1.3)

Hm t tng quan ca tn hiu x(t)

rx() = +

-x(t)x*(t-) dt (1.4)

Tnh cht ca hm t tng quan rx(0) = Ex rx(0) | rx() | , nu x(t) thc rx() l hm chn (theo ) nh l Wiener Khintchine : Mt ph nng lng ca tn hiu x(t)

F{rx()} = +

-rx()e

-j2pif d = +

- +

-x(t)x*(t-) dt e-j2pif d

= +

-x(t)e-j2pift

+

-x*(t-)ej2pif(t-) dt d

= +

-x(t)e-j2pift dt

+

-x*(t)ej2pift dt = X(f)X*(f) = |X(f)|2

Sx(f) = +

-rx()e

-j2pifd (1.5)

Th du : x(t) = 6e-2t1(t) rx(), Sx(f), Ex y(t) = 8e-5t1(t) rxy()

Spectral analysis 2

1.2 TN HIU CNG SUT Cng sut ca tn hiu x(t)

Px = lim T 2T1

T

T-|x(t)|2 dt (1.6)

Hm tng quan cho gia cc tn hiu x(t) v y(t)

rxy() = E{x(t)y*(t-)} = lim T 2T1

T

T- x(t)y*(t-) dt (1.7)

(gi thit ergodic) Hm t tng quan ca tn hiu x(t)

rx() = E{x(t)x*(t-)} = lim T 2T1

T

T- x(t)x*(t-) dt (1.8)

(gi thit ergodic) Tnh cht ca hm t tng quan rx(0) = Px rx(0) | rx() | , nu x(t) thc rx() l hm chn (theo ) Cng sut v hm tng quan ca tn hiu chu k Th du : x(t) = 6sin(2t) rx(), Px y(t) = 8sin(4t) rxy()

1.3 TRUYN TN HIU N TRNG QUA H THNG TUYN TNH n trng (white noise) e(t) n trng re() =

2() (1.9) (() : phn b Dirac) Truyn tn hiu n trng qua h thng tuyn tnh e(t) : n trng vi 2 = 1

Hnh 1.1

x(t) = +

-h(t)e(t-t)dt

rx() = E{x(t)x*(t-)} = E{ +

-h(t)e(t-t)dt

+

-h*(t)e*(t--t)dt}

= +

- +

-h(t) h*(t) E{e(t-t)e*(t--t)} dtdt

Spectral analysis 3

= +

- +

-h(t) h*(t) (+t-t) dtdt =

+

-h(t)h*(t-) dt

Sx(f) = +

-rx()e

-j2pifd = +

- +

-h(t)h*(t-) dte-j2pifd

= +

-h(t) e-j2pift

+

-h*( t-) ej2pif(t-) d dt

= +

-h(t) e-j2pift

+

--h*(v) ej2pifv dv dt (t v = t-)

= +

-h(t) e-j2pift dt

+

-h*(v) ej2pifv dv = H(f)H*(f)

Sx(f) = | H(f) |

2 (1.10)

1.4 MC TIU Xc nh mt ph cng sut ca tn hiu t cc d liu o vi chiu di hu hn {x(n), n = 0, 1, 2, , N-1} Phng php khng thng s : Periodogram da vo (1.2). Correlogram da vo (1.5). Phng php thng s : da vo (1.10) vi cc m hnh AR, MA, ARMA. M hnh Prony v Pisarenko : dng cho ph vch. Vn : phn gii v thi gian quan st. c lng : Lch Khng lch phn tn : variance Bi ging ny c bin son da vo [F1].

[F1] G. Fleury, Analyse Spectrale. Mthodes non-paramtriques et paramtriques, Ellipses, 2001.

Spectral analysis 4

Periodogram ca tn hiu 2cos(400pit), Ts = 1ms

Periodogram ca tn hiu 2cos(400pit) + cos(440pit), Ts = 1ms

Spectral analysis 5

Chng 2 : PERIODOGRAM

2.1 NH NGHA Bin i Fourier

X(f) = +

pi

-

ftj2 dtx(n)e +

=

pi

- n

fnTj2 Tx(n)e (2.1)

T : chu k ly mu (sampling period) T (1.2) mt ph cng sut

P(f) = NT

1|X(f)|2 (2.2)

Periodogram

PPER(f) = NT

121-N

0 n

fnTj2x(n)eT =

pi (2.3)

2.2 TNH CHT

XR(f) =

=

pi1N

0 n

fnTj2x(n)eT = +

=

pi

- n

fnTj2R (n)exT bin i Fourier ri rc

vi xR(n) = x(n)wR(n), wR(n) =

khacn,01Nn0,1 : ca s ch nht (ri rc)

XR(f) = X(f)*WR(f) : chp ri rc

WR(f) = fTj2-

fNT-j2

e-1

e-1pi

pi

= e-jpifNT)fTsin(

)fNTsin(pipi

|WR(f)| = |)fTsin(

)fNTsin(pipi

|

PPER(f) = NT

1|XR(f)|

2 = NT

1| X(f)*WR(f) |

2

c lng khng lch tim cn nhng variance khng 0 khi N

v )fTsin(

)fNTsin(pipi

phn gii Th d : x(t) = cos(2pift) X(f) WR(f) : v ph

Tn hiu e(t) Periodogram ca tn hiu e(t)

Spectral analysis 6

Periodogram ca tn hiu x(t) = cos(400pit)

Periodogram ca tn hiu x(t) = cos(400pit) + 0.1e(t)

Periodogram ca tn hiu x(t) = cos(400pit) + e(t)

2.3 CA S Ca s ch nht

wR(n) =

khacn 0,1Nn 0 ,1

Ca s tam gic (Bartlett)

wT(n) =

khacn ,0

Nn2

N,N

2n2

2Nn0,

N2n

Ca s Hanning

Spectral analysis 7

wV(n) =

khacn , 0

Nn 0 , )N

n20.5cos( - 0.5 pi

Ca s Hamming

wH(n) =

khacn , 0

Nn 0 ),N

n20.46cos( - 0.54 pi

Ca s Blackman

wB(n) =

+

khacn , 0

Nn 0 ),N

n40.08cos( )N

n20.5cos( - 0.42 pipi

So snh cc loi ca s [OS1]

Ca s Peak side-lobe Width of main lobe (Hz) Ch nht -13 dB 2/NT Bartlett -25 dB 4/(N-1)T Hanning -31 dB 4/(N-1)T Hamming -41 dB 4/(N-1)T Blackman -57 dB 6/(N-1)T

Spectral analysis 8

Periodogram ca tn hiu x(n) = cos(400pin) + 0.5cos(440pin) vi cc ca s khc nhau

T = 1ms, N = 512

2.4 BIN TH CA PERIODOGRAM Phng php ca Daniell

PDAN(fk) = 12m

1+

+

=

mk

m- k i PPER(fi)

Phng php ca Bartlett : - Chia khong quan st thnh M on khng ph nhau - Xc nh periodogram ca tng on - Tnh trung bnh

PBAR(f) = M

1

=

1M

0 i PPERi(f)

Phng php ca Welch (thng dng nht) : Tng t Barlett nhng cc an ph nhau (thng 50%) Kt hp s dng ca s Hanning

PPERi(f) = TN

1

i

21-N

0 n

fnTj2i

i(n)ew(n)xT

=

pi , i = 0, , M-1

L = iN

T

=

1N

0 n

i

w2(n)

PWEL(f) = ML

1

=

1M

0 i PPERi(f)

Spectral analysis 9

Periodogram ca tn hiu x(n) = cos(400pin) + 0.5cos(440pin) + 0.2e(n)

T = 1ms, N = 512, M = 8, 50% overlap

Spectral analysis 10

Chng 3 : CORRELOGRAM

3.1 NH NGHA (BLACKMAN TUKEY) Mt ph cng sut

P(f) = +

pi -

fj2x d)e(r

+

=

pi

- m

fmTj2x T(m)er

-fs/2 f fs/2 , T = 1/fs : chu k ly mu (sampling period). Correlogram

PCOR(f) = +

=

piM

M- m

fmTj2x (m)erT

M N/10 (Blackman Tukey)

3.2 C LNG HM TNG QUAN

rx(m) = limN+ 12N

1

+ =N

N- n

x(n+m)x*(n) (ergodic)

Phng php 1

rx(m) = mN

1 =

1-m-N

0 nx(n+m)x*(n) , 0 m N-1

rx(m) = rx*(-m) , -N+1 m 0 Tnh cht

- c lng khng lch (unbiased) : E{rx(m)} = rx(m) - Nu m


c lng hm tng quan cho

rxy(m) =

+

+

+

=

=

1-mN

0 n

1-m-N

0 n

0m 1N- , m)-(n*x(n)ym-N

1

1Nm 0 , (n)*m)yx(nm-N

1

r( xy(m) =

+

+

+

=

=

1-mN

0 n

1-m-N

0 n

0m 1N- , m)-(n*x(n)yN

1

1Nm 0 , (n)*m)yx(nN

1

3.3 C LNG MT PH CNG SUT

Phng php 1

Pcor(f) = T =

M

M- mrx(m)e

-j2pifmT

E{Pcor(f)} = T =

M

M- mrx(m)e

-j2pifmT = F{rx(m)(M

2m)} bin i Fourier ri rc

= Px(f)*fT)

fMT)sin(2pipi

sin( : ca s ch nht ri rc

v )fTsin(

)fNTsin(pipi

phn gii Th d : x(t) = cos(2pift) X(f) WR(f) : v ph Phng php 2

P(

cor(f) = T =

M

M- mr( (m)e-j2pifmT

E{ P(

cor(f)} = T =

M

M- m(1-

N|m|)rx(m)e

-j2pifmT = F{rx(m)(1-N

|m|)}

= Px(f)*F{(1-N

|m|)} : ca s tam gic ri rc

v F{(1-N

|m|)}

phn gii Th d : x(t) = cos(2pift) X(f) WR(f) : v ph B c lng ca Blackman - Tukey

PBT(f) = T =

M

M- mw(m)rx(m)e

-j2pifmT w(m) : ca s vi w(0) = 1


E{PBT(f)} = T =

M

M- mw(m)rx(m)e

-j2pifmT = F{w(m)rx(m)}

= Px(f)*W(f) chn ca s c W(f) > 0 E{PBT(f)} > 0 c lng mt ph cng sut cho

Pxy(f) = T =

M

M- mw(m)rxy(m)e

-j2pifmT


Chng 4 : M HNH AR

4.1 NGUYN L 1) Phn hoch pho

e : n trng vi variance 2e = 1 Tn hiu lin tc

Py(s) = H(s)H*(-s*)Pe(s) = H(s)H*(-s*)2e

Py(f) = |H(j2pif)|2 2

e = |H(j2pif)|2

H(s) n nh v cc tiu pha. Tn hiu ri rc

Py(z) = H(z)H*( *z1)Pe(z) = H(z)H*( *z

1) 2e

Py(f) = |H(ej2pifT)|2 2e = |H(e

j2pifT)|2

H(z) n nh v cc tiu pha.

2) M hnh AR

H(z) = pp

22

11

o

za...zaza1b

++++

y(n) = - a1y(n-1) - a2y(n-2) - - apy(n-p) + boe(n) - c lng cc thng s a1, a2, , ap v bo t cc d liu o y(0), y(1), , y(N-1) - c lng mt ph cng sut Py(f) = |H(e

j2pifT)|2

4.2 D BO TUYN TNH V PHNG PHP BNH PHNG TI THIU 1) Phng php bnh phng ti thiu

T cc d liu o y(0), y(1), , y(N-1), thnh lp h phng trnh sau

+

)1N(y

)1p(y)p(y

M =

)1pN(y)3N(y)2N(y

)1(y)1p(y)p(y)0(y)2p(y)1p(y

L

MLMM

L

L

p

2

1

a

aa

M +

+

)1N(e

)1p(e)p(e

Mbo

y~ = R + bo e~ (N >> p) min || y~ - R ||2 = [RTR]-1RT y~

E{|| y~ - R ||2} = bo2(N-p) c lng bo = pN

||R-y~|| 2

2) ngha hnh hc : min || y~ - R ||2 khong cch t y~ n b mt sinh bi cc vect ct

ca R


3) D bo tuyn tnh

y(n) = - a1y(n-1) - a2y(n-2) - - apy(n-p) + boe(n) y(n) = - a1y(n-1) - a2y(n-2) - - apy(n-p) Sai s d bo (n) = y(n) - y(n) = boe(n) trc giao vi cc quan st qu kh (nu e(n) l n trng) E{(n)y(n-k)} = 0 k > 0

4) Gii thut bnh phng ti thiu quy B nghch o ma trn

(A + BCD)-1 = A-1 - A-1B(C-1 + DA-1B)-1DA-1 Phin bn n gin

(A + bdT)-1 = A-1 - bAd1

AbdA1T

1T1

+

Bnh phng ti thiu

y~ k = Rk + ~ k k = [R Tk Rk]-1R Tk y~ k

+1k

k

yy~

=

+T

1k

kR +

+1k

k~

k+1 = [RTk Rk + k+1

T1k+ ]

-1[R Tk y~ k + k+1yk+1]

[R Tk Rk+k+1T

1k+ ]-1 = [R Tk Rk]

-1 - 1k

1k

Tk

T1k

1k

Tk

T1k1k

1k

Tk

]RR[1]RR[]RR[

+

+

++

+

k+1 = [RTk Rk]

-1R Tk y~ k - 1k

1k

Tk

T1k

1k

Tk

T1k1k

1k

Tk

]RR[1]RR[]RR[

+

+

++

+

R Tk y~ k +

+ [R Tk Rk]-1k+1yk+1 -

1k1

kTk

T1k

1k

Tk

T1k1k

1k

Tk

]RR[1]RR[]RR[

+

+

++

+

k+1yk+1

(ha ng mu s)

= k + 1k

1k

Tk

T1k

1k1k1

kTk

T1k1k

1k

Tk

]RR[1y]RR[]RR[

+

+

++

++

+

+ k

= k + 1k

1k

Tk

T1k

T1k1k1k

1k

Tk

]RR[1]y[]RR[

+

+

+++

+

k

t

Kk+1 = 1k

1k

Tk

T1k

1k1

kTk

]RR[1]RR[

+

+

+

+

Gk+1 = [RT

1k+ Rk+1]-1 = Gk -

1k1

kT

1k

kT

1k1kk

G1GG

+

+

++

+

= Gk - k

T1k1k GK ++

Gk+1 = [I - Kk+1T

1k+ ]Gk


Kk+1 = 1kk

T1k

1kk

G1G

++

+

+

Gii thut quy (0) Khi ng tr : 1 = 0, G1 = aI (a ln) (1) Vi k = 1, 2, 3, 4,

Kk+1 = 1kk

T1k

1kk

G1G

++

+

+

(nx1)

k+1 = k + Kk+1[yk+1 - T

1k+ k] (nx1)

Gk+1 = [I - Kk+1T

1k+ ]Gk (nxn) Quay v (1)

5) c lng bc p ca m hnh :

AIC = )ln 2o(bN2p

+

FPE = 2ob1)(p-N1)(pN

+++

BIC = )ln 2o(bNp.ln(N)

+

4.3 PHNG TRNH YULE-WALKER

y(n) = - a1y(n-1) - a2y(n-2) - - apy(n-p) + boe(n)

e(n) : n trng vi variance = 1 rey(m) =

>

=

0mneu,00mneu,b o

{ }=

p

0kk )mn(y)kn(yEa =

>

=

0mneu,00mneu,b 2o

=

0

0b

a

a1

)0(R)1p(R)p(R

)1p(R)0(R)1(R)p(R)1(R)0(R 2o

p

1

yyyyyy

yyyyyy

yyyyyy

MM

L

MMMM

L

L

c lng ma trn tng quan cc thng s ai bo

4.4 PHNG PHP AUTOCORRELATION V COVARIANCE

E1 =

p

2

1

e

e

e

M, E2 =

+

+

N

2p

1p

e

e

e

M, E3 =

+

+

+

pN

2N

1N

e

e

e

M, Y1 =

p

2

1

y

y

y

M, Y2 =

+

+

N

2p

1p

y

y

y

M, Y3 =

+

+

+

pN

2N

1N

y

y

y

M-


R1 =

0y

0yy

00y

000

1p

12

1

LL

MLLM

L

L

L

, R2 =

++

+

pN2N1N

31p2p

2p1p

11pp

yyy

yyy

yyy

yyy

L

MLMM

L

L

L

R3 =

+

+

+

N

3pN

2pNN

1pN1NN

y00

y00

yy0

yyy

L

MLLM

L

L

L

4.5 CHUYN I LIN TC RI RC 1) Phng php 1 : o hm

nTt

a

dt

(t)dx

=

T

1)-x(n-x(n)

s = T

z-1 -1

2) Phng php 2 : tch phn

=t

d)(x)t(y y(n) = y(n-1) + 2

T[x(n) + x(n-1)]

s

1 =

2T

1

1

z1z1

+ =

2T

1z1z

+

s = T2

1z1z

+

3) Phng php 3 : ly mu p ng xung

G(s) = =

p

1i i

ips

b

h(t) = =

p

1i

tpi

ieb 1(t)

h(n) = =

p

1i

nTpi

ieb 1(nT) H(z) = +

=nh(n)z-n

H(z) = =

p

1i+

=n

nnTpi zeb i

= =

p

1i1Tp

i

ze1b

i

cc pi Tp ie


Chng 5 : M HNH MA

5.1 NGUYN L 1) M hnh MA

e : n trng vi variance 2e = 1

H(z) = =

q

0i

ii zb y(n) =

=

q

0ii )in(eb

2) Cc c trng thng ke E{y(n)} = 0

E{y(n)2} = =

q

0i

2ib

3) Cc phng php c lng mt ph cng sut

Phng php 1 : - c lng cc thng s bo, b1, , bq t cc d liu o y(0), y(1), , y(N-1). - c lng mt ph cng sut

Py(f) = |H(ej2pifT)|2 = |

=

piq

0i

fTi2ji eb |

2 vi -fs/2 < f < fs/2.

Phng php 2 : - c lng hm t tng quan ry(m)

ry(m) =

>

+

=

q|m|,0

q|m|,bb)mq,qmin(

)m,0max(imii , |m| q

- c lng mt ph cng sut dng nh l Wiener-Khintchine

Py(f) = | =

q

qm

fTmj2y (m)er

pi |2 vi -fs/2 < f < fs/2.

5.2 NHN DNG THNG S 1) Phng php moments :

Dng phng php lp, th d Newton-Raphson, gii phng trnh sau c lng cc thng s bo, b1, , bq

000b

0bbbbb

q

21

q1o

MNMM

L

L

q

1

o

b

bb

M =

q

1

o

r

rr

M


Phng php lp n gin : gii phng trnh x = f(x). Phng php Newton-Raphson : gii phng trnh f(x) = 0.

f(xn+1) f(xn) + f(xn)[xn+1 - xn] = 0 xn+1 = xn - )x('f

)x(f

n

n

2) Phng php Durbin

Xp x H(z)

H(z) = =

q

0i

ii zb

=

P

0i

ii z

1 vi N >> P >> q

=

P

0ii y(n-i) = (n)

Hm mc tiu

J = E{(n)2} = E{=

P

0ii y(n-i)

=

P

0kk y(n-k)} =

=

P

0i=

P

0kikE{y(n-i)y(n-k)}

= =

P

0i=

P

0kikry(k-i)

mt khc

ry(m) =

>

+

=

q|m|,0

q|m|,bb)mq,qmin(

)m,0max(imii

Ta c

J = =

P

0i=

P

0k=

q

0likblbl-k+i

min J jb

J

= 0 vi k = 1, 2, , q; bo = 1

jb

J

= =

P

0i=

P

0kikbj-k+i +

=

P

0i=

P

0kikbj+k-I = 2

=

P

0i=

P

0kikbj-k+i

= 2=

P

0i+

+=

ij

qijk

ikbj-k+i (v 0 j-k+i q )

- c lng cc thng s i t d liu : m hnh AR


- Xc nh bi : min J


Chng 6 : M HNH ARMA

6.1 NGUYN L 1) M hnh MA

e : n trng vi trung bnh = 0 v variance 2e = 1

H(z) =

=

=

p

0i

ii

q

0i

ii

za

zb vi ao = 1

y(n) = -=

p

1ii )in(ya +

=

q

0ii )in(eb

2) Cc c trng thng ke

E{y(n)} = 0 (gi thit =

p

0iia 0 h thng n nh tim cn)

E{y(n)2} = E{y(n).[-=

p

1ii )in(ya +

=

q

0ii )in(eb ]} = -

=

p

1iyi )i(ra +

=

q

0iyei )i(rb

p ng xung : H(z) = +

=

0i

iizh

rye(m) = E{y(k)e(k-m)} = E{+

=0i

hie(k-i)e(k-m)} = +

=0i

hiE{e(k-i)e(k-m)} = hm2

rye(m) =

q

=

p

0iyi )ik(ra = 0 k > q ry(k) = -

=

p

1iyi )ik(ra k > q

++

++

+

)q(r)2pq(r)1pq(r

)2pq(r)q(r)1q(r)1pq(r)1q(r)q(r

yyy

yyy

yyy

L

MOMM

L

L

p

2

1

a

aa

M =

+

+

+

)pq(r

)2q(r)1q(r

y

y

y

M ai

- Lc y(n) dng b lc =

p

0i

ii za : w(n) =

=

p

0ii )in(ya

w(n) = =

q

0ii )in(eb

- Dng cc phng php ca m hnh MA : c lng mt ph cng sut ca w(n)

Pw(f) = | =

piq

qm

fTmj2w (m)er |

2

Py(f) = 2p

0i

fTi2ji

w

ea

)f(P

=

pi

hoc c lng cc thng s bi

Py(f) = |H(ej2pifT)|2 =

2

p

0i

fTi2ji

q

0i

fTi2ji

ea

eb

=

pi

=

pi


Chng 7 : M HNH PRONY V M HNH PISARENKO

7.1 M HNH PRONY 1) M hnh Prony

y(n) = =

p

1i

Ymi nie cos(2pifin+i) ==

p

1i

Ymi nie

+ ++

2

ee )npif2j()npif2j( iiii

= =

p

1i

[ ])npif2j()npif2j(nmi iiiii eee2

Y ++ +

= =

2p

1i 2

Ymi )nf2(jn iii ee +pi = =

2p

1i

i ije n)f2j( iie pi+

= =

2p

1i

niizh (7.1)

vi i = i+p , fi = -fi+p , i = -i+p , i = i+p = 2

Ymi , i p

hi = i ije , zi = ii f2je pi+ Vn : c lng hi v zi t cc d liu o v xc nh mt ph cng sut ca tn hiu.

2) a thc c trng nh ngha

P(z) = =

2p

1ii )z(z =

=

2p

0i

i2piza , ao = 1 (7.2)

T (7.1) ta c

=

2p

0ii i)y(ka =

=

2p

0iia

=

2p

1m

ikmmzh =

=

2p

1mmh

=

2p

0i

ikmiza

= =

2p

1mmh

2pkmz

=

2p

0i

i2pmiza = 0 (do P(zi) = 0)

Vy y(n) tha m hnh AR vi p chn

=

2p

0ii i)y(ka = 0 (7.3)

3) c lng m hnh Prony

- c lng cc thng s ai : t (7.3)

1)y(2p3)y(4p2)y(4p

y(1)1)y(2py(2p)

y(0)2)y(2p1)y(2p

L

MLMM

L

L

2p

2

1

a

a

a

M = -

+

1)y(4p

1)y(2p

y(2p)

M (7.4)


- c lng zi : nghim ca phng trnh c trng =

p

0i

ipmi za = 0 zi

- c lng hi : y(n) = =

p

1i

nii zh hi

1pp

1p2

1p1

p21

zzz

zzz111

L

MLMM

L

L

p

2

1

h

hh

M =

)1p(y

)1(y)0(y

M (7.5)

4) Ph ca m hnh Prony

y(n) = =

2p

1i

niizh

Y(z) = =

2p

1i1

i

i

zz1

h (7.6)

Py(f) = |TY(ej2pifT)|2 (7.7)

7.2 M HNH PISARENKO 1) M hnh Pisarenko

y(n) = =

+pip

1iiii )nf2cos(h + e(n) (7.8)

e(n) : n trng vi trung bnh bng 0, variance 2e

i : bin ngu nhin phn b u trn [0, 2pi]

2) Hm t tng quan

Ry(m) = E{y(n)y(n-m)} = )m()mpif2cos(h2

1 p

1i

2ei

2i

=

+

= )m(eh4

1 p

-pi

2e

mpif2j2i

i=

+ vi

==

=

0h,hh

ff

oii-

ii- (7.9)

3) c lng m hnh Pisarenko

Trng hp e(n) 0

yo(n) = =

+p

-pi

)nfj(2i

iieh2

1 pi vi

=

==

=

ii

oii-

ii-

0h,hhff

(7.10)

m hnh Pisarenko l trng hp ring ca m hnh Prony vi zi = exp(-j2pifi) : a thc c trng c 2p nghim trn vng trn n v


P(z) = =

2p

1kk )z(z =

=

2p

0k

k2pkza , ao = 1

P(z)yo(n) = =

+2p

0k

ok k)2p(nya = =

2p

0kka

=

++p

-pi

)k]-2p[nfj(2i

iieh2

1 pi=

= =

+p

-pi

)nfj(2i

iieh2

1 pi =

2p

0k

k)-(2pfj2k

iea pi

= =

+p

-pi

)nfj(2i

iieh2

1 pi =

2p

0k

k-2pikza = 0 (do P(zi) = 0)

=

+2p

0k

ok k)2p(nya = 0 (7.11)

Trng hp e(n) 0 yo(n) = y(n) e(n)

=

+2p

0kk k)2py(na =

=

+2p

0kk k)-2pe(na (7.12)

Vy y(n) tha m hnh ARMA vi cc h s ca phn AR = cc h s ca phn MA.

c lng 2e v cc h s ai

yT(n) = eT(n) (7.13) vi

y(n) =

+

+

y(n)

1)2py(n

2p)y(n

M e(n) =

+

+

e(n)

1)2pe(n

2p)e(n

M =

2p

1

o

a

a

a

M

l cc vect (2p + 1) x 1. E{y(n)yT(n)} = E{y(n)eT(n)}

E{y(n)yT(n)}=

)0(r)1p2(r)p2(r

)1p2(r)0(r)1(r)p2(r)1(r)0(r

yyy

yyy

yyy

L

MOMM

L

L

E{y(n)eT(n)}=

2e

2e

2e

00

0000

L

MOMM

L

L

(y(n) = =

+pip

1iiii )nf2cos(h + e(n) : y(n) khng tng quan vi e(n+k), k0)


Ry = 2e (7.14) 2e v l tr ring v vect ring ca ma trn tng quan Ry.

Ry l ma trn i xng, xc nh khng m c th phn tch thnh Ry = V*D*VT vi D

l ma trn dng cho cha cc tr ring (thc v khng m) ca Ry v V l ma trn vung vi cc ct l cc vect ring (trc chun VTV = VVT = I) ca Ry. Hn na

2e l

tr ring nh nht ca ma trn tng quan Ry. Tht vy, gi i l cc tr ring ca ma trn Ry, ma trn tng quan ca tn hiu khng nhiu (e(n) 0) Ro = Ry -

2e I

c cc tr ring l i - 2e . Do Ro xc nh khng m, i - 2e 0 i i

2e I

c lng 2e v bi tr ring nh nht v vect ring tng ng ca Ry. Do (7.14) khng duy nht, ta c th s dng iu kin buc ao = 1.

c lng fi bng cch gii phng trnh c trng P(z) = 0. c lng hi da vo

Ry(m) = )m()mpif2cos(h2

1 p

1i

2ei

2i

=

+

TI LIU THAM KHO

[F1] G. Fleury, Analyse Spectrale. Mthodes non-paramtriques et paramtriques, Ellipses, 2001.

[OS1] A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal Processing, Prentice-Hall International, 1989.

[JW1] G.M. Jenkins, D.G. Watts, Spectral Analysis and its applications, Holden-Day, 1968. [H1] M.H. Hayes, Statistical Digital Signal Processing and Modelling, John Wiley & Sons,

1996.

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